International Journal of
Turbomachinery
Propulsion and Power
Review
A Review on Turbine Trailing Edge Flow
Claus Sieverding 1
1
2
*
and Marcello Manna 2, *
Turbomachinery and Propulsion Department, von Kàrmàn Institute for Fluid Dynamics,
Chaussée de Waterloo 72, 1640 Rhode-St-Genèse, Belgium; sieverding@vki.ac.be
Dipartimento di Ingegneria Industriale, Università degli Studi di Napoli Federico II, via Claudio 21,
80125 Napoli, Italy
Correspondence: marcello.manna@unina.it; Tel.: +39-081-768-3287
Received: 15 February 2020; Accepted: 11 May 2020; Published: 20 May 2020
Abstract: The paper presents a state-of-the-art review of turbine trailing edge flows, both from an
experimental and numerical point of view. With the help of old and recent high-resolution time
resolved data, the main advances in the understanding of the essential features of the unsteady
wake flow are collected and homogenized. Attention is paid to the energy separation phenomenon
occurring in turbine wakes, as well as to the effects of the aerodynamic parameters chiefly influencing
the features of the vortex shedding. Achievements in terms of unsteady numerical simulations of
turbine wake flow characterized by vigorous vortex shedding are also reviewed. Whenever possible
the outcome of a detailed code-to-code and code-to-experiments validation process is presented and
discussed, on account of the adopted numerical method and turbulence closure.
Keywords: turbine wake flow; vortex shedding; base pressure correlation; energy separation;
numerical simulation
1. Introduction
The first time the lead author came in touch with the problematic of turbine trailing edge flows
was in 1965 when, as part of his diploma thesis, which consisted mainly in the measurement of the
boundary layer development around a very large scale HP steam turbine nozzle blade, he measured
with a very thin pitot probe a static pressure at the trailing edge significantly below the downstream
static pressure. This negative pressure difference explained the discrepancy between the losses obtained
from downstream wake traverses and the sum of the losses based on the momentum thickness of the
blade boundary layers and the losses induced by the sudden expansion at the trailing edge. Pursuing
his curriculum at the von Kármán Institute the author was soon in charge of building a small transonic
turbine cascade tunnel with a test section of 150 × 50 mm, the C2 facility, which was intensively used
for cascade testing for industry and in-house designed transonic bladings for gas and steam turbine
application. These tests allowed systematic measurements of the base pressure as part of the blade
pressure distribution for a large number of cascades which were first presented at the occasion of
a Lecture Series held at the von Kàrmàn Institute (VKI) in 1976 and led to the publication of the
well-known VKI base pressure correlation published in 1980. This correlation has served ever since for
comparison with new base pressure data obtained in other research labs. Among these let us already
mention in particular the investigations carried out on several turbine blades at the University of
Cambridge, published in 1988, at the University of Carlton, published between 2001 and 2004, and at
the Moscow Power Institute, published between 2014 and 2018.
In parallel to these steady state measurements, the arrival of short duration flow visualizations
and the development of fast measurement techniques in the 1970’s allowed to put into evidence the
existence of the von Kármán vortex streets in the wakes of turbine blades. Pioneering work was
performed at the DLR Göttingen in the mid-1970’s, with systematic flow visualizations revealing the
Int. J. Turbomach. Propuls. Power 2020, 5, 10; doi:10.3390/ijtpp5020010
www.mdpi.com/journal/ijtpp
Int. J. Turbomach. Propuls. Power 2020, 5, 10
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existence of von Kármán vortices on a large number of turbine cascades in the mid-seventies. This was
the beginning of an intense research on the effect of vortex shedding on the trailing edge base pressure.
A major breakthrough was achieved in the frame of two European research projects. The first one,
initiated in 1992, Experimental and Numerical Investigation of Time Varying Wakes Behind Turbine Blades
(BRITE/EURAM CT92-0048, 1992–1996) included very large-scale cascade tests in a new VKI cascade
facility with a much larger test section allowing the testing of a 280 mm chord blade in a three bladed
cascade at a moderate subsonic Mach number, M2,is = 0.4, with emphasis on flow visualizations and
detailed unsteady trailing edge pressure measurements. The VKI tests were completed by low speed
tests at the University of Genoa on the same large-scale profile for unsteady wake measurements using
LDV. In the follow-up project Turbulence Modelling of Unsteady Flows on Flat Plate and Turbine Cascades in
1996 (BRITE/EURAM CT96-0143, 1996-1999) VKI extended the blade pressure measurements on a 50%
reduced four bladed cascade model to a high subsonic Mach number, M2,is = 0.79. Both programs not
only contributed to an improved understanding of unsteady trailing edge wake flow characteristics,
of their effect on the rear blade surface and on the trailing edge pressure distribution, but also offered
unique test cases for the validation of unsteady Navier-Stokes flow solvers.
A special and unexpected result of the research on unsteady turbine blade wakes was the discovery
of energy separation in the wake leading to non-negligible total temperature variations within the
wake. This effect was known from steady state tests on cylindrical bodies since the early 1940’s, but its
first discovery in a turbine cascade was made at the NRAC, National Research Aeronautical Laboratory
of Canada, in the mid-1990s within the framework of tests on the performance of a nozzle vane cascade
at transonic outlet Mach numbers. The experimental results of the total temperature distribution in the
wake of cascade at supersonic outlet Mach number served many researchers, in particular from the
University of Leicester, for elaborating on the effect of energy separation.
The paper starts with the evaluation of the VKI base pressure correlation (Section 2) in view of
new experiments. This is followed with a review of the advances in the understanding of unsteady
trailing edge wake flows (Section 3), the observation and explanation of energy separation in turbine
blade wakes (Section 4), the effect of vortex shedding on the blade pressure distribution (Section 5)
and the effect of Mach number and boundary layer state on the vortex shedding frequency (Section 6).
This experimental part is complemented with a review of the numerical methods and modelling
concepts as applied to the simulation of unsteady turbine wake characteristics using advanced
Navier-Stokes solvers. Available numerical data documenting significant vortex shedding affecting
the turbine performance even in a time averaged sense, are collected and compared on a code-to-code
and code-to-experiments basis in Section 7.
2. Turbine Trailing Edge Base Pressure
Traupel [1], was probably the first to present in his book Thermische Turbomaschinen, a detailed
analysis of the profile loss mechanism for turbine blades at subsonic flows conditions. The total losses
comprised three terms: the boundary losses including the downstream mixing losses for infinitely thin
trailing edges, the loss due to the sudden expansion at the trailing edge (Carnot shock) for a blade with
finite trailing edge thickness dte taking into account the trailing edge blockage effect and a third term
which did take into account that the static pressure at the trailing edge differed from the average static
pressure between the pressure side (PS) and the suction side (SS) trailing edges across one pitch. Thus,
the profile loss coefficient ζp reads:

2
 dte 
 sin2 (α2 ) + k dte
ζp = 2 Θ + 

1 − dte
where:
Θ=
Θss + Θps
g sin(α2 )
!
(1)
Int. J. Turbomach. Propuls. Power 2020, 5, 10
3 of 55
is the dimensionless average momentum thickness, and:
dte =
dte
g sin(α2 )
the dimensionless thickness of the trailing edge. The constant k appearing at the right-hand-side of
Equation (1) depends on the ratio:
!
dte
∗
dte =
Θss + Θps
that is, k = 0.1 for d∗te = 2.5 and k = 0.2 for d∗te = 7, while a linear variation of k is used for 2.5 < d∗te < 7.
Terms containing squares and products of Θss + Θps /dte were considered to be negligible.
Most researchers are, however, more familiar with a similar analysis of the loss mechanism by
Denton [2], who introduced in the loss coefficient expression ζp , the term cpb dte quantifying the trailing
edge base pressure contribution, with:
cpb =
p2 − pb
(2)
2
1/2ρVre
f
Forcommodity
commodityVre f may
maybebetaken
takenasasthe
theisentropic
isentropicdownstream
downstreamvelocity
velocityV2,is, . . However,
However,there
there
For
wasaabig
biguncertainty
uncertainty
regards
magnitude
ofterm,
this term,
although
it appeared
that become
it could
was
as as
regards
the the
magnitude
of this
although
it appeared
that it could
become
very important
in the range
transonic
range and
of aloss
strong
local loss
very
important
in the transonic
and explain
the explain
presencethe
of apresence
strong local
maximum
as
maximum
as
demonstrated
in
Figure
1,
which
presents
a
few
examples
of
early
transonic
cascades
demonstrated in Figure 1, which presents a few examples of early transonic cascades measurements
measurements
performed
at VKI and the DLR. (performance of VKI blades B and C are unpublished).
performed
at VKI
and the DLR.
Pioneering experimental
experimental research
research concerning
concerning the
the evolution
evolution of
of the
the turbine
turbine trailing
trailing edge
edge base
base
Pioneering
pressure
from
subsonic
to
supersonic
outlet
flow
conditions
was
carried
out
at
the
von
Kármán
pressure from subsonic to supersonic outlet flow conditions was carried out at the von Kármán Institute.
Institute.
In 1976,
at of
thetheoccasion
of the
VKITransonic
LectureFlows
Series
Transonic
Flows
in Axialpresented
Turbines,
In
1976, at the
occasion
VKI Lecture
Series
in Axial
Turbines,
Sieverding
Sieverding
presented
base
pressure
data
for
eight
different
cascades
for
gas
and
steam
turbine
blade
base pressure data for eight different cascades for gas and steam turbine blade profiles over a wide
profiles
over anumbers
wide range
of Mach
[3] et
and
Sieverding
al. [4] published
base
range
of Mach
[3] and
in 1980numbers
Sieverding
al. in
[4]1980
published
a baseetpressure
correlationa(also
pressuretocorrelation
(alsoon
referred
on a total of 16 blade profiles.
referred
as BPC) based
a totalto
ofas
16BPC)
bladebased
profiles.
A
B
C
∗
Blade
D
A
B
C
D
30°
60°
66°
156°
22°
25°
18°
19.5°
g/c
Ref
0.75
0.75
0.70
0.85
[5]
VKI
VKI
[6]
Figure 1. Blade profile losses versus isentropic outlet Mach number for four transonic turbines. Blade A
Figure 1. Blade profile losses versus isentropic outlet Mach number for four transonic turbines. Blade
data from [5], blade B and C unpublished data from VKI, blade D data from [6].
A data from [5], blade B and C unpublished data from VKI, blade D data from [6].
Int. J. Turbomach. Propuls. Power 2020, 5, 10
4 of 55
All tests were performed with cascades containing typically 8 blades and care was taken to ensure
in all cases, and over the whole Mach range, a good periodicity. The latter was quantified to be 3%,
in the supersonic range, in terms of the maximum difference between the pitch-wise averaged Mach
number (based on 10 wall pressure tappings per pitch) of each of the three central passages and the
mean value computed over the same three passages. The correlation covered blades with a wide range
of cascade parameters, as outlined in Table 1:
Table 1. Parameters range for Sieverding’s correlation.
Parameter
Pitch to Chord Ratio
Trailing edge thickness to throat ratio
Inlet flow angle
Outlet flow angle
Trailing edge wedge angle
Rear suction side turning angle
Symbol
Value
g/c
dte /o
α1
α2
δte
ε
0.32–0.84
0.04–0.16
45◦ –156◦
18◦ –34◦
2◦ –16◦
0◦ –18◦
Of all cascade parameters only the rear suction side turning angle ε and the trailing edge wedge
angle δte appeared to correlate convincingly the available data, although the latter were insufficient to
differentiate their respective influence. In fact, in many blade designs both parameters are closely linked
to each other and, for two thirds of all convergent blades with convex rear suction side, both ε and δte
were of the same order of magnitude. For this reason, it was decided to use the mean value (ε + δte )/2
as parameter. The relation pb /p01 = f (ps2 /p01 ), is graphically presented in Figure 2. The curves cover
a range from M2,is ≈ 0.6 to M2,is ≈ 1.5, but flow conditions characterized by a suction side shock
interference with the trailing edge wake region are not considered. Comparing the experiments with
the correlation (results not shown herein), it turned out that 80% of all data fall within a bandwidth
Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW
5 of 65
±5% and 96% within ±10%.
129
130
Figure 2. Sieverding’s base pressure correlation; solid lines (resp. dashed lines) denote convergent
Figure
2. Sieverding’s
base pressure
correlation;
solid lines (resp. dashed lines) denote convergent
blades (resp.
convergent-divergent
blades)
[4].
blades (resp. convergent-divergent blades) [4].
131
132
133
134
135
136
137
138
139
140
An explanation for the significance of ε for the trailing edge base pressure is seen in Figure 3,
An
explanation
the significance
for the
trailing edge
base
pressure
is seen
in Figure
3,
presenting
the blade for
velocity
distributionoffor two
convergent
blades
with
different
rear suction
side
◦
◦
presenting
the of
blade
distribution
convergent
with different rearblade
suction
side
turning angles
ε = velocity
20 and 4.5
, blade A for
andtwo
B, together
withblades
a convergent/divergent
with
an
∗ = 1.05,
turning
20° and of
4.5°,
blade
A and
B, together
with aend
convergent/divergent
blade
with
internal angles
passageofarea=increase
A/A
blade
C. The curves
at x/c = 0.95 because
beyond,
∗
an
area isincrease
of by
/ the
= acceleration
1.05, blade around
C. The curves
end edge.
at / = 0.95 because
the internal
pressurepassage
distribution
influenced
the trailing
beyond, the pressure distribution is influenced by the acceleration around the trailing edge.
The rear suction side turning angle ε has a remarkable effect on the pressure difference across
the blade near the trailing edge. For blade A one observes a strong difference between the SS and PS
isentropic Mach numbers, respectively pressures, while the difference is very small for blade B. On
the contrary, for blade C the pressure side curve crosses the SS curve well ahead of the trailing edge
and the PS isentropic Mach number near the trailing edge exceeds considerably that of the SS. The
130
131
132
133
134
135
136
137
138
139
140
141
blades (resp. convergent-divergent blades) [4].
An explanation for the significance of for the trailing edge base pressure is seen in Figure 3,
presenting the blade velocity distribution for two convergent blades with different rear suction side
turning angles of = 20° and 4.5°, blade A and B, together with a convergent/divergent blade with
Int. J. Turbomach. Propuls. Power 2020, 5, 10
5 of 55
an internal passage area increase of / ∗ = 1.05, blade C. The curves end at / = 0.95 because
beyond, the pressure distribution is influenced by the acceleration around the trailing edge.
Therear
rearsuction
suctionside
sideturning
turningangle
angleεεhas
hasaaremarkable
remarkableeffect
effecton
onthe
thepressure
pressuredifference
differenceacross
across
The
theblade
bladenear
nearthe
thetrailing
trailingedge.
edge.For
Forblade
bladeAAone
oneobserves
observesaastrong
strongdifference
differencebetween
betweenthe
theSS
SSand
andPS
PS
the
isentropic
Mach
numbers,
respectively
pressures,
while
the
difference
is
very
small
for
blade
B.
On
isentropic Mach numbers, respectively pressures, while the difference is very small for blade B. On the
the contrary,
for blade
the pressure
side curve
crosses
SS curve
well ahead
the trailing
edge
contrary,
for blade
C theCpressure
side curve
crosses
the SSthe
curve
well ahead
of theof
trailing
edge and
and
PS isentropic
number
near
the trailing
edge exceeds
considerably
SS.
The
the
PSthe
isentropic
Mach Mach
number
near the
trailing
edge exceeds
considerably
that ofthat
the of
SS.the
The
base
base
pressure
is
function
of
the
blade
pressure
difference
upstream
of
the
trailing
edge.
pressure is function of the blade pressure difference upstream of the trailing edge.
Blade A
Convergent
blade ε = 20°
Blade B
Convergent
blade ε = 4.5°
Blade C
Convergent
divergent blade
(A/A*) = 1.05
142
143
Figure
isentropic
Mach number
two convergent
convergent/divergent
Figure3. Surface
3. Surface
isentropic
Machdistribution
number for
distribution
for and
twooneconvergent
and one
blades
at M2,is = 0.9, based
on at
data from
[3]. based on data from [3].
convergent/divergent
blades
, = 0.9,
144
145
ItItisisalso
worthwhile
mentioning
thatthat
ε also plays
an important
role forrole
the for
optimum
blade design
also
worthwhile
mentioning
also plays
an important
the optimum
blade
indesign
function
of
the
outlet
Mach
number.
Figure
4
presents
design
recommendations
for
the
rear
suction
in function of the outlet Mach number.
side curvature with increasing Mach number from subsonic to low supersonic Mach numbers as
successfully used at VKI.
(a)
(b)
(c)
(d)
Figure 4. Recommended values of ε (a) and l/L (b) for the design of the blade rear suction side for
increasing outlet Mach numbers. Full (c) and close-up (d) view of the parameters characterizing the
turbine geometry.
(a)
(b)
Int. J. Turbomach. Propuls. Power 2020, 5, 10
6 of 55
The rear suction turning angle ε for convergent blades should decrease with increasing Mach
number reaching a minimum of ∼ 4 at M2,is ∼ 1.3 (maximum Mach number for convergent blades).
Note that similar trends can be derived from the loss correlation by Craig and Cox [7]. They showed
that in order to minimize the blade profile losses the rear suction side curvature, expressed by the ratio
(d)
g/e, where g represents the pitch (c)
and e the radius of a circular arc approximating
the rear suction side
curvature,
decrease
with increasing
154 should
Figure
4. Recommended
values of (a)Mach
and / number.
(b) for the design of the blade rear suction side for
Commented [M17]: Please add explan
155a given increasing
outlet Mach
numbers.
(c) and
close-up (d)is
view
of the
the
For
rear suction
side
angleFull
ε the
designer
free
as parameters
regardscharacterizing
the evolution
of the surface
add “a,b,c,d” in the figure. Please unify
156
turbine geometry.
angle from the throat to the trailing edge. It appears to be a good design practice to subdivide the
haverear
subgraph like figure 51.
ForLa into
giventwo
rear suction
angle
ε thealong
designer
is freethe
as regards
evolution
of the surface decreases to
suction157
side length
parts,side
a first
part
which
bladethe
angle
asymptotically
Commented [MM18R17]: Done.
158of the
angle from the throat to the trailing edge. It appears to be a good design practice to subdivide the
the value
trailing edge angle, followed by a second entirely straight part of length l, see Figure 4.
159 rear suction side length into two parts, a first part along which the blade angle asymptotically
With increasing
outlettoMach
number,
the edge
length
of followed
the straight
part,entirely
that isstraight
the ratio
l/L
increases, but it
160 decreases
the value
of the trailing
angle,
by a second
part of
length
161 extend
, see Figure
4. With
increasing outlet Mach number, the length of the straight part, that is the ratio
does never
up to
the throat.
162calculating
/ increases, but it does never extend up to the throat.
For
the trailing edge losses induced by the difference between the base pressure and
163
For calculating the trailing edge losses induced by the difference between the base pressure and
the downstream
pressure,pressure,
FabryFabry
& Sieverding
presented
the for
data
the convergent
blades in
164 the downstream
& Sieverding[8],
[8], presented
the data
the for
convergent
blades in
Figureof
2 inthe
terms
of the
base pressure
coefficientcpb , , defined
defined byby
Equation
(2), see(2),
Figure
Figure 165
2 in terms
base
pressure
coefficient
Equation
see5. Figure 5.
166
167
Figure
5. Base pressure
coefficients
corresponding toto
thethe
basebase
pressure
curves ofcurves
Figure 2of
[8].Figure 2 [8].
Figure 5. Base
pressure
coefficients
corresponding
pressure
Since the base pressure losses are proportional to the base pressure coefficient cpb , the curves give
immediately an idea of the strong variation of the profile losses in the transonic range. As regards the
low Mach number range, the contribution of the base pressure loss is implicitly taken into account by
all loss correlations. Therefore the base pressure loss is not to be added straight away to the profile
losses as predicted for example with the methods by Traupel [1] and Craig and Cox [7] but rather as a
difference with respect to the profile losses at M2,is = 0.7:
ζbp = cpb − cpb,M2,is =0.7
dte
g sin(α2 )
!
Martelli and Boretti [9], used the VKI base pressure correlation for verifying a simple procedure
to compute losses in transonic turbine cascades. The surface static pressure distribution for a given
downstream Mach number is obtained from an inviscid time marching flow calculation.
An integral boundary layer calculation is used to calculate the momentum thickness at the trailing
edge before separation. The trailing edge shocks are calculated using the base pressure correlation.
Two examples are shown in Figure 6. Calculation of eight blades showed that 80% of the predicted
losses were within the range of the experimental uncertainty.
Besides the data reported by Sieverding et al. in [4,6], the only authors who published recently
a systematic investigation of the effect of the rear suction side curvature on the base pressure were
Granovskij et al. Of the Moscow Power Institute [10]. The authors investigated 4 moderately loaded
rotor blades (g/c = 0.73, dte /o = 0.12, β1 ≈ 85, β2 ≈ 22) with different unguided turning angles (ε = 2
to 16◦ ) in the frame of the optimization of cooled gas turbine blades. A direct comparison with the VKI
base pressure correlation is difficult because the authors omitted to indicate the trailing edge wedge
Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW
168
7 of 58
Since the base pressure losses are proportional to the base pressure coefficient
, the curves
Int. J. Turbomach.
Propuls.
Power 2020,
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10the strong variation of the profile losses in the transonic range. As regards
169 give
immediately
an idea
7 of 55
170 the low Mach number range, the contribution of the base pressure loss is implicitly taken into account
171 by all loss correlations. Therefore the base pressure loss is not to be added straight away to the profile
losses as predicted
for example with
the methods
[1] Figure
and Craig7and
Cox [7] but
as
angle δ172
a comparison
appeared
toby
beTraupel
useful.
presents
therather
comparison,
after
te . Nevertheless,
173 a difference with respect to the profile losses at , = 0.7:
conversion, of the base pressure coefficient:
=
174
175
176
−
,
.
)
pb − psin(
2
cpbase
Martelli and Boretti [9], used the VKI
b =pressure correlation for verifying a simple procedure
p2 static pressure distribution for a given
02 −
to compute losses in transonic turbine cascades. p
The
surface
,
Commented [M19]: Please confirm a
number of equation in order..
Commented [MM20R19]: No need to
downstream Mach number is obtained from an inviscid time marching flow calculation.
number here, and wherever it was n
2 , used by Fabry
used by
Granovskij
et al. [10], to the base pressure coefficient (2) based on V2,is
and
177
An integral boundary layer calculation is used to calculate the momentum thickness
at the
original manuscript
Sieverding
at
VKI
[8].
The
data
of
Granovskij
et
al.
[10]
(dashed
lines)
confirm
globally
the
overall
178 trailing edge before separation. The trailing edge shocks are calculated using the base pressure
examples
are shown in(solid
Figure 6.
Calculation
of eight blades
showed
of the
trends 179
of the correlation.
VKI baseTwo
pressure
correlation
lines).
However,
the peaks
inthat
the80%
transonic
range are
180 predicted losses were within the range of the experimental uncertainty.
much more pronounced.
181
(b)
(a)
182
Figure 6. Example of profile loss prediction for transonic turbine cascade, adapted from [9]; (a) low
Commented [M21]: Please add explan
Figure 6. Example of profile loss prediction for transonic turbine cascade, adapted from [9]; (a) low
183
pressure steam turbine tip section, (b) high pressure gas turbine guide vane.
Int.
J.
Turbomach.
Propuls.
Power
2018,
3,
x
FOR
PEER
REVIEW
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58
pressure steam turbine tip section, (b) high pressure gas turbine guide vane.
Commented [MM22R21]: Done
184
Besides the data reported by Sieverding et al. in [6] and [4], the only authors who published
185 recently a systematic investigation of the effect of the rear suction side curvature on the base pressure
186 were Granovskij et al. of the Moscow Power Institute [10]. The authors investigated 4 moderately
187 loaded rotor blades ( ⁄ = 0.73, / = 0.12, ≈ 85°, ≈ 22°) with different unguided turning
188 angles ( = 2° to 16°) in the frame of the optimization of cooled gas turbine blades. A direct
189 comparison with the VKI base pressure correlation is difficult because the authors omitted to indicate
190 the trailing edge wedge angle
. Nevertheless, a comparison appeared to be useful. Figure 7
191 presents the comparison, after conversion, of the base pressure coefficient:
=
−
−
192 used by Granovskij et al. [10], to the base pressure coefficient (2) based on , , used by Fabry and
193 Sieverding at VKI [8]. The data of Granovskij et al. [10] (dashed lines) confirm globally the overall
194 trends of the VKI base pressure correlation (solid lines). However, the peaks in the transonic range
195 are much more pronounced.
196 7. Comparison
Figure 7. Comparison
of Granovskij’s
base
pressure data
(dashed
lines) lines)
with thewith
VKI base
Figure
of Granovskij’s
base
pressure
data
(dashed
thepressure
VKI base pressure
197
correlation (solid lines).
correlation
(solid
lines).
198
Also, cascade data reported by Dvorak et al. in 1978 [11] on a low pressure steam turbine rotor
216
217
218
219
220
221
222
boundary layers the base pressure would increase with increasing momentum thickness. On the
contrary, supersonic flat plate model tests simulating the overhang section of convergent turbine
cascades with straight rear suction sides showed that for fully expanded flow along the suction side
(limit loading condition) an increase of the ratio of the boundary layer momentum to the trailing edge
thickness by a factor of two, obtained roughening the blade surface, did not affect the base pressure,
Sieverding et al. [18]. Note, that for both the smooth and rough surface the boundary layer was
turbulent. Similarly, roughening the blade surface in case of shock boundary layer interactions on the
Also,
data
et al.
In 1978
[11]
on arotor
low
pressure
steam
199 cascade
tip section,
andreported
by Jouini et by
al. inDvorak
2001 [12] for
a relatively
high
turning
blade
(∆ = 110°,
and aturbine rotor
200 and
smaller
to chord
/ = 0.73),
agreement high
with the
VKI baserotor
pressure
correlation,
tip section,
by pitch
Jouini
et al.ratio
In 2001
[12] are
forinafair
relatively
turning
blade
(∆β = 110, and a
201 although the latter authors state that below / = 0.45, their data drop below those of the BPC.
smaller202
pitch to chord
ratio g/c = 0.73), are in fair agreement with the VKI base pressure correlation,
However, some other cascade measurements deviate very significantly from the VKI curves.
although
latter authors
state
that
below
p2 /p
= 0.45,
data
those
of the BPC.
203the Deckers
and Denton
[13], for
a low
turning
blade
andtheir
Gostelow
et drop
al. [14] below
for a high
turning
01 model
204
nozzle
guide
vane,
report
base
pressure
data
far
below
those
of
Sieverding’s
BPC,
while
Xu
and
However, some other cascade measurements deviate very significantly from the VKI curves.
205 Denton [15], for a very highly loaded HP gas turbine rotor blade (∆ = 124° and / = 0.84) report
Deckers
Denton
[13], for a low turning blade model and Gostelow et al. [14] for a high turning
206and base
pressure data far above those of the BPC.
nozzle207
guide vane,
report base
pressure
farcorrelation
below those
of criticized
Sieverding’s
BPC,
while Xu and
The simplicity
of Sieverding’s
basedata
pressure
was often
because it
was felt
208
that
aspects
as
important
as
the
state
of
the
boundary
layer,
the
ratio
of
boundary
layer
momentum
Denton [15], for a very highly loaded HP gas turbine rotor blade (∆β = 124 and g/c = 0.84) report
209 to trailing edge thickness and the trailing edge blockage effects (trailing edge thickness to throat
base pressure
data far
above those of the BPC.
210 opening)
should play an important role.
The
simplicity
of
Sieverding’s
base
pressure
wastests
often
because
it was felt
211
As regards
the state of the
boundary
layercorrelation
and its thickness,
on acriticized
flat plate model
at
212
moderate
subsonic
Mach
numbers
in
a
strongly
convergent
channel
by
Sieverding
and
Heinemann
that aspects as important as the state of the boundary layer, the ratio of boundary layer momentum to
213 [16], showed that the difference of the base pressure for laminar and turbulent flow conditions was
trailing214
edge only
thickness
andofthe
trailing
edge blockage
(trailing
edge
thickness
throat opening)
of the order
1.5–2%
of the dynamic
head of theeffects
flow before
separation
from
the trailingto
edge.
should215
play an
role.
For important
the case of supersonic
trailing edge flows, Carriere [17], demonstrated, that for turbulent
Int. J. Turbomach. Propuls. Power 2020, 5, 10
8 of 55
As regards the state of the boundary layer and its thickness, tests on a flat plate model at moderate
subsonic Mach numbers in a strongly convergent channel by Sieverding and Heinemann [16], showed
that the difference of the base pressure for laminar and turbulent flow conditions was only of the order
of 1.5–2% of the dynamic head of the flow before separation from the trailing edge. For the case of
supersonic trailing edge flows, Carriere [17], demonstrated, that for turbulent boundary layers the base
pressure would increase with increasing momentum thickness. On the contrary, supersonic flat plate
model tests simulating the overhang section of convergent turbine cascades with straight rear suction
sides showed that for fully expanded flow along the suction side (limit loading condition) an increase
of the ratio of the boundary layer momentum to the trailing edge thickness by a factor of two, obtained
roughening the blade surface, did not affect the base pressure, Sieverding et al. [18]. Note, that for
both the smooth and rough surface the boundary layer was turbulent. Similarly, roughening the blade
surface in case of shock boundary layer interactions on the blade suction side did not affect the base
pressure as compared to the smooth blade, Sieverding and Heinemann [16]. However, a comparison
of the base pressure for the same Mach numbers before separation at the trailing edge for a fully
expanding flow and a flow with shock boundary layer interaction on the suction side before the TE
showed an increase of the base pressure by 10–25 % in case of shock interaction before the TE. Since
it was shown before that an increase of the momentum thickness did not affect the base pressure,
the difference may be attributed to (a) different total pressures due to shock losses for the shock
interference curve, (b) differences in the boundary layer shape factor and (c) differences in pressure
gradients in stream-wise direction in the near wake region.
A systematic investigation of possible effects of changes in shape factor and boundary layer
momentum thickness on the base pressure in cascades is difficult. Hence, the investigations are mostly
confined to variations of the incidence angle which, via a modification of the blade velocity distribution,
should have an impact on both the shape factor and the boundary layer momentum thickness. Based
on linear transonic cascade tests on two high turning rotor blades Jouini et al. [19] at Carlton University,
(blade HS1A: g/c = 0.73, dte /o = 0.082, β1 = 39.5, β2 = 31, δte = 6, ε = 11.5; blade HS1B is similar to
HS1A, but with less loading on the front side and β1 = 29) concluded that discrepancies in the base
region did not appear to be strongly related to changes of the inlet angle by ±14.5, however in broad
terms the weakest base pressure drop in the transonic range were obtained for high positive incidence.
Similarly, experiments at VKI on a high turning rotor blade (g/c = 0.49, dte /o = 0.082, β1 = 45, β2 = 28,
δte = 10, ε = 10) did not show any effect on the base pressure for incidence angle changes of ±10 [3].
In conclusion it appears that for conventional blade designs, changes in the boundary layer
thickness alone, as induced by incidence variations, do not affect significantly the base pressure.
Therefore, we need to look for possible other influence factors.
Figure 3 showed that the effect of the blade rear suction side blade turning angle ε on the base
pressure was in fact function of the pressure difference across the blade near the trailing edge. Inversely,
one should be able to deduct from the rear blade loading the tendency of the base pressure. The higher
the blade loading at the trailing edge, the higher the base pressure. Corollary, a low or even negative
blade loading near the trailing edge causes increasingly lower base pressures. This might help in
explaining the large differences with respect to the BPC as found by Xu and Denton [15] on one side
and Deckers et al. [13] and Gostelow et al. [14], mentioned before, on the other side.
To illustrate this, Figure 8 presents the base pressure data of Xu and Denton [15] for three of a
family of four very highly loaded gas turbine rotor blades with a blade turning angle of ∆β = 124
and a pitch-to-chord g/c = 0.84, tested with three different trailing edge thicknesses. The blades are
referred to as blade RD, for the datum blade, and blades DN and DK for changes of 0.5 and 1.5 times
the trailing edge thickness with respect to the datum case.
The base pressures are overall much higher than those of the BPC which are indicated in the figure
by the dashed line for a mean value of (ε + δte )/2 = 9.
A possible explanation for the large differences is given by comparing the blade Mach number
distribution of the datum blade with that of a VKI blade with a (ε + δte )/2 = 16 taken from [6],
253 [15] on one side and Deckers et al. [13] and Gostelow et al. [14], mentioned before, on the other side.
254
To illustrate this, Figure 8 presents the base pressure data of Xu and Denton [15] for three of a
255 family of four very highly loaded gas turbine rotor blades with a blade turning angle of ∆ = 124°
256 and a pitch-to-chord / = 0.84, tested with three different trailing edge thicknesses. The blades are
257 referred to as blade RD, for the datum blade, and blades DN and DK for changes of 0.5 and 1.5 times
258 the trailing edge thickness with respect to the datum case.
Int. J. Turbomach.
Propuls.
Power
2020, are
5, 10overall much higher than those of the BPC which are indicated in the
9 of 55
259
The base
pressures
260 figure by the dashed line for a mean value of ( + )/2 = 9°.
261
A possible explanation for the large differences is given by comparing the blade Mach number
262 9.distribution
of the
the datum
blade withthe
thatblade
of a VKIMach
blade with
a ( +distribution
)/2 = 16° taken
from&[6],
see
see Figure
To enable
comparison,
number
of Xu
Denton
(solid line)
263 Figure 9. To enable the comparison, the blade Mach number distribution of Xu & Denton (solid line)
presented
originally
in
function
of
the
axial
chord
x/c
,
had
to
be
replotted
in
function
of x/c.
264 presented originally in function of the axial chord / , hadaxto be replotted in function of / . The
The comparison
is done
forforananisentropic
outlet
number
M2,is = 0.8.
265 comparison
is done
isentropic outlet
MachMach
number
, = 0.8.
266
Commented [M23]: Please add explanat
267 8. Base
Figure
8. Base pressure
variation
blades of
of Xu
Denton;
blade blade
RD datum
blade case,
DK thick
Figure
pressure
variation
forfor
blades
Xu& &
Denton;
RDcase,
datum
blade DK thick
268
trailing edge, blade DN thin trailing edge. Adapted from [15].
trailing edge, blade DN thin trailing edge. Adapted from [15].
Commented [MM24R23]: This is not a s
269
Note that the geometric throat for the Xu & Denton blade is situated at
/ ≈ 0.34, while for the
geometries are well explained by the capt
Note
geometric
theedge,
Xu &
bladedifference
is situated
at x/c
≈ 0.34,
for theis inappropriate.
270 that
VKI the
blade
at / = 0.5.throat
At the for
trailing
theDenton
Mach number
between
pressure
and whilereferencing
VKI blade at x/c = 0.5. At the trailing edge, the Mach number difference between pressure and suction
J. Turbomach.
Power
2018,same,
3, x FOR but
PEER contrary
REVIEW
10 of 58 for the VKI
side for both Int.
blades
are Propuls.
exactly
the
to the nearly constant Mach number
blade downstream
of the
throat,
the
and
characterized
a very
strong adverse
271 suction side
for both
blades
areblade
exactlyof
theXu
same,
butDenton
contrary is
to the
nearly constantby
Mach
number
272gradient
for the in
VKI
blade
downstream
of the throat,
Xu and Denton
is characterized
by a very
pressure
this
region.
As pointed
outthe
byblade
the ofauthors,
this causes
the suction
side boundary
strong adverse pressure gradient in this region. As pointed out by the authors, this causes the suction
layer to273
be either
separated or close to separation up-stream of the trailing edge. Clearly, Sieverding’s
274 side boundary layer to be either separated or close to separation up-stream of the trailing edge.
correlation
with blade
designs
by very
strong by
adverse
pressure
275 cannot
Clearly,deal
Sieverding’s
correlation
cannot characterized
deal with blade designs
characterized
very strong
adverse gradients on
pressure
the rear suction
side
causing boundary
layer
the rear276
suction
sidegradients
causingonboundary
layer
separation
before
theseparation
trailingbefore
edge.the trailing
277
edge.
Figure
of blade
Mach
distribution
RD
of Xu
278 9. Comparison
Figure 9. Comparison
of blade
Machnumber
number distribution
for bladefor
RD blade
of Xu and
Denton
[15],and
(solid Denton [15],
5
6
279
curve,
=
0.8,
=
8
10
)
with
VKI
blade
(dashed
curve,
=
0.8,
=
10
).
, curve, M2,is = 0.8, Re2 = 10 ).
(solid curve, M2,is = , 0.8, Re2 = 8 × 10 ) with VKI blade (dashed
Commented [M25]: Please add explan
280
The possible effect of boundary layer separation resulting from high rear suction side diffusion
Commented [MM26R25]: This is not
The
effect of boundary layer separation resulting from high rear suction side diffusion
281possible
resulting in high base pressures was also mentioned by Corriveau and Sjolander in 2004 [20],
geometries are well explained by the ca
resulting
base pressures
was
also mentioned
Corriveau
and
Sjolander
2004
282in high
comparing
their nominal
mid-loaded
rotor bladeby
HS1A,
mentioned
already
before,in
with
an [20],
aft- comparing
referencing is inappropriate.
283 loaded
blade HS1C
withblade
an increase
of the
suction side already
unguided before,
turning angle
to 14.5°. blade HS1C
their nominal
mid-loaded
rotor
HS1A,
mentioned
withfrom
an 11.5°
aft- loaded
284 It appears that the increased turning angle could cause, in the transonic ◦range, shock
induced
with an285
increase
of the suction side unguided turning angle from 11.5 to 14.5◦ . It appears that the
boundary layer transition near the trailing edge with, as consequence, a sharp increase of the base
increased
angle
in base
the transonic
range,asshock
boundary
286 turning
pressure,
i.e. a could
sudden cause,
drop in the
pressure coefficient
seen in induced
Figure 10. Note
that the layer transition
287
reported
in with,
the figure
been convertedato
−
of the original
data.base pressure, i.e., a sudden drop in
near the
trailing
edge
as has
consequence,
sharp
increase
of the
288
As regards the base pressure data by Deckers and Denton [13] for a low turning blade model
the base
coefficient
Figure
10.guide
Note
that
cpbase
in far
thebelow
figure has been
b reported
289pressure
and Gostelow
et al. [14]as
forseen
a highin
turning
nozzle
vane,
whothe
report
pressure data
converted
−cpb ofofSieverding’s
the original
data.
290 to those
BPC,
their blade pressure distribution resembles that of the convergent/
divergent
bladepressure
C in Figuredata
3 with
negative blade
near
thefor
trailing
edge
which would
As291
regards
the base
byaDeckers
and loading
Denton
[13]
a low
turning
blade model and
292 explain the very low base pressures. In addition, the blade of Deckers and Denton has a blunt trailing
Gostelow
et
al.
[14]
for
a
high
turning
nozzle
guide
vane,
who
report
base
pressure
data far below
293 edge, and there is experimental evidence that, compared to a circular trailing edge, the base pressure
for blades BPC,
with blunt
trailing
might bedistribution
considerably lower.
Sieverding
[16]
those of294
Sieverding’s
their
bladeedge
pressure
resembles
thatand
of Heinemann
the convergent/divergent
report 3forwith
flat plate
tests at moderate
numbers
a drop ofedge
the base
pressure
coefficient
blade C295
in Figure
a negative
blade subsonic
loadingMach
near
the trailing
which
would
explain the very
296 by 11% for a plate with squared trailing edge compared to that with a circular trailing edge.
low base pressures. In addition, the blade of Deckers and Denton has a blunt trailing edge, and there is
This image cannot currently be displayed.
289
289
290
290
291
291
292
292
293
293
294
294
295
295
296
296
and
andGostelow
Gostelowetetal.
al.[14]
[14]for
foraahigh
highturning
turningnozzle
nozzleguide
guidevane,
vane,who
whoreport
reportbase
basepressure
pressuredata
datafar
farbelow
below
those
those ofof Sieverding’s
Sieverding’s BPC,
BPC, their
their blade
blade pressure
pressure distribution
distribution resembles
resembles that
that ofof the
the convergent/
convergent/
divergent
divergentblade
bladeCCininFigure
Figure33with
withaanegative
negativeblade
bladeloading
loadingnear
nearthe
thetrailing
trailingedge
edgewhich
whichwould
would
explain
explainthe
thevery
verylow
lowbase
basepressures.
pressures.In
Inaddition,
addition,the
theblade
bladeofofDeckers
Deckersand
andDenton
Dentonhas
hasaablunt
blunttrailing
trailing
edge,
isisexperimental
edge,and
andJ.there
there
experimental
evidence
that,compared
comparedtotoaacircular
circulartrailing
trailingedge,
edge,the
thebase
basepressure
pressure10 of 55
Int.
Turbomach.
Propuls. Powerevidence
2020,
5, 10 that,
for
forblades
bladeswith
withblunt
blunttrailing
trailingedge
edgemight
mightbe
beconsiderably
considerablylower.
lower.Sieverding
Sieverdingand
andHeinemann
Heinemann[16]
[16]
report
reportfor
forflat
flatplate
platetests
testsatatmoderate
moderatesubsonic
subsonicMach
Machnumbers
numbersaadrop
dropofofthe
thebase
basepressure
pressurecoefficient
coefficient
experimental
evidence
that,
compared
to
a circular
trailing
edge,
the base
pressure
by
for
squared
trailing
edge
totothat
aacircular
trailing
edge.
by11%
11%
foraaplate
platewith
with
squared
trailing
edgecompared
compared
thatwith
with
circular
trailing
edge.for blades with
blunt trailing edge might be considerably lower. Sieverding and Heinemann [16] report for flat plate
tests at moderate subsonic Mach numbers a drop of the base pressure coefficient by 11% for a plate
with squaredInt.trailing
to that
with a circular trailing edge.
J. Turbomach.edge
Propuls.compared
Power 2018, 3, x FOR
PEER REVIEW
13 of 65
297
297
298
298
298Base
Figure
10.
Base pressure
coefficient
for mid-loaded
(solid
line)
and
aft-loaded
(dashed
line)line)
rotor
Figure
10.
coefficient
for
mid-loaded
(solid
line)
and
aft-loaded
(dashed
FigureFigure
10.
Base
pressure
coefficient
for
mid-loaded
(solid
line)
and
aft-loaded
(dashed
line)
rotor
10.pressure
Base
pressure
coefficient
for mid-loaded
(solid
line)
and
aft-loaded
(dashed
line)rotor
rotor blade.
299
blade. Symbols:
HS1A geometry,
HS1C geometry. Adapted from [20].
blade.
HS1C
geometry.
Adapted
from
[20].
Symbols: HS1A
HS1A
geometry,
geometry.
Adapted
from
[20].
blade.Symbols:
HS1Ageometry,
geometry, HS1C
HS1C
geometry.
Adapted
from
[20].
300
It is important to remember that the measurement of the base pressure carried out with a single
330
perform some systematic schlieren visualizations on transonic flat plate and cascades with different
It301
is important
to remember
thatedge
thebase
measurement
of assumption
the base pressure
carried
out with a single
pressure tapping
in the trailing
region implies the
of an isobaric
trailing edge
302tapping
pressure
However,
2003region
Sieverding
et al. [21]
thatof
at an
highisobaric
subsonic trailing edge
pressure
indistribution.
the trailing
edge in
base
implies
thedemonstrated
assumption
303distribution.
Mach numbers the pressure distribution could be highly non-uniform with a marked pressure
pressure
However, in 2003 Sieverding et al. [21] demonstrated that at high subsonic Mach
304 minimum at the center of the trailing edge base, as will be shown later in Section 5. Under these
numbers
the
pressure
could
be highly
non-uniform
with ahole
marked
305 conditions it isdistribution
likely that the base
pressure
measured
with a single pressure
does notpressure
reflect the minimum at
306
true
mean
pressure.
In
addition,
the
measured
pressure
would
depend
on
the
ratio
of
the
pressure
the center of the trailing edge base, as will be shown later in Section 5. Under these conditions
it is likely
307
hole to trailing edge diameter / , which is typically in the range / = 0.15 − 0.50. This fact was
that the base pressure measured with a single pressure hole does not reflect the true mean pressure.
308 also recognized by Jouini et al. [12], who mentioned the difficulties for obtaining representative
In addition,
the measured
pressure
would depend
ratio
ofatthe
hole
to trailing edge
309 trailing
edge base pressures
measurements:
“It shouldon
alsothe
be noted
that
high pressure
Mach numbers
the base
310 d/D,
pressure
varies
location
on the
trailing
edge and the single
gives
a somewhat
limited
diameter
which
is considerably
typically with
in the
range
d/D
= 0.15–0.50.
This tap
fact
was
also recognized
by Jouini
311 who
picture of the base pressure behavior”. It is probably correct to say that differences between experimental
et al. [12],
mentioned the difficulties for obtaining representative trailing edge base pressures
312 base pressure data and the base pressure correlation may at least partially be attributed to the use of
measurements:
“It should
noted edge
that diameters
at high Mach
numbers
theresearchers.
base pressure varies considerably with
313 different
pressurealso
hole be
to trailing
/ by
the various
314
Finally, edge
it is important
mention
the trailing
edge pressure
is sensitive
edge
location
on the trailing
and the tosingle
tapthat
gives
a somewhat
limited
picture to
ofthe
thetrailing
base pressure
behavior”.
315 shape as demonstrated by El Gendi et al. [22] who showed with the help of high fidelity simulation
It is probably
correct to say that differences between experimental base pressure data and the base
316 that the base pressure for blades with elliptic trailing edges was higher than for blades with circular
pressure
mayMelzer
at least
attributed
to thethat
use
of different
pressure
hole to trailing
317correlation
trailing edges.
and partially
Pullan [23] be
proved
experimentally
designing
blades with
elliptical
318 trailing
edges
the blade
performance. The reason is that an elliptic trailing edge reduces
edge diameters
d/D
byimproved
the various
researchers.
319 not only the wake width but causes also an increase of the base pressure compared to that of blades
Finally,
it is important to mention that the trailing edge pressure is sensitive to the trailing edge
320 with a circular trailing edge. This suggests that inaccuracies in the machining of blades with thin
shape 321
as demonstrated
by El
Gendi
al. [22] from
whotheshowed
with the
help
ofshape
highand
fidelity
trailing edges could
easily
lead toetdeviations
designed circular
trailing
edge
thus simulation
322basecontribute
to the
in the elliptic
base pressure.
that the
pressure
fordifferences
blades with
trailing edges was higher than for blades with circular
trailing
edges.
Melzer and Pullan [23] proved experimentally that designing blades with elliptical
323
3. Unsteady Trailing Edge Wake Flow
trailing
edges
improved
the blade performance. The reason is that an elliptic trailing edge reduces
324
The mixing process of the wake behind turbine blades has been viewed for a long time as a
not only
the
wake
width
butalthough
causesit also
an increase
base pressure
compared
325 steady state process
was well
known thatof
thethe
separation
of the boundary
layers atto
thethat of blades
326
trailing
edge
is
a
highly
unsteady
phenomenon
which
leads
to
the
formation
of
large
coherent
with a circular trailing edge. This suggests that inaccuracies in the machining of blades with thin
327 structures, known as the von Kármán vortex street. The unsteady character of turbine blade wakes is
trailing
edges could easily lead to deviations from the designed circular trailing edge shape and thus
328 best illustrated by flow visualizations.
contribute
differences
in the base
329 to theLawaczeck
and Heinemann
[24],pressure.
and Heinemann and Bütefisch [25], were probably the first to
331 trailing
edgeEdge
thicknesses
using
a flash light of 20 nano-seconds only, and deriving from the photos
3. Unsteady
Trailing
Wake
Flow
332
the vortex shedding frequencies and Strouhal numbers. The schlieren picture in Figure 11 shows
333mixing
impressively
thatof
the
shedding
each vortex
from the
trailinghas
edgebeen
generates
a pressure
which
The
process
the
wakeofbehind
turbine
blades
viewed
for wave
a long
time as a steady
334 travels upstream.
state process
although it was well known that the separation of the boundary layers at the trailing edge
is a highly unsteady phenomenon which leads to the formation of large coherent structures, known as
the von Kármán vortex street. The unsteady character of turbine blade wakes is best illustrated by
flow visualizations.
Lawaczeck and Heinemann [24], and Heinemann and Bütefisch [25], were probably the first to
perform some systematic schlieren visualizations on transonic flat plate and cascades with different
trailing edge thicknesses using a flash light of 20 nano-seconds only, and deriving from the photos
the vortex shedding frequencies and Strouhal numbers. The schlieren picture in Figure 11 shows
impressively that the shedding of each vortex from the trailing edge generates a pressure wave which
travels upstream.
328
329
330
331
332
333
Lawaczeck and Heinemann [24], and Heinemann and Bütefisch [25], were probably the first to
perform some systematic schlieren visualizations on transonic flat plate and cascades with different
trailing edge thicknesses using a flash light of 20 nano-seconds only, and deriving from the photos
the vortex shedding frequencies and Strouhal numbers. The schlieren picture in Figure 11 shows
impressively that the shedding of each vortex from the trailing edge generates a pressure wave which
Int. J. Turbomach. Propuls. Power 2020, 5, 10
11 of 55
travels upstream.
334
Figure 11.
11. Schlieren
Schlierenpicture
pictureof
of turbine
turbine rotor
rotor blade
blade wake
wake at
at M2,is
= 0.8
0.8 [24].
Figure
[24].
, =
335
336
337
338
Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW
12 of 58
In 1982 Han and Cox [26], performed smoke visualizations on a very large-scale nozzle blade
at
In
1982 Han and Cox [26], performed smoke visualizations on a very large-scale nozzle blade at
low speed (Figure 12). The authors found much sharper and well-defined contours of the vortices
low speed
(Figureside
12). and
The concluded
authors found
much
sharper
and well-defined
contours
ofthis
theside
vortices
from
the
pressure
that this
implied
stronger
vortex
shedding
from
and
Figure
40. Blade
distributions
front and
rear-loaded
blades;
adapted
from
[16].
from the
pressure
sideMach
and number
concluded
that thisfor
implied
stronger
vortex
shedding
from
this
side and
attributed this to the circulation around the blade.
attributed this to the circulation around the blade.
A
B
339
Figure 12.
12. Smoke
Smoke visualization
visualization of
of the
the vortex
vortex shedding
shedding from
from aa low
low speed
speed nozzle
nozzle blade
blade [26].
[26].
Figure
340
341
342
343
344
345
346
347
Beretta-Piccoli [27] (reported by Bölcs and Sari [28]) was possibly the first to use interferometry to
Beretta-Piccoli [27] (reported by Bölcs and Sari [28]) was possibly the first to use interferometry
visualize the vortex formation at the blunt trailing edge of a blade at transonic flow conditions.
to visualize the vortex formation at the blunt trailing edge of a blade at transonic flow conditions.
C Besides the problem of time resolution for measuring high frequency phenomena, there was also
Besides the problem of time resolution for measuring high frequency
phenomena, there was also
the problem of spatial resolution for resolving
the vortex
structures
behind
the usually
rather thin
𝜶∗𝟐 behind
Bladethe
𝜶𝟏 structures
Ref rather
g/cthe
the problem of spatial resolution for resolving
vortex
usually
thin
turbine blade trailing edges. First tests on a large
scale
flat
late
model
simulating
the
overhang
section
A
30°
22°
0.75
[5]
turbine blade trailing edges. First tests on a large scale flat late model simulating the overhang section
of a cascade allowed to visualize impressively details of the vortex
shedding at transonic
outlet Mach
25°
VKIoutlet
of a cascade allowed to visualize impressivelyBdetails of60°
the vortex
shedding0.75
at transonic
Mach
number (Figure 13a,b).
C
66°
18°
0.70
VKI
number
(Figure 13a,b).
D
Following Hussain and Hayakawa [29],Dthe wake156°
vortex structures
can
be described
19.5°
0.85
[6] by a set of
centers which characterize the location of a peak of coherent span-wise vortices and saddles located
between the coherent vorticity structures and defined by a minimum of coherent span-wise vorticity.
Figure 1.span-wise
Blade profile
losses versus
isentropicbyoutlet
number
for four transonic
turbines.
The successive
vortices
are connected
ribs,Mach
that are
longitudinal
smaller scale
vortices of
alternating signs.
(a) turbine blade flat plate model.
(b) schlieren photograph.
(c) topology of wake vortex structure
behind a cylinder [29].
348
Figure 13. Vortex shedding at transonic exit flow conditions [30].
349
350
351
352
353
354
355
356
(c) topology
vortex structure
Following Hussain and Hayakawa [29], the wake vortex structures
can of
bewake
described
by a set of
(a) turbine blade flat plate model.
(b) schlieren photograph.
behind a cylinder
[29]. located
centers which characterize the location of a peak of coherent span-wise vortices
and saddles
between the coherentFigure
vorticity Vortex
structures and defined
by a minimum
of coherent
span-wise vorticity.
conditions [30].
[30].
Figure 13.
13. Vortex shedding
shedding at
at transonic
transonic exit
exit flow
flow conditions
The successive span-wise vortices are connected by ribs, that are longitudinal smaller scale vortices
of alternating signs.
A significant progress was made in the 1990’s
1 in the frame of two European Research Projects,
i.e. Experimental and Numerical Investigations of Time Varying Wakes behind Turbine Blades
(BRITE/EURAM CT-92-0048) and Turbulence Modeling for Unsteady Flows in Axial Turbines
Int. J. Turbomach. Propuls. Power 2020, 5, 10
12 of 55
A significant progress was made in the 1990’s in the frame of two European Research Projects, i.e.,
Experimental and Numerical Investigations of Time Varying Wakes behind Turbine Blades (BRITE/EURAM
CT-92-0048) and Turbulence Modeling for Unsteady Flows in Axial Turbines (BRITE/EURAM CT-96-0143) in
which the von Kármán Institute, the University of Genoa and the ONERA of Lille used short duration
flow visualizations and fast response instrumentation in combination with large scale blade models
to improve the understanding of the formation of the vortical structures at the turbine blade trailing
edges and their impact on the unsteady wake flow characteristics. Results of these research projects
are reported by Ubaldi et al. [31], Cicatelli and Sieverding [32], Desse [33], Sieverding et al. [34], Ubaldi
and Zunino [35] and Sieverding et al. [21,36].
The large-scale turbine guide vane used in these experiments was designed at VKI (Table 2) and
released in 1994 [37]. The blade design features a front-loaded blade with an overall low suction side
turning in the overhang section and, in particular, a straight rear suction side from halfway downstream
of the throat, Figure 14. Due to mass flow restrictions in the VKI blow down facility, the three-bladed
cascade with a chord length c = 280 mm was limited to investigations at a relatively low subsonic
outlet Mach number of M2,is = 0.4. The suction side boundary layer undergoes natural transition at
Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW
13 of 58
x/cax ∼ 0.6. On the pressure side the boundary layer was tripped at x/cax ∼ 0.61. The boundary
cascade
with ashape
chord length
280
wasand
limited
to investigations
at a relatively
layers368
at thethree-bladed
trailing edge
with
factors=H
ofmm
1.64
1.41
for the pressure
and suction sides
369 low subsonic outlet Mach number of , = 0.4. The suction side boundary layer undergoes natural
respectively,
were clearly turbulent. The schlieren photographs in Figure 14 were taken with a Nanolite
370 transition at / ~0.6. On
the pressure side the boundary layer was tripped at / ~0.61. The
−9 s. The dominant vortex shedding frequency was 2.65 kHz and the
spark 371
source,
with ∆t
= 20
boundary
layers
at ×
the10
trailing
edge with shape factors
of 1.64 and 1.41 for the pressure and
372 suction
sides respectively,
clearly as:
turbulent. The schlieren photographs in Figure 14 were taken
corresponding
Strouhal
number,were
defined
373
374
with a Nanolite spark source, with ∆ = 20 10
. The dominant vortex shedding frequency was
2.65 kHz and the corresponding Strouhal number, defined
fvs dte as:
St =
=
was
was St375
= 0.27.
,
V2,is
(3)
(3)
= 0.27.
376
(a)
(b)
(c)
377 14. Very
Figure 14. Very large-scale turbine nozzle guide vane (VKI LS-94);
2 x 10 case. (a)
, = 0.4,
Figure
large-scale turbine nozzle guide vane (VKI LS-94);
M2,is= =
0.4, Re2 = 2 × 106 case.
378
test section, (b) surface isentropic Mach number distribution, (c) schlieren photographs. Adapted
(a)
test section,
379
from(b)
[32].surface isentropic Mach number distribution, (c) schlieren photographs. Adapted
from [32].
380
Figure 14c presents two instances in time of the vortex shedding process. The left flow
381 14c
visualization shows the enrolment of the pressure side shear layer into a vortex, the right one the
Figure
presents two instances in time of the vortex shedding process. The left flow visualization
382 formation of the suction side vortex. Note that the pressure side vortex appears to be much stronger
shows383
the enrolment
of the
side shear
layer intomade
a vortex,
theCox
right
than the suction
side pressure
one, which confirms
the observations
by Han and
[26].one the formation of the
suction side vortex. Note that the pressure side vortex appears to be much stronger than the suction
384
Table 2. VKI LS-94 large scale nozzle blade geometric characteristics [32].
side one, which confirms the observations made by Han and Cox [26].
Parameter
Symbol
Value
Gerrard [38], describes the
vortex formation for the flow
behind
a cylinder as follows, Figure 15.
Chord
c
280 mm
The growing vortex (A) is fed Pitch
by the
circulation
existing
in
the
upstream
shear layer until the vortex is
to chord ratio
0.73
/
ratio
strong enough to entrain fluid Blade
fromaspect
the opposite
shear layer ℎ/
bearing0.7
vorticity of the opposite circulation.
Stagger angle
−49.83°
When the quantity of entrained
fluid is sufficient to cut off /the supply
of circulation to the growing
0.053
Trailing edge thickness to throat ratio
Trailing edge wedge angle
Gauging angle (arcsin( / ))
385
386
387
∗
7.5°
19.1°
Gerrard [38], describes the vortex formation for the flow behind a cylinder as follows, Figure 15.
The growing vortex (A) is fed by the circulation existing in the upstream shear layer until the vortex
is strong enough to entrain fluid from the opposite shear layer bearing vorticity of the opposite
Int. J. Turbomach. Propuls. Power 2020, 5, 10
13 of 55
vortex—the opposite vorticity of the fluid in both shear layers cancel each other—then the vortex is
shed off.
Table 2. VKI LS-94 large scale nozzle blade geometric characteristics [32].
Parameter
388
389
Symbol
Chord
Pitch to chord ratio
Blade aspect ratio
Int. J. Turbomach. Propuls.Stagger
Power 2018,
3, x FOR PEER REVIEW
angle
Trailing edge thickness to throat ratio
Trailingopposite
edge wedge
angle of the fluid in
the growing vortex—the
vorticity
(o/g))
Gauging
angle
(arcsin
the vortex is shed off.
Value
c
280 mm
g/c
0.73
h/c
0.7
14 of 58
γ
−49.83◦
dte /o
0.053
δte shear layers
7.5◦ cancel each other—then
both
∗
α
19.1◦
390
adapted from
from [38].
[38].
Figure 15. Vortex formation mechanism; adapted
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
Contrary to the blow down tunnel at VKI, the Istituto di Macchine e Sistemi Energetici (ISME)
at the University of Genoa
Genoa used
used aa continuous
continuous running
running low
low speed
speed wind
wind tunnel.
tunnel. Miniature cross-wire
hot-wire probe and a four-beam
four-beam laser Doppler
Doppler velocimeter are used for the measurements
measurements of the
unsteady wake. An example of the instantaneous patterns of the ensemble averaged periodic wake
characteristics is presented in Figure 16. A detailed
detailed description
description is
is given
given by
by Ubaldi
Ubaldi and
and Zunino
Zunino [35].
[35].
es − Us in−Figurein16Figure
The
component
of the
U
(upper 16
left),
showsleft),
asymmetric
The streamwise
streamwiseperiodic
periodic
component
ofvelocity,
the velocity,
(upper
shows
periodic
patterns
of alternating
positive
and negative
components
issued
from theissued
pressure
to
asymmetric
periodic
patterns of
alternating
positive velocity
and negative
velocity
components
from
the suction
side.
As already
schematically
in Figure
13, saddlein
points
separating
groups
of
pressure
to the
suction shown
side. As
already shown
schematically
Figure
13, saddle
points
four
cores, are
located
along
the wake
center along
line. On
contrary,
periodic
of thethe
transverse
separating
groups
of four
cores,
are located
thethe
wake
centerthe
line.
On the parts
contrary,
periodic
e
component
right) appear
cores of
positive
andasnegative
n − Un (upper
parts of the U
transverse
component
− as(upper
right)
appear
cores ofvalues,
positiveapproximately
and negative
centered
in the wake which
alternate,
in streamwise
direction.inThe
combination
of the two
values, approximately
centered
in theenlarging
wake which
alternate, enlarging
streamwise
direction.
The
velocity
components
give
rise
to
the
rolling
up
of
the
periodic
flow
into
a
row
of
vortices
rotating
in
combination of the two velocity components give rise to the rolling up of the periodic flow into a row
opposite
direction
by direction
the velocity
vector plots
left).
of vortices
rotatingasinshown
opposite
as shown
by the(lower
velocity
vector plots (lower left).
As illustrated by
by Gerrard
Gerrard [38]
[38](see
(seeFigure
Figure15),
15),the
thevortex
vortexformation
formationis is
driven
vorticity
driven
byby
thethe
vorticity
in
e
in
suction
pressure
boundary
layers.
The vorticity
ω and
inwake
the wake
thethe
suction
andand
pressure
sideside
boundary
layers.
The vorticity
termsterms
and
in ωthe
have have
been
been
respectively
the
curl
the phase
averaged
and averaged
time averaged
velocity
determined
taking
respectively
the curl
theofphase
averaged
and time
velocity
field: field:
=
determined
taking
of
Us
∂U n
∂Us
∂Un
e = −∂∂n
ω
− and
and
, Figure
16 (lower
Themaxima
local maxima
and minima
and
= ω = − ∂n −, Figure
16 (lower
right).right).
The local
and minima
and saddle
∂s
∂s
saddle
(the points
the vorticity
changes
its sign)
the location,
extension,
rotation
regionsregions
(the points
where where
the vorticity
changes
its sign)
definedefine
the location,
extension,
rotation
and
and
intensity
of
the
vortices.
intensity of the vortices.
With increasing downstream Mach number, the vortices become much more intense as
demonstrated in Figure 17 on a half scale model of the blade already presented in Figure 14 and Table 2,
at an outlet Mach number M2,is = 0.79 in a four bladed cascade, Sieverding et al. [36]. Contrary to
schlieren photographs which visualize density changes, the smoke visualizations in Figure 17 show
the instantaneous flow patterns and are therefore particular well suited to visualize the enrolment of
the vortices. A close look at the vortex structures reveals that the distances between successive vortices
change. In fact, the distance between a pressure side vortex and a suction side vortex is always smaller
than the distance between two successive pressure side vortices. A possible reason is that the pressure
side vortex plays a dominant role and exerts an attraction on the suction side vortex as already found
by Han and Cox [26].
e
e
Int. J. Turbomach. Propuls. Power 2020, 5, 10
Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW
14 of 55
15 of 58
(a)
(b)
(c)
(d)
411
412
Figure 16.
Instantaneous realization
realization of the ensemble averaged streamwise velocity (a), transversal
16. Instantaneous
transversal
velocity (b), velocity vector
vector (c)
(c) and
and vorticity
vorticity patterns
patterns (d)
(d) of
of the
the periodic
periodic flow
flow [35].
[35].
413
414
415
416
417
418
419
420
421
422
With increasing downstream Mach number, the vortices become much more intense as
demonstrated in Figure 17 on a half scale model of the blade already presented in Figure 14 and Table
2, at an outlet Mach number
, = 0.79 in a four bladed cascade, Sieverding et al. [36]. Contrary to
schlieren photographs which visualize density changes, the smoke visualizations in Figure 17 show
the instantaneous flow patterns and are therefore particular well suited to visualize the enrolment of
the vortices. A close look at the vortex structures reveals that the distances between successive
vortices change. In fact, the distance between a pressure side vortex and a suction side vortex is
always smaller than the distance between two successive pressure side vortices. A possible reason is
that the pressure side vortex plays a dominant role and exerts an attraction on the suction side vortex
as already found by Han and Cox [26].
(a)
(b)
(c)
Figure 17. VKI LS94 turbine blade, M2,is = 0.79, Re2 = 2.8 × 106 case. (a) four blades cascade, (b) surface
isentropic Mach number distribution and (c) smoke visualizations. Adapted from [36].
The vortex formation and subsequent shedding is accompanied by large angle fluctuations of
the separating shear layers which does not only lead to large pressure fluctuations in the zone of
separations but also induces strong acoustic waves. The latter travel upstream on both the pressure
and suction side as shown in the corresponding schlieren photographs obtained this time with a
continuous light source, a high speed rotating drum and rotating prism camera from ONERA with a
maximum frame rate of 35,000 frames per second (see Figure 18), as reported by Sieverding et al. [21].
(a)
(b)
(c)
Int. J. Turbomach. Propuls. Power 2020, 5, 10
15 of 55
425
426
427
428
429
430
In image 1 of Figure 18 the suction side shear layer has reached its farthest inward position
and
Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW
16 of 58
the local pressure just upstream of the separation point has reached its minimum value. Conversely,
on the
pressure
sideLS94
the separating
shear, layer
has reached
its most
position.
A pressure
Figure
17. VKI
turbine blade,
= 0.79,
= 2.8 x 10
case. outward
(a) four blades
cascade,
(b)
wavesurface
denoted
Pi
originates
from
the
point
where
the
boundary
layer
separates
from
the
trailing
edge.
isentropic Mach number distribution and (c) smoke visualizations. Adapted from [36].
Upstream of Pi is the pressure wave from the previous cycle. It interferes with the suction side of
the neighboring
fromand
where
it is reflected.
In image
4 of Figure 18,
suction
shear layer
The vortex blade
formation
subsequent
shedding
is accompanied
bythe
large
angleside
fluctuations
of
is
atseparating
its most outward
position.
A pressure
originates
at pressure
the pointfluctuations
of separation,
denoted
the
shear layers
which
does not wave
only lead
to large
in the
zone Si.
of
The
pressurebut
wave
upstream
is due towaves.
the previous
cycle.
Onupstream
the pressure
side the
separations
alsofurther
induces
strong acoustic
The latter
travel
on both
the pressure
pressure
wave
Pi extends
to the in
suction
side of the neighboring
blade. The wave
interference
and suction
sidenow
as shown
the corresponding
schlieren photographs
obtained
this timepoint
withofa
the
previouslight
cyclesource,
has moved
It can
therefore
be expected
the suction
side pressure
continuous
a highup-stream.
speed rotating
drum
and rotating
prismthat
camera
from ONERA
with a
maximum frame
ratethroat
of 35,000
frames
per second
(see Figure 18), as reported by Sieverding et al. [21].
distribution
near the
region
is highly
unsteady.
431
432
Figure
Figure 18.
18. Schlieren
Schlieren photographs
photographs of
of vortex
vortex shedding
shedding at
at two
twoinstances
instancesinintime;
time;M2,is, == 0.79,
0.79,Re2 =
=
6 [21].
2.8
×
10
2.8 x 10 [21].
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
Holographic
density measurements,
at VKI
at M2,is inward
= 0.79 by
Sieverding
In image 1 ofinterferometric
Figure 18 the suction
side shear layerperformed
has reached
its farthest
position
and
Int.
J.
Turbomach.
Propuls.
Power
2018,
3,
x
FOR
PEER
REVIEW
17
of
58
et
al.
[36],
give
further
information
about
the
formation
and
the
shedding
process
of
the
von
Kármán
the local pressure just upstream of the separation point has reached its minimum value. Conversely,
vortices.
The reference
density
is
from
pressure
with
a position.
fast
response
needle
The interferogram
in evaluated
Figure
19 shows
thehas
suction
sidemeasurements
vortex
surface)
in its
out A pressure
on the 451
pressure
side
the
separating
shear
layer
reached
its (upper
most blade
outward
452 most
outward
position i.e.
at the
start of the
shedding
phase.
On the pressure
side the
density patterns
static
pressure
probe
positioned
just
outside
of
the
wake
assuming
the
total
temperature
to
be
constant
wave denoted Pi originates from the point where the boundary layer separates from the trailing edge.
453
point to the start of the formation of a new pressure side vortex. The pressure side vortex of the
outside
the wake.
Upstream
is thecycle
pressure
wave
previous
cycle.
It
interferes
the suction
⁄
454 of Pi
previous
is situated
at a from
trailingthe
edge
distance of
≈ 2.
This vortexwith
is defined
by ten side of the
The
interferogram
Figure
suction
side
vortex
surface)
inshear
its out
most
⁄ blade
455
fringes. With
a relative
density
change the
between
two successive
fringes(upper
of18,
∆( the
) suction
= 0.0184
the
total
neighboring
blade
fromin
where
it19isshows
reflected.
In image
4 of
Figure
side
layer
is
⁄
456
relative
density
change
from
the
outside
to
the
vortex
center
is
∆(
)
=
0.184.
The
minimum
in
outward
position
i.e.,position.
at the start
of
the shedding
phase.
On the
pressure
side
the densitydenoted
patternsSi.point
at its most
outward
A
pressure
wave
originates
at
the
point
of
separation,
The
457 the vortex center is ⁄ = 0.552 compared to an isentropic downstream static to total density ratio
to
the start
of the
formation
of a new
pressure
side vortex.
The
side side
vortex
the previous
pressure
upstream
is due
to the previous
cycle.
Onpressure
the pressure
theofpressure
wave
⁄
458wave
of further
= 0.745.
cycle
is
situated
at
a
trailing
edge
distance
of
x/d
≈
2.
This
vortex
is
defined
by
ten
fringes.
a
te
459 now Based
on asuction
large number
withneighboring
holographic
interferometry
and white
light
interferometry, pointWith
Pi extends
to the
sideof tests
of the
blade. The
wave
interference
of the
460
see
Desse
[33],
Figure
19
shows
the
variation
of
the
vortex
density
minima
non-dimensionalized
by
relative
density
between
two successive
fringes ofbe
∆(expected
ρ/ρ0 ) = 0.0184
thesuction
total relative
density
previous
cycle change
has moved
up-stream.
It can therefore
that the
side pressure
461 the upstream total density / , in function of the trailing edge distance / . There are two distinct
(
)
change
from
the
outside
to
the
vortex
center
is
∆
ρ/ρ
=
0.184.
The
minimum
in
the
vortex
center
is
0
distribution
the
region
is vortex
highly
unsteady.
462 near
regions
forthroat
the evolution
of the
minima:
a rapid linear density rise-up to distance / = 1.7
ρ/ρ0 Holographic
=463
0.552 followed
compared
to
an
isentropic
downstream
static
to
total
density
ratio
of
ρ
/ρ
=
0.745.
by a much slower rise
further downstream.
interferometric
density
measurements, performed at VKI at 2 01 = 0.79 by
423
424
464
,
Sieverding et al. [36], give further information about the formation and the shedding process of the
von Kármán vortices. The reference density is evaluated from pressure measurements with a fast
response needle static pressure probe positioned just outside of the wake assuming the total
temperature to be constant outside the wake.
The interferogram in Figure 19 shows the suction side vortex (upper blade surface) in its out
most outward position i.e. at the start of the shedding phase. On the pressure side the density patterns
point to the start of the formation of a new pressure side vortex. The pressure side vortex of the
≈ 2. This vortex is defined by ten
previous cycle is situated at a trailing edge distance of ⁄
fringes. With a relative density change between two successive fringes of ∆( ⁄ ) = 0.0184 the total
(a)
relative density change from the
outside to the vortex center is ∆( (b)⁄ ) = 0.184. The minimum in
465 center
Figure
distribution
(a)and
and
variation
of downstream
density
minimaminima
with
trailing
Commented
Figure
19. Instantaneous
distribution
variation
of density
with
trailing
edge
= density
0.552 density
compared
to(a)an
isentropic
static
toedge
total
density
ratio [M39]: Can it be in term
the vortex
is ⁄19. Instantaneous
466
distance (b) at
= 2.8 6x 10 ; adapted from [36].
, = 0.79,
Please
check all figure captions format
at M2,is = 0.79, Re2 = 2.8 × 10 ; adapted from [36].
⁄
= (b)
0.745.
of distance
467 on a large
Comparing
the vortex
formation
at
0.79 shows thatand
withwhite
increasing
Based
number
of tests
with holographic
interferometry
lightMach
interferometry,
, = 0.4 and
Commented [MM40R39]: Done
468 number the vortices form much closer to the trailing edge. This tendency goes crescendo with further
see Desse [33], Figure 19 shows the variation of the vortex density minima non-dimensionalized by
469 increase of the downstream Mach number as already shown in Figure 13 where normal shocks
the upstream
total density
, in function
of the
distance
. Thereofare
470 oscillate
close to the/ trailing
edge forward
and trailing
backward edge
with the
alternating/shedding
the two distinct
471
472
473
vortices. A further increase of the outlet flow leads gradually to the formation of an oblique shock
system at the convergence of the separating shear layers at short distance behind the trailing edge,
causing a delay of the vortex formation to this region as demonstrated by Carscallen and Gostelow
Int. J. Turbomach. Propuls. Power 2020, 5, 10
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
16 of 55
Based on a large number of tests with holographic interferometry and white light interferometry,
(b)non-dimensionalized by
see Desse [33], Figure 19(a)
shows the variation of the vortex density minima
the upstream
density ρ/ρ
function of (a)
theand
trailing
edge
x/D. There
are twoedge
distinct
0 , in distribution
Figure 19.total
Instantaneous
density
variation
ofdistance
density minima
with trailing
regions
for
the
evolution
of
the
vortex
minima:
a
rapid
linear
density
rise-up
to
distance
x/D
= 1.7
distance (b) at
= 2.8 x 10 ; adapted from [36].
, = 0.79,
followed by a much slower rise further downstream.
Comparing
formation
at M
that with
Mach number
Comparingthe
thevortex
vortex
formation
at2,is =, 0.4
= and
0.4 0.79
and shows
0.79 shows
thatincreasing
with increasing
Mach
the
vortices
much
closer
to the
trailing
edge.
Thisedge.
tendency
crescendo
further
increase
of
number
theform
vortices
form
much
closer
to the
trailing
Thisgoes
tendency
goes with
crescendo
with
further
the
downstream
numberMach
as already
shown
Figure shown
13 where
oscillate
closeshocks
to the
increase
of the Mach
downstream
number
as in
already
innormal
Figureshocks
13 where
normal
trailing
forward
backward
the alternating
shedding
thealternating
vortices. Ashedding
further increase
oscillateedge
close
to the and
trailing
edge with
forward
and backward
with of
the
of the
of
the outlet
flow leads
gradually
the formation
an obliquetoshock
system atof
the
of
vortices.
A further
increase
of thetooutlet
flow leadsofgradually
the formation
anconvergence
oblique shock
the
separating
shear layers of
at the
short
distance behind
the trailing
causing
a delay
of the vortex
system
at the convergence
separating
shear layers
at shortedge,
distance
behind
the trailing
edge,
formation
to thisof
region
as demonstrated
bythis
Carscallen
and
Gostelow [39],
the high speed
cascade
causing a delay
the vortex
formation to
region as
demonstrated
by in
Carscallen
and Gostelow
facility
of the
NRC
Canada.
highofspeed
schlieren
pictures
revealed
some very
unusual
types
[39], in the
high
speed
cascadeThe
facility
the NRC
Canada.
The high
speed schlieren
pictures
revealed
of
wake
vortex
patterns
asof
shown
Figurepatterns
20. Besides
the regular
von 20.
Kármán
vortex
street (left),
some
very
unusual
types
wakeinvortex
as shown
in Figure
Besides
the regular
von
the
authors
visualized
other vortex
patterns,
such as other
e.g. couples
doublets,
on the
right.
In other
Kármán
vortex
street (left),
the authors
visualized
vortex or
patterns,
such
as e.g.
couples
or
moments
in time
they observed
what they
hybrid
or random
nocalled
patterns.
Theorschlieren
doublets, on
the right.
In other moments
in called
time they
observed
what or
they
hybrid
random
photos
in Figure The
20 show
the existence
of Figure
an unexpected
emanating
the trailing shock
edge
or no patterns.
schlieren
photos in
20 showshock
the existence
of from
an unexpected
pressure
side
at
the
beginning
of
the
trailing
edge
circle.
Questioning
Bill
Carscallen
[40]
recently
emanating from the trailing edge pressure side at the beginning of the trailing edge circle.
about
the origin
this shock[40]
it appeared
that the
simply
dueit to
an inaccuracy
the blade
Questioning
Billof
Carscallen
recently about
theshock
originwas
of this
shock
appeared
that theinshock
was
manufacturing
of inaccuracy
the trailingin
edge
simply due to an
the circle.
blade manufacturing of the trailing edge circle.
(a)
(b)
(c)
479
480
Figure
Figure 20. Occurrence of different vortex patters in wake of transonic blade at M2,is
= 1.07.
1.07. (a) regular
, =
vortex
vortex street,
street, (b)
(b) couples,
couples, (c)
(c) doublets
doublets [39].
[39].
481
482
483
484
The
question whether
whetherinindistinction
distinctionofofthe
the
conventional
von
Kármán
vortex
street,
a double
The question
conventional
von
Kármán
vortex
street,
a double
row
row
vortex
street
of
unequal
vortex
strength
may
exist,
was
treated
by
Sun
in
1983
[41].
Figure
21
vortex street of unequal vortex strength may exist, was treated by Sun in 1983 [41]. Figure 21 presents
presents
an example
of a double
row vortex
streetunequal
with unequal
strength
and vortex
distances.
an example
of a double
row vortex
street with
vortexvortex
strength
and vortex
distances.
The
Int.
J.
Turbomach.
Propuls.
Power
2018,
3,
x
FOR
PEER
REVIEW
18
of 58
The
author
demonstrated
that
such
configurations
are
basically
unstable.
author demonstrated that such configurations are basically unstable.
485
486
Figure
Figure 21.
21. An
An example
example of
of vortex
vortex street
street with
with unequal
unequal vortex
vortex strengths
strengths and
and unequal
unequal vortex
vortex spacings;
spacings;
adapted
adapted from
from [41].
[41].
487
4. Energy Separation in the Turbine Blade Wakes
488
489
490
491
In the course of a joint research program between the National Research Council of Canada and
Pratt & Whitney Canada of the flow through an annular transonic nozzle guide vane in the 1980s
certain experiments revealed a non-uniform total temperature distribution downstream of the
uncooled blades, Carscallen and Oosthuizen [42]. Considering the importance of the existence of a
485
486
Figure 21. An example of vortex street with unequal vortex strengths and unequal vortex spacings;
adapted from
[41].
Int. J. Turbomach.
Propuls.
Power 2020, 5, 10
17 of 55
487
4. Energy Separation in the Turbine Blade Wakes
4. Energy Separation in the Turbine Blade Wakes
In the course of a joint research program between the National Research Council of Canada and
course of
a jointof
research
program
between
the National
Research
Council
of Canada
and
PrattIn&the
Whitney
Canada
the flow
through
an annular
transonic
nozzle guide
vane
in the 1980s
Pratt
&
Whitney
Canada
of
the
flow
through
an
annular
transonic
nozzle
guide
vane
in
the
1980s
certain experiments revealed a non-uniform total temperature distribution downstream of the
certain
experiments
revealed a and
non-uniform
total[42].
temperature
distribution
downstream
of existence
the uncooled
uncooled
blades, Carscallen
Oosthuizen
Considering
the importance
of the
of a
blades,
Carscallen
andtemperature
Oosthuizen [42].
Considering
the exit
importance
of the existence
of a non-uniform
non-uniform
total
distribution
at the
of uncooled
stator blade
row for the
total
temperature
distribution
at the exitrotor,
of uncooled
stator blade
row for the
aerothermal
aspects
aerothermal
aspects
of the downstream
the Gasdynamics
Laboratory
of the
National Research
ofCouncil,
the downstream
rotor, the
Gasdynamics
Laboratory
of the
National
Council,
Canada decided
to build
a continuously
running
suction
typeResearch
large scale
planarCanada
cascade
decided
to
build
a
continuously
running
suction
type
large
scale
planar
cascade
tunnel
(chord
length
tunnel (chord length 175.3 mm, turning angle 76°, trailing edge diameter 6.35 mm) and launched
an
◦ , trailing edge diameter 6.35 mm) and launched an extensive research
175.3
mm,
turning
angle
76
extensive research program aiming at the understanding of the mechanism causing the occurrence
program
aiming at thevariations
understanding
of the mechanism
the occurrence
of total
temperature
of total temperature
downstream
of a fixed causing
blade row,
determine their
magnitude
and
variations
downstream
of
a
fixed
blade
row,
determine
their
magnitude
and
evaluate
their
significance
evaluate their significance for the design of the downstream rotor. Downstream traverses with copper
for
the design
of the downstream
rotor.
thermocouples
constantan
thermocouples
reported
byDownstream
Carscallen ettraverses
al. [43] inwith
1996copper
showedconstantan
that the total
temperature
reported
Carscallenperfectly
et al. [43]with
in 1996
that thewake
total profiles,
temperature
contours
contoursbycorrelated
the showed
total pressure
Figure
22. correlated perfectly
with the total pressure wake profiles, Figure 22.
488
489
490
491
492
493
494
495
496
497
498
499
500
501
(b) Total pressure coefficient
(c) Total temperature difference
(a) Test section
502
503
Figure
contours
downstream
of of
a nozzle
guide
vane
at aat
Figure22.
22.Total
Totalpressure
pressurecoefficient
coefficientand
andtemperature
temperature
contours
downstream
a nozzle
guide
vane
pressure
ratio
PR
=
1.9;
adapted
from
[42].
a pressure ratio PR = 1.9; adapted from [42].
504
505
506
507
In
Inthe
thewake
wakecenter
centerthe
thetotal
totaltemperature
temperaturedropped
droppedsignificantly
significantlybelow
belowthe
theinlet
inlettotal
totaltemperature
temperature
while
higher
values
were
recorded
near
the
border
of
the
wake.
The
differences
increased
while higher values were recorded near the border of the wake. The differences increasedwith
withMach
Mach
number
and
reached
a
maximum
at
sonic
outlet
conditions.
The
question
was
then
to
elucidate
number and reached a maximum at sonic outlet conditions. The question was then to elucidatethe
the
reasons
reasonsfor
forthese
thesetemperature
temperaturevariations.
variations.The
Theresearch
researchon
onflows
flowsacross
acrosscylinders
cylinderswas
wasalready
alreadymore
more
advanced in this respect. Measurements of the temperature distribution around a cylinder for a flow
normal to the axis of the cylinder, performed at the Aeronautical Institute of Braunschweig in the
late 1930’s and reported by Eckert and Weise [44] in 1943, showed that the recovery temperature at
the base of the cylinder reduced below the true (static) temperature of the incoming flow, so that the
recovery factor:
Tr − T
r0 =
T0 − T
attained negative values in the base region (see Figure 23).
The authors suspected that the low values were possibly due to the intermittent separation of
vortices from the cylinder. These results were confirmed by Ryan [45] in 1951 at Ackeret’s Institute in
Zürich who clearly related this low temperature to the periodic vortex shedding behind the cylinder as
(a) Test section
506
507
Figure 22. Total pressure coefficient and temperature contours downstream of a nozzle guide vane at
a pressure ratio PR = 1.9; adapted from [42].
Commented [M43]: Please add ex
Commented [MM44R43]: Done.
508 Propuls.
In Power
the wake
center
the total temperature dropped significantly below the inlet total temperature
Int. J. Turbomach.
2020,
5, 10
18 of 55
509 while higher values were recorded near the border of the wake. The differences increased with Mach
510 number and reached a maximum at sonic outlet conditions. The question was then to elucidate the
reasons for these temperature variations. The research on flows across cylinders was already more
cause for511
the energy
separation in the fluctuating wake. He also noticed that the energy separation
512 advanced in this respect. Measurements of the temperature distribution around a cylinder for a flow
was particularly
large
sound
was generated
by the
flow.of Braunschweig in the late
513 normal towhen
the axisaofstrong
the cylinder,
performed
at the Aeronautical
Institute
514
1930’s
and
reported
by
Eckert
and
Weise
[44]
in
1943,
showed
that
thewas
recovery
theSieverding
The existence of a low temperature field at the base of a cylinder
alsotemperature
observedatby
515 base of the cylinder reduced below the true (static) temperature of the incoming flow, so that the
in 1985, who used an infrared camera to visualize through a germanium window in the side wall of a
516 recovery factor:
514
515
blow down wind tunnel the wall temperature field around
a 15 mm diameter cylinder at M = 0.4,
−
=
Figure
23.
Evolution
of
the
recovery
factor
in
the
azimuthal
direction
the stagnation
see Figure 24. Unfortunately, due to a lack of time it−was not
possiblefrom
to determine
thepoint
absolute
( = 517
0°) to
the
rear
side
of
the
cylinder
(
=
180°);
adapted
from
[44].
temperature
values.
attained negative values in the base region (see Figure 23).
516
517
518
519
520
521
522
523
524
525
The authors suspected that the low values were possibly due to the intermittent separation of
vortices from the cylinder. These results were confirmed by Ryan [45] in 1951 at Ackeret’s Institute
in Zürich who clearly related this low temperature to the periodic vortex shedding behind the
cylinder as cause for the energy separation in the fluctuating wake. He also noticed that the energy
separation was particularly large when a strong sound was generated by the flow.
The existence of a low temperature field at the base of a cylinder was also observed by Sieverding in
1985, who used an infrared camera to visualize through a germanium window in the side wall of a
blow down wind tunnel the wall temperature field around a 15 mm diameter cylinder at
= 0.4,
see Figure 24. Unfortunately, due to a lack of time it was not possible to determine the absolute
51823. Evolution
Figure 23.
of the recovery
in the
azimuthal direction
the stagnation
point
Figure
of Evolution
the recovery
factorfactor
r0 in the
azimuthal
directionfrom
α from
the stagnation
point
temperature
519 values.
( = 0°) to the rear side of the cylinder ( = 180°); adapted from [44].
(α = 0◦ ) to the rear side of the cylinder (α = 180); adapted from [44].
520
521
522
The authors suspected that the low values were possibly due to the intermittent separation of
vortices from the cylinder. These results were confirmed by Ryan [45] in 1951 at Ackeret’s Institute
in Zürich who clearly related this low temperature to the periodic vortex shedding behind the
Commented [M45]: Please add ex
Commented [MM46R45]: These t
considered as a single entity.
526
Figure 24. Side wall temperature field around a cylinder recorded by an infrared camera.
Figure 24. Side wall temperature field around a cylinder recorded by an infrared camera.
527
528
Eckert [46] explained the mechanism of energy separation along a flow path with the help of the
Eckert [46] explained the mechanism of energy separation along a flow path with the help of the
unsteady energy equation:
unsteady energy equation:
!
∂p
DT0
∂
∂T
∂ =
+
+
ρ cp
k
νi σij
(4)
Dt
∂t
∂xi ∂xi
∂x j
|{z} | {z } | {z }
(a)
(b)
(c)
The change of the total temperature with time depends on: (a) the partial derivative of the pressure
with time, (b) on the energy transport due to heat conduction between regions of different temperatures
and (c) on the work due to viscous stresses between regions of different velocities. As regards the flow
behind bluff bodies the two latter terms are considered small compared the pressure gradient term and
Equation (4) then reduces to:
∂p
DT0
ρ cp
=
Dt
∂t
The occurrence of total temperature variations in the vortex streets behind cylinders was e.g.
extensively described by Kurosaka et al. [47], Ng et al. [48] and Sunduram et al. [49].
The progress in the understanding of the mechanism was boosted with the arrival of fast
temperature probes as for example the dual sensor thin film platinum resistance thermometer probe
developed by Buttsworth and Jones [50] in 1996 at Oxford. Using their technique Carscallen et al. [51,52]
were the first to measure the time varying total pressure and temperature in the wake of their turbine
534
535
536
537
538
539
540
541
542
543
544
The occurrence of total temperature variations in the vortex streets behind cylinders was e.g.
extensively described by Kurosaka et al. [47], Ng et al. [48] and Sunduram et al. [49].
The progress in the understanding of the mechanism was boosted with the arrival of fast
temperature probes as for example the dual sensor thin film platinum resistance thermometer probe
developed by Buttsworth and Jones [50] in 1996 at Oxford. Using their technique Carscallen et al.
Int. J. Turbomach. Propuls. Power 2020, 5, 10
19 of 55
[51,52] were the first to measure the time varying total pressure and temperature in the wake of their
turbine vane. Figure 25 presents the results for an isentropic outlet Mach number
, = 0.95 and a
vortex
shedding
frequency
of
the
order
of
10
kHz.
The
probe
traverse
plane
was
normal
thea wake
vane. Figure 25 presents the results for an isentropic outlet Mach number M2,is = 0.95 to
and
vortex
atshedding
a distancefrequency
of 5.76 trailing
from
vanetraverse
trailing plane
edge. was
In a normal
later paper
concerning
of theedge
orderdiameters
of 10 kHz.
Thethe
probe
to the
wake at a
the
same cascade,
Gostelow
anddiameters
Rona [53],from
published
alsotrailing
the corresponding
entropy
variations
from
distance
of 5.76 trailing
edge
the vane
edge. In a later
paper
concerning
the
the
Gibb’s
relation:
same
cascade,
Gostelow and Rona [53], published also the corresponding entropy variations from the
Gibb’s relation:
545
p02
T02−
− =
s2 − s1 = cp ln
− R ln
T01
p01
The results
are
presented
in
Figure
26.
The results are presented in Figure 26.
(a)
(b)
546
547
Figure 25. Contour plots of phase averaged total pressure (a) and total temperature (b) behind turbine
vane; adapted from [51].
vane; adapted from [51].
25. Contour
plots 2018,
of phase
averaged
pressure (a) and total temperature (b) behind turbine 21 of 58
Int. J.Figure
Turbomach.
Propuls. Power
3, x FOR
PEER total
REVIEW
548
Figure 26.
26. Time
Time resolved
resolved measurements
measurements of
of entropy
entropy increase
increase [53].
Figure
[53].
549
550
551
552
The variation of the maxima and minima of the total temperature in the center of the wake vary
The variation of the maxima
and minima of the total temperature in the center of the wake vary
between a minimum of −15◦ to a maximum of −4◦ with respect to the inlet ambient temperature,
between a minimum of −15° to a maximum of −4° with respect to the inlet ambient temperature,
Figure 27. At the border of the wake the temperature raises considerably above the inlet temperature,
Figure 27. At the border of the wake the temperature raises considerably
above the inlet temperature,
while the time averaged temperature in the wake center is about −10◦ .
while the time averaged temperature in the wake center is about −10°.
553
Figure 27. Variation of total temperature maxima and minima through the wake; adapted from [51].
548
Figure 26. Time resolved measurements of entropy increase [53].
549
550
551
552
The variation of the maxima and minima of the total temperature in the center of the wake vary
between a minimum of −15° to a maximum of −4° with respect to the inlet ambient temperature,
Figure 27. At the border of the wake the temperature raises considerably above the inlet temperature,
Int. J. Turbomach. Propuls. Power 2020, 5, 10
20 of 55
while the time averaged temperature in the wake center is about −10°.
553
Figure 27. Variation of total temperature maxima and minima through the wake; adapted from [51].
Figure 27. Variation of total temperature maxima and minima through the wake; adapted from [51].
554
555
556
557
558
559
560
561
562
563
564
In 2004, Sieverding et al. [36] published very similar results for the turbine vane shown in Figure 17.
In 2004,
Sieverding
et al. [36] published
very
similar
results
for the
vane shown
Figure
The wake
traverse
was performed
at a trailing
edge
distance
of only
2.5turbine
dte in direction
of theintangent
17.
Theblade
wakecamber
traverse
was
performed
at aangle
trailing
edge
distance
of direction.
only 2.5 The
intraverse
direction
the
◦ with
to the
line,
which
forms an
of 66
the axial
is of
made
tangent
to
the
blade
camber
line,
which
forms
an
angle
of
66°
with
the
axial
direction.
The
traverse
is
normal to this tangent.
madeThe
normal
to
this
tangent.
steady
state total pressure and total temperature measurements are presented
in Figure 28.
Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW
22 of 58
Thetosteady
state obtained
total pressure
total
temperature
measurements
are presented
Figure 28.
Similar
the results
at theand
NRC
Canada,
the wake
center is characterized
by a in
pronounced
562
The steady
state total
pressure
andCanada,
total temperature
measurements
arecharacterized
presented in Figure
28.
Similar
to
the
results
obtained
at
the
NRC
the
wake
center
is
by
a
pronounced
◦
total temperature drop of 3% of the inlet value of 290 K which corresponds to about −9 , a variation
563 Similar drop
to the results obtained at the NRC Canada, the wake center is characterized by a pronounced
total
of 3%
of the
inlet value
of 290
KOn
which
corresponds
to
about
−9°,temperature
a variation
whichtemperature
is of the
same
order
as that
Figure
the
borders
the−9°,
wake,
total
564
total
temperature
drop
of 3%reported
of the inletin
value
of 29027.
K which
corresponds
toof
about
a variation
which
is
of
the
same
order
as
that
reported
in
Figure
27.
On
the
borders
of
the
wake,
total
temperature
whichof
is of
theinlet
same order
as that reported
Figurerecorded.
27. On the borders
the wake,
total temperature
peaks 565
in excess
the
temperature
areinalso
The ofmass
integrated
total temperature
566
peaksof
in the
excessinlet
of thetemperature
inlet temperature
are also recorded.
The mass mass
integrated total temperature
peaks
in excess
recorded.
total
value across
the wake
(denoted
with a < are
∗ >)also
should
be suchThe
that <T02integrated
> /< T01 >
= 1,temperature
but lack of
567
value across the wake (denoted with a <∗ ) should be such that <
/<
= 1, but lack of
value
across
the
wake
(denoted
with
a
<∗
)
should
be
such
that
<
/<
=
1,
but
lack of
information
on the local
velocity
did not
allow
perform
integration.
568 information
on the
local velocity
did not
allowto
to perform
thisthis
integration.
information on the local velocity did not allow to perform this integration.
(a) Steady total pressure
(b) Steady total temperature
(a) Steady total pressure
(c) Unsteady total pressure
(b) Steady total temperature
(d) Unsteady total temperature
569
Figure 28. Steady and time varying phase lock averaged total pressure and total temperature through
570
Figure 28. Steady and time varying phase lock averaged total pressure and total temperature through
Commented [M49]: Please add explana
turbine vane wake at M2,is = 0.79; adapted from [36].
571
turbine vane wake at
, = 0.79; adapted from [36].
Commented [MM50R49]: Done. Pleas
For
of the of
time
varying
temperature
a fast
2 µm
coldprobe,
wiredeveloped
probe, developed
by
572the measurement
For the measurement
the time
varying
temperature a fast
2 μm
cold wire
figure on two pages.
by Denos and[54],
Sieverding
wasNumerical
used. Numerical
compensation allowed
to extend
the naturally
Denos573
and Sieverding
was [54],
used.
compensation
allowed
to extend
the naturally low
574 low frequency response of the probe to much higher ranges for adequate restitution of the nearly
frequency
response of the probe to much higher ranges for adequate restitution of the nearly sinusoidal
575 sinusoidal temperature variation associated with the vortex shedding frequency of 7.6 kHz at a
temperature
variation isentropic
associated
with
the ofvortex
shedding frequency of 7.6 kHz at a downstream
576 downstream
Mach
number
, = 0.79 . As regards the total pressure variation
⁄ number
577 Mach
, minimumof
values
are reached
in the wake
while
at the wake
border maximum
isentropic
M2,isof 0.768
= 0.79.
As regards
thecenter
total
pressure
variation
p02 /p01 , minimum
578 values of 1.061 are recorded. As regards the total temperature the authors quote maximum and
values579
of 0.768
are
reached
in
the
wake
center
while
at
the
wake
border
maximum
values of 1.061
⁄
= 1.046 and 0.96, respectively. With a
minimum total temperature ratios of
= 290
are recorded.
regards
total temperature
the
quote
maximum
and minimum
total
580 the As
maximum
totalthe
temperature
variations are of
theauthors
order of 24°,
similar
to those reported
by
581
582
583
Carscallen et al. [51]. However, the flow conditions were different:
, = 0.79 at VKI, versus 0.95
at NRC Canada, and a distance of the wake traverses with respect to the trailing edge of 2.5 diameters
at VKI, versus 5.76 at NRC.
584
5. Effect of Vortex Shedding on Blade Pressure Distribution
585
The previous section focused on the unsteady character of turbine blade wake flows, the
Int. J. Turbomach. Propuls. Power 2020, 5, 10
21 of 55
temperature ratios of T02 /T01 = 1.046 and 0.96, respectively. With a T01 = 290 K the maximum
total temperature variations are of the order of 24◦ , similar to those reported by Carscallen et al. [51].
However, the flow conditions were different: M2,is = 0.79 at VKI, versus 0.95 at NRC Canada, and a
distance of the wake traverses with respect to the trailing edge of 2.5 diameters at VKI, versus 5.76
at NRC.
5. Effect of Vortex Shedding on Blade Pressure Distribution
The previous section focused on the unsteady character of turbine blade wake flows,
the visualization of the von Kármán vortices through smoke visualizations, schlieren photographs and
interferometric techniques. The measurement of the instantaneous velocity fields using LDV and PIV
techniques allowed to determine the vorticity distribution and the measurement of the unsteady total
pressure and temperature distribution putting into evidence the energy separation effect in the wakes
due to the von Kármán vortices.
Naturally the vortex shedding affects also the trailing edge pressure distribution and, beyond that,
the suction side pressure distribution. The following is entirely based on research work carried out at
the VKI by the team of the lead author, who was the only one to measure with high spatial resolution
the pressure distribution around the trailing edge of a turbine blade.
5.1. Effect on Trailing Edge Pressure Distribution
The very large-scale turbine guide vane designed and tested at the von Kármán Institute with
a trailing edge thickness of 15 mm did allow an innovative approach for obtaining a high spatial
resolution for the pressure distribution around the trailing edge. Cicatelli and Sieverding [32], fitted
the blade with a rotatable 20 mm long cylinder in the center of the blade (Figure 29). The cylinder was
equipped with a single Kulite fast response pressure sensor side by side with an ordinary pneumatic
pressure tapping. The pressure sensor was mounted underneath the trailing edge surface with a slot
width of only 0.2 mm to the outside, the same width as the pressure tapping, reducing the angular
sensing area to only 1.53◦ . To control any effect of the rear facing step between the blade lip and
the rotatable trailing edge, a second blade was equipped with additional pressure sensors placed at,
and slightly up-stream of, the trailing edge.
The time averaged base pressure distribution, non-dimensionalized by the inlet total pressure,
is presented in Figure 30. The circles denote data obtained with the rotatable trailing edge cylinder
on blade A, while the triangles are measured with pressure tappings on blade B (see Figure 29),
except for the two points “a” and “e” which are taken from the pressure tappings positioned aside
the rotating cylinder on blade A (see Figure 30, left panel). The flow approaching the trailing edge
undergoes, both on the pressure and suction side, a strong acceleration before separating from the
trailing edge circle. The authors attribute the asymmetry to differences in the blade boundary layers
and to the blade circulation, which, following Han and Cox [26], strengthens the pressure side vortex
shedding. Compared to the downstream Mach number M2,is = 0.4, the local peak numbers are as high
as Mmax = 0.49 and 0.47, respectively. These high over-expansions are incompatible with a steady state
boundary layer separation and are attributed to the effect of the vortex shedding.
606
607
608
609
610
611
undergoes, both on the pressure and suction side, a strong acceleration before separating from the
trailing edge circle. The authors attribute the asymmetry to differences in the blade boundary layers
and to the blade circulation, which, following Han and Cox [26], strengthens the pressure side vortex
shedding. Compared to the downstream Mach number
, = 0.4, the local peak numbers are as
high as
= 0.49 and 0.47, respectively. These high over-expansions are incompatible with a
Int. J. Turbomach. Propuls. Power 2020, 5, 10
22 of 55
steady state boundary layer separation and are attributed to the effect of the vortex shedding.
(b)
(a)
612
613
Figure
Blade
instrumented
with
rotatable
trailing
edge
blade
and
control
blade
blade
Figure
29.29.
Blade
instrumented
with
rotatable
trailing
edge
(a:(a:
blade
A)A)
and
control
blade
(b:(b:
blade
B);B);
adapted
from
[32].
adapted from [32].
614
615
616
617
618
619
620
Figure30,30,right
rightpanel,
panel,presents
presentsthe
thecorresponding
corresponding
root
mean
square
pressure
signal.
Figure
root
mean
square
of of
thethe
pressure
signal.
Maximum
pressure
fluctuations
of
the
order
of
8%
occur
near
the
locations
of
the
pressure
minima,
i.e.,
Maximum pressure fluctuations of the order of 8% occur near the locations of the pressure minima,
close
toto
the
boundary
layer
separation
points,
while
the
RMS/Q
drops
toto
nearly
half
this
value
inin
the
i.e.
close
the
boundary
layer
separation
points,
while
the
RMS/Q
drops
nearly
half
this
value
central
region
the
trailing
edge
base.
It
is
also
worth
noting
that
the
pressure
fluctuations
affect
also
Int.of
J. Turbomach.
Propuls. Power
2018,
3, x FOR
PEER
REVIEW
24 of 58
the central region of the trailing edge base. It is also worth noting that the pressure fluctuations affect
thethe
flow
upstream
of the trailing
edge.edge.
In theIncenter
of the base
regionregion
there is an extended
constant
also
flow
upstream
the trailing
the
is an extended
624
also
the flow of
upstream
of the trailing
edge. In
thecenter
center ofof
thethe
basebase
region there is there
an extended
pressure
plateau
(Figure
The(Figure
baseThe
pressure
coefficient
corresponding
to tothis
agrees
well
625
constant
pressure30).
plateau
30).
The
base
pressure coefficient
corresponding
thisplateau
plateau
constant
pressure
plateau
(Figure
30).
base
pressure
coefficient
corresponding
to this
plateau
626Sieverding’s
agrees well with
thepressure
Sieverding’s
base pressure correlation.
with
the
base
correlation.
agrees well with the Sieverding’s base pressure correlation.
(a)
(b)
627 30. Phase
Figurelock
30. Phase
lock averaged
trailing
edgepressure
pressure distribution
(a) and(a)
root
mean
square
values
of
Figure
averaged
trailing
edge
distribution
and
root
mean
square
values of
628
the pressure fluctuations (b) for the VKI turbine blade at
, = 0.4; adapted from [32].
the pressure fluctuations (b) for the VKI turbine blade at M2,is = 0.4; adapted from [32].
629
The base pressure distribution changes dramatically at high subsonic downstream Mach
(b)
(a) by Sieverding
The
pressure
distribution
changes
dramatically
at high
subsonic
downstream
Mach numbers
630base
numbers
as illustrated
et al.
[21], Figure 31. The
pressure
distribution
is characterized
631 byby
the Sieverding
presence of three
theFigure
two pressure
associated with
the over-expansion
of the
as illustrated
etminima:
al. [21],
31. minima
The pressure
distribution
is characterized
by the
632 suction and pressure side flows before separation from the trailing edge, and an additional minimum
presence
of
three
minima:
the
two
pressure
minima
associated
with
the
over-expansion
of
the
suction
633 around the center of the trailing edge circle. The pressure minima related to the overexpansion from
⁄ trailing
and pressure
side flows
before
separation
fromof the
edge,
around
634 suction
and pressure
sides
are of the order
= 0.52 for
both and
sides,an
i.e. additional
the local peak minimum
Mach
635 ofnumbers
are close
to circle.
1. Contrary
the low Mach
number
flow condition
of Figure 30, the from suction
the center
the trailing
edge
Thetopressure
minima
related
to the overexpansion
636 recompression following the over-expansion does not lead to a pressure plateau but gives way to a
and pressure
sides
are pressure
of the order
of p/p01
= 0.52 for
i.e.,
⁄
637 new
strong
drop reaching
a minimum
of both
=sides,
0.485 at
+7°.The
Thislocal
is thepeak
result Mach
of the numbers are
enrolmentto
ofthe
the low
separating
layers into
vortex right of
at the
trailing
vortex core
close to638
1. Contrary
Machshear
number
flowa condition
Figure
30,edge;
the the
recompression
following
639 approaches
the wake
its distanceplateau
to the trailing
becomes
half the
trailing
the over-expansion
does
notcenterline
lead to and
a pressure
butedge
gives
wayless
tothan
a new
strong
pressure drop
640 edge diameter, see smoke visualization and
◦ interferogram in Figure 17 and Figure 19.
reaching a minimum of p/p01 = 0.485 at +7 . This is the result of the enrolment of the separating shear
layers into a vortex right at the trailing edge; the vortex core approaches the wake centerline and its
distance to the trailing edge becomes less than half the trailing edge diameter, see smoke visualization
and interferogram in Figures 17 and 19.
633 around the center of the trailing edge circle. The pressure minima related to the overexpansion from
634 suction and pressure sides are of the order of ⁄ = 0.52 for both sides, i.e. the local peak Mach
635 numbers are close to 1. Contrary to the low Mach number flow condition of Figure 30, the
636 recompression following the over-expansion does not lead to a pressure plateau but gives way to a
637 new strong pressure drop reaching a minimum of ⁄ = 0.485 at +7°. This is the result of the
638 enrolment of the separating shear layers into a vortex right at the trailing edge; the vortex core
639 approaches
the wake
centerline
Int. J. Turbomach.
Propuls. Power
2020,
5, 10 and its distance to the trailing edge becomes less than half the trailing
640 edge diameter, see smoke visualization and interferogram in Figure 17 and Figure 19.
(a)
23 of 55
(b)
641 Figure 31. Steady state trailing edge pressure distribution (a) and phase locked average pressure fluctuation (b)
Figure 31. Steady state trailing edge pressure distribution (a) and phase locked average pressure
642 around trailing edge; adapted from [21].
fluctuation (b) around trailing edge; adapted from [21].
643
The phase lock averaged pressure fluctuations around the trailing edge in Figure 31 are
644phase
impressive.
They are naturally
near the separation
points
of the boundary
from
The
lock averaged
pressurehighest
fluctuations
around the
trailing
edge in layer
Figure
31the
are impressive.
They are naturally highest near the separation points of the boundary layer from the trailing edge
where maximum values of around 100% of the dynamic pressure (p01 − p2 ) are recorded. The minimum
pressure in a given position corresponds to the maximum inward motion of the separating shear layer,
the maximum pressure to the maximum outward motion of the separating shear layer. The maximum
local instantaneous Mach number at the point of the most inward position can be as high as Mmax = 1.25.
The authors assumed that the curvature driven supersonic trailing edge expansion is the real reason
for the formation of the vortex so close to the trailing edge, with the entrainment of high-speed free
stream fluid into the trailing edge base region. In the center of the trailing edge the fluctuations drop
to 20% of the dynamic head.
The authors provide also some interesting information on the evolution of the pressure signal on
the trailing edge circle over one complete vortex shedding cycle. This is demonstrated in Figure 32
showing the evolution for the phase locked average pressure at the angular position of 60◦ on the
pressure side of the trailing edge circle. A decrease of the pressure indicates an acceleration of the flow
around the trailing edge i.e., The separating shear layer moves inwards, the vortex is in its formation
phase. An increase of the pressure indicates on the contrary an outwards motion of the shear layer,
the vortex is in its shedding phase. Surprisingly, the pressure rise time is much shorter than the
pressure fall time, i.e., The time for the vortex formation is longer than that for the vortex shedding.
The same was observed for the pressure evolution on the opposite side of the trailing edge, but of
Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW
25 of 58
course with 180◦ out of phase.
657
Figure 32. Phase locked average pressure variation at trailing edge at an angular position of −60◦ [21].
Figure 32. Phase locked average pressure variation at trailing edge at an angular position of −60° [21].
658
659
660
661
662
663
664
The change of an isobaric pressure zone over an extended region at the base of the trailing edge at
The
change
of anMisobaric
pressure zone over an extended region at the base of the trailing edge
an exit
Mach
number
2,is = 0.4 to a highly non-uniform pressure distribution with a strong pressure
at an exit Mach number
, = 0.4 to a highly non-uniform pressure distribution with a strong
pressure minimum at the center of the trailing edge circle at
, = 0.79, did of course raise the
question about the evolution of the trailing edge pressure distribution over the entire Mach number
range, from low subsonic to transonic Mach numbers. To respond to this lack of information a
research program was carried out by Mateos Prieto [55] at VKI as part of his diploma thesis in 2003.
For various reasons, the data were not published at that time but only in 2015, as part of the paper of
Int. J. Turbomach. Propuls. Power 2020, 5, 10
24 of 55
Int. J. Turbomach.
Propuls.
Power
2018,trailing
3, x FORedge
PEER circle
REVIEW
of 58
minimum
at the
center
of the
at M2,is = 0.79, did of course raise the question 25
about
the evolution of the trailing edge pressure distribution over the entire Mach number range, from low
subsonic to transonic Mach numbers. To respond to this lack of information a research program was
carried out by Mateos Prieto [55] at VKI as part of his diploma thesis in 2003. For various reasons,
the data were not published at that time but only in 2015, as part of the paper of Vagnoli et al. [56] on
the prediction of unsteady turbine blade wake flow characteristics and comparison with experimental
data, see Figure 33.
The figure puts clearly into evidence the effect of the vortex shedding on the trailing edge pressure
distribution. Up to about M2,is ≈ 0.65 the trailing edge base region is characterized by an extended,
nearly isobaric, pressure plateau which implies that the vortex formation occurs sufficiently far
downstream not to affect the trailing edge base region. With increasing Mach number, the enrolment
of the shear layers into vortices occurs gradually closer to the trailing edge causing an increasingly
non-uniform pressure distribution with a marked pressure minimum at the center of the trailing edge.
To characterize the degree of non-uniformity the authors define a factor Z:
Z = pb,max − pb,min /(p01 − p2 )
(5)
657
Figure 32. Phase locked average pressure variation at trailing edge at an angular position of −60° [21].
658
659
660
661
662
663
664
665
666
where pb,max is the maximum pressure following the recompression after the separation of the shear
change
of an isobaric
pressure
zoneminimum
over an extended
at the
baseofofthe
thetrailing
trailing edge.
edge
layerThe
from
the trailing
edge and
pb,min the
pressureregion
near the
center
at
an
exit
Mach
number
=
0.4
to
a
highly
non-uniform
pressure
distribution
with
a
strong
,
The maximum degree of non-uniformity
is reached at M2,is = 0.93 with a Z value of 21%. At this
pressure
minimum
at the center
of the
trailing
edge
at =, 0.325
= 0.79,
of course raise
the
Mach
number
the minimum
pressure
reaches
a value
of circle
pb,min /p
for did
a downstream
pressure
01
about the evolution of the trailing edge pressure distribution over the entire Mach number
pquestion
2 /p01 = 0.572. With further increase of the Mach number, Z starts to decrease rapidly. It decreases to
range,
from
low=subsonic
to transonic
Tothis
respond
to this lack
of information
Z = 12% at M2,is
0.99 and drops
to zero Mach
at M2,isnumbers.
= 1.01. For
Mach number
the local
trailing edgea
research program
out
by Mateos
Prieto
VKI as
of his diploma
thesis
in 2003.
conditions
are suchwas
thatcarried
oblique
shocks
emerge
from[55]
theat
region
of part
the confluence
of the
suction
and
For
various
reasons,
the
data
were
not
published
at
that
time
but
only
in
2015,
as
part
of
the
paper
of
pressure side shear layers and the vortex formation is delayed to after this region as shown e.g. In the
Vagnoli
et
al.
[56]
on
the
prediction
of
unsteady
turbine
blade
wake
flow
characteristics
and
schlieren picture by Carscallen and Gostelow [39] at the NRC Canada in Figure 34, left, and another
comparison
withtaken
experimental
see34,
Figure
schlieren
picture
at VKI indata,
Figure
right33.
(unpublished).
667
Figure 33. Effect of downstream Mach number on trailing edge Mach number distribution [56].
Figure 33. Effect of downstream Mach number on trailing edge Mach number distribution [56].
668
669
670
671
672
673
674
The figure puts clearly into evidence the effect of the vortex shedding on the trailing edge
pressure distribution. Up to about
, ≈ 0.65 the trailing edge base region is characterized by an
extended, nearly isobaric, pressure plateau which implies that the vortex formation occurs
sufficiently far downstream not to affect the trailing edge base region. With increasing Mach number,
the enrolment of the shear layers into vortices occurs gradually closer to the trailing edge causing an
increasingly non-uniform pressure distribution with a marked pressure minimum at the center of the
trailing edge. To characterize the degree of non-uniformity the authors define a factor :
679
680
681
682
683
684
= 0.572. With further increase of the Mach number,
starts to decrease rapidly. It
pressure ⁄
decreases to
= 12% at
=
0.99
and
drops
to
zero
at
=
1.01.
For this Mach number the
,
,
local trailing edge conditions are such that oblique shocks emerge from the region of the confluence
of the suction and pressure side shear layers and the vortex formation is delayed to after this region
as
e.g. Propuls.
in the schlieren
by Carscallen and Gostelow [39] at the NRC Canada in Figure
Int.shown
J. Turbomach.
Power 2020, picture
5, 10
25 of 55
34, left, and another schlieren picture taken at VKI in Figure 34, right (unpublished).
(b)
(a)
Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
27 of 58
Figure
AtAs
fully
established
oblique
trailing
edge
shock
system
the
vortex formation
formation
is delayed
delayed to
to
Figure
34. At
fully
established
trailing
edge
system
vortex
is
693 34.
already
pointed outoblique
at the end
of Section
2 theshock
departure
from the
the generally
assumed isobaric
after
the point
confluence
of
the
blade
layers.
NRC
blade
(a)
[39],
VKI
694the
trailingof
base region
the differences
of baseblade
pressure
by (b).
different
after
point
ofedge
confluence
ofmay
the explain
blade shear
shear
layers. NRC
(a) data
[39],published
VKI blade
blade
(b).
695 authors at high subsonic/transonic downstream Mach numbers. The scatter between experimental
datapointed
from different research
organizations
may 2
bethe
partially
due to the
use ofthe
verygenerally
different ratios
of
As696
already
at the
the end
end
of Section
Section
departure
from
assumed
isobaric
As
already
pointedofout
out
at
of
2 the
thetrailing
departure
from
the/ generally
isobaric
697
the diameter
the trailing
edge pressure
hole to
edge diameter,
. Small / assumed
ratios
trailing698
edgemay
base
region
may explain
explain
the
differences
of
base
pressure
data
by different
lead
to an overestimation
of thethe
basedifferences
pressure effect.of
Hence,
base
pressure measurements
should by different
trailing
edge
base
region
may
base
pressure
data published
published
authors
at
high
subsonic/transonic
downstream
699
be taken
with a / ratio as large
as possible. Mach numbers. The scatter between experimental
authors at high subsonic/transonic downstream Mach numbers. The scatter between experimental
700 different
Theresearch
existence oforganizations
an isobaric base pressure
region
for supersonic
edgeof
flows,
i.e.different
for blades ratios of the
data from
may
be
partially
due
totrailing
thethe
use
very
data
from
research organizations
may
be
partially
due
use
ofalready
very
different ratios of
701 different
with a well-established
oblique trailing edge
shock
system as those
into
Figure 34,
was
known
diameter
edge tests
pressure
hole
to
the
edgeedge
diameter,
d/D. Small
d/D ratios
may
702of the
from
flat plate
modeledge
simulating
the
overhang
section
of convergent
turbine blades with
the
diameter
of trailing
the
trailing
pressure
hole
to trailing
the
trailing
diameter,
/ .straight
Small
/ ratios
lead
to
an
overestimation
of
the
base
effect.
Hence,
base
pressure
measurements
should
be
703
rear suction side since
1976
[18], pressure
see Figure 35.
The tests
were performed
for a gauging
angle ∗ =
may lead to an overestimation of the base pressure effect. Hence, base pressure measurements should
704
30°.
The
inclination
of
the
tail
board
attached
to
the
lower
nozzle
block
allows
to
increase
the
taken
with
a d/D
as large
as possible.
be
taken
a /ratioratio
large
as possible.
705withdownstream
Machasnumber
which
entails of course the displacement of the suction side shock
The
existence
of
an
isobaric
base
pressure
region
for
trailing
for blades
blades
706existence
boundary
along
the blade
suctionregion
side towards
the trailing edge.
The
of interaction
an isobaric
base
pressure
for supersonic
supersonic
trailingedge
edgeflows,
flows,i.e.,
i.e. for
with
a
well-established
oblique
trailing
edge
in
Figure
34,
was
already
known
707
The schlieren
photograph
on the
right shock
in Figuresystem
34 showsas
thethose
occurrence
of so-called
lip
shocks
with a well-established oblique trailing edge shock system as those in Figure 34, was already known
708plate
at model
the separation
the shear layers
the trailing
edge due
to a slight overturning
a non-with straight
from flat
flat
tests of
simulating
the from
overhang
section
of convergent
convergent
turbineand
blades
from
modelseparation
tests
simulating
of
turbine
709plate
tangential
of the flowthe
fromoverhang
the trailing section
edge surface.
In a later test series
withblades
a denserwith straight
rear suction
since
[18],
see
Figure
35.
tests
were
performed
ashock
gauging
angle
β∗2 =∗30.
710 side
instrumentation
of the
trailing
Sieverding
et al.
[4]
showed
that
the trailing for
edge
strength
rear
suction
side
since1976
1976
[18],
seeedge,
Figure
35.The
The
tests
were
performed
for
a gauging
angle
=
⁄ block
The
inclination
the
tail
board
attached
the
lower
nozzle
allows
to
increase
the
711
wasof
however
weak.
In Figure
36 theto
pressure
increase
across the
lip shock
is presented
in downstream
30°. The inclination of the tail board attached to the lower nozzle block allows to increase the
⁄ , where
712 function
the expansion
ratio around
the trailing edgeof the
is the pressure
before the interaction
Mach number
whichofnumber
entails
ofwhich
course
the displacement
suction side
shock
boundary
downstream
Mach
entails
of course
the
displacement
theAll
suction
713 start
of the expansion around
the
trailing edge
and
the
pressure
before the lip of
shock.
data are side shock
along the
suction
side the
towards
the
edge.
⁄ = 1.1
714blade
within
a bandwidth
of
− trailing
1.2. side
boundary
interaction
along
blade
suction
towards the trailing edge.
The schlieren photograph on the right in Figure 34 shows the occurrence of so-called lip shocks
at the separation of the shear layers from the trailing edge due to a slight overturning and a nontangential separation of the flow from the trailing edge surface. In a later test series with a denser
instrumentation of the trailing edge, Sieverding et al. [4] showed that the trailing edge shock strength
was however weak. In Figure 36 the pressure increase ⁄ across the lip shock is presented in
function of the expansion ratio around the trailing edge ⁄ , where
is the pressure before the
start of the expansion around the trailing edge and
the pressure before the lip shock. All data are
within a bandwidth of ⁄ = 1.1 − 1.2.
(a)
(b)
715 35. Surface
Figure 35.
(a) and trailing
pressure distribution
for flat
tests;
adapted
Figure
(a)Surface
and trailing
edgeedge
(b)(b)
pressure
distribution
forplate
flatmodel
plate
model
tests; adapted
Commented [M57]: Please add the add
716
from [18].
subgraph(a, left)
from [18].
Commented [MM58R57]: Done.
The schlieren photograph on the right in Figure 34 shows the occurrence of so-called lip shocks at
the separation of the shear layers from the trailing edge due to a slight overturning and a non-tangential
separation of the flow from the trailing edge surface. In a later test series with a denser instrumentation
of the trailing edge, Sieverding et al. [4] showed that the trailing edge shock strength was however
weak. In Figure 36 the pressure increase p3 /p2 across the lip shock is presented in function of the
expansion ratio around the trailing edge p2 /p1 , where p1 is the pressure before the start of the expansion
Figure
36. Strength
edge lipbefore
shocks; adapted
from
[4].
around717
the trailing
edge
and pof2 the
thetrailing
pressure
the lip
shock.
All data are within a bandwidth
of [M59]: Please add explan
Commented
p3 /p2 =
1.1–1.2.
718
Raffel and Kost [57] performed similar large-scale tests on a flat plate with a slotted trailing edge
Commented [MM60R59]: The graph a
719
720
to investigate the effect of trailing edge blowing on the formation of the trailing edge vortex street.
Their trailing edge pressure distributions taken at zero coolant flow ejection for downstream Mach
be considered as a single entity.
(a)
(b)
715
FigurePower
35. Surface
Int. J. Turbomach.
Propuls.
2020,(a)
5, and
10 trailing edge (b) pressure distribution for flat plate model tests; adapted
716
from [18].
26 of 55 [M57]: Please add the ad
Commented
subgraph(a, left)
Commented [MM58R57]: Done.
717
Figure
36. Strength
of the trailing
lip shocks;edge
adapted
[4].
Figure
36. Strength
of edge
the trailing
lipfrom
shocks;
adapted from [4].
Commented [M59]: Please add explan
718
Raffel and Kost [57] performed similar large-scale tests on a flat plate with a slotted trailing edge
Commented [MM60R59]: The graph
Raffel
and Kost
[57] performed similar large-scale tests on a flat plate with a slotted trailing edge
to
719 to investigate the effect of trailing edge blowing on the formation of the trailing edge vortex street.
be considered as a single entity.
investigate
effect
of trailing
edgedistributions
blowing taken
on the
formation
of the
trailing
edge vortex
720 the
Their
trailing
edge pressure
at zero
coolant flow
ejection
for downstream
Mach street. Their
trailing edge pressure distributions taken at zero coolant flow ejection for downstream Mach numbers
of M2,is = 1.01 − 1.45 confirm the existence of an isobaric pressure trailing edge base region, but the
measurements are unfortunately not dense enough to extract consistent data about the strength of the
lip shock. A few data allow to conclude that in their experiments the maximum lip shock strength is of
the order of Int.
p3J./p
2 = 1.08.
Turbomach.
Propuls. Power 2018, 3, x FOR PEER REVIEW
28 of 58
721 onnumbers
of
1.01Pressure
− 1.45 confirm
the existence
5.2. Effect
Blade Suction
Side
Distribution
, =
of an isobaric pressure trailing edge base
722 region, but the measurements are unfortunately not dense enough to extract consistent data about
In723
the discussion
ofthe
thelipschlieren
photographs
in Figure
was
shown that
the outwards motion
the strength of
shock. A few
data allow to conclude
that18
in it
their
experiments
the maximum
724
lip
shock
strength
is
of
the
order
of
/
=
1.08.
of the oscillating shear layers at the blade trailing edge does not only lead to large pressure fluctuations
in the 725
zone of
but itSide
does
alsoDistribution
induce strong acoustic pressure waves travelling upstream
5.2.separations,
Effect on Blade Suction
Pressure
on both
pressure
of the
blade. To
of the suction side
726the suction
In theand
discussion
of theside
schlieren
photographs
in facilitate
Figure 18 itthe
wasunderstanding
shown that the outwards
727 fluctuations
motion of thein
oscillating
layers
at the
bladeof
trailing
edge does not
only leadin
to large
pressure
pressure
Figureshear
37, the
left
photo
the schlieren
pictures
Figure
18 is reproduced
fluctuations
in the zone
of separations,
butPi
it generated
does also induce
strong
acoustic side
pressure
at the 728
right of
the pressure
curves.
The wave
at the
pressure
willwaves
interact with the
729 travelling upstream on both the suction and pressure side of the blade. To facilitate the understanding
suction
side of
ofthe
the
neighboring
blade
causing
significant
variations
as measured by
730
suction
side pressure
fluctuations
in Figure
37, the leftunsteady
photo of thepressure
schlieren pictures
in Figure
731 18 pressure
is reproduced
at the right
of the pressure curves.
The wave
Pi generated
the pressure
will of this blade,
fast response
sensors
implemented
between
the throat
andatthe
trailingside
edge
732 37.
interact with the suction side of the neighboring blade causing significant unsteady pressure
see Figure
The pressure wave Pi induced by the outwards motion of the pressure side shear
733 variations as measured by fast response
pressure sensors implemented between the throat and the
layer of
the
neighboring
blade
intersects
theThe
suction
sensors
3 motion
and 4.ofIt moves then
734 trailing edge of this
blade,
see Figure 37.
pressureside
wavebetween
inducedthe
by the
outwards
735 the
pressure side
shearthe
layersensors
of the neighboring
side between
the sensors
successively
upstream
across
3 and 2.blade
Theintersects
signalsthe
aresuction
asymmetric,
characterized
by a sharp
736
3 and 4. It moves then successively upstream across the sensors 3 and 2. The signals are asymmetric,
pressure rise followed by a slow decay. The amplitude of the pressure fluctuations is important with
737 characterized by a sharp pressure rise followed by a slow decay. The amplitude of the pressure
∆p = ±12%
up to 15%isof
(p01 − with
p2 ) at
±10%
sensor
2, while
pressure
738 fluctuations
important
Δ sensor
= 12%3,upand
to 15%
of ( at −
) at sensor
3, andthe10%
at sensorsignal is flat at
2, whileslightly
the pressure
signal is flat at
1 situated slightly
up-stream
of the
geometric
where
sensor739
1 situated
up-stream
ofsensor
the geometric
throat
where
the
blade throat
Mach
number reaches
740
the blade Mach number reaches
, = 0.95. The pressure waves observed at sensor 4 and further
M2,is = 0.95. The pressure waves observed at sensor 4 and further downstream at sensors 5 and 6 are
741 downstream at sensors 5 and 6 are more sinusoidal in nature and of smaller amplitude. The authors
more sinusoidal
in nature
and
of smaller
amplitude.
The
authors
suggested
that
these
fluctuations are
742 suggested
that these
fluctuations
are likely
to be caused
by the
downstream
travelling
vortices
of the
743
neighboring
blade.
likely to be caused by the downstream travelling vortices of the neighboring blade.
744
(b)
(a)
(c)
745 37. Unsteady
Figure 37. Unsteady pressure variations along rear suction side (a); schlieren photograph (b); kulite
Commented [M61]: Please add explana
Figure
pressure variations along rear suction side (a); schlieren photograph (b); kulite
746
sensor positioning (c). Adapted from [21].
sensor positioning (c). Adapted from [21].
Commented [MM62R61]: Done.
747
The periodicity of the pressure signal at position 7, slightly upstream of the trailing edge, is
748 rather poor and only phase lock averaging provides useful information on its periodic character. The
749 reason is most likely the result of a superposition of waves induced by the von Kármán vortices in
750 the wake of the neighboring blade and upstream travelling waves induced by the oscillation of the
751 suction side shear layer designated by “S” in the schlieren photographs. Right at the trailing edge,
752 position 11, we have, as expected, strong periodic signals associated with the oscillating shear layers.
Int. J. Turbomach. Propuls. Power 2020, 5, 10
27 of 55
The periodicity of the pressure signal at position 7, slightly upstream of the trailing edge, is rather
poor and only phase lock averaging provides useful information on its periodic character. The reason
is most likely the result of a superposition of waves induced by the von Kármán vortices in the wake
of the neighboring blade and upstream travelling waves induced by the oscillation of the suction side
shear layer designated by “S” in the schlieren photographs. Right at the trailing edge, position 11,
we have, as expected, strong periodic signals associated with the oscillating shear layers.
6. Turbine Trailing Edge Vortex Frequency Shedding
Besides the importance of trailing edge vortex shedding for the wake mixing process and the
trailing edge pressure distribution discussed before, vortex shedding deserves also special attention
due to its importance as excitation for acoustic resonances and structural vibrations. Heinemann
& Bütefisch [25], investigated 10 subsonic and transonic turbine cascades: two flat plate turbine tip
sections, three mid-sections with nearly axial inlet (one blade tested with three different trailing edge
thicknesses) and 3 high turning hub sections. The trailing edge thickness varied from 0.8% to 5%.
The vortex shedding frequency was determined with an electronic-optical method developed at the
DFVLR-AVA by Heinemann et al. [58].
The corresponding Strouhal numbers defined in (3) covered a wide range: 0.2 ≤ St ≤ 0.4 for
a Reynolds number range based on trailing edge thickness and downstream isentropic velocity of
0.3 × 104 ≤ Red ≤ 1.6 × 105 . The Strouhal numbers for flows from cylinders over the same Reynolds
Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW
29 of 58
number range are of the order of St = 0.19 and 0.21 as shown in Figure 38.
761
762
Figure
Strouhal numbers
Reynolds number
number range
range for
for flow
flow over
adapted
Figure 38.
38. Strouhal
numbers in
in sub-critical
sub-critical Reynolds
over cylinders;
cylinders; adapted
from
[59].
from [59].
763
764
765
766
767
768
769
770
Figure 39
Figure
39 presents
presents the
the Strouhal
Strouhal numbers
numbers for
for 33 of
of the
the 10
10 turbine
turbine blade
blade sections
sections investigated
investigated by
by
Heinemann
and
Bütefisch
[25].
The
comparison
with
the
flow
across
cylinders
is
limited
to
the
subsonic
Heinemann and Bütefisch [25]. The comparison with the flow across cylinders is limited to the
range.
The
Strouhal
for the flat
tip section
are of T2
theare
order
of St
= 0.2ofin the= Mach
subsonic
range.
The numbers
Strouhal numbers
forplate
the flat
plate tipT2section
of the
order
0.2 in
range
M
=
0.2
to
0.8
and
therewith
very
close
to
the
those
of
the
flow
over
cylinders.
On the other
the Mach2,isrange
=
0.2
to
0.8
and
therewith
very
close
to
the
those
of
the
flow
over
cylinders.
On
,
side,
the
hub
section
H2
with
a
high
rear
suction
side
curvature
distinguishes
itself
by
Strouhal
numbers
the other side, the hub section H2 with a high rear suction side curvature distinguishes itself by
as
high asnumbers
0.38 − 0.3,
decreasing
from M2,is tendency
= 0.2 to 0.9.
For the mid-section
Strouhal
aswith
highaas
0.38 – 0.3,tendency
with a decreasing
from
= 0.2 to 0.9.M2
Forwith
the
,
low
rear
suction
side
turning,
the
authors
report
Strouhal
numbers
increasing
from
St
=
0.22
to
0.29
for
mid-section M2 with low rear suction side turning, the authors report Strouhal numbers increasing
afrom
Mach range
0.8.for a Mach range 0.2 to 0.8.
= 0.220.2
to to
0.29
292
293
294
295
296
769
mid-section
M2Figure
with 39low
rearthe
suction
turning,
authors
report
Strouhal
numbers
increasing
768
presents
Strouhalside
numbers
for 3 ofthe
the 10
turbine blade
sections
investigated
by
explain
the
very
low
base
pressures.
In
addition,
the
blade
of
Deckers
and
Denton
has
a
blunt
trailing
769
Heinemann
and
Bütefisch
[25].
The
comparison
with
the
flow
across
cylinders
is
limited
to
the
770
from
= 0.22 to 0.29 for a Mach range 0.2 to 0.8.
770is experimental
subsonic range. The
Strouhal that,
numbers
for the flat to
plate
tip section trailing
T2 are of the
orderthe
of base
= 0.2pressure
in
edge, and there
evidence
compared
a circular
edge,
771 blunt
the Mach range
= 0.2 to 0.8 and therewith very close to the those of the flow over cylinders. On
,
for blades with
trailing edge
might be considerably lower. Sieverding and Heinemann [16]
772 the other side, the hub section H2 with a high rear suction side curvature distinguishes itself by
report for flat773
plate Strouhal
tests atnumbers
moderate
subsonic
Mach
drop offrom
the base
coefficient
as high
as 0.38 – 0.3,
withnumbers
a decreasinga tendency
= pressure
0.2 to 0.9. For
the
,
Propuls.
Power
2020,
5, rear
10edge
28 of 55
774 with
mid-section
M2 with
low
suction
side turning,
authors
Strouhaltrailing
numbers edge.
increasing
by 11%Int.
forJ. aTurbomach.
plate
squared
trailing
compared
tothe
that
withreport
a circular
775
from
= 0.22 to 0.29 for a Mach range 0.2 to 0.8.
(a) steam turbine tip section T2
(a) steam turbine tip section T2
297
298
771
772
773
774
775
776
777
778
779
780
781
782
783
(b) steam turbine mid section M2
(b) steam turbine mid section M2
(c) gas turbine hub section H2
(c) gas turbine hub section H2
776
Figure 39.
Strouhal number
3 blade
sections.
andand
on isentropic
downstream
Strouhal
number
for
blade
sections.
and based
M2 based
on
Figure
39.
3for
blade
sections.
: St
based
on isentropic
isentropic
downstream
Figure Figure
10.
Base39.
pressure
coefficient
for 3mid-loaded
(solid :line)
aft-loaded
(dashed
line) downstream
rotor
777
velocity. :
and
based on homogeneous downstream velocity. Adapted from [25].
velocity.
St and
M2 based
onon
homogeneous
downstream
from
[25].
velocity. : :HS1A
and
based
homogeneous
downstream
velocity.
Adapted
from
[25].
blade. Symbols:
geometry,
HS1C
geometry.
Adapted velocity.
from
[20].Adapted
Additional
Additional information
information on
on turbine
turbine blade
blade trailing
trailing edge
edge frequency
frequency measurements
measurements were
were published
published
by
response
pressure
sensors
implemented
in the
trailing
edgeedge
and
bySieverding
Sieverding[60]
[60]who
whoused
usedfast
fast
response
pressure
sensors
implemented
in blade
the blade
trailing
in
a
total
pressure
probe
positioned
at
short
distance
from
the
trailing
edge,
while
Bryanston-Cross
and
and in a total pressure probe positioned at short distance from the trailing edge, while BryanstonCamus
[61] made
use[61]
of a 20
MHzuse
bandwidth
combined
conventional
schlieren
Cross and
Camus
made
of a 20digital
MHz correlator
bandwidth
digital with
correlator
combined
with
optics.
The Strouhal
numbers
Sieverding’s
rotor blade
with a straight
rearblade
suction
side
were in rear
the
conventional
schlieren
optics.of
The
Strouhal numbers
of Sieverding’s
rotor
with
a straight
lower
part
of
the
band
width
of
the
DFVLR-AVA
data,
while
those
of
Bryanston-Cross
and
Camus
suction side were in the lower part of the band width of the DFVLR-AVA data, while those of
rotor
blades with higher
suction
sideblades
curvature
the upper
Bryanston-Cross
and Camus
rotor
withresided
higher in
suction
sidepart.
curvature resided in the upper
The
large
range
of
Strouhal
numbers
were
possibly
due
to
differences
in the state of the boundary
part.
layersThe
at the
point
of
separation.
Besides
that,
the
vortex
shedding
frequency
does in
notthe
simply
large range of Strouhal numbers were possibly due to differences
statedepend
of the
on
the trailing
edge
thickness
augmented
the boundary
displacement
thickness,does
which,
boundary
layers
at the
point of
separation.byBesides
that, thelayer
vortex
shedding frequency
not
however,
is
in
general
not
known,
but
rather
on
the
effective
distance
between
the
separating
shear
simply depend on the trailing edge thickness augmented by the boundary layer displacement
layers which could be significantly smaller than the trailing edge thickness. Patterson & Weingold [62],
simulating a compressor airfoil trailing edge flow field on a flat plate, concluded that, compared to the
effective distance between the separating upper and lower shear layers, the state of the boundary layer
before separation played a much more important role.
The influence of the boundary layer state and of the effective distance of the separating shear
layers was specifically addressed in a series of cascade and flat plate tests investigated by Sieverding
& Heinemann [16], at VKI and DLR. Figure 40 shows the blade surface isentropic Mach number
distributions of a front loaded blade, with the particularity of a straight rear suction side (blade
A), and a rear loaded blade (blade C), characterized by a high rear suction side turning angle, at a
downstream Mach number of M2is ≈ 0.8.
FigureFigure
40. Blade
Mach
number
distributions
for
blades;adapted
adapted
from
[16].
40. Blade
Mach
number
distributions
forfront
frontand
and rear-loaded
rear-loaded blades;
from
[16].
The early suction side velocity peak on blade A will cause early boundary layer transition. On the
contrary, considering the weak velocity peak on the rear suction side followed by a very moderate
recompression, the suction side boundary layer of blade C is likely to be laminar at the trailing edge
over a large range of Reynolds numbers. As regards the pressure sides of both blades, the strong
A
B
Int. J. Turbomach. Propuls. Power 2020, 5, 10
29 of 55
acceleration over most part of the surface is likely to guarantee laminar conditions at the trailing edge
on both blades and trip wires had to be used to enforce transition and turbulent boundary layers at the
trailing edge, if desired so. The blades were tested from low subsonic to high subsonic outlet Mach
numbers. Due to the use of blow down and suction tunnels at VKI and DLR, respectively, the Reynolds
number increases
with Mach
number
as shown
in Figure 41.
Int. J. Turbomach.
Propuls. Power
2018, 3, x FOR
PEER REVIEW
31 of 58
Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW
31 of 58
810 41. Variation
Figure 41.of
Variation
of Reynolds
number
in function
downstream Mach
number;
adapted
from
Figure
Reynolds
number
in function
ofofdownstream
Mach
number;
adapted
from [16].
Commented [MM65]: Please do not cu
811
[16].
over two pages.
The
the
front-loaded blade A are presented in Figure 42. In case of forced transition on
812tests forThe
tests for the front-loaded blade A are presented in Figure 42. In case of forced transition on
the pressure
through
trip wire
at 24%
of24%
theofchord
length,
thetheStrouhal
number
is nearly constant
813 side
the pressure
sidea through
a trip
wire at
the chord
length,
Strouhal number
is nearly
814 constant
and
equal
to the
~0.195
over
the entire
Mach range.
In absence
trip from
wire,
810
Figure
41.
Variation
of Reynolds
number
in function
ofrange.
downstream
number;
adapted
and roughly
equal
to
Stroughly
∼ 0.195
over
entire
Mach
In Mach
absence
ofof
aatrip
wire,thethe evolution
Commented [MM65]: Please do not cu
815 evolution
= (
) is quite different. Starting from the low Mach number and Reynolds
811
[16]. of
overend,
two pages.
)
of St =816
St(Mnumber
is
quite
different.
Starting
from
the
low
Mach
number
and
Reynolds
number
2is
end, the Strouhal number decreases from ~0.34 at
= 0.2 to ~0.26 at
= 0.53.
812
Thepoint
tests
for thedrops
front-loaded
A are
Figure
42.
In
of at
forced
on At this point
the Strouhal
decreases
from
Stblade
∼
0.34
atpresented
Mall
=in0.2
to
St This
∼case
0.26
M2istransition
=
0.53.
817 number
At this
the
suddenly
to the
level
of
cases.
sudden
change
obviously
2isturbulent
813
the
pressure
side
through
aof
trip
at has
24%taken
ofcases.
theplace
chord
the Strouhal
number
nearlyindicates that
818 suddenly
indicates
thatto
boundary
layer
transition
onlength,
thesudden
pressure
side. The
slow isdecrease
the St drops
the
level
allwire
turbulent
This
change
obviously
814
and
roughly
equal
to to~0.195
over thechange
entire Mach
In absence
of a trip boundary
wire, the
819 constant
before the
sudden
jump
points
a progressive
from arange.
laminar
to a transitional
boundary
transition
place
on theStarting
pressure
side.
The
slow
decrease
before the sudden
815
evolution
of is obviously
=has
( taken
) is quite
different.
fromnumber.
the low
Mach
number
and Reynolds
820 layer
layer which
related
to the
increasing
Reynolds
816
number
end, the
Strouhal
number
from to
~0.34
= 0.2
to ~0.26
at
=at0.53.
jump points
a progressive
change
from
a laminar
a transitional
boundary
layer
821 to
Cascade
C was
tested
with
adecreases
circular
trailing
edge
at at
DLR and
a squared
trailing
edgewhich
VKI is obviously
this
point theReynolds
to the
level
turbulent
This
sudden
change
822
over
a range
=drops
0.2 to suddenly
0.9.number.
The two
series
of of
testalldiffered
notcases.
only by
their
trailing
edgeobviously
geometry
,
related817
to theAt
increasing
818
that
layer
transition
taken
place onnumber
the pressure
The slow
decrease
823 indicates
but also, at
theboundary
same Mach
number,
by ahas
higher
Reynolds
in the side.
VKI tests,
see Figure
41.
Cascade
C was
with
a circular
trailing
atfrom
DLR
and atheto
squared
trailing
edge
819
thetested
sudden
jump
points
to a trailing
progressive
change
a laminar
a transitional
boundary
824 before
Note,
that
in
the case
of the
squared
edgeedge
the distance
between
separating
shear
layers
is at VKI over
which
is However,
obviously
related
to the
increasing
number.
a range820
M2,islayer
= 0.2
to 0.9.
The two
series
test
not
onlyedge
by their
trailing
geometry but
825
well
defined.
this
is
not
theof
case
fordiffered
theReynolds
rounded
trailing
in which
case the edge
distance
Cascade
C was
with
circular
trailing
at=DLR
arun
squared
trailing
edge
at VKI
826
should
be in any
waytested
smaller.
But
one
single
test, atedge, number
0.59, and
wasin
at VKI
also
with
asee
rounded
also, at821
the same
Mach
number,
by
a ahigher
Reynolds
the
VKI
tests,
Figure 41. Note,
822
a range
= 0.2 to any
0.9. The
seriesthe
of tests
test differed
827 over
trailing
edge to ,eliminate
bias two
between
at DLR not
andonly
VKI.by their trailing edge geometry
that in823
the case
squared
trailing
edge
distance
separating
shear
but of
also,the
at the
same Mach
number,
by a the
higher
Reynoldsbetween
number in the
the VKI
tests, see Figure
41.layers is well
828
824However,
Note, that
in the
thecase
squared
edge the distance
between
shearthe
layers
is
defined.
this
is case
notofthe
fortrailing
the rounded
trailing
edgethe
inseparating
which case
distance
should
825 way
well defined. However, this is not the case for the rounded trailing edge in which case the distance
be in any
smaller. But one single test, at M2,is = 0.59, was run at VKI also with a rounded trailing
826 should be in any way smaller. But one single test, at , = 0.59, was run at VKI also with a rounded
edge to827
eliminate
between
thebetween
tests attheDLR
and
trailingany
edgebias
to eliminate
any bias
tests at
DLRVKI.
and VKI.
828
829
830
Figure 42. Strouhal number variation with downstream Mach number for cascade A; adapted from
[16].
831
Figure 43 presents the Strouhal number for blade C both in function of the downstream Mach
832 number and the Reynolds number. Both data sets show a plateau of = 0.36 at low Mach number
833 and Figure
Reynolds
numbernumber
whichvariation
is characteristic
for a fully
trailing
boundary
829
42. Strouhal
with downstream
Mach laminar
number for
cascadeedge
A; adapted
from layer
Figure
Strouhal
number
variation
with to
downstream
number
for cascade
834 42.separation.
The
Strouhal
number starts
decrease withMach
increasing
Reynolds
number, A;
theadapted
drop of from [16].
830
[16].
835 occurring earlier at
= 0.35 × 10 for the squared trailing edge, instead of 0.6 10 for the circular
831
Figure
43At
presents
the×Strouhal
numberfor
for blade
blade
CC
both
inreach
function
of thewith
downstream
Mach
836 43
trailing
edge.
= 1.1
10 number
the squared
trailing
edge
data
plateau
= 0.24.
Note
Figure
presents
the Strouhal
both
inafunction
of the
downstream
Mach
832
number
therounded
Reynolds
number.
Both
setsindicated
show a plateau
of in the
= 0.36
at is
low
Mach
number
837
that the and
single
trailing
edge
testdata
at VKI
by a star
graph
right
in line
with
number and the Reynolds number. Both data sets show a plateau of St = 0.36 at low Mach number and
833 and Reynolds number which is characteristic for a fully laminar trailing edge boundary layer
Reynolds
which
is characteristic
a fullywith
laminar
trailing
edge
boundary
layer separation.
834 number
separation.
The Strouhal
number startsfor
to decrease
increasing
Reynolds
number,
the drop of
835 occurring
earlier
at to=decrease
0.35 × 10 for
the squared
trailing
edge, instead
of 0.6 10 the
for the
circular
The Strouhal
number
starts
with
increasing
Reynolds
number,
drop
of St occurring
836 trailing edge. At
= 1.1 × 10 the squared trailing edge data reach a plateau6with
= 0.24. Note
6
earlier at Re = 0.35 × 10 for the squared trailing edge, instead of 0.6 × 10 for the circular trailing edge.
837 that the single rounded trailing edge test at VKI indicated by a star in the graph is right in line with
At Re = 1.1 × 106 the squared trailing edge data reach a plateau with St = 0.24. Note that the single
rounded trailing edge test at VKI indicated by a star in the graph is right in line with the squared
Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW
32 of 58
838 the squared trailing edge data. Extrapolating the DLR data to higher Reynolds number one may
839 expect that they will reach the plateau of
= 0.24 at
≈ 1.1 → 1.2 10 .
840
Comparing the two curves in Figure 43 raises of course the question as to the reasons for the
841 differences between them. The possible influence of the different distance between the separating
842 shear
layers
was already
before, but, if this would be the case, then the Strouhal number
Int. J. Turbomach.
Propuls.
Power
2020, 5, mentioned
10
30 of 55
843 for the VKI tests with squared trailing edge should be higher than those of the DLR tests with
844 rounded trailing edge. There must be therefore a different reason. The key for the understanding
845 comes from flat plate tests presented in [16], see Figure 44, which showed that the difference of the
trailing846
edge Strouhal
data. Extrapolating
the DLR data to higher Reynolds number one may expect that they
number between a full laminar and full turbulent flow conditions was much bigger for tests
6.
will reach
the
plateau
of
St
=
0.24
Resquared
≈ 1.1 trailing
→ 1.2edges,
× 1030%
847 with rounded trailing edgesat
than
instead of 13%.
848
Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW
32 of 58
(a)
(b)
838 the squared trailing edge data. Extrapolating the DLR data to higher Reynolds number one may
849
Figure
Strouhal
number
with
Mach
number
(a)
and
Reynolds
(b) for cascade
839 43.expect
that 43.
they
will reach
thevariation
plateau
of
= 0.24
at
≈(a)
1.1and
→ 1.2
10 number
.
[M66]: Please add explan
Figure
Strouhal
number
variation
with
Mach
number
Reynolds
number
(b)C;for cascade Commented
C;
850
adapted
fromthe
[16].two curves in Figure 43 raises of course the question as to the reasons for the
840
Comparing
adapted from [16].
Commented [MM67R66]: Done.
841 differences between them. The possible influence of the different distance between the separating
842 shear layers was already mentioned before, but, if this would be the case, then the Strouhal number
Comparing
two
in Figure
raises
the those
question
totests
thewith
reasons for the
843 for thethe
VKI
testscurves
with squared
trailing43
edge
shouldofbecourse
higher than
of the as
DLR
844 between
rounded them.
trailing edge.
There mustinfluence
be thereforeofa the
different
reason.distance
The key for
the understanding
differences
The possible
different
between
the separating shear
845 comes from flat plate tests presented in [16], see Figure 44, which showed that the difference of the
layers was
already mentioned before, but, if this would be the case, then the Strouhal number for the
846 Strouhal number between a full laminar and full turbulent flow conditions was much bigger for tests
VKI tests
with
squared
edgethan
should
betrailing
higher
than
those
ofofthe
DLR tests with rounded trailing
847 with
roundedtrailing
trailing edges
squared
edges,
30%
instead
13%.
848 must be therefore a different reason. The key for the understanding comes from flat plate
edge. There
tests presented in [16], see Figure 44, which showed that the difference of the Strouhal number between
a full laminar and full turbulent flow conditions was much bigger for tests with rounded trailing edges
than squared trailing edges, 30% instead of 13%.
(b)
(a)
This
behavior
can
be explained
if one
assumes
thatrounded
the shape
the (b)
trailing edge may
851different
Figure
44. Strouhal
numbers
for vortex shedding
from
flat plates with
(a) and of
squared
852affect the
trailing
edges; adapted
fromshear
[16].
strongly
evolution
of the
layer, and that it is the state of the shear layer rather than
that of 853
the boundary
layer which plays the most important role in the generation of the vortex street.
This different behavior can be explained if one assumes that the shape of the trailing edge may
Of course,
a
sharp
corner
not necessarily
induce
immediately
transition,
butthan
transition will
854 strongly affect thewill
evolution
of the shear layer,
and that
it is the state offull
the shear
layer rather
855
that
of
the
boundary
layer
which
plays
the
most
important
role
in
the
generation
of
the
vortex
street.
(a)
(b)
occur over a certain length, and this length affects the length of the enrolment of the vortex and
856 Of course, a sharp corner will not necessarily induce immediately full transition, but transition will
therewith
frequency.
The transition
length
ofaffects
the
shear
layer
willnumber
be affected
by
both
849 its occur
Figure
Strouhal
numberand
variation
with Mach
number
and
Reynolds
(b)of
forthe
cascade
C; andthe Reynolds
Commented [M66]: Please add explan
857
over 43.
a certain
length,
this length
the (a)
length
of the enrolment
vortex
850
adapted
from
[16].
number858
and the
Mach
number.
therewith its frequency. The transition length of the shear layer will be affected by both the Reynolds
859
860
861
862
863
864
number and the Mach number.
Contrary to the vortex shedding for subsonic flow conditions discussed above, where the
vortices are generated by the enrolment of the separating shear layers close to the blade trailing edge,
the situation changes with the emergence of oblique shocks from the region of the confluence of the
pressure and suction side shear layers for transonic outlet Mach numbers. In this case the vortex
formation is delayed to after this region as shown already in the schlieren pictures in Figure 34.
(a)
Commented [MM67R66]: Done.
(b)
851
Figure 44. Strouhal numbers for vortex shedding from flat plates with rounded (a) and squared (b)
Figure
numbers
for [16].
vortex shedding from flat plates with rounded (a) and squared
852 44. Strouhal
trailing edges;
adapted from
(b) trailing edges; adapted from [16].
853
This different behavior can be explained if one assumes that the shape of the trailing edge may
854 strongly affect the evolution of the shear layer, and that it is the state of the shear layer rather than
Contrary to the vortex shedding for subsonic flow conditions discussed above, where the
855 that of the boundary layer which plays the most important role in the generation of the vortex street.
vortices856
are generated
the
enrolment
of the separating
shear full
layers
closebut
to transition
the blade
Of course, a by
sharp
corner
will not necessarily
induce immediately
transition,
willtrailing edge,
857 occur
over awith
certain
length,
and this length
affects the
length from
of the the
enrolment
of the
vortex
and
the situation
changes
the
emergence
of oblique
shocks
region
of the
confluence
of the
858 therewith its frequency. The transition length of the shear layer will be affected by both the Reynolds
pressure
and suction side shear layers for transonic outlet Mach numbers. In this case the vortex
859 number and the Mach number.
formation
to after
region
as shown
already
in the schlieren
in Figure
34.
860 is delayed
Contrary
to the this
vortex
shedding
for subsonic
flow conditions
discussed pictures
above, where
the
861
vortices
are
generated
by
the
enrolment
of
the
separating
shear
layers
close
to
the
blade
trailing
edge,
This is even more clearly demonstrated in Figure 45 presenting the evolution of the wake density
862 the situation changes with the emergence of oblique shocks from the region of the confluence of the
gradients
predicted with a LES by Vagnoli et al. [56], for the turbine blade shown in Figure 17, from
863 pressure and suction side shear layers for transonic outlet Mach numbers. In this case the vortex
high subsonic
to low issupersonic
outlet
Machasnumbers.
For
M2is
= 1.05
the vortex
frequency
864 formation
delayed to after
this region
shown already
in the
schlieren
pictures
in Figureshedding
34.
Int. J. Turbomach. Propuls. Power 2020, 5, 10
31 of 55
Int. J. conditioned
Turbomach. Propuls. Power
2018, 3,
x FOR PEER
REVIEW
33 of 58
is not any more
by the
trailing
edge
thickness but by the distance between
the feet of the
33 of 58
trailing865
edge shocks
emanating
from
the
region
of
the
confluence
of
the
two
shear
layers.
This is even more clearly demonstrated in Figure 45 presenting the evolution of the wake density
Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW
866
867
868
869
870
gradients predicted with a LES by Vagnoli et al. [56], for the turbine blade shown in Figure 17, from
high subsonic to low supersonic outlet Mach numbers. For
= 1.05 the vortex shedding
frequency is not any more conditioned by the trailing edge thickness but by the distance between the
feet of the trailing edge shocks emanating from the region of the confluence of the two shear layers.
(a)
865
866
867
868
,
= 0.79
(b)
,
= 0.97
(c)
,
= 1.05
=density
0.79
(b)from, subsonic
= 0.97
(c)
= 1.05
(a) wake
, exit
Figure45.
45.Change
Changeofof
gradientsfrom
supersonic
Machnumbers
numbersasas
Figure
wake, density
gradients
subsonic
totosupersonic
exit
Mach
871
Figure
45.
Change
of
wake
density
gradients
from
subsonic
to
supersonic
exit
Mach
numbers
as
predictedby
byLES
LES[56].
[56].
Commented [M68]: Please add explana
predicted
872
predicted by LES [56].
Commented [MM69R68]: Done.
Consequently,
oneobserves
observes
sudden
increase
ofthe
the
vortex
shedding
frequency
forexample
example
Consequently,
one
aasudden
increase
the
vortex
shedding
frequency
asasfor
873
Consequently,
one observes
a sudden
increaseof
of
vortex
shedding
frequency
as for example
874
recorded by Carscallen
et al.
[43],
on their
nozzleguide
guide vane,
seesee
Figure
46. 46.
recorded
byCarscallen
Carscallen
al.[43],
[43],
on
their
nozzle
guide
vane,
see
Figure
46.
recorded
by
etetal.
on
their
nozzle
vane,
Figure
875
Figure 46. Figure
Strouhal
number
versus
Mach
number;
46. Strouhal
number
versusdownstream
downstream Mach
number;
adaptedadapted
from [43]. from [43].
876 7.inAdvances
in the Numerical
Simulation
Unsteady Turbine
WakeWake
Characteristics
7. Advances
the Numerical
Simulation
ofofUnsteady
Turbine
Characteristics
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
877
The numerical simulation of unsteady turbine wake flow is relatively young, and the first
Figure 46.simulation
Strouhal number
versus downstream
Machflow
number;
adapted from
[43]. and the first
The
of unsteady
turbine wake
is relatively
young,
878numerical
contributions appeared in the mid-80s. The decade 1980–1990 has in fact seen the final move from the
contributions
appeared
in thetomid-80s.
TheNavier-Stokes
decade 1980–1990
has in
fact seen
the were
final move from
879 potential
flow models
the Euler and
equations whose
numerical
solutions
7. Advances
in
the Numerical
Simulation
of
Unsteady
Turbine
Wake
Characteristics
880
tackled
with
new,
revolutionary
for
the
time,
techniques.
Those
were
also
the
years
of
the
first
vector
the potential flow models to the Euler and Navier-Stokes equations whose numerical solutions were
881 and parallel super-computers capable of a few sustained gigaflops (CRAY YMP, IBM SP2, NEC SXThe882
numerical
simulation offorunsteady
wake
flowwere
is relatively
young,
andfirst
thevector
first
tackled
with
new, revolutionary
the time, turbine
techniques.
Those
also the years
of the
3, to quote a few examples), and of the beginning of the massive availability of computing resources
contributions
appeared
in the
The
1980–1990
hasSince
in fact
the final
move
fromSX-3,
the
and parallel
capablecount
of a decade
few sustained
(CRAY
IBMhave
SP2, NEC
883 super-computers
obeying
Moore’s
lawmid-80s.
(transistor
doubling
every twogigaflops
years).
thenseen
theYMP,
progresses
884
been
huge
both
on
the
numerical
techniques
and
on
the
turbulence
modelling
side.
Indeed,
the
most
potential
to the
Euler
and
Navier-Stokes
whose numerical
solutions
were
to quote flow
a fewmodels
examples),
and
of the
beginning
of the equations
massive availability
of computing
resources
885 advanced option, that is the Direct Numerical Simulation (DNS) approach, where all turbulent scales
tackled
with
new,
revolutionary
for
the
time,
techniques.
Those
were
also
the
years
of
the
first
vector
obeying
Moore’s
law
(transistor
count
doubling
every
two
years).
Since
then
the
progresses
have
886 are properly space-time resolved down to the dissipative one, has also recently entered the
and
parallel
super-computers
capable
of
a
few
sustained
gigaflops
(CRAY
YMP,
IBM
SP2,
NEC
SXbeen
huge
both
on
the
numerical
techniques
and
on
the
turbulence
modelling
side.
Indeed,
the
most
887 turbomachinery community starting from the pioneering work of Jan Wissink in 2002 [63].
888
Unfortunately,
because
of
the
very
severe
resolution
requirements,
there
is
still
no
DNS
study
of
3,advanced
to quote option,
a few examples),
and ofNumerical
the beginning
of the massive
availability
of computing
that is the Direct
Simulation
(DNS) approach,
where
all turbulentresources
scales are
889 turbine wake flow (TWF) at realistic Reynolds and Mach numbers, that is Re ~ 10 and high subsonic
obeying
Moore’s
law
(transistor
count
doubling
every
two
years).
Since
then
the
progresses
have
properly
space-time
resolved
down
to
the
dissipative
one,
has
also
recently
entered
the
turbomachinery
890 and transonic outlet Mach numbers with shocked flow conditions, although improvements have
been
huge
onrecently
thefrom
numerical
techniques
andofon
turbulence
modelling
side.
Indeed,
the mostof
community
starting
the [64].
pioneering
work
Jan
Wissink
in 2002
[63].
891 both
been
attained
With the development
ofthe
highly
parallelizable
codes
andUnfortunately,
the
help of
very because
892severe
large-scale
hardware
such there
a simulation
is likely
to appear
soon,
the
resultwake
of turbulent
some
advanced
option,
that computing
is the Direct
Numerical
Simulation
approach,
where
all
scalesat
the very
resolution
requirements,
is still
no(DNS)
DNS
study
ofasturbine
flow (TWF)
are
properly
space-time
resolved
down
one,
has also
the
realistic
Reynolds
and Mach
numbers,
thattois the
Re ~dissipative
106 and high
subsonic
and recently
transonicentered
outlet Mach
turbomachinery
community
from
the pioneering
work have
of Jan
Wissink
2002 [63].
numbers with shocked
flow starting
conditions,
although
improvements
been
recentlyinattained
[64].
Unfortunately,
because of
veryparallelizable
severe resolution
is still
no DNScomputing
study of
With the development
of the
highly
codesrequirements,
and the help there
of very
large-scale
turbine
wake
flowa (TWF)
at realistic
Reynolds
and Mach
thatof
is Re
~ 10
and high subsonic
hardware
such
simulation
is likely
to appear
soon, numbers,
as the result
some
cutting-edge
scientific
and
transonic
outlet
Mach numbers
with
flow
conditions,
although
improvements
have
research.
In the
meantime,
and within
the shocked
foreseeable
future,
the industrial
world
and the designers
been
recently
[64]. With
the development
of highly parallelizable
and the run
helpunsteady
of very
interested
in attained
tangled aspects
of TWF
for stage performance
enhancement codes
will certainly
large-scale
computing
such
a simulation
is advanced
likely to appear
soon,
as the
result
of some
flow simulations
wherehardware
turbulence
is handled
through
modeling.
Many
of those
simulations
cutting-edge
scientific developed
research. In
the meantime,
and
within the foreseeable
the industrial
will rely on in-house
research
codes and
turbomachinery
oriented future,
commercial
packages,
world
the designers
interested
in tangled
aspects
of TWF
stage performance
enhancement
which,and
indeed,
have improved
significantly
since
the very
firstfor
unsteady
TWF simulation.
Yet, there
will
certainly
unsteady
flow
simulations
is handled
through
are two
areas run
where
important
challenges
still where
need toturbulence
be satisfactorily
faced before
theadvanced
presently
modeling. Many of those simulations will rely on in-house developed research codes and
turbomachinery oriented commercial packages, which, indeed, have improved significantly since the
very first unsteady TWF simulation. Yet, there are two areas where important challenges still need to
be satisfactorily faced before the presently available (lower fidelity) computations could be
Int. J. Turbomach. Propuls. Power 2020, 5, 10
32 of 55
available (lower fidelity) computations could be considered reliable and successful. They can be,
loosely speaking, termed of numerical and modeling nature. We shall try to review both, in the context
of the presently discussed unsteady turbine wake flow subject category, presenting a short overview of
the available technologies. A more specialized review study on high-fidelity simulations as applied to
turbomachinery components has recently been published by Sandberg et al. [65].
7.1. Numerical Aspects
Most of the available turbine wake flow computations have been obtained with eddy viscosity
closures and structured grid technologies, although a few examples documenting the use of fully
unstructured locally adaptive solvers are available [66,67]. In the structured context turbomachinery
blades gridding is considered a relatively simple problem, and automated mesh generators of
commercial nature producing appreciable quality multi-block grids, are available [68]. The geometrical
factors most affecting the grid smoothness are the cooling holes, the trailing edge shape, the sealing
devices and the fillets. Of those the trailing edge thickness and its shape are the most important
in TWF computations. Low and intermediate pressure turbines (LPT and IPT, respectively) have
relatively sharp trailing edges, while the first and second stages of the high-pressure turbines (HPT),
often because of cooling needs, have thicker trailing edges. Typically, the trailing edge thickness to
chord ratio D/C, is a few percent in LPTs and IPTs, and may reach values of 10% or higher in some
HPTs. Thus, the ratio of the trailing edge wet area to the total one may easily range from 1/200 to
1/20, having roughly estimated the blade wet area as twice the chord. Therefore, resolving the local
curvature of the trailing edge area is extremely demanding in terms of blade surface grid, that is,
in number of points on the blade wall. Curvature based node clustering may only partially alleviate
this problem. In addition, preserving grid smoothness and orthogonality in the trailing edge area is
difficult, if not impossible with H or C-type grids, even with elliptic grid generators relying on forcing
functions [69]. Wrapping an O-type mesh around the blade is somewhat unavoidable, and in any event
the use of a multi-block or multi-zone meshing is highly desirable. Unstructured hybrid meshes would
also typically adopt a thin O mesh in the inner wall layer. Non-conformal interfaces of the patched or
overlapped type would certainly enhance the grid quality, at the price of additional computational
complexities and some local loss of accuracy occurring on the fine-to-coarse boundaries [70]. Local grid
skewness accompanied by a potential lack of smoothness will pollute the numerical solution obtained
with low-order methods, introducing spurious entropy generation largely affecting the features of
the vortex shedding flow. In those conditions, the base pressure is typically under-predicted as a
consequence of the local flow turning and separation mismatch, with a higher momentum loss and
an overall larger unphysical loss generation in the far wake. The impact of those grid distorted
induced local errors on the quality of the solution is hard to quantitatively ascertain both a-priori
and a-posteriori, and often grid refinement will not suffice, as they frequently turn out to be order 1,
rather than order hp with h the mesh size and p the order of accuracy. Nominally second order schemes
have in practice 1 < p < 2. In this context, higher order finite difference and finite volume methods,
together with the increasingly popular spectral-element methods, offer a valid alternative to standard
low order methods [71–75]. This is especially true for those techniques capable of preserving the
uniform accuracy over arbitrarily distorted meshes, a remarkable feature that may significantly relieve
the grid generation constraints, besides offering the opportunity to resolve a wider range of spatial
and temporal scales with a smaller number of parameters compared to the so called second order
methods (rarely returning p = 2 on curvilinear grids). The span of scales that needs to be resolved
and the features of the coherent structures associated to the vortex shedding depend upon the blade
Reynolds number, the Mach number (usually built with the isentropic downstream flow conditions)
and the D/C ratio. This is equivalent to state that the Reynolds number formed with the momentum
thickness of the turbulent boundary layer at the trailing edge (Reθ ) and the Reynolds number defined
using the trailing edge thickness (ReD ), are independent parameters. For thick trailing edge blades the
vortex shedding is vigorous and the near wake development is governed by the suction and pressure
Int. J. Turbomach. Propuls. Power 2020, 5, 10
33 of 55
side boundary layers which differ. Thus, the early stages of the asymmetric wake formation chiefly
depend upon the local grid richness, the resolution of the turbulent boundary layers at the TE and the
capabilities of the numerical method to properly describe their mixing process. Well-designed turbine
blades operate with an equivalent diffusion factor smaller than 0.5 yielding a θ/C ratio less than 1%
according to Stewart correlation [76]. This effectively means that the resolution to be adopted for the
blade base area will have to scale like the product θ/C × C/D which may be considerably less than
one; in order words the base area region needs more points that those required to resolve the boundary
layers at the trailing edge. Very few simulations have complied with this simple criterion as today.
Compressibility effects present additional numerical difficulties, especially in scale resolving
simulations. It is a known fact that transonic turbulent TWF calculations require the adoption of
special numerical technologies capable to handle time varying discontinuous flow features like shock
waves and slip lines without affecting their physical evolution. Unfortunately, most of the numerical
techniques with successful shock-capturing capabilities rely on a local reduction of the formal accuracy
of the convection scheme whether or not based on a Riemann solver. Since at grid scale it is hard
to distinguish discontinuities from turbulent eddies, and even more their mutual interaction, Total
Variation Diminishing (TVD) and Total Variation Bounded (TVB) schemes [77–79] are considered too
dissipative for turbulence resolving simulations, and they are generally disregarded. At present, in the
framework of finite difference and finite volume methods, there is scarce alternative to the adoption
of the class of ENO (Essentially Non Oscillatory) [80–82] and WENO (Weighted Essentially Non
Oscillatory) [83–87] schemes developed in the 90s. A possibility is offered by the Discontinuous Galerkin
(DG) methods [88]. The DG is a relatively new finite element technique relying on discontinuous
basis functions, and typically on piecewise polynomials. The possibility of using discontinuous basis
functions makes the method extremely flexible compared to standard finite element techniques, in as
much arbitrary triangulations with multiple hanging nodes, free independent choice of the polynomial
degree in each element and an extremely local data structure offering terrific parallel efficiencies are
possible. In their native unstructured framework, opening the way to the simulation of complex
geometries, h and p-adaptivity are readily obtained. The DG method has several interesting properties,
and, because of the many degrees of freedom per element, it has been shown to require much coarser
meshes to achieve the same error magnitudes when compared to Finite Volume Methods (FVM) and
Finite Difference Methods (FDM) of equal order of accuracy [89]. Yet, there seem to persist problems
in the presence of strong shocks requiring the use of advanced non-linear limiters [90] that need to
be solved. This is an area of intensive research that will soon change the scenario of the available
computational methods for high fidelity compressible turbulence simulations.
7.2. Modeling Aspects
The lowest fidelity level acceptable for TWF calculations is given by the Unsteady Reynolds
Averaged Navier-Stokes Equations (URANS) or, better, Unsteady Favre Averaged Navier-Stokes
Equations (UFANS) in the compressible domain. URANS have been extensively used in the
turbomachinery field to solve blade-row interaction problems, with remarkable success [91,92].
The pre-requisite for a valid URANS (here used also in lieu of UFANS) is that the time scale of the
resolved turbulence has to be much larger than that of the modeled one, that is to say the characteristic
time used to form the base state should be sufficiently small compared to the time scale of the unsteady
phenomena under investigation. This is often referred to as the spectral gap requirement of URANS [93].
Therefore, we should first ascertain if TWF calculations can be dealt with this technology, or else if a
spectral gap exists. The analysis amounts at estimating the characteristic time τvs , or frequency fvs ,
of the wake vortex shedding, and compare it with that of the turbulent boundary layer at the trailing
edge, τbll , or fbl . The wake vortex shedding frequency is readily estimated from:
fvs = St
V2,is
= f (geometry, Reynolds, Mach)
dte
Int. J. Turbomach. Propuls. Power 2020, 5, 10
34 of 55
which has been shown to depend upon the turbine blade geometry and the flow regimes (see Figures 39,
42–44 and 46). For the turbulent boundary layers the characteristic frequency can be estimated,
using inner scaling variables, as:
u2
fbl ≈ τ
ν
q
with uτ = τρw the friction velocity, and ν the kinematic viscosity. Assuming the boundary layer to be
fully turbulent from the leading edge, and using the zero pressure gradient incompressible flat plate
correlation of Schlichting [59]:
0.059
2 τw
= 1/5
C f ,x =
2
ρ u∝
Rex
one gets:
u2τ
u2
u2
= C f ,x ∝ = 0.0295 Rex −1/5 ∝ .
ν
2ν
ν
At the turbine trailing edge x = C, and u∞ = V2,is so that:
fbl = 0.0295
V2,is 4/5
Re2,is
C
Therefore, the ratio of the turbulent boundary layer characteristic frequency to the wake vortex
shedding one is, roughly:
4/5
fbl
dte Re2,is
τvs
= 0.0295
(6)
=
fvs
τbl
C St
The explicit dependence of the Strouhal number upon the geometry term dte /C is unknown,
although clear trends have been highlighted in the previous section. However, taking dte /C ≈ 0.05 and
St ≈ 0.3 as reasonable values, Equation (6) returns:
fbl
4/5
≈ 0.005 Re2,is
fvs
(7)
The estimates obtained from the above Equation are reported in Table 3, for a few Reynolds
numbers.
Table 3. Turbulent boundary layer to vortex shedding frequency ratio; Equation (7).
Re2,is
fbl
fvs
5 × 105
180
106
310
2 × 106
550
3 × 106
760
From the above table it is readily inferred that, for the problem under investigation, a neat spectral
gap exists, and, thus, URANS calculations can be carried out with some confidence. The results
reported in the foregoing confirm that this is indeed the case.
Formally, RANS are obtained from URANS dropping the linear unsteady terms, and, therefore,
the closures developed for the steady form of the equations apply to the unsteady ones as well.
Whether the abilities of the steady models broaden to the unsteady world is controversial, even though
the limited available literature seem to indicate that this is rarely the case. A review of the existing
RANS closures is out of the scope of the present work, and the relevant literature is too large to
be cited here, even partially. In the turbomachinery field, turbulence and transition modelling
problems have been extensively addressed over the past decades, and significant advances have been
achieved [94–96]. Here, we will mainly stick to those models which have been applied in the TWF
simulations presently reviewed.
In the RANS context Eddy Viscosity Models (EVM) are by far more popular than Reynolds
Stress Models (RSM), whether differential (DRSM) or algebraic (ARSM). Part of the reasons are to be
Int. J. Turbomach. Propuls. Power 2020, 5, 10
35 of 55
found with the relatively poor performance of DRS and ARS when compared to the computational
effort required to implement these models, especially for unsteady three-dimensional problems. Also,
the prediction of pressure induced separation and, more in general, of separated shear layers is,
admittedly, disappointing, so that the expectations of advancing the fidelity level attainable with
EVM has been disattended. This explains why most of the engineering applications of RANS,
and thus of URANS, are routinely based on EVM, and typically on algebraic [97], one equation [98]
and two equations (k- of Jones and Launder [99], k-ω of Wilcox [100], Shear Stress Transport (SST)
of Menter [101]) formulations. In the foregoing we shall see that the TWF URANS computations
reviewed herein all adopted the above closures. A few of those were based on the k-ω model of Wilcox.
This closure, and its SST variant, has gained considerable attention in the past two decades and it is
widely used and frequently preferred to the k- models, as it is reported to perform better in transitional
flows and in flows with adverse pressure gradients. Further, the model is numerically very stable,
especially its low-Reynolds number version, and considered more “friendly” in coding and in the
numerical integration process, than the k- competitors [100].
On the scale resolved simulations the scenario is rather different. Wall resolved Large Eddy
Simulations (LES) are now recognized as unaffordable for engineering applications because of the very
stringent near wall resolution requirements and of the inability of all SGS models to account for the
effects of the near wall turbulence activity on the resolved large scales [102,103]. On the wall modeled
side, the most successful approaches rely on hybrid URANS-LES blends, and in this framework the
pioneering work of Philip Spalart and co-workers should be acknowledged [104,105]. Already 20 years
ago this research group introduced the Detached Eddy Simulation (DES), a technique designed to
describe the boundary layers with a URANS models and the rest of the flow, particularly the separated
(detached) regions, with an LES. The switching or, better, the bridging between the two methods takes
place in the so called “grey area” whose definition turned out to be critical, because of conceptual
and/or inappropriate, though very frequent, user decisions. The latter are particularly related to the
erroneous mesh sizes selected for the model to follow the URANS and the LES branches.
Nevertheless, the original DES formulation suffered from intrinsic to the model deficiencies
leading to the appearance of unphysical phenomena in thick boundary layers and thin separation
regions. Those shortcomings appear when the mesh size in the tangent to the wall direction, i.e.,
parallel to it, ∆|| , becomes smaller than the boundary layer thickness δ, either as a consequence of a
local grid refinement, or because of an adverse pressure gradient leading to a sudden rise of δ. In those
instances, the local grid size, i.e., The filter width in most of the LES, is small enough for the DES
length scale to fall in the LES mode, with an immediate local reduction of the eddy viscosity level
far below the URANS one. The switching to the LES mode, however, is inappropriate because the
super-grid Reynolds stresses do not have enough energy content to properly replace the modeled one,
a consequence of the mesh coarseness. The decrease in the eddy viscosity, or else the stress depletion,
reduces the wall friction and promotes an unphysical premature flow separation. This is the so-called
Model Stress Depletion (MSD) phenomenon, leading to a kind of grid induced separation, which is not
easy to tackle in engineering applications, because it entails the unknown relation between the flow to
be simulated and the mesh spacing to be used. In recent years, however, two new models offering
remedies to the MSD phenomenon have been proposed, one by Philip Spalart and co-workers [106],
the other by Florian Menter and co-workers [107]. Before proceeding any further, let us briefly mention
the physical idea underlying the DES approach. In its original version based on the Spalart and
Allmaras turbulence model [98] the length scale d˜ used in the eddy viscosity is modified to be:
d˜ ≡ min(d, CDES ∆)
(8)
where d is the distance from the wall, ∆ a measure of the grid spacing (typically ∆ ≡ max(∆x, ∆y, ∆z)
in a Cartesian mesh), and CDES a suitable constant of order 1. The URANS and the wall modeled
LES modes are obtained when d˜ ≡ d and d˜ ≡ CDES ∆, respectively. The DES formulation based on the
two equations Shear Stress Transport turbulence model of Menter [101] is similar. It is based on the
Int. J. Turbomach. Propuls. Power 2020, 5, 10
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introduction of a multiplier (the function FDES ) in the dissipation term of the k-equation of the k-ω
model which becomes:
β∗ ρkωFDES
with:
FDES = max
Lt
CDES ∆
!
,1
(9)
In the above equations Lt is the turbulent length scale as predicted by the k-ω model, β∗ = 0.09 the
model equilibrium constant and CDES a calibration constant for the DES formulation:
√
k
Lt = ∗
βω
Both the DES-SA (DES based on the Spalart and Allmaras model) and the DES-SST (DES based on
Menter’s SST model) models suffer from the premature grid induced separation occurrence previously
discussed. To overcome the MSD phenomenon Menter and Kuntz [107] introduced the FSST blending
functions that were designed to reduce the grid influence of the DES limiter (9) on the URANS part of
the boundary layer that was “protected” from the limiter, that is, protected from an uncontrolled and
undesired switch to the LES branch. This amounts to modify Equation (9) as follows:
"
#
Lt
(1 − FSST ), 1
FDES-SST-zonal = max
CDES ∆
√
with FSST selected from the blending functions
√ of the SST model, whose argument is k/(ωd), that
is the ratio of the k-ω turbulent length scale k/ω and the distance from the wall d. The blending
functions are 1 in the boundary layer and go to zero towards the edge.
The proposal of Spalart et al. [106] termed DDES is similar to the DES-SST-zonal proposal of
Menter et al. [107], and, while presented for the Spalart and Allmaras turbulence model it can be readily
extended to any EVM. In the Spalart and Allmaras model a turbulence length scale is not solved for
through a transport equation. It is instead built from the mean shear and the turbulent viscosity:
rd =
L2t
νt
= p
(κd)
2Sij Sij (κd)2
2
with Sij = ∂Ui /∂x j + ∂U j /∂xi /2 the rate of strain tensor, νt the eddy viscosity and κ the von Kàrmàn
constant. This quantity, actually a length scale squared, is 1 in the outer portion of the boundary layer
and goes to zero towards its edge. The term νt is often augmented of the molecular viscosity ν to
ensure that rd remains positive in the inner layer. This dimensionless length scale squared is used in
the following function:
fd = 1 − tanh [8rd ]3
reaching 1 in the LES region where Lt < κd and 0 in the wall layer. It plays the role of 1 − FSST in the
DES-SST-zonal model. Additional details on the design and calibration of the model constants can be
found in [106]. The Delayed DES (DDES), a surrogate of the DES, is obtained replacing d˜ in Equation
(8) with the following expression:
d˜ ≡ d − fd max(0, d − CDES ∆)
(10)
The URANS and the original DES model are retrieved when fd = 0 and fd = 1, respectively,
corresponding to d˜ ≡ d and d˜ ≡ CDES ∆. This new formulation makes the length scale (10) depending
on the resolved unsteady velocity field rather than on the grid solely. As the authors stated the model
prevents the migration on the LES branch if the function fd is close to zero, that is the current point is in
the boundary layer as judged from the value of rd . If the flow separates fd increases and the LES mode
is activated more rapidly than with the classical DES approach. As for DES this strategy, designed to
Int. J. Turbomach. Propuls. Power 2020, 5, 10
37 of 55
tackle the MSD phenomenon, does not relieve the complexity of generating adequate grids, that is
grids capable of properly resolving the energy containing scales of the LES area. Thus, unlike a proper
grid assessment study is conducted, it will be difficult to judge the quality of those scale resolving
models especially in the present context of TWF.
7.3. Achievements
Unsteady turbine wake flow simulation is a relatively new subject and the very first pioneering
works appeared in the mid-90s [66,108–110]. The reason is twofold; on one side the numerical and
modelling capabilities were not yet ready to tackle the complexities of the physical problem, and on the
other side, the lack of detailed experimental measurements discouraged any attempt to simulate the
wake flow. This until the workshop held at the von Kàrmàn Institute in 1994 during a Lecture Series [37],
where the first detailed time resolved experimental data of a thick trailing edge turbine blade where
presented and proposed for experiment-to-code validation in an open fashion. The turbine geometry
was also disclosed. As mentioned in Section 3 those tests referred to a low Mach, high Reynolds number
case (M2,is = 0.4, Re2 = 2 × 106 ). The numerical efforts of [108,110–113] addressing this test case and
listed in Table 4, were devoted at ascertaining the capabilities of the state-of-the-art technologies to
predict the main unsteady features of the flow, namely the wake vortex shedding frequency and the
time averaged blade surface pressure distribution, particularly in the base region.
Table 4. Available computations of the M2,is = 0.4, Re2 = 2 × 106 VKI LS-94 turbine blade.
Authors
Manna
et al. [110]
Arnone
et al. [111]
Sondak
et al. [112]
Ning et al.
[113]
Eqs.
Numerical
Method
Grid
Closure
URANS
CC-FVM
Structured
Multi-Block (H-O)
URANS
CC-FVM
Structured C-grid
URANS
FDM
Overset grids (H-O)
EVM (Baldwin &
Lomax [97])
EVM (Baldwin &
Lomax [97])
EVM (Deiwert
et al. [114])
URANS
CV-FVM
Structured
Multi-Block
(H-O-H-H)
EVM (Roberts
[115])
Space/Time
Discretization
Grid
Density
2nd/2nd
44k
2nd/2nd
36k
3nd/2nd
60k
2nd/2nd
42k
All of the above contributors solved the URANS with a Finite Volume (FVM) or Finite Difference
Method (FDM) and adopted simple algebraic closures. Both Cell Vertex (CV) and Cell Centered (CC)
approaches where used. The more recent computations of Magagnato et al. [116] referred to a similar
test case, though with rather different flow conditions, and will not be reviewed. Appropriate resolution
of the trailing edge region and the adoption of O grids turned out to be essential to reproduce the basic
features of the unsteady flow in a time averaged sense. The use of C grids with their severe skewing
and distortion of the base region affected the resolved flow physics and required computational and
modelling tuning to fit the experiments. The time mean blade loading could be fairly accurately
predicted (see Figure 47) by nearly all authors listed in Table 4, although discrepancies with the
experiments and among the computations exist. They have been attributed to stream-tube contraction
effects and to the tripping wire installed on the pressure side at x/Cax = 0.61 in the experiments [112].
292
explain the very low base pressures. In addition, the blade of Deckers and Denton has a blunt trailing
293 Propuls.
edge,
and
there
evidence that, compared to a circular trailing edge,
the
Int.
Propuls.
Power
2018,
3,3,3,
PEER
REVIEW
40
of
58
Int.
J.Turbomach.
Turbomach.
Propuls.
Power
2018,
xFOR
FOR
PEER
REVIEW
40
of
58base pressure
Int.
Turbomach.
Propuls.
Power
2018,
xFOR
FOR
PEER
REVIEW
40
of
58
Int.
J.J.J.Turbomach.
Power
2018,
3,is
xxexperimental
PEER
REVIEW
40
of
58
294
for blades with blunt trailing edge might be considerably lower. Sieverding and Heinemann [16]
Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW
40 of 58
295
report
for flat plate tests at moderate subsonic Mach numbers a drop of the
base pressure coefficient
Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW
40 of 58
296
11%
for Power
acontraction
plate
with
trailing
compared
to thatside
with
1139by stream-tube
effects
and to the
tripping edge
wire installed
on the pressure
at a/ circular
= 0.61 trailing
Int.
J. Turbomach.
Propuls.
2020,
5,
10squared
38edge.
of 55
1140 stream-tube
in the experiments
[112].
1139
contraction
effects and to the tripping wire installed on the pressure side at /
1140 in the experiments [112].
1135
1135
1135
1135
1136
1136
1136
1136
1137
1137
1137
1137
= 0.61
6Time
Figure
47.
VKI
LS94
turbine
blade,
0.4,
22210
case.
Time
mean
blade
surface
isentropic
Figure
47.
VKI
LS94
turbine
blade,
=
0.4,
=
210
10
case.
Time
mean
blade
surface
isentropic
1141
Figure
47.
LS94
turbine
= 0.4,
=
2210
case.
Time
mean
blade
surface
isentropic
Figure
47.
VKI
LS94
turbine
blade,
0.4,
=
10
case.
Time
mean
blade
surface
isentropic
Figure
47.
VKI
turbine
blade,
===
0.4,
==
2for
case.
mean
blade
surface
isentropic
47.LS94
VKI
LS94
turbine
blade,
=, 0.4,
Re
=
×
10
case.
Time
mean
blade
surface
isentropic
, ,M
, , blade,
2,is
297Figure
Figure
10.VKI
Base
pressure
coefficient
mid-loaded
(solid
line)
and
aft-loaded
(dashed line) rotor
1142
Mach
number
distribution.
experiments
[32];
[110];
1141
Figure
47. VKI LS94 experiments
turbine
blade,
=
0.4,
= 2 10
case. Time mean[111];
blade
surface[112];
isentropic
Mach
number
distribution.
experiments
[32];
[110];
[111];
[112];
Mach
number
distribution.
experiments
[32];
[110];
[111];
[112];
Mach
number
distribution.
experiments
[32];
[110];
[111];
[112];
Mach
number
distribution.
[32];
[110];
[111];
[112];
,
Mach
number
distribution.
experiments
[32];
[110];
[111];
[112];
[113].
2981143
blade.
Symbols:
HS1A
geometry,
HS1C
geometry.
Adapted
from
[20].
[113].number distribution.
1142
Mach
experiments [32];
[110];
[111];
[112];
[113].
[113].
[113].
[113].
1143
[113].
The
pressure
regionregion
was also
wellwell
reproduced
available numerical
1144time averaged
The time base
averaged
base pressure
was fairly
also fairly
reproduced by
by the
the available
1145
numerical
data,
although
the
differences
among
the
computations
and
the
experiments
are
generally
The
time
averaged
base
pressure
region
was
also
fairly
well
reproduced
by
the
available
data,
although
thetime
differences
among
theregion
computations
and
the
experiments
are
generally
larger
The
averaged
base
pressure
was
alsofairly
fairly
well
reproduced
by theby
available
The1144
timeaveraged
averaged
base
pressure
region
was
also
fairly
well
reproduced
by
the
available
The
time
averaged
base
pressure
region
was
also
fairly
well
reproduced
by
the
available
The
time
base
pressure
region
was
also
well
reproduced
the
available
1146 numerical
larger than
those
reported
in Figure 48.
Indeed,
the underlying
physics
is more complex,
as the
data,
although
the
differences
among
the
computations
and
thethe
experiments
are generally
numerical
data,
although
the
differences
among
the
computations
and
experiments
are
generally
than1145
those
reported
in
Figure
48.
Indeed,
the
underlying
physics
is
more
complex,
as
the
presence
numerical
data,
although
the
differences
among
the
computations
and
the
experiments
are
generally
numerical
data,
although
the
differences
among
the
computations
and
the
experiments
are
generally
numerical
data,
although
the
differences
among
the
computations
and
the
experiments
are
generally
1147 larger
presence
the two
pressure
suction
side sharp
at theislocations
of the boundary
1146
thanofthose
reported
in and
Figure
48. Indeed,
theover-expansions
underlying physics
more complex,
as the
larger
than
those
reported
in
Figure
48.
Indeed,
the
underlying
physics
more
complex,
as
the
ofthan
the
two
pressure
and
suction
side
sharp
the
locations
the
boundary
layer
larger
than
those
reported
in
Figure
48.
Indeed,
theunderlying
underlyingat
physics
ismore
more
complex,
asthe
the
larger
than
those
reported
in
Figure
48.
Indeed,
the
underlying
physics
more
complex,
as
the
larger
those
reported
in
Figure
48.
Indeed,
the
physics
isisis
complex,
as
1148
layer
separation
suggests.
1147
presence
of the
two
pressure
and
suction
sideover-expansions
sharp
over-expansions
at
the locations
of
theof
boundary
1148
layer
separation
suggests.
presence
of
the
two
pressure
and
suction
side
sharp
over-expansions
at
the
locations
of
the
boundary
separation
suggests.
presence
of
the
two
pressure
and
suction
side
sharp
over-expansions
at
the
locations
of
the
boundary
presence
of
the
two
pressure
and
suction
side
sharp
over-expansions
at
the
locations
of
the
boundary
presence
of
the
two
pressure
and
suction
side
sharp
over-expansions
at
the
locations
of
the
boundary
1138
1138
1138
1138
1139
1139
1139
1139
1140
1140
1140
1140
1141
1141
1141
1141
1142
layer
separation
suggests.
1142 layer
layer
separation
suggests.
1142
layer
separation
suggests.
1142
separation
suggests.
1149
Figure 48. VKI LS94 turbine blade,
= 2 10 case. Time mean base pressure
, = 0.4,
Commented [M73]: This figure has no
1150 48. VKI
distribution.
ForLS94
symbols
seeM
Figure
47.0.4, Re
1149
Figure
48. turbine
VKI
turbine
blade,
=
0.4,
=
210
106 case.
case. Time
Time mean
mean base
base pressure
Figure
LS94
blade,
=
=
2
×
pressure distribution.
,
2
2,is
within the text
of the
manuscript.
Commented
[M73]:
This
figure hasPlease
not b
1150
distribution.
For 47.
symbols see Figure 47.
For
symbols
see
Figure
subgraph.
within
the
text
of
the
manuscript.
Please
a
1151
The location and the magnitude of these two accelerations seem within the reach of the adopted
1143
1143
1143
1143
1144
1144
1144
1144
1145
1145
1145
1145
1146
1146
1146
1146
1147
1147
1147
1147
1148
1148
1148
1148
1149
1149
1149
1149
1150
1150
1150
1150
1151
1151
1151
1151
1152
1152
1152
1152
1153
1153
1153
1153
1154
1154
1154
1154
1155
1155
1155
1155
1156
1156
1156
1156
1157
1157
1157
1157
1158
1158
1158
1158
1152 closure,
as well and
as the
plateau
of the
region. The
predicted
basereach
pressure
subgraph.
1151
The location
thepressure
magnitude
of these
twobase
accelerations
seem
within the
of thecoefficients
adopted
Commented [MM74R73]: You are rig
The
and
the
magnitude
ofwell
these
accelerations
seem
within
the
of the adopted
1153location
definedas
by
Equation
(2)
agree fairly
the experimental
as well
aspressure
with the
onereach
obtained
1152
closure,
well
as the
pressure
plateau
ofwith
thetwo
base
region. The value,
predicted
base
coefficients
corrected. The
subgraph explanation
is r
Commented
[MM74R73]:
You are right,
1154
fromas
the
VKI
correlation
[110].
The
success
ofbase
these
simple
models
ispredicted
attributed
tothe
the
proper
space- coefficients
1153
defined
bythe
Equation
(2) agree
fairly
well
experimental
value,
well as with
one
obtained
closure,
as well
pressure
plateau
ofwith
thethe
region.
Theas
base
pressure
because
the
graph
and
the
associated
ske
corrected.
The
subgraph
explanation
is
rea
1155
timethe
resolution
of the boundary
layers
at separation
pointsmodels
in the trailing
edge region.
Again,spacethis has
1154
from
VKI correlation
[110].
The
success
of
these simple
is attributed
to
the proper
Figure
48.
VKI
LS94
turbine
blade,
==
0.4,
case.
Time
mean
base
pressure
Figure
48.VKI
VKILS94
LS94
turbine
blade,
=
0.4,
=
210
10 case.
case.
Time
mean
basepressure
pressure
Figure
48.
VKI
LS94
turbine
blade,
0.4,
10
case.
Time
mean
base
pressure
Figure
48.
turbine
blade,
0.4,
===
22210
Time
defined
by
Equation
(2)
agree
fairly
well
the
experimental
value,
asmean
well
asbase
with
the
one obtained
,,=
, ,with
1156
been
documented
by
Manna
et
al.
[110]
and
by
Sondak
et
al.
[112]
(see
Figure
49)
who
could
show
a
considered
as
a
single
entity
because
the
graph
and
the
associated
sketc
1155 time resolution of the boundary layers at separation points in the trailing edge region. Again, this has
distribution.
For
symbols
see
Figure
47.
distribution.
For
symbols
see
Figure
47. of theof
distribution.
For
symbols
see
Figure
47.
distribution.
For
symbols
see
Figure
47.
1157VKI
more
than satisfactory
agreement
computed
timeetaveraged
velocity
profiles
with
thethe
measured
from
the
correlation
[110].
The
these
simple
is attributed
to
proper
space-time
1156
been
documented
by Manna
etsuccess
al. [110] and
by Sondak
al. models
[112] (see
Figure
49) who
could
show
a
considered as a single entity
1158
one,
both
on
the
pressure
and
sides atpoints
1.75
diameters
the trailing
edge
( / = this has been
1157 of
more
satisfactory
agreement
of the computed
time
averaged
velocity of
profiles
with
the
measured
resolution
thethan
boundary
layers
at suction
separation
in theupstream
trailing
edge
region.
Again,
1159
1.75
with
=
0pressure
at the trailing
edge,
and
=
).
The thinner
pressure
side
boundary
layer
and
theadopted
The
location
and
the
magnitude
of
these
two
accelerations
seem
within
the
reach
adopted
1158
one,
both
onmagnitude
the
and
suction
sides
ataccelerations
1.75
diameters
upstream
of
the trailing
edge
( of
/of
=the
The
location
and
the
magnitude
of
these
two
accelerations
seem
within
the
reach
of
the
adopted
The
location
and
the
magnitude
of
these
two
accelerations
seem
within
the
reach
of
the
adopted
The
location
and
the
of
these
two
seem
within
the
reach
the
documented
by
Manna
etstrengthening
al. [110]
and
by
Sondak
et al.shedding
[112]
(see
Figure
49)
could
1160 1.75
blade
circulation
theand
pressure
side
vortex
were
estimated
to who
be theand
cause
ofshow a more
1159
with
=
0
at
the
trailing
edge,
=
).
The
thinner
pressure
side
boundary
layer
the
closure,
as
well
as
the
pressure
plateau
of
the
base
region.
The
predicted
base
pressure
coefficients
closure,
as
wellas
asthe
the
pressure
plateau
ofthe
the
base
region.
The
predicted
base
pressure
coefficients
closure,
as
well
as
the
pressure
plateau
of
the
base
region.
The
predicted
base
pressure
coefficients
closure,
as
well
pressure
of
base
region.
The
predicted
base
pressure
coefficients
1161
the
higher
local strengthening
overplateau
expansion
at
the
trailing
edge
[32].shedding
The
very
consistent
grid
refinement
study
of
than
satisfactory
agreement
of
the computed
time
averaged
velocity
profiles
the
one,
1160
blade
circulation
the
pressure
side
vortex
were
estimated
to be with
the cause
ofmeasured
defined
by
Equation
(2)
agree
fairly
well
with
the
experimental
value,
as
well
as
with
the
one
obtained
defined
by
Equation
(2)
agree
fairly
well
with
the
experimental
value,
as
well
as
with
the
one
obtained
defined
by
Equation
(2)
agree
fairly
well
with
the
experimental
value,
as
well
as
with
the
one
obtained
defined
by
Equation
(2)
agree
fairly
well
with
the
experimental
value,
as
well
as
with
the
one
1162
[112]
brought
some
improvements
in
the
thinner
and
fuller
pressure
side
boundary
layers
higher local
over
expansion
at the at
trailing
[32]. The very
consistentof
grid
refinement
of obtained
both1161
on thethe
pressure
and
suction
sides
1.75edge
diameters
upstream
the
trailingstudy
edge
(s/D
= ±1.75
1162
[112] brought
someThe
improvements
in
the
thinner
and
fuller pressure
side boundary
layers
from
the
VKI
correlation
[110].
The
success
of
these
simple
models
attributed
to
the
proper
spacefrom
the
VKI
correlation
[110].
The
success
of
these
simple
models
is
attributed
to
the
proper
spacefrom
the
VKI
correlation
[110].
The
success
of
these
simple
models
attributed
to
the
proper
spacefrom
the
VKI
correlation
[110].
success
of
these
simple
models
isisis
attributed
to
the
proper
space-
with s = 0 at the trailing edge, and D = dte ). The thinner pressure side boundary layer and the blade
time
resolution
of
the
boundary
layers
at
separation
points
in
the
trailing
edge
region.
Again,
this
has
time
resolution
of
the
boundary
layers
at
separation
points
in
the
trailing
edge
region.
Again,
this
has
time
resolution
of
the
boundary
layers
at
separation
points
in
the
trailing
edge
region.
Again,
this
has
time
resolution
of
the
boundary
layers
at
separation
points
in
the
trailing
region.
has
circulation
strengthening
the
pressure
side vortex
shedding
were edge
estimated
toAgain,
be
the this
cause
of the
been
documented
by
Manna
et
al.
[110]
and
by
Sondak
et
al.
[112]
(see
Figure
49)
who
could
show
been
documented
by
Manna
et
al.
[110]
and
by
Sondak
et
al.
[112]
(see
Figure
49)
who
could
show
a
been
documented
by
Manna
et
al.
[110]
and
by
Sondak
et
al.
[112]
(see
Figure
49)
who
could
show
been
documented
by
Manna
et
al.
[110]
and
by
Sondak
et
al.
[112]
(see
Figure
49)
who
could
show
higher local over expansion at the trailing edge [32]. The very consistent grid refinement study ofaaa[112]
more
than
satisfactory
agreement
of
the
computed
time
averaged
velocity
profiles
with
the
measured
more
than
satisfactory
agreement
of
the
computed
time
averaged
velocity
profiles
with
the
measured
more
than
satisfactory
agreement
the
computed
time
averaged
velocity
profiles
with
the
measured
more
than
satisfactory
agreement
ofof
the
computed
averaged
velocity
profiles
with
the
measured
brought
some improvements
in
the
thinner time
and
fuller
pressure
side
boundary
layers
predictions.
one,
both
on
the
pressure
and
suction
sides
at
1.75
diameters
upstream
of
the
trailing
edge
(
/// ===
±±
one,
both
on
the
pressure
and
suction
sides
at
1.75
diameters
upstream
of
the
trailing
edge
=±
±
one,
both
on
the
pressure
and
suction
sides
at
1.75
diameters
upstream
of
the
trailing
edge
one,
both
on
the
pressure
and
suction
sides
at
1.75
diameters
upstream
of
the
trailing
edge
(
It is no surprise that with a proper characterization of the boundary layers and of the((/base
region,
1.75
with
=
0
at
the
trailing
edge,
and
=
).
The
thinner
pressure
side
boundary
layer
and
the
1.75
with
=
0
at
the
trailing
edge,
and
=
).
The
thinner
pressure
side
boundary
layer
and
the
1.75
with
=
0
at
the
trailing
edge,
and
=
).
The
thinner
pressure
side
boundary
layer
and
the
1.75
with
=
0
at
the
trailing
edge,
and
=
).
The
thinner
pressure
side
boundary
layer
and
the
the computed and measured losses agreed well.
blade
circulation
strengthening
the
pressure
side
vortex
shedding
were
estimated
to
be
the
cause
of
bladecirculation
circulationstrengthening
strengtheningthe
thepressure
pressureside
sidevortex
vortexshedding
shedding
wereestimated
estimatedto
tobe
bethe
thecause
causeof
of
blade
circulation
strengthening
the
pressure
side
vortex
shedding
were
estimated
to
be
the
cause
of
blade
were
the
higher
local
over
expansion
at
the
trailing
edge
[32].
The
very
consistent
grid
refinement
study
of
the
higher
local
over
expansion
at
the
trailing
edge
[32].
The
very
consistent
grid
refinement
study
of
the
higher
local
over
expansion
at
the
trailing
edge
[32].
The
very
consistent
grid
refinement
study
of
the
higher
local
over
expansion
at
the
trailing
edge
[32].
The
very
consistent
grid
refinement
study
of
[112]
brought
some
improvements
in
the
thinner
and
fuller
pressure
side
boundary
layers
[112]brought
broughtsome
someimprovements
improvementsin
inthe
thethinner
thinnerand
andfuller
fullerpressure
pressureside
sideboundary
boundarylayers
layers
[112]
brought
some
improvements
in
the
thinner
and
fuller
pressure
side
boundary
layers
[112]
predictions.
no
surprise
that
with
proper
characterization
of
the
boundary
layers
and
of
the
predictions.ItItItItisisis
isno
nosurprise
surprisethat
thatwith
withaaaaproper
propercharacterization
characterizationof
ofthe
theboundary
boundarylayers
layersand
andof
ofthe
the
predictions.
no
surprise
that
with
proper
characterization
of
the
boundary
layers
and
of
the
predictions.
base
region,
the
computed
and
measured
losses
agreed
well.
base
region,
the
computed
and
measured
losses
agreed
well.
base
region,
the
computed
and
measured
losses
agreed
well.
base
region,
the
computed
and
measured
losses
agreed
well.
Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW
41 of 58
Int. J. Turbomach.
Power
5, 10 that with a proper characterization of the boundary layers and of the
1163 Propuls.
predictions.
It is2020,
no surprise
1164
39 of 55
base region, the computed and measured losses agreed well.
(b)
(a)
1165 49.
Figure
on
1166
pressure side (a) and suction side (b) at s/D = ±1.75. For symbols see Figure 47.
1167
The correct prediction of the vortex shedding frequency within experimental uncertainty proved
1168
to be
more difficult,
to this aim,
the near wake
physics has
to be captured
in terms ofuncertainty
the largeThe
correct
prediction
of since,
the vortex
shedding
frequency
within
experimental
proved
1169
scale
coherent
structures
formation,
development
and
propagation.
This
is
probably
outside
the
reach
to be more difficult, since, to this aim, the near wake physics has to be captured in terms of the
1170 of any eddy viscosity closure, and most likely of the URANS approach. Also, it has been shown
large-scale
structures
formation,
development
and propagation.
This is probably
1171coherent
experimentally
that the
dominant frequency
does not appear
as a single sharp amplitude
peak in the outside the
Fourier
transform,closure,
but ratherand
as a small
frequency
band-width
[32].
reach of1172
any eddy
viscosity
mostsize
likely
of the
URANS
approach. Also, it has been shown
1173
This is best seen with the help of Table 5, comparing the predicted Strouhal number with the
experimentally
that
the dominant frequency does not appear as a single sharp amplitude peak in the
1174 experimental datum. Computations are assumed to report the dimensionless frequency in terms of
Fourier1175
transform,
but
rather
as a ,small
size frequency
band-width
isentropic exit
velocity
. The experimental
Strouhal
value of 0.27,[32].
has been rescaled using the
1176
nominal
shedding
frequency
of
2.65
kHz
and
the
isentropic
velocity
corresponding
to thenumber
This is best seen with the help of Table 5, comparing the
predicted
Strouhal
with the
, =
1177
0.409 value (Cicatelli et al. [32]). Despite the use of the same simplistic closure the scatter is rather
experimental datum. Computations are assumed to report the dimensionless frequency in terms of
1178 large both among the computations and with the experiments. The predicted Strouhal number of
isentropic
velocity
Theperfectly
experimental
Strouhal value.
value of 0.27, has been rescaled using the
2,is .agrees
1179exitSondak
et al. V
[112]
with the experimental
6 case.
VKI LS94blade,
turbine M
blade, = ,0.4,
= 0.4,
1010case.
TimeTime
mean velocity
profiles on profiles
VKIFigure
LS9449.turbine
Re2 ==22 ×
mean velocity
2,is
pressure side (a) and suction side (b) at / = 1.75. For symbols see Figure 47.
nominal shedding frequency of 2.65 kHz and the isentropic velocity corresponding to the M2,is = 0.409
1180
Table 5. VKI LS94 turbine blade,
= 2 10 case. Strouhal numbers.
, = 0.4,
value (Cicatelli et al. [32]). Despite the use of the
same simplistic closure the scatter is rather large
Authors
Method
both among the computations and with the experiments. Closure
The predictedStrouhal
Strouhal number of Sondak
Cicatelli et al. [32] Experiments
/
0.285
et al. [112] agrees perfectlyManna
withet the
experimental
value.
al. [110]
URANS
EVM: Baldwin & Lomax [97]
0.253
Arnone et al. [111]
URANS
EVM: Baldwin & Lomax [97]
0.210
6
Sondak
et al. [112]
EVM:
et al. [114]
0.285
Table 5. VKI LS94
turbine
blade,URANS
M2,is = 0.4,
ReDeiwert
2 = 2 × 10 case. Strouhal numbers.
Ning et al. [113]
Authors
URANS
Method
EVM: Roberts [115]
0.245
Closure
Strouhal
1181
Those results obtained at a relatively low Mach number pushed the VKI group to extend the
1182 Cicatelli
experimental
investigation
into the high subsonic/transonic range
in 2003 [21] and 2004
[36] as
et al. [32]
Experiments
/
0.285
1183 Manna
already
3. This was a new breakthrough,
as it&offered
once
more, and0.253
again for
et discussed
al. [110] in Section URANS
EVM: Baldwin
Lomax
[97]
1184Arnone
the first
a set of highly URANS
resolved experimental
data
documenting
the effects
et time,
al. [111]
EVM:
Baldwin
& Lomax
[97] of compressibility
0.210
1185Sondak
on the unsteady wake formation and development process, throwing some considerable light on the
et al. [112]
URANS
EVM: Deiwert et al. [114]
0.285
1186 relation between the base pressure distribution and the vortex shedding phenomenon. In the next
Ning et al. [113]
URANS
EVM: Roberts [115]
0.245
1187 ten, fifteen years a number of research groups attempted to simulate this flow setup, mostly with
1188 higher fidelity approaches and the results were again rather satisfactory. The nominal Mach and
1189 Reynolds numbers were increased considerably ( , = 0.79,
= 2.8 10 ), and a variety of
Those
obtained
at a relatively
low Mach
pushed
the
VKI group
to extend the
1190 results
additional
flow conditions
including supersonic
outletnumber
regimes were
tested, as
discussed
in § 5. Table
1191 investigation
6 summarizes theinto
relevant
experimental
thecontributions.
high subsonic/transonic range in 2003 [21] and 2004 [36] as already
Commented [MM75]: OK for displa
discussed in Section 3. This was a new breakthrough, as it offered once more, and again for theTable
first
6.
time, a set of highly resolved experimental data documenting the effects of compressibility on the
unsteady wake formation and development process, throwing some considerable light on the relation
between the base pressure distribution and the vortex shedding phenomenon. In the next ten, fifteen
years a number of research groups attempted to simulate this flow setup, mostly with higher fidelity
approaches and the results were again rather satisfactory. The nominal Mach and Reynolds numbers
were increased considerably (M2,is = 0.79, Re2 = 2.8 × 106 ), and a variety of additional flow conditions
including supersonic outlet regimes were tested, as discussed in Section 5. Table 6 summarizes the
relevant contributions.
Int. J. Turbomach. Propuls. Power 2020, 5, 10
40 of 55
Table 6. Available computations of the M2,is = 0.79, Re2 = 2.8 × 106 VKI LS-94 turbine blade.
Authors
Eqs.
Numerical
Method
Grid
Closure
Space/Time
Discretization
Grid
Density
y+
Structured
Multi-Block
(O-H)
EVM (Baldwin
Lomax [97], Spalart
and Allmaras [98],
Wilcox [100])
2nd/2nd
NA
5–10
EVM (Wilcox [100])
2nd/2nd
0.63 M
5
SRS (Smagorinsky
[119])
2nd/2nd
0.63 M
5
Mokulys
et al. [117]
URANS
CC-FVM
Leonard
et al. [118]
URANS
CC-FVM
Leonard
et al. [118]
LES
CC-FVM
LES
CV-FVM
Unstructured
SRS (Smagorinsky
[119])
3nd/3nd
0.4 M
40
DDES
CV-FVM
Structured
(O)
SRS (Spalart et al.
[104])
2nd/2nd
4M
1
CV-FEM
Unstructured
EVM (Wilcox [100])
1st–2nd/2nd
1.3 M
1
CC-FVM
Unstructured
SRS (Wall dumped
Smagorinsky [122])
1st/2nd
2.53 M
0.4
CC-FVM
Structured
Multi-Block
(H-O-H-H)
SRS (Spalart et al.
[104])
2nd/2nd
4.3 M
0.4–1
Leonard
et al. [118]
El-Gendi
et al.
[120,121]
Kopriva
et al. [67]
Vagnoli
et al. [56]
URANS
(CFX)
LES
(OpenFOAM)
Wang et al.
[123]
DDES
(Fluent)
Structured
Multi-Block
(H-O-H-H-H)
Structured
Multi-Block
(H-O-H-H-H)
For the structured meshes the number of nodes and the number of cells is similar. In the
unstructured cases the difference is rather large, and typically there is a factor 5 more cells than nodes.
The URANS simulations should have been carried on a two-dimensional mesh, since there is no reason
for transversal modes to develop with 2D inflow conditions in a perfectly cylindrical geometry extruded
by some percentage of the chord in the spanwise direction. The URANS computations of Leonard
et al. [118] and those of Kopriva et al. [67] were carried out on a 3D mesh obtained expanding the 2D
domain in the third direction by a fraction of the chord length (5.7% in [118] and 8% in [67]). None of
the authors discussed the appearance of spanwise modes in the URANS data. Conversely, the scale
resolving simulations (LES and DDES) need to be carried out on a 3D domain, with a homogeneous
spanwise direction, to allow for the appropriate description of the most relevant energy carrying
turbulent eddies, which are inherently three dimensional in nature. Occasionally, some authors
reported two dimensional pseudo-DDES and pseudo-LES, that is, unsteady computations obtained
on a purely two dimensional mesh, none of which has been included in Table 6. On the resolution
side, the URANS simulations of Kopriva et al. [67] seem to have gone through some grid refinement
study, while those of Leonard et al. [118] did not. On the LES and DDES side the situation is far more
involved. At a Reynolds number of about three million the grid point requirements for a wall resolved
LES is about 5 × 108 [124] which is a couple of orders of magnitude higher than the most refined LES of
Table 6. Thus, the very neat inertial subrange of this HPT flow is likely not to be resolved at all by
any of the available simulations, and consequently the cut-off is poorly placed. These deficiencies will
seriously impact on the quality of the simulations as they undermine the essential prerequisites upon
which LES relies. For the DDES simulation this inconsistency is only partially relieved. Detached
Eddy Simulation and similar hybrid URANS-LES approaches have somewhat met certain expectations,
even though they are routinely overlooked as a means of achieving a LES-like quality at the cost of a
URANS setup. Instead, DES and its evolved version DDES, should be categorized as Wall Modeled
LES, and thus they can by no means be considered as a coarser grid version of LES [106]. In the
present context modelling the boundary layer via URANS all the way down to the point of incipient
separation will not return any of the key features the true turbulent boundary layer should possess to
properly form the wake and determine its correct space time development. And relieving the Modeled
Stress Depletion of DES by better addressing the URANS-LES migration in the grey area, will only
281
resulting in high base pressures was also mentioned by Corriveau and
282
comparing their nominal mid-loaded rotor blade HS1A, mentioned alrea
283
loaded blade HS1C with an increase of the suction side unguided turning a
284
It appears that the increased turning angle could cause, in the transoni
285
boundary layer transition near the trailing edge with, as consequence, a sh
Int. J. Turbomach. Propuls. Power 2020, 5, 10
41 of 55
286
pressure,
as seen in Figu
Int.J.J.Turbomach.
Turbomach.Propuls.
Propuls.Power
Power2018,
2018,3,3,x xFOR
FORPEER
PEER
REVIEW i.e. a sudden drop in the base pressure coefficient
Int.
REVIEW
4343ofof5858
287
reported in the figure has been converted to −
of the original data.
288
As regards
the
base
pressure
data
Deckers
and Denton [13] for a l
alleviate
the grid induced
separation
of
these
hybrid
methods.
All
in by
all,
the
two
DDES
1216 partially
addressing
theURANS-LES
URANS-LES
migration
theissue
grey
area,
will
only
partially
alleviate
the
grid
induced
1216
addressing
the
migration
ininthe
grey
area,
will
only
partially
alleviate
the
grid
induced
289
and
Gostelow
et
al.
[14]
for
a
high
turning
nozzle
guide
vane,
simulations
of
El-Gendi
et
al.
[120,121]
and
Wang
et
al.
[123]
are
also
to
be
considered
as
unresolved,
1217 separation
separationissue
issueofofthese
thesehybrid
hybridmethods.
methods.All
Allininall,
all,the
thetwo
twoDDES
DDESsimulations
simulationsofofEl-Gendi
El-Gendietet
al. who report base
1217
al.
Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW
43 of 58
290
of Sieverding’s
BPC,
their
distribution resembles
ofand
the Wang
previously
mentioned
cut-off
misplacement.
We
shall
return
toblade
this point
later
on.
1218 because
[120,121]
and
Wangetetal.
al.
[123]are
are
alsotothose
tobe
beconsidered
considered
asunresolved,
unresolved,
because
ofthe
the
previously
1218
[120,121]
[123]
also
as
because
ofpressure
previously
1220
all
the subsonic
way
down toregime,
the point
of incipient
separation
will
not
return
any
of
the
features
the true
291
divergent
blade
C inlater
Figure
3keywith
adistribution
negative
blade
loading near the tra
At
thiscut-off
high
the
experimental
time
mean
blade
pressure
already
1219 mentioned
mentioned
cut-off
misplacement.
We
shall
return
this
point
later
on.
1219
misplacement.
We
shall
return
totothis
point
on.
1221 turbulent boundary layer should possess to properly form the wake and determine its correct space
292
explain
the
very
low
base
pressures.
In
addition,
the
blade
presented
in
Figure
17
in
terms
of
local
isentropic
Mach
number,
reveals
that
the
flow
is
subsonic
all of Deckers and De
1220
Atthis
thishigh
highsubsonic
subsonicregime,
regime,the
thethe
experimental
time
mean
blade
pressuredistribution
distribution
already
1220
At
experimental
mean
blade
already
1222
time development.
And relieving
Modeled Stresstime
Depletion
of DES
by pressure
better addressing
the
293
edge,
and
there
is 3,
experimental
evidence
that,
to
thein
blade.
1221 around
presented
inURANS-LES
Figure17
17migration
terms
of
local
isentropic
Mach
number,
reveals
that
theflow
flow
subsonicall
alla circular trailing
1221
presented
Figure
ininterms
isentropic
Mach
number,
reveals
that
the
isiscompared
subsonic
1223
in of
the
grey
area,
will
only
partially
alleviate
grid
induced
separation
issue
Int.
J. local
Turbomach.
Propuls.
Power
2018,
xthe
FOR
PEER
REVIEW
1224
of
these
hybrid
methods.
All
in
all,
the
two
DDES
simulations
of
El-Gendi
et
al.
[120,121]
and
Wang
294
for
blades
with
blunt
trailing
edge
might
be
considerably
lower.TheSieverdi
The
computations
compared
in
Figure
50
predict
fairly
well
the
continuous
acceleration
of
the
Commented
[MM80]:
reference o
1222
around
the
blade.
1222
around
the PEER
blade.
omach.
PowerREVIEW
2018,
3, x FOR
REVIEW
40 of 58
018, 3, Propuls.
x FOR
PEER
40 of 58
1225 et al. [123] are also to be considered as unresolved,
because of the previously mentioned cut-off
Table
6.
295
report
for
flat
plate
tests
at
moderate
subsonic
Mach
numbers
a
drop
of the
flow both
the suction
(till the throat location at x/Cax =0.61) and on the pressure side (till the
1226 onmisplacement.
We shall return to this point later on.
296 the experimental
by 11%
athroat
plate
squared
trailing
edge
comparedby
to that with a circul
trailing
edge). Also,
sudden
fromfor
the
to the
trailing
edge isalready
well predicted
1227
At thisthe
high
subsonicdeceleration
regime,
time
mean with
blade
pressure
distribution
1228 presented in Figure 17 in terms of local isentropic Mach number, reveals that the flow is subsonic all
all simulations.
1229
1223
1223
around the blade.
1230
1224
Figure50.
50.VKI
VKILS94
LS94
turbine
blade,
=
0.79,
2.8
10
case.
Time
mean
blade
surface
1224
Figure
50.
VKI
LS94
turbine
blade,
0.79,
==
2.8
case.
Time
mean
blade
surface
1135
Figure
47., =
LS94
blade,
=
0.4,
=2
10 case.
Time
mean
blade
is
1231
Figure
50.
VKI
LS94 turbine
blade,
=
0.79,
==
2.8
10
case.
blade
surface
,VKI
297
Figure
10.
Base
pressure
coefficient
for
mid-loaded
(solid
line)
and surface
aft-loade
Figure
turbine
blade,
M2,is
0.79,
Re2turbine
2.8
×10
106 Time
case.
Time
mean
blade
surface
, =
, mean
1232
isentropic
Mach
number
distribution;
eddy
viscosity
models.
experiments
[21];
Baldwin
&
1225
isentropic
Mach
number
distribution;
eddy
viscosity
models.
experiments
[21];
Baldwin
&
1225
isentropic
Mach
number
distribution;
eddy
viscosity
models.
experiments
[21];
Baldwin
&
ure
47.
VKI
LS94
turbine
blade,
=
0.4,
=
2
10
case.
Time
mean
blade
surface
isentropic
1136
Mach
number
distribution.
[32];
[110];
[111]; from [20].
[112]
bine blade,
= 2 10 , Mach
case. Time
mean
blade298
surface
isentropic
number
distribution;
eddy
viscosity
models.
experiments
[21];
Baldwin
blade.
Symbols:
HS1A geometry,
HS1C
geometry. Adapted
, = 0.4, isentropic
1233
Lomax [117];
Spalart & Allmaras [117];
Wilcox - [117];
Wilcox - [118];
1226
Lomax
[117];
Spalart
&
Allmaras
[117];
Wilcox
[117];
Wilcox
[118];
1226
Lomax
[117];
Spalart
&
Allmaras
[117];
Wilcox
[117];
Wilcox
h number
distribution.
experiments
[32];
[110];
[111];
[112];
1137
[113].
ion.
experiments
[32];
[110];
[111];
[112];
&
Lomax
[117];
Spalart
&
Allmaras
[117];
Wilcox
k-ω
[117];
Wilcox
k-ω
[118];
1234
Wilcox - [67].
1227
Wilcoxk-ω
[67].
1227
Wilcox
- - [67].
[67].
].
1235
The computations
compared
Figure averaged
50 predict fairly
wellpressure
the continuous
acceleration
the fairly well reproduced by t
1138
Theintime
base
region
was ofalso
1236 flow both on the suction (till the throat location at / = 0.61) and on the pressure side (till the
Leonard
et
al. [118]
and
later
Kopriva
et
al.
[67] have
clearly
demonstrated
a steadyof
state
1228region
Thecomputations
computations
compared
inFigure
Figure
50
predict
fairly
well
thecontinuous
continuous
acceleration
of
thethe experiments a
1228
The
compared
in
50
predict
fairly
well
the
acceleration
the
timepressure
averaged
base was
pressure
region
was
also
fairly
well
reproduced
by
the
available
base
fairly
well
reproduced
bydata,
the
available
1139
although
the
among
the that
computations
and
1237alsotrailing
edge).
Also, thenumerical
sudden deceleration
from the throat
todifferences
the trailing edge
is well predicted
by
solution
will
propose
supersonic
flow
conditions
and
a
normal
shock
on
the
suction
side
at
x/C
≈
1229
flow
both
on
the
suction
(till
the
throat
location
at
/
=
0.61)
and
on
the
pressure
side
(till
the
1229 among
flow
both
onallthe
suction
(till
the throat
location
at / in= Figure
0.61)
and
the pressure
side (tillaxphysics
the
al
although
the
differences
among
the computations
and
thegenerally
experiments
are
generally
hedata,
differences
the
computations
and
the larger
experiments
are
1140
than
those
reported
48. on
Indeed,
the underlying
is more com
1238
simulations.
1239
Leonard
et
al.
[118]
and
later
Kopriva
et
al.
[67]
have
clearly
demonstrated
that
a
steady
state
0.61,
anthe
artefact
of the
wrong
modelling
which
disappears
into
the
unsteady
approach
(see
Figure 51).
1230
trailing
edge).
Also,
the
sudden
deceleration
from
the
throat
tosuction
the
trailing
edgeisis
well
predicted
bythe locations of th
1230
trailing
edge).
the
sudden
the
throat
the
trailing
well
predicted
by
han
reported
in Figure
48.Also,
Indeed,
the
underlying
physics
ispressure
more
complex,
as
the
d in those
Figure
48. Indeed,
underlying
physics
isdeceleration
more of
complex,
as
the
1141
presence
the from
two
and
side edge
sharp
over-expansions
at
1240 solution will propose supersonic flow conditions and a normal shock on the suction side at / ≈
The
trailing
edge
induced
unsteadiness,
upstream
propagation
is significant (see Figure 37),
1231
all
simulations.
1231
all
simulations.
ereofand
the suction
two
pressure
and
suction
side
sharp
over-expansions
at
the
locations of
the boundary
side
sharp
over-expansions
thewrong
locations
ofwhose
the
boundary
1142at
layer
separation
suggests.
1241
0.61, an artefact
of the
modelling
which
disappears in the unsteady approach (see Figure 51).
the shock
to
flap
and
down
on the
rear
part
ofis demonstrated
the
suction
side,
a aphenomenon
1232 causes
Leonard
al.[118]
[118]
and
later
Kopriva
al.
[67]have
have
clearly
demonstrated
that
asteady
steadystate
state
1232
Leonard
etettrailing
al.
and
later
Kopriva
etetstraight
al.
[67]
clearly
that
paration suggests.
1242
The
edgeup
induced
unsteadiness,
whose
upstream
propagation
significant
(see Figure
37),
1243will
the shock
to flap up of
and
down
on the straight
rearaa
part
of theat
suction
phenomenon
that
causes
acauses
spatial
smoothing
the
pressure
discontinuity
the side,
wall
and
the disappearance
1233 that
solution
will
propose
supersonic
flow
conditions
and
normal
shock
onathe
thesuction
suction
sideatat // of
1233
solution
propose
supersonic
flow
conditions
and
normal
shock
on
side
≈≈
1244 causes a spatial smoothing of the pressure discontinuity at the wall and the disappearance of the
supersonic
pocket
in
a
time
averaged
sense.
In
fact,
it
is
likely
that
the
lack
of
sharpness
of
many
1234 the
0.61,
an
artefact
of
the
wrong
modelling
which
disappears
in
the
unsteady
approach
(see
Figure
51).
1234
0.61,
an
artefact
of
the
wrong
modelling
which
disappears
in
the
unsteady
approach
(see
Figure
51).
1245 supersonic pocket in a time averaged sense. In fact, it is likely that the lack of sharpness of many
experimentally
measured
surface
pressure
distributions
obtained
slow
response
sensors,
1235 transonic
Thetrailing
trailing
edgeinduced
induced
unsteadiness,
whose
upstream
propagation
iswith
significant
(seeFigure
Figure
37),
1235
The
edge
unsteadiness,
whose
upstream
propagation
significant
(see
37),
1246
transonic
experimentally
measured
surface
pressure
distributions
obtained is
with
slow
response
1247
sensors,
to implicit
be
to the
temporal
averaging
resulting
from the
unresolved
to be
attributed
to
the
temporal
averaging
resulting
from
thesuction
unresolved
unsteadiness.
Eddy
1236 iscauses
causes
theshock
shock
toisflap
flap
upattributed
anddown
down
onimplicit
thestraight
straight
rear
part
the
suction
side,
phenomenon
that
1236
the
to
up
and
on
the
rear
part
ofof
the
side,
aaphenomenon
that
1248 unsteadiness. Eddy viscosity and scale resolving models (Figure 52) seem to yield comparable results
scale
resolving models
(Figure
52)
seem to yield
results
in
a time mean of
sense
1237 viscosity
causes
spatial
smoothing
thepressure
pressure
discontinuity
thewall
walland
and
thedisappearance
disappearance
ofthe
the
1237
causes
aaand
spatial
smoothing
ofofthe
discontinuity
atatcomparable
the
the
1249
in a time mean sense all along the blade, while the proper prediction of the base flow appears more
along
blade,
the
proper
prediction
of
the
base
appears
more
cumbersome.
Yet,
there
1238 all
supersonic
pocketwhile
time
averaged
sense.
Infact,
fact,
likely
thatthe
the
lack
ofwith
sharpness
many
1238
supersonic
ininaaYet,
time
averaged
sense.
In
ititisisflow
likely
that
of
sharpness
ofofmany
1250thepocket
cumbersome.
there
are appreciable
differences
among
the
computations,
aslack
well
as
the
1251 experimentally
experiments,
in themeasured
leading
edge
for 0 <pressure
/
< 0.2,
whose
origin
is unclear.
Potential
sources
appreciable
differences
among
thearea
computations,
as distributions
well
as
with
the
experiments,
in
theresponse
leading
1239 are
transonic
experimentally
measured
surface
pressure
distributions
obtained
with
slow
response
1239
transonic
surface
obtained
with
slow
1252 of discrepancies are the inflow angle setting (purely axial) yielding some leading edge de-loading in
areaisis
for
<
< 0.2,
whose
origin istemporal
unclear. averaging
Potential
of from
discrepancies
are the
1240 edge
sensors,
to0be
bex/C
attributed
the implicit
implicit
temporal
averagingsources
resulting
from the
the unresolved
unresolved
1240
sensors,
to
attributed
toto
the
resulting
ax the
1253
the experiments,
low Mach number effects on the accuracy of compressible flow solver not relying
angle on
setting
(purely
axial)
yielding
some errors
leading
edge
de-loading
in
experiments,
the
low
1241 inflow
unsteadiness.
Eddyviscosity
viscosity
andscale
scale
resolving
models
(Figure
52)seem
seem
yield
comparable
results
1241
unsteadiness.
Eddy
and
resolving
models
(Figure
52)
totothe
yield
comparable
results
1254
pre-conditioning
techniques,
larger
relative
of the
pressure
sensors
in this
incompressible
1255
flow
region,
some
geometry
the
remaining
partflow
of the
blade, trailing
edge
area
excluded,
number
effects
on
the
accuracy
ofIn compressible
solver
notofof
relying
on
pre-conditioning
1242 Mach
time
mean
senseall
allalong
along
theeffects.
blade,
while
theproper
proper
prediction
thebase
base
flow
appearsmore
more
1242
ininaatime
mean
sense
the
blade,
while
the
prediction
the
flow
appears
1256 i.e. 0.2 < / < 0.9 , the agreement among all computations and experiments is very good.
1243 techniques,
cumbersome.
Yet,relative
thereare
are
appreciable
differences
among
thecomputations,
computations,as
aswell
well
with
the
errors
of the pressure
sensors
in this
incompressible
flow
region,
some
1243
cumbersome.
Yet,
there
appreciable
differences
among
the
asaswith
the
1257 larger
Surprisingly, the difficult region of the unguided turning in the rear part of the suction side (0.6 <
1244 geometry
experiments,
inthe
the
leading
edge
area
for
< boundary
0.2,interaction
whose
origin
unclear.
Potential
sources
In
the
remaining
part
of
the
edge
area
excluded,
i.e., 0.2
< x/Cax
< 0.9,
1244
experiments,
leading
edge
area
for
00<blade,
// trailing
<<layer
0.2,
whose
origin
is
unclear.
Potential
sources
1258 effects.
/in
< 0.8)
where
the
shock
wave
turbulent
occurs,
isiswell
predicted
in
1259 a time
averaged
sense
byangle
all
closures.
Figure
VKI LS94
blade,
= 0.4,edge
=de-loading
2
10 region
case.inin
Time mean base
1245 the
discrepancies
are1143
the
inflow
angle
setting
(purely
axial)
yielding
some
edge
de-loading
agreement
among
all inflow
computations
and48.
experiments
isturbine
very
good.
Surprisingly,
the
difficult
1245
ofofdiscrepancies
are
the
setting
(purely
axial)
yielding
some
leading
,leading
ure
48. VKI
LS94
turbine
blade,
=
0.4,
=
2
10
case.
Time
mean
base
pressure
1144
distribution.
For
symbols
see
Figure
47.
turbine
blade,
=
0.4,
=
2
10
case.
Time
mean
base
pressure
, low
1246 of
experiments,
the
lowin
Mach
number
effects
onthe
theaccuracy
accuracy
compressible
flowsolver
solver
notrelying
relying
the
unguided turning
the rear
parteffects
of
the on
suction
side (0.6ofof
<compressible
x/C
the shock
wave
1246
the
experiments,
the
Mach
number
flow
not
, the
ax < 0.8) where
ribution.
For
symbols
see
Figure
47.
s see Figure
47.
1247
on
pre-conditioning
techniques,
larger
relative
errors
thepressure
pressure
sensorsinin
thisincompressible
incompressible
turbulent
boundary
layer
interaction
occurs,
is well
predicted
in
a time sensors
averaged
sense
by
all closures.
1247
on
pre-conditioning
techniques,
larger
relative
errors
ofofthe
this
1145 effects.
The
and thepart
magnitude
of these
two accelerations
seem within the reach of
1248 flow
flowregion,
region,some
somegeometry
geometry
effects.
theremaining
remaining
part
theblade,
blade,
trailing
edgearea
areaexcluded,
excluded,
1248
InInlocation
the
ofofthe
trailing
edge
location1249
and
the
magnitude
of
these
two
accelerations
seem
within
the
reach
of
the
adopted
magnitude
of
these
two
accelerations
seem
within
the
reach
of
the
adopted
1146
closure,
as
well
as
the
pressure
plateau
of
the
base
region.
The
1249 i.e.
i.e. 0.2
0.2<< // <<0.9
0.9, , the
the agreement
agreement among
among all
all computations
computations and
and experiments
experiments isis very
verypredicted
good. base pressure
good.
as well
as
the
pressure
plateau
of
the
base
region.
The
predicted
base
pressure
coefficients
ssure
plateau
of
the
base
region.
The
predicted
base
pressure
coefficients
1147
defined
by
Equation
(2)
agree
fairly
well
with
the
experimental
value,
as
1250 Surprisingly,
Surprisingly,the
thedifficult
difficultregion
regionofofthe
theunguided
unguidedturning
turningininthe
therear
rearpart
partofofthe
thesuction
suctionside
side(0.6
(0.6<
<well as with the o
1250
by Equation
(2)
agree
well
with
the
experimental
value,
well
as with
the one
obtained
ree
fairly1251
well
with
the
experimental
value,
as well
as with
theas
one
obtained
1148
from
the
VKI
correlation
[110].
The
success occurs,
of
theseissimple
models isinin
attributed to the p
1251
0.8)where
where
the
shock
wave
turbulent
boundary
layer
interaction
occurs,
iswell
wellpredicted
predicted
//fairly
<<0.8)
the
shock
wave
turbulent
boundary
layer
interaction
VKI The
correlation
[110].
The
success
of
these
simple
models
is
attributed
to
the
proper
space110].
success
of
these
simple
models
is
attributed
to
the
proper
space1149
time
resolution
of
the
boundary
layers
at
separation
points
in
the
trailing
edge region. Ag
1252 aatime
timeaveraged
averagedsense
senseby
byall
allclosures.
closures.
1252
olutionlayers
of theatboundary
layers
at in
separation
points
in
the
trailing
edge
region.
Again,
this
has
ndary
separation
points
the 1150
trailing
edge
region.
Again,
this
has
been documented by Manna et al. [110] and by Sondak et al. [112] (see Figure 49) who c
cumented
by Manna
al. [110]
by1151
Sondak
et
al.49)
[112]
(see
Figure
49)
show
a
na et al. [110]
and by et
Sondak
et and
al. [112]
(see Figure
who
could
show
awho could
more
than
satisfactory
agreement
of the
computed
time averaged velocity profiles with th
an
satisfactory
agreementtime
of the
computed
time averaged
velocity
profiles with
the measured
eement
of the computed
averaged
velocity
profiles
with
1152
one, both
on the
themeasured
pressure
and suction
sides at 1.75 diameters upstream of the trailing ed
h onsuction
the pressure
and
suction
sidesupstream
at 1.75
upstream
/ = ±=
and
sides at
1.75
diameters
of the
edge
( the
/thetrailing
=
± edge
1153diameters
1.75trailing
with
= 0ofat
trailing
edge,( and
). The thinner pressure side boundary l
hing=
0 atand
the trailing
). Theside
thinner
pressure
side
boundary
layer
and the side vortex shedding were estimated to be
edge,
= edge,
). Theand
thinner=pressure
boundary
layer
and
the
1154
blade
circulation
strengthening
the pressure
rculation
the pressure
side
vortex
shedding
estimated
the
cause
of edge [32]. The very consistent grid refinem
ening
the strengthening
pressure side vortex
shedding
were
estimated
towere
be the
cause
of to be at
1155
the
higher
local
over
expansion
the
trailing
er local
expansion
the trailing
[32]. The
very
consistent
grid
refinement study
of thinner and fuller pressure side boun
nsion
at over
the trailing
edgeat[32].
The veryedge
consistent
grid
refinement
study
of
1156
[112]
brought
some
improvements
in the
Int.
J.Turbomach.
Turbomach.
Propuls.
Power
2018,
xFOR
FOR
PEER
REVIEW
Int.
J.J.Turbomach.
Propuls.
Power
2018,
3,3,x3,
PEER
REVIEW
Int.
Propuls.
Power
2018,
xFOR
PEER
REVIEW
278
44of58
of
4444of
5858
Figure 9. Comparison of blade Mach number distribution for blade RD of Xu and Den
42 of 55
= 0.8,
= 8 10 ) with VKI blade (dashed curve,
= 10 )
, = 0.8,
Int. J. Turbomach. Propuls. Power 2020, 5, 10
279
curve,
,
Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW
44 of 58
280
The possible effect of boundary layer separation resulting from high rear suc
281
resulting in high base pressures was also mentioned by Corriveau and Sjola
282
comparing their nominal mid-loaded rotor blade HS1A, mentioned already be
Int. J. Turbomach. Propuls. Power
2018,
3,
x FOR PEERblade
REVIEW HS1C with an increase of the suction
44 of 58 side unguided turning angle f
283
loaded
284
It appears that the increased turning angle could cause, in the transonic rang
285
boundary layer transition near the trailing edge with, as consequence, a sharp in
286
pressure, i.e. a sudden drop in the base pressure coefficient as seen in Figure 10.
287
reported in the figure has been converted to −
of the original data.
288
As regards the base pressure data by Deckers and Denton [13] for a low tu
289
and Gostelow et al. [14] for a high turning nozzle guide vane, who report base press
253
290
those of Sieverding’s BPC, their blade pressure distribution resembles that o
53
53
291 , , =,, =divergent
blade
C case.
in
Figure
3 with
a negative
blade loading near the trailing e
254
Figure
51.
VKI
LS-94
turbine
blade,
=0.79,
0.79, ==
=2.8
2.8
case.
Density
gradient
based
54
Figure
51.
VKI
LS-94
turbine
blade,
0.79,
2.8
10
Density
gradient
based
54
Figure
51.
VKI
LS-94
turbine
blade,
1010
case.
Density
gradient
based
292
explain
the
very
low
base
pressures.
In
addition,
the blade of Deckers and Denton
255
contours;
RANS,- - ,- (b)
, (b)
URANS,- - ,- (c)
, (c)
LES,
(d)
Experiments,
schlieren
photograph.
Adapted
55
contours;
(a)
RANS,
URANS,
LES,
(d)
Experiments,
schlieren
photograph.
Adapted
55
contours;
(a)(a)
RANS,
, (b)
URANS,
, (c)
LES,
(d)
Experiments,
schlieren
photograph.
Adapted
293
edge,
and
there
is
experimental
evidence
that,
compared
to a circular trailing edge
256
from
Leonard
etal.al.
[118].
56
from
Leonard
etetal.
[118].
56 Int. J. Turbomach.
from
Leonard
[118].
Propuls.
Power
2018,
3, x FOR PEER REVIEW
40 of 58
294
for blades with blunt trailing edge might be considerably lower. Sieverding an
1253
1260
6at moderate
295
report
flat
plate tests
subsonic
Machcontours;
numbers
a drop of the base p
Figure
LS-94
blade,
M2,is
=for
0.79,
Re0.79,
gradient
based
1254
51.VKI
VKI
LS-94
blade,
=102.8
10
case. Density
gradient
based
2 = 2.8=×
1261 51.
Figure
51. turbine
VKIturbine
LS-94
turbine
blade,
2.8
10 case.
case. Density
Density
gradient
based
, , == 0.79,
by 11%
aLES,
plate
with
squared
trailing
edge
compared
to that with a circular trai
1262
contours;
(a) -RANS,
- ,(c)
(b)LES,
URANS,
, (c)LES,
(d)
schlieren
photograph.
Adapted
(a)
RANS, (a)
k-ω,
(b) URANS,
k-ω,
(d)
schlieren
photograph.
Adapted
fromAdapted
Leonard
1255
contours;
RANS,
, 296
(b)
URANS,
- Experiments,
,- for
(c)
(d)Experiments,
Experiments,
schlieren
photograph.
1263
from Leonard et al. [118].
et
al.
[118].
1256
from Leonard et al. [118].
257
57
57
258
58
58
259
59
59
260
60
60
Figure
52.
VKI
LS94
turbine
blade,
=
0.79,
=
2.8
10
case.
Time
mean
blade
surface
Figure
52.
VKI
LS94
blade,
=
0.79,
=
case.
mean
blade
surface
Figure
52.
VKI
LS94
turbine
blade,
0.79,
=
2.8
10
case.
Time
mean
blade
surface
1264
Figure
VKI
blade,
=
0.79,
=
case.coefficient
Time
mean
blade
surface
Figure
47. VKI
LS94
turbine
blade,
0.4,
=
2 10
case.
Time
mean
blade
surface
isentropic
,, =M
Figure
52. turbine
VKI
LS9452.
turbine
blade,
=Figure
0.79,
Re
=Base
2.8
×2.8
10106Time
case.
Time
mean
blade
surface isentropic
, 2.8
, ,turbine
, =LS94
210
2,is
297
10.
pressure
for
mid-loaded
(solid line) and aft-loaded (das
1265
isentropic
Mach
number
distribution;
scale
resolving
simulations.
experiments
[21];
DDES
isentropic
Mach
number
distribution;
scale
resolving
simulations.
experiments
[21];
DDES
isentropic
number
distribution;
scale
resolving
simulations.
experiments
[21];
DDES
isentropic
Mach
number
distribution;
scale
resolving
simulations.
experiments
[21];
DDES
Mach
numberMach
distribution.
experiments
[32];
[110];
[111];
[112];
Mach
number
distribution;
scale
resolving
simulations.
experiments
[21];
DDES
[120,121];
298
blade.
Symbols:
HS1A
geometry,
HS1C geometry. Adapted from [20].
1266
[120,121];
DDES [123];
structured LES [118];
unstructured LES [118];
[120,121]; 1267
DDES
[123];
structured
LES
[118];
unstructured
LES
[118];
[120,121];
DDES
[123];
structured
LES
[118];
unstructured
LES
[118];
[120,121];
DDES
[123];
LES
[118];
unstructured
LES
[118];
[113].
DDES
[123];
structured
LES
unstructured
LES
[118];
unstructured
unstructured
LESstructured
[56].
unstructured
LES
[56].
unstructured
LES
[56].
unstructured
LES
[56].
LES [56].
1268 base In
the base flow
regionwas
the scatter
instead remarkable,
as shown inby
Figure
and Figure 54.
The
pressure
region
alsois fairly
well reproduced
the53 available
1257time averaged
Figure
52.
LS94
blade,
2.8 10
case.
Time
mean
blade surface
, = 0.79,
1269
The VKI
physics
of the turbine
time averaged
base pressure,
consisting =
of three
pressure
minima
and
two maxima,
In
the
base
flow
region
the
scatter
is
instead
remarkable,
as
shown
in
Figures
53
and 54.
The physics
261
In
the
base
flow
region
the
scatter
is
instead
remarkable,
as
shown
in
Figure
53
and
Figure
54.
61
In
the
base
flow
region
the
scatter
is
instead
remarkable,
as
shown
in
Figure
53
and
Figure
54.
61 numerical
In
the
base
flow
region
the
scatter
is
instead
remarkable,
as
shown
in
Figure
53
and
Figure
54.
data,
although
the
differences
among
the
computations
and
the
experiments
are
generally
1270 has
already
been explained
before,scale
and will
not be repeated
here. What experiments
is worth mentioning
1258
isentropic
Mach
number
distribution;
resolving
simulations.
[21]; is that
DDES
of
the
time
averaged
base
pressure,
consisting
of
three
pressure
minima
and
two
maxima,
262
The
physics
the
time
averaged
base
pressure,
consisting
three
minima
and
two
maxima,
62
The
physics
ofofof
the
time
averaged
base
consisting
ofofof
three
pressure
and
two
maxima,
62 larger
The
physics
the
time
averaged
base
pressure,
consisting
three
pressure
minima
and
two
maxima,
1271
the
physical
explanation
offered
for
the disappearance
ofpressure
the pressure
plateau
at
the
trailing
edge has already
than
those
reported
in
Figure
48.pressure,
Indeed,
the
underlying
physics
is minima
more
complex,
as
the
1259
[120,121];
DDES
[123];
structured
LES
[118];
unstructured
LES
[118];
1272
center
atsuction
higher
Mach
number
is
thoroughly
supported
by
thelocations
numerical
results
ofboundary
Leonard
et
al.that
explained
before,
and
will
not
be
repeated
here.
What
isis
worth
mentioning
isisis
that
the physical
263
has
already
been
explained
before,
and
will
not
be
repeated
here.
What
isworth
worth
mentioning
isthat
63
has
already
been
explained
before,
and
will
not
be
repeated
here.
What
worth
mentioning
that
63 presence
has
already
been
explained
before,
and
will
not
be
repeated
here.
What
is
mentioning
of thebeen
two
pressure
and
side
sharp
over-expansions
at
the
of
the
1260
unstructured
LES
[56].
1273 [118] and
Kopriva et al. [67] (results not shown herein). In fact, when the simulations are performed
explanation
offered
forfor
the
disappearance
of of
the
pressure
plateau
at the
trailing
edgeedge
center
264
the
physical
explanation
offered
for
the
disappearance
the
pressure
plateau
the
trailing
edge at higher
64
the
physical
explanation
offered
the
disappearance
the
pressure
plateau
atatat
the
trailing
64 layer
the
physical
explanation
offered
for
the
disappearance
ofof
the
pressure
plateau
the
trailing
edge
separation
suggests.
1274 with
a steady-state approach there is no sudden pressure drop originated by the enrolment of the
Mach
number
is flow
thoroughly
supported
byainstead
the
numerical
results
of
Leonard
et al. [118]
and
Kopriva
265
center
higher
Mach
number
is
thoroughly
supported
by
the
numerical
results
of
Leonard
et
al.Figure
65
higher
Mach
is
thoroughly
supported
by
the
numerical
results
Leonard
etetand
al.
65 center
center
higher
Mach
number
isregion
thoroughly
supported
by
the
results
of
Leonard
al.
1275
unsteady
separating
shear
layers
into
vortex
right
atnumerical
the
trailingas
edge,
andof
the
over-expansions
1261atatat
In
the number
base
the
scatter
is
remarkable,
shown
in
Figure
53
54.
1276
occurring
atshown
thenot
separation
points
are
followed
by
awhen
marked
and
unphysical
recompression
leading
et
al.
[67]
(results
not
herein).
In
fact,
when
the
simulations
are
performed
with
a
steady-state
266
[118]
and
Kopriva
et
al.
[67]
(results
not
shown
herein).
In
fact,
when
the
simulations
are
performed
66
and
Kopriva
et
al.
[67]
(results
shown
herein).
In
fact,
the
simulations
are
performed
66 [118]
[118]
and
Kopriva
et
al.
[67]
(results
not
shown
herein).
In
fact,
when
the
simulations
are
performed
1262
The physics of the time averaged base pressure, consisting of three pressure minima and two maxima,
1277 there
to a nearly constant pressure zone. Conversely, all unsteady simulations reproduce, at least
approach
is explained
no
sudden
pressure
drop
originated
by
thehere.
enrolment
of
the unsteady
separating
267
with
asteady-state
steady-state
approach
there
isno
no
sudden
pressure
drop
originated
by
the
enrolment
the
67
aasteady-state
approach
there
isisno
sudden
pressure
originated
by
the
enrolment
of
the
67 with
with
approach
there
sudden
pressure
drop
originated
by
the
enrolment
ofof
the
1263
has already
been
before,
will
notdrop
be
repeated
What
is
worth
is that
1278
qualitatively,
the correct
baseand
pressure
footprint.
There is some
scatter
in the
positionmentioning
of the
shear
layers
into
a
vortex
right
at
the
trailing
edge,
and
the
over-expansions
occurring
at
the
separation
268
unsteady
separating
shear
layers
into
a
vortex
right
at
the
trailing
edge,
and
the
over-expansions
68
separating
shear
layers
into
a
vortex
right
at
the
trailing
edge,
and
the
over-expansions
68 unsteady
unsteady
separating
shear
layers
into
a
vortex
right
at
the
trailing
edge,
and
the
over-expansions
1279
separating
shear
layers
as
predicted
by
the
eddy
viscosity
closures,
a
phenomenon
that
is
related
to
1264
the physical explanation offered for the disappearance of the pressure plateau at the trailing edge
1280
the correct
characterization
of the
turbulent
boundary
layers
at the point
ofrecompression
incipient
Both
points
are
followed
by
anumber
marked
unphysical
recompression
to a separation.
nearly
constant
pressure
269
occurring
at
the
separation
points
are
followed
by
amarked
marked
and
unphysical
leading
69
the
separation
points
are
followed
by
aamarked
and
unphysical
recompression
leading
69 occurring
occurring
the
separation
points
are
followed
by
and
unphysical
recompression
leading
1265 atat
center
at
higher
Mach
isand
thoroughly
supported
by
theleading
numerical
results
of
Leonard
et al.
1281
experiments and computations have shown in fact that there is little or no motion of the separation
zone.
Conversely,
all
unsteady
simulations
reproduce,
at
least
qualitatively,
the
correct
base
pressure
270
nearly
constant
pressure
zone.
Conversely,
all
unsteady
simulations
reproduce,
at
least
70
aa anearly
constant
pressure
zone.
Conversely,
all
unsteady
simulations
reproduce,
at
least
70 tototo
nearly
constant
pressure
zone.
Conversely,
all
unsteady
simulations
reproduce,
at
least
1266
[118]
and
Kopriva
et
al.
[67]
(results
not
shown
herein).
In
fact,
when
the
simulations
are
performed
1282 point along the blade surface so that the position of the over-expansion is neat both in a time averaged
There
is some
scatter
in is
thenoposition
of
the
separating
layers
as
predicted
271
qualitatively,
the
correct
base
pressure
footprint.
There
is
some
scatter
in
the
position
of
the by
71
the
correct
pressure
footprint.
There
isis
some
scatter
ininshear
the
position
ofof
the
71 qualitatively,
qualitatively,
the
correct
base
pressure
footprint.
There
some
scatter
the
position
the
andbase
instantaneous
sense.
Conversely,
the
intensity
of
the
over-expansion
strongly
depends
upon
the
1267 footprint.
with
a1283
steady-state
approach
there
sudden
pressure
drop
originated
by the
enrolment
of the
the
eddy
viscosity
closures,
a
phenomenon
that
is
related
to
the
correct
characterization
of
the
turbulent
272
separating
shear
layers
predicted
by
the
eddy
closures,
aphenomenon
phenomenon
that
isrelated
related
72
layers
asasas
predicted
by
the
eddy
that
isisrelated
tototo
72 separating
separating
shear
layers
predicted
by
the
eddy
viscosity
closures,
aphenomenon
that
1268 shear
unsteady
separating
shear
layers
intoviscosity
aviscosity
vortexclosures,
right
at athe
trailing edge,
and
the
over-expansions
boundary
layers
at
the
point
of incipient
separation.
Both
experiments
and
computations
haveleading
shown
273
the
correct
characterization
the
turbulent
boundary
layers
at
the
point
incipient
separation.
Both
73
correct
characterization
the
turbulent
boundary
layers
atatthe
point
ofand
incipient
separation.
Both
73 the
the
correct
characterization
ofof
turbulent
boundary
layers
point
ofof
incipient
separation.
Both
1269
occurring
at theofseparation
points
are followed
by
athe
marked
unphysical
recompression
in
fact
that
there
is
little
or
no
motion
of
the
separation
point
along
the
blade
surface
so
that
the
274
experiments
and
computations
have
shown
in
fact
that
there
is
little
or
no
motion
of
the
separation
74
and
computations
have
shown
in
fact
that
there
is
little
or
no
motion
of
the
separation
74 experiments
experiments
and
computations
have
shown
in
fact
that
there
is
little
or
no
motion
of
the
separation
1270
to a nearly constant pressure zone. Conversely, all unsteady simulations reproduce, at least
position
of
the
over-expansion
is
neat
both
in
a
time
averaged
and
instantaneous
sense.
Conversely,
275
point
along
the
blade
surface
so
that
the
position
the
over-expansion
isneat
neat
both
atime
time
averaged
75
along
the
blade
surface
that
the
position
ofofof
the
over-expansion
neat
both
ininin
aatime
75 point
point
along
the
blade
surface
so
that
the
position
the
over-expansion
both
averaged
1271
qualitatively,
theso
correct
base
pressure
footprint.
Thereisisis
some
scatter
inaveraged
the
position of the
the
intensity
of
the
over-expansion
strongly
depends
upon
the
pitchwise
flapping
motion
ofrelated
the shear
276
and
instantaneous
sense.
Conversely,
the
intensity
of
the
over-expansion
strongly
depends
upon
the
76
instantaneous
sense.
Conversely,
the
intensity
of
the
over-expansion
strongly
depends
upon
the
76 and
and
instantaneous
sense.
Conversely,
the
intensity
of
the
over-expansion
strongly
depends
upon
the
1272
separating shear layers as predicted by the eddy viscosity closures, a phenomenon that
is
to
layers,
which,
as
shown
by
the
experiments,
is
vigorous.
This
is
necessarily
smeared
by
the
Reynolds
1273
the correct characterization of the turbulent boundary layers at the point of incipient separation. Both
by computations
theblade,
time averaging.
small
ofTime
scales
resolved
the
eddy viscosity
closures
Figure
VKI LS94and
turbine
0.4, The
=
2 in
10 range
case.
base
pressure
, =
1274 48.averaging
experiments
and
have
shown
fact
that
theremean
is
little
orby
no
motion
of the separation
distribution.
For
symbols
see
Figure
47.
causing
large
discrepancies
between
the computations
and the experiments.
1275 is
point
alongthe
the
blade
surface
so that
the position
of the over-expansion
is neat both in a time averaged
1276
and instantaneous sense. Conversely, the intensity of the over-expansion strongly depends upon the
The location and the magnitude of these two accelerations seem within the reach of the adopted
closure, as well as the pressure plateau of the base region. The predicted base pressure coefficients
defined by Equation (2) agree fairly well with the experimental value, as well as with the one obtained
from the VKI correlation [110]. The success of these simple models is attributed to the proper space-
Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW
45 of 58
1284
of the shear
which, as=shown
the experiments,
is vigorous.
1288 pitchwise
Figureflapping
53. VKImotion
LS94 turbine
blade, layers,
2.8 ∙ 10bycase.
Time mean base
pressure This
, = 0.79,
1285
smeared
by themodels.
Reynolds
by50.
the time averaging. The small range of
1289 is necessarily
distribution;
eddy viscosity
For averaging
symbols seeand
Figure
Int. J. Turbomach.
Propuls.
Power by
2020,
10 viscosity closures is causing the large discrepancies between the
43 of 55
1286 scales
resolved
the5,eddy
1290 computations
Remarkably
closure, implemented in a similar numerical technology returns very large
1287
andthe
thesame
experiments.
1291 scatters in the time averaged base pressure region (Leonard et al. [118] and Kopriva et al. [67]), a
1292 phenomenon that should be traced back to the inadequate grid resolution, both in the normal to the
1293 wall and in the streamwise direction of nearly all computations. None of the presented simulations
1294 did undergo a consistent grid refinement study in an unsteady sense, and the effects of the lack of
1295 resolution are evident from the improper prediction of the near trailing edge pressure data, that is
1296 the region at /
2. As a matter of fact, only one out the three - contributions has an adequate
1297 first cell y+ value [67], and has attempted to investigate the effects of the grid size in an unstructured
1298 approach. The authors claimed that the coarsest grid achieved grid convergence, but, on account of
1299 the adopted technology, this conclusion is uncertain. Scale resolving simulations presented in Figure
1300 54, produce significant improvements in the base pressure distribution predictions, and the quality
1301 of the LES and DDES data should be considered comparable, despite the differences in modelling
1302 and grid densities, the latter playing a key role. The general trend is to under-predict the pressure
1303 level, while the shape of the wall signal, with its characteristic peak-valley structure, is well
1304 53.represented
by all simulations. Inspection of the boundary layer profiles
extracted one diameter
1288
Figure
53. VKI
LS94 turbine
0.79,Re2 ==
2.82.8
∙ 10 × case.
basemean
pressure
Figure
VKI
LS94
turbine
blade,blade,
M2,is =, =0.79,
106 Time
case.mean
Time
base pressure
1305
upstream of the trailing edge circle on both sides of the blades is helpful to understand the scatter in
1289
distribution;
eddy viscosity
models.
For
symbolssee
see Figure
50.50.
distribution;
eddy
viscosity
models.
For
symbols
Figure
1306 the base pressure data. Those data are presented later on.
Commented [MM81]: The reference t
1290
Remarkably the same closure, implemented in a similar numerical technology returns very large
causing the problem has been eliminate
1291 scatters in the time averaged base pressure region (Leonard et al. [118] and Kopriva et al. [67]), a
1292 phenomenon that should be traced back to the inadequate grid resolution, both in the normal to the
1293 wall and in the streamwise direction of nearly all computations. None of the presented simulations
1294 did undergo a consistent grid refinement study in an unsteady sense, and the effects of the lack of
1295 resolution are evident from the improper prediction of the near trailing edge pressure data, that is
1296 the region at /
2. As a matter of fact, only one out the three - contributions has an adequate
1297 first cell y+ value [67], and has attempted to investigate the effects of the grid size in an unstructured
1298 approach. The authors claimed that the coarsest grid achieved grid convergence, but, on account of
1299 the adopted technology, this conclusion is uncertain. Scale resolving simulations presented in Figure
1300 54, produce significant improvements in the base pressure distribution predictions, and the quality
1301 of the LES and DDES data should be considered comparable, despite the differences in modelling
1302
andVKI
grid
densities,
the
latter
playing
a key
The
trend
under-predict
the
pressure
Figure
LS94
turbine
blade,
M2,is
=, role.
0.79,
Regeneral
×case.
10is6 to
case.
Time
base pressure
1307 54.
Figure
54. VKI
LS94
turbine
blade,
= 0.79,
10
Time
mean
basemean
pressure
2 ==2.82.8
1303
level,
while
the
shape
of
the
wall
signal,
with
its
characteristic
peak-valley
structure,
is well
1308
distribution;
scale resolving
simulations.
symbols see
52. 52.
distribution;
scale
resolving
simulations.
ForFor
symbols
seeFigure
Figure
1304 represented by all simulations. Inspection of the boundary layer profiles extracted one diameter
1305 upstream of the trailing edge circle on both sides of the blades is helpful to understand the scatter in
Remarkably
samedata.
closure,
implemented
inon.
a similar numerical technology returns very
1306 the basethe
pressure
Those data
are presented later
Commented [MM81]: The reference t
large scatters in the time averaged base pressure region (Leonard et al. [118] and Kopriva et al.causing
[67]),
the problem has been eliminate
a phenomenon that should be traced back to the inadequate grid resolution, both in the normal to the
wall and in the streamwise direction of nearly all computations. None of the presented simulations
did undergo a consistent grid refinement study in an unsteady sense, and the effects of the lack of
resolution are evident from the improper prediction of the near trailing edge pressure data, that is
the region at S/D ± 2. As a matter of fact, only one out the three k-ω contributions has an adequate
first cell y+ value [67], and has attempted to investigate the effects of the grid size in an unstructured
approach. The authors claimed that the coarsest grid achieved grid convergence, but, on account of
the adopted technology, this conclusion is uncertain.
Scale
simulations presented in Figure
54,=produce
significant improvements in the base
1307 resolving
Figure 54. VKI LS94 turbine blade,
2.8 10 case. Time mean base pressure
, = 0.79,
pressure
predictions,
the quality
ofsee
the
LES
1308distribution
distribution;
scale resolvingand
simulations.
For symbols
Figure
52.and DDES data should be considered
comparable, despite the differences in modelling and grid densities, the latter playing a key role.
The general trend is to under-predict the pressure level, while the shape of the wall signal, with its
characteristic peak-valley structure, is well represented by all simulations. Inspection of the boundary
layer profiles extracted one diameter upstream of the trailing edge circle on both sides of the blades is
helpful to understand the scatter in the base pressure data. Those data are presented later on.
Before proceeding with the analysis of the boundary layers, let us briefly discuss the numerical
results of Vagnoli et al. [56], whose simulations are the only one documenting the capabilities of
scale resolving simulations to cope with the difficulties associated to the base flow prediction in the
transonic regime, all the way up to mildly supersonic exit Mach numbers. Those data are reported
in Figure 55, where some of the experimental data already presented in Figure 33 (see Section 5),
are compared with the LES results obtained with the numerical setup and technology previously
described. The agreement is, generally speaking, good at all Mach numbers. The shape of the
static pressure traces and level of the base pressure is fairly well captured, although discrepancies
exist. At M2,is = 0.79 and M2,is = 0.97 the peak-valley structure of the pressure signal with the neat
pressure minimum at the center of the trailing edge is essentially reproduced, and the position of
the separating shear layers is reasonable. The maximum differences appear to be in order of 10%.
1311
1311
1312
1312
1313
1313
1314
1314
1315
1315
1316
1316
1317
1317
1318
1318
1319
1319
1309
proceeding
withpeak-valley
the analysis of the
boundaryof
layers,
us briefly signal
discuss
numerical
=
0.79
structure
the
pressure
with
the
=
0.79 and
andBefore
= 0.97
0.97 the
the
peak-valley
structure
of
thelet
pressure
signalthe
with
the neat
neat pressure
pressure
,, =
1310 results of Vagnoli et al. [56], whose simulations are the only one documenting the capabilities of scale
minimum
minimum
at
at
the
the
center
center
of
of
the
the
trailing
trailing
edge
edge
is
is
essentially
essentially
reproduced,
reproduced,
and
and
the
the
position
position
of
of the
the
1311 resolving simulations to cope with the difficulties associated to the base flow prediction in the
separating
separating
shear
layers
layers
isisreasonable.
reasonable.
The
The
maximum
maximum
differences
appear
appear
to
be
beare
in
inreported
order
orderof
of10%.
10%.When
When
1312 shear
transonic
regime,
all
the way up to
mildly
supersonic differences
exit
Mach numbers.
Thoseto
data
in
1313 number
Figure 55,is
where
some of the
data already
presented
in
33 (see Section 5),
Commented [M82]: Figures should be
the
the Mach
Mach
number
is
increased
increased
to
toexperimental
= 1.047
1.047
the
the degree
degree of
ofFigure
non-uniformity
non-uniformity
of
ofarethe
the pressure
pressure
,, =
1314 compared with the LES results obtained with the numerical setup and technology previously
order.
distribution,
distribution,
quantified
quantified
through
the
the
parameter
parameter
(see
(seeatEquation
Equation
(5))
(5)) reduces
reduces
drastically,
drastically,
ending
in55
aa
Int.
J. Turbomach.
Propuls. Power
2020, 5, 10
44 in
of
1315 described.
The through
agreement
is,
generally
speaking, good
all Mach numbers.
The shape
of the static ending
1316plateau.
pressure
traces
and level of the base
is fairly well of
captured,
although
discrepancies
exist. At in
pressure
pressure
plateau.
The
The
disappearance
disappearance
of
ofpressure
the
the enrollment
enrollment
of the
the shear
shear
layers
layers
into
into vortices
vortices
in the
the
base
base
Commented [MM83R82]: The text ha
and
the peak-valley
of
thethe
pressure
signal
neat pressure
, = 0.79 the
, = 0.97
region
region1317
characterizing
characterizing
the lower
lower
Mach
Mach
numbers
numbersstructure
cases,
cases, and
and
the effects
effects
of
ofwith
the
thethe
shock
shock
patterns
patterns delaying
delaying
1318
minimum
at theiscenter
of the trailing
edge
is1.047
essentially
reproduced,
and the position of of
the the pressure
When
the Mach
number
increased
M2,is
=edge
the degree
non-uniformity
the
the vortex
vortex
formation
downstream
downstream
the
thetotrailing
trailing
edge
appears
appears
very
veryof
well
well
predicted,
at least
least in
in aa time
time
1319 formation
separating shear
layers is reasonable.
The maximum
differences
appear
to
be inpredicted,
order of 10%. at
When
distribution,
quantified
through
the
parameter
Z
(see
Equation
(5))
reduces
drastically,
ending
in a
averaged
sense.
Those
are
indeed
remarkable
results,
still
representing
the
state-of-the-art
in
the
field.
1320
the
Mach
number
is
increased
to
=
1.047
the
degree
of
non-uniformity
of
the
pressure
averaged sense. Those are indeed remarkable
results, still representing the state-of-the-art in the field.
,
pressure
The quantified
disappearance
ofparameter
the enrollment
of the
intoending
vortices
1321plateau.
distribution,
through the
(see Equation
(5))shear
reduceslayers
drastically,
in a in the base
pressure plateau.
The disappearance
of the enrollment
of the
the shear
layers
the base delaying the
region1322
characterizing
the lower
Mach numbers
cases, and
effects
of into
the vortices
shock in
patterns
1323 region characterizing the lower Mach numbers cases, and the effects of the shock patterns delaying
vortex1324
formation
downstream the trailing edge appears very well predicted, at least in a time averaged
the vortex formation downstream the trailing edge appears very well predicted, at least in a time
sense.1325
Those averaged
are indeed
results,
stillresults,
representing
the state-of-the-art
field.
sense.remarkable
Those are indeed
remarkable
still representing
the state-of-the-art inin
thethe
field.
1320
1320
1321
1321
1322
1322
Figure
Figure
55. VKI
LS94
turbine
turbine
blade.
blade.
Comparison
Comparison
ofnumerical
numerical
mean base
base
Figure
VKI
LS94
turbine
blade.
Comparison
numerical
and experimental
experimental
time
mean
base
1326 55.
Figure
55. VKI
LS94 turbine
blade.
Comparison of
of
and and
experimental
time meantime
base mean
1327
pressure distribution
at transonic
exit
flow
conditions. Symbols—experiments,
lines—computations
pressure
pressure
distribution
distribution
at
attransonic
transonic
exit
exitflow
flow
conditions.
conditions.
Symbols—experiments,
Symbols—experiments,
lines—computations
lines—computations
lines—computations
1328
(LES).
, = 0.79,
, = 0.97,
, = 1.05. Adapted from [56].
(LES).
=
(LES).
=
0.79,
= 0.97,
0.97,
= 1.05.
1.05.Adapted
M,2,is
=0.79,
0.79
M2,is
0.97,
M2,is
=
1.05.
Adapted from
from [56].
[56].
, =
,, =
,, =
1329
Returning now to the numerical prediction of the boundary layer profiles at the trailing edge,
1330 Figure
56 shows
the eddy viscosity
simulations
differ
considerably,
both profiles
on the pressure
and
Returning
now
to the
the
numerical
prediction
of the
the
boundary
layer
profiles
at the
the
trailing edge,
edge,
Returning
Returning
now
to
thethat
numerical
numerical
prediction
of
boundary
layer
at
trailing
1331
suction sides. Again, the two - of models of Mokulys et al. [117] and Kopriva et al. [67], disagree
Figure
56
shows
that
the
eddy
viscosity
simulations
differ considerably,
considerably,
both
on
the
pressure
and
Figure
Figure
56
56
shows
shows
that
that
the
the
eddy
eddy
viscosity
viscosity
simulations
simulations
differ
considerably,
both
both
on
on
the
the
pressure
pressure
and
and
1332 to some considerable extent. The results of Kopriva et al. [67] are closer to the measurements, and
suction
sides.
Again,
the
two
k-ω
of
models
of
Mokulys
et
al.
[117]
and
Kopriva
et
al.
[67],
disagree
to
suction
suction
sides.similar
Again,
Again,
two and
-- Lomax
of
ofmodels
models
ofMokulys
Mokulys
et
etal.
al.[117]
[117]
and
and
Kopriva
Kopriva
et
etal.
al.[67],
[67],disagree
disagree
1333
to thethe
Baldwin
values of of
[117].
This last agreement
seems
fortuitous,
and probably
1334
related
to
the
insufficient
grid
resolution
of
Mokulys
et
al.
[117].
The
already
mentioned
grid
some
considerable
extent.
TheThe
results
of Kopriva
et al.et
[67]
closer
to the to
measurements,
and similar
to
to some
some
considerable
considerable
extent.
extent.
The
results
results
of
of Kopriva
Kopriva
et
al.
al.are
[67]
[67]
are
are closer
closer
to the
the measurements,
measurements,
and
and
refinement study of Kopriva et al. [67] is based on three unstructured grids characterized by element
to
the 1335
Baldwin
and Lomax
values
of
[117].of
This
last
agreement
seems fortuitous,
and probably
related
similar
similar
to
to
the
the
Baldwin
Baldwin
and
and
Lomax
Lomax
values
values
of
[117].
[117].
This
This
last
last
agreement
agreement
seems
seems
fortuitous,
fortuitous,
and
and
probably
probably
1336 edge length change in the wake region of approximately 15–20%. Results presented in their study
to
the1337
insufficient
resolution
of Mokulys
et
al. [117].
already
mentioned
grid
related
related
to
to the
the
insufficient
insufficient
grid
grid
resolution
resolution
of
Mokulys
Mokulys
et
etThe
al.
al.a[117].
[117].
The
already
already
mentioned
mentioned
grid
grid
refer
to grid
near wake
time
averaged
pressureof
data
collected
through
traverseThe
across
the wake in
the refinement
1338
direction
normal
to the
camber
line
the trailing
edge.
The traverse
2.5by
trailing
edge edge
study
of Kopriva
et
[67]toistheet
based
on
three
unstructured
grids
characterized
element
length
refinement
refinement
study
study
of
ofal.
Kopriva
Kopriva
ettangent
al.
al.[67]
[67]
is
isbased
based
on
onat
three
three
unstructured
unstructured
grids
gridsischaracterized
characterized
by
byelement
element
1339 diameters downstream the trailing edge itself. Since velocity and rate of strains in the boundary layers
change
in the
wake region
of
approximately
15–20%. Results
presented
inpresented
their studyin
near
edge
change
in
region
15–20%.
Results
their
edge length
length
change
in the
the wake
wake
region of
of approximately
approximately
15–20%.
Results
presented
inrefer
theirtostudy
study
1340
are known to be more sensitive quantities than pressure, and on account of the convection scheme
wake
time
averaged
pressure
data
a traverse
acrossadifferencing
wakeand
in
the order
direction
refer
refer to
to near
near
wake
wake time
time
averaged
pressure
pressure
data
collected
through
through
athe
traverse
traverse
across
across
the
the wake
wakenormal
in
in the
the
1341
adopted
in
the averaged
solver,
whichcollected
is based on through
adata
blendcollected
of second
order
central
first
upwinding,
the
achievement
of
independence
coarsest
mesh is
uncertain.
Yet, their
to
the1342
tangent
toto
the
line
atgrid
the
trailing
edge.
The
traverse
is
2.5
trailing
edge
diameters
direction
direction
normal
normal
to
the
thecamber
tangent
tangent
to
tothe
the
camber
camber
line
lineat
atwith
the
thethe
trailing
trailing
edge.
edge.
The
The
traverse
traverse
isis2.5
2.5-trailing
trailing
edge
edge
1343
is by far the best eddy viscosity result available as today. It is not a coincidence that the
downstream
thesimulation
trailingthe
edge
itself.edge
Sinceitself.
velocity
and
rate ofand
strains
boundary
are known
diameters
diameters
downstream
the
trailing
trailing
edge
itself.
Since
Since
velocity
velocity
and
rate
rateinof
ofthe
strains
strains
in
inthe
thelayers
boundary
boundary
layers
layers
1344 downstream
appropriate resolution of the boundary layers at the point of incipient separation warrants a more
to
more
sensitive
quantities
than
pressure,
and
on account
of
convection
scheme
adopted
in
are
arebe
known
known
tothan
be
besatisfactory
more
more
sensitive
sensitive
quantities
quantities
than
than
pressure,
pressure,
and
and on
onthe
account
account
of
of the
the convection
convection
scheme
scheme
1345 to
base pressure
region
prediction.
1323
1323
1324
1324
1325
1325
1326
1326
1327
1327
1328
1328
1329
1329
1330
1330
1331
1331
1332
1332
1333
1333
1334
1334
1335
1335
1336
1336
1337
1337
,,
the
solver,
is based
on is
aisblend
order
central
differencing
and first order
upwinding,
adopted
adopted
in
inwhich
the
thesolver,
solver,
which
which
based
basedof
on
onsecond
aablend
blend
of
ofsecond
second
order
order
central
centraldifferencing
differencing
and
and
first
firstorder
order
the
achievement
of
grid
independence
with
the
coarsest
mesh
is
uncertain.
Yet,
their
k-ω
simulation
upwinding,
upwinding,the
theachievement
achievementof
ofgrid
gridindependence
independencewith
withthe
thecoarsest
coarsestmesh
meshisisuncertain.
uncertain.Yet,
Yet,their
their -is by
far the best
eddy
viscosity
result
available
as today.
It isas
not
a coincidence
the appropriate
simulation
simulation
isisby
by
far
farthe
the
best
besteddy
eddy
viscosity
viscosity
result
result
available
available
as
today.
today.
ItItisisnot
notaathat
coincidence
coincidence
that
thatthe
the
resolution of the boundary layers at the point of incipient separation warrants a more than satisfactory
base pressureInt.region
prediction.
J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW
47 of 58
(a)
(b)
1346 56. VKI
Figure
56. VKI
LS94 turbine
= 0.79,Re =
2.8 10 case.
6 Time mean velocity profiles on
Figure
LS94
turbine
blade,blade,
M2,is =, 0.79,
2 = 2.8 × 10 case. Time mean velocity profiles on
1347
pressure (a) and suction (b) side of the blade at ⁄ = 1.75. Eddy viscosity models. For symbols see
pressure
(a) and
suction
(b) side of the blade at S/D = ±1.75. Eddy viscosity models. For symbols see
1348
Figure
50.
Figure 50.
1349
The scale resolving simulations here exhibit the largest differences, Figure 57. The two DDESs
Commented [MM84]: Text has been am
1350 of El-Gendi et al. [120,121] and Wang et al. [123] predict remarkably well the suction and pressure
1351 side velocity profiles. Conversely the two LESs of Leonard et al. [118] and Vagnoli et al. [56],
1352 completely miss both profiles. There is a factor 10 in the number of grid nodes between the two
1353 DDESs and the LES of Leonard et al. [118], and a factor of 2 for that of Vagnoli et al. [56]. Furthermore,
1354 the inner layer of the LESs is either bypassed (first y+ at 5 or 40) or fully unresolved (spacing of 48
1355 wall units along the blade height, in [56]). The major shortcoming of wall resolving LESs is precisely
1356 the inability of all subgrid scale models to reproduce the effects of the dynamics of the low speed
Int. J. Turbomach. Propuls. Power 2020, 5, 10
45 of 55
(a)
(b)
(a)
(b)
The scale resolving simulations here exhibit the largest differences, Figure 57. The two DDESs of
1346
Figure 56. VKI LS94 turbine blade,
2.8 10 case. Time mean velocity profiles on
, = 0.79,
El-Gendi
et al. [120,121]
and Wang et al. [123]
predict= remarkably
well the suction and pressure side
1347
pressure (a) and suction (b) side of the blade at ⁄ = 1.75. Eddy viscosity models. For symbols see
velocity
profiles.Figure
Conversely
the two LESs of Leonard et al. [118] and Vagnoli et al. [56], completely
1348
50.
miss both profiles. There is a factor 10 in the number of grid nodes between the two DDESs and the
1349
The scale resolving simulations here exhibit the largest differences, Figure 57. The two DDESs
LES of1350
Leonard
et al. et
[118],
and aand
factor
2 for
ofremarkably
Vagnoli well
et al.
theCommented
inner [MM84]: Text has been a
of El-Gendi
al. [120,121]
Wangof
et al.
[123]that
predict
the[56].
suctionFurthermore,
and pressure
+ atLESs
1351
side velocity
profiles.
Conversely
theytwo
Leonard
et al.unresolved
[118] and Vagnoli
et al. [56],
layer of
the LESs
is either
bypassed
(first
5 orof40)
or fully
(spacing
of 48 wall units
1352
completely
miss
both
profiles.
There
is
a factor 10 in the
number
of grid nodes
between
the two the inability
along the blade height, in [56]). The major shortcoming
of wall
resolving
LESs
is precisely
1353 DDESs and the LES of Leonard et al. [118], and a factor of 2 for that of Vagnoli et al. [56]. Furthermore,
of all subgrid
toLESs
reproduce
the effects
of the low
speed
+ atthe
1354 thescale
inner models
layer of the
is either bypassed
(first yof
5 or dynamics
40) or fully unresolved
(spacing
of 48streaks, their
1355
wall unitsand
alongthe
the blade
in [56]). generation
The major shortcoming
wall resolving LESs
precisely
growth,
breakdown
wall height,
turbulence
processof[95,102,124].
Theis consequence
of this
1356 the
inability of all subgrid scale models to reproduce the effects of the dynamics of the low speed
shortcoming
is that
the only successful wall resolving LESs are those whose inner layer resolution is
1357 streaks, their growth, breakdown and the wall turbulence generation process [95,102,124]. The
sufficient
toofsome
extent the
streaks
The
requirements
are
rather
1358to describe
consequence
this shortcoming
is that
the onlydynamics.
successful wall
resolving
LESs are those
whose
innersevere, since
1359 wall
layer
resolutionstructures
is sufficient tohave
describe
to some length
extent theof
streaks
The requirements
these near
coherent
a typical
1000dynamics.
wall units,
a width ofare
30, while their
1360
rather severe, since these near wall coherent structures have a typical length of 1000 wall units, a
average
lateral
spacing is in the order of 100 wall units [103,124]. They are responsible for the sweep
1361 width of 30, while their average lateral spacing is in the order of 100 wall units [103,124]. They are
and ejection
the
inward/outward
motion
respectmotion
to the(with
wall)
of high
1362 phenomena,
responsible for the
sweep
and ejection phenomena,
the (with
inward/outward
respect
to the energy fluid
1363
wall) of highthey
energyare
fluidenergy
lumps, and
therefore they
are energy carrying
structures. Their numerical
appropriate resolution
lumps,
and therefore
carrying
structures.
Their appropriate
1364
numerical resolution usually qualified in terms of mesh spacing in inner coordinates, is rather
usually
qualified
in terms of mesh spacing in inner coordinates, is rather demanding, and also heavily
1365 demanding, and also heavily depends upon the accuracy of the numerical procedure used to solve
depends
upon
the
accuracy
of the numerical procedure used to solve the governing equations.
1366 the governing
equations.
6 case.
1367 57. VKI
Figure
57. turbine
VKI LS94 blade,
turbine blade,
= 0.79,
= 2.8
case.
Time mean
on profiles on
Figure
LS94
M2,is =, 0.79,
Re2 =
2.810× 10
Timevelocity
meanprofiles
velocity
1368
pressure (a) and suction (b) side of the blade at ⁄ = 1.75. Scale resolving models. For symbols
pressure
suction (b) side of the blade at S/D = ±1.75. Scale resolving models. For symbols see
1369 (a) and
see Figure 52.
Figure 52.
For higher order methods, viz those with spectral error decay, they can be estimated to be
∆x+ ≈ 50–100, ∆y+ ≈ 10–20 in the streamwise and spanwise directions, respectively, while in the
normal to the wall direction there should be some 10–20 points in the first 30 wall units.
These requirements are not respected by any of the two LESs clearly highlighting the inability of the
SGS model to provide the correct energy contribution of the sub-grid scales to the super-grid one; it is
also no surprise that the two DDESs perform better than the two LESs, thanks to the properly modelled
(via k-ω) inner wall layer. Indeed, their suction and pressure side boundary layer predictions are by far
the most accurate among the available data. This is clearly shown in Figure 57. Also, the first order
time integration scheme of Vagnoli et al. [56] is inadequate for a scale resolving simulation requiring a
minimum time accuracy of order two. The benefit of the considerably more refined DDES meshes,
allowing for the resolution of larger number of turbulent scales, should become evident elsewhere.
We next compare in Figure 58 the wake shape as predicted by the available closures.
The comparison is based on a wake traverse located at 2.5 trailing edge diameters downstream
the trailing edge itself, as already previously described. The prediction of a turbulent wake behind a
turbine blade is a rather challenging task which is complicated by the trailing edge bluntness promoting
the shedding of large-scale vortex structures. Essential for the correct prediction of the wake formation
and development is the proper description of the boundary layers at the point of incipient separation.
At the current Reynolds number, the scale separation is huge, and the boundary between modelled
and resolved scales is uncertain, so that the extent of the grey area and the filter width may become a
1381 become evident elsewhere.
1382
We next compare in Figure 58 the wake shape as predicted by the available closures. The
1383 comparison is based on a wake traverse located at 2.5 trailing edge diameters downstream the trailing
1384 edge itself, as already previously described. The prediction of a turbulent wake behind a turbine
1385 blade is a rather challenging task which is complicated by the trailing edge bluntness promoting the
1386 shedding of large-scale vortex structures. Essential for the correct prediction of the wake formation
Int. J. Turbomach.
Propuls.
Power 2020,
10 description of the boundary layers at the point of incipient separation.
46 of 55
1387 and
development
is the 5,
proper
1388 At the current Reynolds number, the scale separation is huge, and the boundary between modelled
1389 and resolved scales is uncertain, so that the extent of the grey area and the filter width may become
1390Yet all
a concern.
Yetseem
all closures
seem
to reproduce
the essential
features
of the
large-scaleunsteadiness
concern.
closures
capable
to capable
reproduce
the essential
features
of the
large-scale
1391 unsteadiness associated with the vortex shedding process; the agreement is a little more than
associated
the vortex
process;
the agreement
is atotal
little
moreprofiles
than qualitative.
This is
1392 with
qualitative.
This is shedding
best seen in Figure
58 comparing
the numerical
pressure
with the
1393in Figure
experimental
data.
best seen
58 comparing
the numerical total pressure profiles with the experimental data.
(a)
(b)
1394 58. VKI
Figure 58. VKI LS94 turbine blade,
= 0.79,
= 2.8 10 case.
Time mean total pressure wake
Figure
LS94 turbine blade, M2,is =, 0.79,
Re2 =
2.8 × 106 case.
Time mean total pressure wake
1395
traverses at 2.5 diameters downstream the trailing edge. (a): eddy viscosity models, (b): scale resolving
traverses
at
2.5
diameters
downstream
the
trailing
edge.
(a):
eddy
viscosity
models, (b): scale resolving
1396
simulations. For symbols see Figure 50 and Figure 52.
simulations. For symbols see Figures 50 and 52.
1397
While the wake width seems fairly well predicted by all closures, the wake velocity deficit is not,
1398 the
by some appreciable quantities. Surprisingly, the - results of Mokulys et al. [117] look better than
While
wake width seems fairly well predicted by all closures, the wake velocity deficit is
1399 those of Kopriva et al. [67] despite the grid refinement study of the latter and the superior agreement
not, by
someinappreciable
quantities.
Surprisingly,
k-ω results
ofofMokulys
[117] look better
1400
terms of boundary
layer features
on both sides ofthe
the blade.
The DDES
El-Gendi etet
al. al.
[120,121]
1401 ofis by
far the worst
of all[67]
simulations
in terms
closeness
to the experiments.
Thisthe
is surprising
giventhe superior
than those
Kopriva
et al.
despite
the of
grid
refinement
study of
latter and
1402 inthe
good quality of the other results extracted from the same simulation. The authors discuss in some
agreement
terms
of boundary layer features on both sides of the blade. The DDES of El-Gendi
1403 details the potential reasons for those discrepancies, addressing numerical issues, turbulence
et al. [120,121]
is by far
theand
worst
of effects.
all simulations
inthe
terms
ofwas
closeness
to the
This is
1404 modelling
issues
grid size
Unfortunately,
analysis
inconclusive,
and experiments.
a more in1405 given
depththe
inspection
the dataof
would
been
necessary
to identifyfrom
the root
reasons
forsimulation.
the deviations The authors
surprising
good of
quality
thehave
other
results
extracted
the
same
1406 documented in Figure 58. As previously detailed in this section, a DDES is characterized by three
discuss in some details the potential reasons for those discrepancies, addressing numerical issues,
turbulence modelling issues and grid size effects. Unfortunately, the analysis was inconclusive, and a
more in-depth inspection of the data would have been necessary to identify the root reasons for the
deviations documented in Figure 58. As previously detailed in this section, a DDES is characterized by
three zones, namely a URANS, a LES and a hybrid one, and the extent of the latter dominates to some
remarkable extent the quality of the whole simulation. The in-depth analysis of the spatial distribution
of the fd function (see Equation (10)) (or equivalently of the FSST in the DES-SST-zonal model) would
have been of great help to identify the responsibilities of the turbulence modelling and of the filter
width. What can be conjectured here, is that at the location of the wake traverses the DDES simulation
is in the grey area, or, worse, in the LES one with a too large filter width. Conversely, in the base region
and all around the blade in the boundary layers, the URANS mode is properly working. This can be
inferred from the nearly identical boundary layer profiles as predicted by the k-ω results of Kopriva
et al. [67], and the DDES data of El-Gendi et al. [120,121], both of which qualified through an identical
eddy viscosity model in the wall region (see Figures 56 and 57). Thus, while the very near wake and
the base region features heavily depend upon the characteristics of the boundary layers at the point
of incipient separation, already a few diameters downstream the trailing edge the dynamics of the
vortex shedding formation scheme is too complicate for an eddy viscosity closure as well as for an
unresolved LES.
The total temperature results reported in Figure 59 are similar to the total pressure ones. All models
reproduce approximately well the occurrence of the Eckert-Weiss effect, with its characteristic flow
heating (respectively cooling) at the wake edges (respectively center). The magnitude of the positive
and negative (compared to the inlet value) total temperature peaks, as well as their locations is only
marginally well predicted by the eddy viscosity closures, while some improvements can be appreciated
in the DDES of El-Gendi et al. [120,121].
1420 viscosity closure as well as for an unresolved LES.
1421
The total temperature results reported in Figure 59 are similar to the total pressure ones. All
1422 models reproduce approximately well the occurrence of the Eckert-Weiss effect, with its characteristic
1423 flow heating (respectively cooling) at the wake edges (respectively center). The magnitude of the
1424 positive and negative (compared to the inlet value) total temperature peaks, as well as their locations
1425 is only marginally well predicted by the eddy viscosity closures, while some improvements can be
Int. J. Turbomach. Propuls. Power 2020, 5, 10
1426 appreciated in the DDES of El-Gendi et al. [120,121].
(a)
47 of 55
(b)
1427 59. VKI
Figure
59. turbine
VKI LS94 turbine
= 0.79,Re =
= 2.8
Time mean
total
temperature
Figure
LS94
blade,blade,
M2,is =, 0.79,
2.810× case.
106 case.
Time
mean
total temperature
2
1428
traverse at 2.5 diameters downstream the trailing edge. (a): eddy viscosity models, (b): scale resolving
traverse
diameters
downstream the trailing edge. (a): eddy viscosity models, (b): scale resolving
1429 at 2.5simulations.
For symbols see Figure 50 and Figure 52.
simulations. For symbols see Figures 50 and 52.
1430
Finally, the Strouhal numbers as predicted by all numerical models are presented in Table 7.
Finally, the Strouhal numbers as predicted by all numerical models are presented in Table 7.
1431
Table 7. VKI LS94 turbine blade,
,
Authors
Method
Sieverding et al. [21]
Experiments
Mokulys et al. [117]
URANS
Authors Mokulys et al. [117] Method
URANS
URANS
Sieverding et al. Mokulys
[21] et al. [117]Experiments
Leonard et al. [118]
URANS
Mokulys et al. [117]
URANS
Kopriva et al. [67]
URANS
Mokulys et al. [117]
URANS
El Gendi et al. [120,121]
DDES
Mokulys et al. [117]
URANS
Wang et al. [123]
DDES
Leonard et al. [118]
Leonard et al. [118] URANS
LES
= 0.79,
= 2.8 10 case. Strouhal numbers.
Closure
Strouhal
/
0.219
Baldwin and Lomax [97]
0.206
Closure
Spalart and Allmaras [98]
0.177
Wilcox - [100]
0.199
/
Wilcox - [100]
Baldwin and Lomax [97] 0.276
Wilcox - [100]
0.212
Spalart and Allmaras [98] 0.215
Spalart et al. [104]
Wilcox
k-ω
[100]
Spalart
et al.
[104]
0.216
Wilcox
k-ω
[100]
Smagorinsky [119]
0.228
Table 7. VKI LS94 turbine blade, M2,is = 0.79, Re2 = 2.8 × 106 case. Strouhal numbers.
Kopriva et al. [67]
El Gendi et al. [120,121]
Wang et al. [123]
Leonard et al. [118]
Vagnoli et al. [56]
URANS
DDES
DDES
LES
LES
Wilcox k-ω [100]
Spalart et al. [104]
Spalart et al. [104]
Smagorinsky [119]
Wall damped Smagorinsky [122]
Strouhal
0.219
0.206
0.177
0.199
0.276
0.212
0.215
0.216
0.228
0.220
Recall that the proper evaluation of the vortex shedding frequency requires a correct modelling
of the near wake mixing process, that is of the interaction between the unsteady separating shear
layers [38]. The differences between the experiments and the EVM solutions are definitely larger
than those pertaining to the SRS, all of which predict rather well the dominant shedding frequency.
However, on account of the complexity and cost of the SRS the results obtained with the simple EVM
closures are to be considered appealing. Inspection of the higher pressure modes in the near wake,
both in terms of amplitude and phase, would probably underline larger differences and discrepancies.
8. Conclusions
This review manuscript has addressed in full details the flow peculiarities occurring at the trailing
edge of steam and gas turbine blades, with the help of experimental and numerical data. The study is
started presenting the achievements of the 40 years old VKI base pressure correlation as applied to
old and new turbine blades. While the simple architecture of the formula returns satisfactorily base
pressure estimates and thus loss predictions for conventional turbine blade designs, the correlation
appears to fail in cases of blade designs characterized by very strong adverse pressure gradients on the
rear suction side causing possibly boundary layer separation before the trailing edge. An additional
weakness of the correlation resides in the fact that all experimental base pressure data are recorded by
a single pressure tap in the blade trailing edges which implies the assumption of an isobaric trailing
edge base pressure. This assumption is unfortunately only valid for low subsonic and supersonic
Mach numbers as demonstrated recently by large scale cascade experiments.
Indeed, about twenty years after the publication of the base pressure correlation, experiments
carried out at the von Kàrmàn Institute on large scale turbine blades both at subsonic and transonic
Int. J. Turbomach. Propuls. Power 2020, 5, 10
48 of 55
outlet Mach numbers allowed major advances in the understanding of the mechanism of vortex
formation and shedding in the near trailing edge wake region. Thanks to the large size of the test
article, specifically designed for providing time resolved data at high spatial resolution, it has been
shown that the flow approaching the trailing edge undergoes a strong acceleration both on the pressure
and suction side, before leaving the blade. Two over-expansions of different strength because of the
differences in the boundary layers state and of the blade circulation, have been documented and
attributed to the effects of the vortex shedding. At those locations remarkable pressure fluctuations
occur, reaching 80% of the outlet dynamic pressure. While at subsonic flow conditions the central
trailing edge base region exhibits a rather constant pressure area, at higher Mach numbers the base
pressure is characterized by the appearance of a steadily growing pressure minimum which, at the
transition from a normal to an oblique trailing edge shock system, does give suddenly again way to
an isobaric region. A physically consistent explanation of the departure from the assumed isobaric
trailing edge base region has been proposed, and the implications with the VKI correlation outlined.
The dynamic of the shear layers has also been identified as the root cause of the formation of the
acoustic wave systems occurring in the trailing edge region and their impact on the rear suction side
pressure distribution.
The energy separation phenomenon, since long known to occur in cylinder flow, has been
documented to also exist at the exit of transonic uncooled stator blades, causing major concerns
for the mechanical integrity of the following blade row when invested by uneven total temperature
distributions. Important achievements were obtained by the Canadian research group of the NRC
who first measured time resolved pressure and total temperature distribution in the wake of transonic
turbine blades. The data, corroborated by successive experiments, highlighted the relation between
the vortex street formation and propagation with the energy separation phenomenon. High resolution
experimental data were released for code-to-experiment validation and the outcome of the available
simulations, presented in a dedicated section, has been discussed at length.
The turbine trailing edge frequency features, as measured on a number of blades, have been
analyzed and their relations with the geometry, the boundary layer state at the point of incipient
separation and the governing dimensionless parameters, clarified.
In spite of the considerable progress made so far for a better understanding of unsteady
trailing edge flows and their effects on the blade performance, there is clearly room for further
experimental research on unsteady trailing edge flows. The main objective should be the conception
and preparation of additional large-scale cascade tests allowing high resolution spatial and temporal
measurements. New benchmark test cases would then be available for experiment-to-experiment and
code-to-experiment validation. The benchmark test cases presented in this paper were characterized
by turbulent boundary layers on both suction and pressure sides at the point of separation from
the trailing edge. It would be certainly interesting to dispose of a large-scale test case with mixed
turbulent/laminar (suction side/pressure side) trailing edge flow conditions. It would also be desirable
to apply high resolution fast optical measurement techniques to determine the time varying wake
velocity field for the evaluation of the rate of strain and the vorticity tensors. Long time and phase
averaged turbulence data will naturally come out, thus enhancing the actual knowledge of the wake
mixing process. This may ultimately require the measurement of the three-dimensional time-varying
velocity field. It would also be highly desirable to have more test data for the downstream evolution of
the wake total temperature profile, the knowledge of which is of prime importance for the evaluation
of the mechanical integrity of the downstream blade row. The reduction of the trailing edge vortex
intensity and therewith the profile losses by appropriate trailing edge shaping, as e.g. elliptic trailing
edges, deserves certainly further attention. Again, large scale test set up will be needed to highlight
the differences in the wake mixing process.
On the numerical side the progresses achieved over 40 years of unsteady turbine wake flow
computations have been impressive. This is equally due to the advances in numerical methods and
modelling concepts. The authors have put together all available computations of the VKI LS94 turbine
Int. J. Turbomach. Propuls. Power 2020, 5, 10
49 of 55
blade, whose geometry and experimental data have been previously presented. Within the bounds of
the limited published material, a few concluding remarks on the ability of the adopted turbulence
closures can be put forward. While the freely available turbine geometry is relatively simple, the flow
conditions are not, mainly because of the large Reynolds number. Very few of the URANS contributions
did achieve grid convergence in the sense of the local truncation error, in order to confine those errors
at values smaller than the modelling ones. The problem of the inadequate number of numerical
parameters becomes particularly offending for the scale resolving simulations (DES, DDES and LES)
for which the interaction between the space-time numerical integration procedure and the turbulence
closure is known to be warring, especially when implicit filtering and low order methods are used.
In addition, and unlike URANS, the resolution requirements are far more stringent, and hard to satisfy.
It has been shown that the URANS calculations presently reviewed comply with the spectral gap
requirement, and, therefore, the expectations of predictivity are legitimate. In fact, although a systematic
grid convergence study was rarely achieved, the general quality of the numerical solutions obtained
with eddy viscosity models can be rated satisfactory. Algebraic, one equation and two equations models
proved capable to predict reasonably well the time averaged blade pressure distribution, even in
the difficult base region, both in the subsonic and transonic regimes. Time averaged boundary layer
profiles in the near trailing edge region, and even more, wake features are more problematic, especially
in the high Mach number cases. Particularly, the total pressure and total temperature profiles did not go
further than a qualitative agreement with the experiments, although the energy separation phenomenon
was correctly represented. Scale resolving simulations improved the predictivity level of the URANS,
but not as expected. Most of the deficiencies have been traced back to an inadequate sub-grid filter
positioning often causing severe deviations from the experiments. The hybrid simulations were
more performant than the pure LESs, mainly because of the larger number of parameters of the
former. Boundary layers and even more the near wake region were poorly predicted by the scale
resolving simulations, partly because of the already mentioned inertial subrange reproduction failure,
a consequence of the insufficient spatial resolution, and partly because of known SGS limitations,
that is their inability to provide the appropriate energy contribution of the unresolved scales to the
resolved ones in regions of strong shears. The unsteady features of the flow have not been fully
exploited, and thus the judgment on their quality is uncertain. The Strouhal number was reasonably
well predicted by all closure.
What appears to be needed to improve the quality of the available high-fidelity TWF simulations
is a more detailed and conscious selection of the spatial resolution. At high Reynolds number and
even more in transonic flow conditions, this turns out to be the most difficult objective to comply
with, especially because in the DES/LES world there is no equivalent of the grid convergence concept
routinely applied in the URANS world to isolate the modelling errors. DES/LES have an indissoluble
cut-off placement - modelling error relationship which is difficult to identify, especially when the cut-off,
that is the filter width, is implicitly defined by the mesh size. In those instances, the mixture of numerical
and modelling errors cannot be unraveled. Also, the presence of an inertial subrange, a pre-requisite
for the correct application of the LES concept, is difficult to ascertain a-priori. Nevertheless, to acquire
more credibility, the future class of numerical computations will have to provide more and more details
of the resolved turbulence, presenting spectra, spatial correlations and stress tensor components of the
computed fields at key locations. Those data will hopefully convince the reader of the quality of the
simulations and give more confidence in the collected data. Hybrid methods will have to systematically
offer quantitative details of the boundaries of the so-called grey area, to give a precise idea of what
and where was modeled and resolved by the simulations. Databases of scale resolving simulations
respecting certain properly defined quality criteria should be made openly available to the whole
turbomachinery community for code-to-code and code-to-experiment validation.
Author Contributions: Conceptualization, methodology, formal analysis, investigation, data curation,
writing—original draft preparation, writing—review and editing: C.S. and M.M. All authors have read and agreed
to the published version of the manuscript.
Int. J. Turbomach. Propuls. Power 2020, 5, 10
50 of 55
Funding: This research received no external funding. The APC were afforded by the Dipartimento di Ingegneria
Industriale of the Università degli Studi di Napoli Federico II.
Conflicts of Interest: The authors declare no conflict of interest.
List of Symbols
Roman
c
Cf
cp
cv
cpb
d
dte , D
f
g
κ
k
M
o
p
PR
Q
R
Re
r0
s
Sij
St, S
t
T
u, v
uτ
V
chord
skin friction coefficient
specific heat at constant pressure
specific heat at constant volume
base pressure coefficient
distance from the wall
trailing edge thickness
frequency
pitch
von Kàrmàn constant
thermal conductivity, cp /cv
Mach number
throat, gauging angle
pressure
pressure ratio
dynamic pressure
perfect gas constant
Reynolds number
recovery coefficient
specific entropy
rate of strain tensor
Strouhal number
time
temperature
velocity components
friction velocity
velocity magnitude
Greek letters
α, β
flow angles from tangential direction
α∗ , β ∗
gauging angles from tangential direction
δ
trailing edge wedge angle
∆
difference operator, grid spacing
rear suction side turning angle
ε
turbulent dissipation
θ
momentum thickness
ν
kinematic viscosity
νt
turbulent viscosity
ω
vorticity, specific dissipation
ρ
density
σij
stress tensor
τ
characteristic time
τw
wall shear stress
ζ
loss coefficient
Subscripts
0
stagnation value
1
at blade inlet
2
at blade outlet
ax
in the axial direction
bl
boundary layer
b
at the base of the profile
∞
at infinity
is
isentropic value
n, t
in the normal, tangential direction
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