CHIVES
MASSACHUSETTS INSTITUTE
OF TECHNOLOLGY
Turbine Inlet Non-Uniformities
JUN 23 2015
and Unsteady Mechanisms
LIBRARIES
by
Devon Jedamski
Submitted to the Department of Aeronautics and Astronautics
in partial fulfillment of the requirements for the degree of
Master of Science in Aeronautics and Astronautics
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2015
Massachusetts Institute of Technology 2015. All rights reserved.
Author
Signature redacted
Department of Aeronautics and Astronautics
May 26, 2015
Certified by. ..
Signature redacted
Edward M. Greitzer
H.N. Slater Professor of Aeronautics and Astronautics
Signature redacted
Thesis Supervisor
...........
.
Certified by.
Choon S. Tan
Senior Research Engineer
Thesis Supervisor
Accepted by ....
Signature redacted
Paulo C. Lozano
Associate Professor of Aeronautics and Astronautics
Chair, Graduate Program Committee
2
Turbine Inlet Non-Uniformities
and Unsteady Mechanisms
by
Devon Jedamski
Submitted to the Department of Aeronautics and Astronautics
on May 26, 2015, in partial fulfillment of the
requirements for the degree of
Master of Science in Aeronautics and Astronautics
Abstract
The effect of axial turbine stage inlet non-uniformities are examined through two
model problems: wake attenuation and hot streak processing. In the first, twodimensional calculations (RANS and URANS) are used to identify the mechanisms
contributing to upstream stator wake attenuation through a turbine blade row. For
a representative turbine rotor, pitch and time-averaged wake attenuation by 74%
percent is demonstrated at one quarter chord downstream of the trailing edge. Near
the pressure surface, the wake stagnation pressure increases by up to 42% above the
freestream stagnation pressure. The mechanisms identified are a localized reduction in
flow-through time for wake fluid near the pressure surface, compared to the freestream,
and an unsteady pressure field (Op/&t) in the rotor reference frame that increases work
extraction in the freestream relative to wake fluid. For the second model problem,
three-dimensional calculations (RANS and URANS) identify a difference in turbine
efficiency sensitivity to thermal distortion between a geometry with no tip gap and
a geometry with a finite tip gap. The turbine with a tip clearance is 2.5 times less
sensitive, in terms of efficiency decrease, to an inlet hot streak. For the tip gap and
no tip gap geometries, the efficiency drops by 0.75% and 1.86% respectively for a
peak temperature non-uniformity equal to 0.6 times the combustor temperature rise.
The difference in efficiency decrease, due to hot streak, between the two geometries is
linked to a reduction in tip leakage mixing losses caused by changes in relative rotor
inlet flow angle with and without hot streak.
Thesis Supervisor: Edward M. Greitzer
Title: H.N. Slater Professor of Aeronautics and Astronautics
Thesis Supervisor: Choon S. Tan
Title: Senior Research Engineer
3
4
Acknowledgments
"Trust in the Lord with all your heart, and lean not on your own understanding. In
all your ways acknowledge Him, and He shall direct your paths."
- Proverbs 3:5-6
The following research was conducted with the generous support of the RollsRoyce Whittle Fellowship. The work represents a close collaboration between the
MIT Gas Turbine Laboratory and the Turbine Aerodynamics group at Rolls-Royce
Corporation in Indianapolis.
This work could not have been possible without the support and guidance of
my advisors, Professor Greitzer and Dr. Choon Tan. I would also like to thank
Arthur Huang, a previous Rolls-Royce Whittle Fellow, for his direction and insightful
feedback throughout my first year at MIT.
The engineers of the Rolls-Royce Turbine Aerodynamics group have been an immense help. In particular, Jon Ebacher, Eugene Clemens, Ed Turner, Chong Cha,
and Steve Mazur provided assistance throughout the entirety of this thesis that was
invaluable to its completion. Steve Mazur laid the framework for the beginning of
my research and was always willing to assist with valuable discussions and direction.
In addition, Tyler Gillen and the members of the Rolls-Royce Methods group, Todd
Simons, Kurt Weber, and Moujin Zhang were always willing to help in implementing
HYDRA for my research.
To Mom and Dad, I would not be here today without your love, support, and sacrifice. I would also like to thank Derek and Katie. You are the best siblings I could
have asked for; never let me forget to stay humble. Finally, and most importantly, I
would like to thank my wife, Christiana, who has been my biggest supporter throughout graduate school. You are kind, loving, and most importantly, you've taught me
to find joy in all seasons.
5
6
Contents
Introduction
20
1.1.1
Wake Attenuation . .
20
1.1.2
Hot Streak Processing
22
1.2
Research Questions . . . . .
23
1.3
Methodology
. . . . . . . .
23
1.3.1
Wake Attenuation. .
23
1.3.2
Hot Streak Processing
24
1.4
Contributions . . . . . . . .
24
1.5
Organization of Thesis
25
.
.
.
.
.
.
Background . . . . . . . . .
27
2.1
Introduction . . . . . . . . . . . . . . .
27
2.2
Background . . . . . . . . . . . . . . .
28
2.3
Kelvin's Theorem . . . . . . . . . . . .
28
2.4
Computational Methodology . . . . . .
30
2.4.1
Blade Design Specification . . .
31
2.4.2
Boundary Conditions and Mesh
31
.
.
.
31
. . .
32
.
R esults. . . . . . . . . . . . . . . . . .
2.5.1
Wake Attenuation Metric
2.5.2
Attenuation Quantification
.
2.5
.
Wake Kinematics: Kelvin's Theorem and Wake Convection
.
. .
.
1.1
2
19
.
1
7
32
3 Assessing Wake Attenuation in a Turbine Blade Row
4
5
6
37
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
3.2
B ackground . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
3.3
Computational Methodology . . . . . . . . . . . . . . . . . . . . . . .
38
3.3.1
Boundary Conditions and Mesh . . . . . . . . . . . . . . . . .
38
3.3.2
Blade Parameters and Mesh . . . . . . . . . . . . . . . . . . .
40
3.4
Averaging
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
3.5
R esults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
3.5.1
42
Reversible Wake Attenuation
. . . . . . . . . . . . . . . . . .
Separation of the Wake Attenuation Mechanisms
47
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
4.2
Governing Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
4.3
Linearization
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
4.4
R esults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
4.4.1
53
Attenuation Mechanism Interpretation . . . . . . . . . . . . .
Effects of Hot Streaks on High Pressure Turbine Efficiency
55
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
5.2
Computational Methodology . . . . . . . . . . . . . . . . . . . . . . .
55
5.2.1
Turbine Geometry and Mesh Specification
. . . . . . . . . . .
57
5.2.2
Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . .
58
5.2.3
Unsteady Phase Averaging . . . . . . . . . . . . . . . . . . . .
61
5.3
Generalized Efficiency Definition for Unsteady Non-Uniform Flows . .
63
5.4
Effect of Tip Gap and Hot Streak on Efficiency
65
. . . . . . . . . . . .
Hot Streak Loss Mechanisms and Utility of Entropy as a Turbine
Loss Metric
67
6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
6.2
Spatial Averaging and Efficiency . . . . . . . . . . . . . . . . . . . . .
68
6.2.1
69
Averaging Non-Uniform Flow
8
. . . . . . . . . . . . . . . . . .
6.2.2
6.3
7
Effect of Efficiency Definition on Computed Turbine Stage Sensitiv ity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
6.3.1
Entropy Generation Rate . . . . . . . . . . . . . . . . . . . . .
73
6.3.2
Blade Incidence Angle . . . . . . . . . . . . . . . . . . . . . .
76
6.3.3
Tip Leakage Massflow
. . . . . . . . . . . . . . . . . . . . . .
76
6.3.4
Tip Gap Loss Scaling . . . . . . . . . . . . . . . . . . . . . . .
77
6.3.5
Efficiency Sensitivity to Inlet Hot Streak . . . . . . . . . . . .
79
Summary, Conclusions, and Recommendations for Future Work
7.1
7.2
83
Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . .
83
7.1.1
Reversible Wake Attenuation
. . . . . . . . . . . . . . . . . .
83
7.1.2
Effects of Hot Streaks on Turbine Stage Efficiency . . . . . . .
84
Recommendations for Future Work . . . . . . . . . . . . . . . . . . .
85
9
10
List of Figures
1-1
Compressor rotor wake evolution through a stator passage at peak
pressure rise condition; lines present analysis, symbols present data 1201. 21
Chopping and stretching of an incoming wake [181 . . . . . . . . . .
22
2-1
Computed compressor wake transport,
=-0.85 [8]. . . . . . . . . .
29
2-2
Computed turbine wake transport, q = 0.34 [8]. . . . . . . . . . . .
2-3
Convected wake geometry evolution through passage, 2 (p - po) /p
2-4
Upstream and downstream wake velocity triangles . . . . . . . . . .
2-5
Stagnation pressure defect far downstream for a convected wake segment 34
3-1
Inlet boundary condition, c = 0.25 . . . . . . . . . . . . . . . . . . .
40
3-2
Stagnation pressure defect ratio contour, E= 0.1 . . . . . . . . . . .
42
3-3
Stagnation pressure defect ratio at axial locations, c = 0.1
. . . . .
43
3-4
Axial velocity defect in wake, e = 0.1 . . . . . . . . . . . . . . . . .
44
3-5
Difference between convected and finite wake analyses .
45
3-6
Difference in Op/Dt between wake and freestream fluid at axial slices,
.
.2
33
33
.
.
.
.
.
29
.
.
.
1-2
. . . . . .
46
4-1
Stagnation pressure defect, e = 0.025 . . . . . . . . . .
. . . . . .
48
4-2
Wake attenuation for four wake depths, x/cx = 1.25 . .
. . . . . .
49
4-3
Time averaged velocity, ftx/ux,o
. . . . . . . . . . . . .
. . . . . .
50
4-4
Velocity perturbation,
. . . . . .. . . . . .
. . . . . .
50
4-5
Non-dimensionalized unsteady pressure perturbation
.
. . . . . .
52
4-6
Non-dimensionalized axial velocity perturbation . . . .
. . . . . .
52
.
.
.
'/(EU)
.
.
.
.
6 = 0.1 . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
4-7
Computed D(Apt)/Dx from unsteady CFD . . . . . . . . . . . . . . .
53
4-8
Computed D(Apt)/Dx from linearized analysis . . . . . . . . . . . . .
53
4-9
Description of unsteady pressure mechanism
. . . . . . . . . . . . . .
54
4-10 Description of flow-through time mechanism
. . . . . . . . . . . . . .
54
5-1
Vane and rotor surface mesh, TG-2% . . . . . . . . . . . . . . . . . .
56
5-2
First stage meridional view comparison for TG-0% and TG-2% . . . .
57
5-3
Circumferentially averaged inlet temperature distribution, OTDF = 0.6 60
5-4
Inlet temperature profile, OTDF = 0.6
5-5
OTDF contour, circumferential clocking of the hot streak
5-6
Instantaneous efficiency over two cycles, OTDF = 0.4
5-7
Power spectrum of instantaneous efficiency
5-8
Hot streak sensitivity for TG-0% and TG-2%
6-1
Hot streak sensitivity as a function of efficiency definition
6-2
Viscous entropy generation rate within control volume
. . . . . . . . . . . . . . . . .
60
. . . . . .
61
. . . . . . . .
62
. . . . . . . . . . . . . .
63
. . . . . . . . . . . . .
65
. . . . . .
72
. . . . . . . .
74
6-3
Thermal entropy generation rate within control volume . . . . . . . .
75
6-4
Interstage rotor relative incidence angle
. . . . . . . . . . . . . . . .
77
6-5
Pressure contour at 95% span, TG-2%
. . . . . . . . . . . . . . . . .
78
6-6
Tip leakage mass flow rate reduction with OTDF
. . . . . . . . . . .
79
6-7
Turbine stage loss components . . . . . . . . . . . . . . . . . . . . . .
80
6-8
Entropy generation in the tip gap, TG-2% . . . . . . . . . . . . . . .
81
6-9
Efficiency with control volume model correction
. . . . . . . . . . . .
82
6-10 Efficiency sensitivity to entropy production . . . . . . . . . . . . . . .
82
12
List of Tables
2.1
Turbine stage parameters. . . . . . . . . . . . . . . . . . . . . . . . .
31
5.1
Turbine stage parameters, OTDF
0 .0 . . . . . . . . . . . . . . . . .
58
5.2
CFD inlet boundary conditions and solver
6.1
Isentropic expansion temperature definitions . . . . . . . . . . . . .
.
. . . . . . . . . . . . . . .
13
59
72
14
Nomenclature
Letters
cP
Specific heat at constant pressure
cX
Axial chord
d(
)
Differential quantity
D/Dt
Convective derivative
ht
Stagnation enthalpy
HPT
High pressure turbine
k
Thermal conductivity
Line element vector
Af
Differential wake length
L
Wake segment
r4
Mass flow rate
M
Mach number
N
Feature count (vane, rotor, hot streak, etc.)
OTDF
Overall temperature distribution function
p
Static pressure
Pt
Stagnation (or "total") pressure
R
Gas constant
RTDF
Radial temperature distribution function
s
Entropy per unit mass
Entropy production per unit volume
S
Entropy production within a control volume (=
15
f
a dV)
t
Time
T
(1) Blade passing time
(2) Static temperature
(3) Time of periodicity
T
Stagnation (or "total") temperature (T +
TG
Tip gap
U
Velocity magnitude
U 2 /2
Velocity components in cartesian coordinates
Blade translational speed
V
Volume
AV
Wake velocity defect
W
Pitch
(x, r, 0)
Polar coordinates
(x, y, z)
Cartesian coordinates
y
Non-dimensional wall cell distance
+
U
Symbols
Ratio of specific heats
17
Circulation
(f
w dA)
Wake thickness
Difference or change
/1
Wake strength (1 - umax/Umin)
Isentropic efficiency
p
Dynamic viscosity
Loss coefficient
Density
Work coefficient (Aht/U 2
)
pI
Flow coefficient (UX/U)
Radian frequency (27f)
16
c,)
w
Vorticity
Q
Blade rotational speed
Subscripts
fs
Free stream average
main
Quantity representative of mainstream
max
Maximum value
min
Minimum value
irr
Denotes an irreversible process
1
Denotes quantity due to laminar process
S
Denotes process at constant entropy
t
(1) Stagnation quantities
(2) Denotes quantity due to turbulent process
th
Thermal component
tip
Quantity representative of tip-gap
V
Viscous component
w
Wake average
(x, y, z)
Components in x, y, z directions
0
Reference station
0, 1, 2, etc.
Station numbers
Superscripts
( )'
-
(e.g. ft)
Perturbation quantity
Mean or background flow variable
aa
Availability average
A
Area average
M
Mass average
wa
Work average
17
X
Mixed out average
18
Chapter 1
Introduction
Inlet non-uniformities to a turbine stage have the potential to either increase or
decrease stage performance. This thesis examines two model problems in the context
of inlet non-uniformity processing. The first is wake attenuation in a turbine blade
row. The second is the effect of a hot streak on turbine stage performance and the
aerodynamic mechanisms that influence stage efficiency sensitivity to hot streak.
In an axial compressor, there is consensus that wake fluid receives more work
than freestream fluid due to a longer flow-through time for wake particles.
The
unsteady, reversible work transfer to the wake decreases wake mixing losses and thus
increases component efficiency. Various descriptions of the phenomenon and of the
mechanism have appeared in the literature 18, 17, 18].
In contrast, there is not
consensus concerning the different amounts of work extracted from the freestream
and wake fluid that pass through a turbine blade row. This thesis addresses that
issue.
There has been exploration of hot streak migration in the context of blade cooling
and thermal loads [16, 11]. Flow structures such as the "positive jet" induce circumferential migration of the hot streak and increase the heat load on the pressure surface
of the first stage rotor 110]. The swirling flow in the rotor can create a radial pressure gradient leading to radial migration of the hot streak toward the rotor tip [11].
Questions still remain however, regarding the effect of hot streak on stage efficiency.
Addressing this issue is the objective of this thesis.
19
The thesis does the following:
" Identifies two mechanisms for wake attenuation: a pitchwise variation in flowthrough time of the wake particles compared to the freestream and an unsteady
pressure in the relative coordinate system that increases local freestream work
extraction.
" Relates wake deformation to downstream wake stagnation pressure 18] to quantify the effect of the first mechanism.
" Identifies and explains the relationship between turbine efficiency sensitivity to
inlet hot streak magnitude and turbine tip leakage flow.
" Examines and compares existing efficiency definitions and presents a suggested
efficiency metric for flows with hot streaks and large thermal dissipation.
1.1
1.1.1
Background
Wake Attenuation
Viscous mixing of upstream wakes represents a substantial portion of losses in turbomachinery. Hall et al. report wake mixing losses representing 20% and 13% of the
profile loss for a baseline compressor and uncooled turbine stage respectively, at peak
efficiency conditions [5]. For a steady flow at uniform pressure, wake attenuation is
achieved by an irreversible mixing process. When the decrease in wake depth occurs
reversibly, without mixing, it is referred to as reversible attenuation or wake recovery.
Irreversible wake mixing losses may be reduced by reversible attenuation of the wake
downstream of the blade row before full mixing of the wake occurs, as shown in [17].
For steady flow, attenuation of the velocity defect has been shown by Denton
13]
to
be produced by favorable pressure gradients in a contraction. Attenuation of a stagnation pressure defect by unsteady mechanisms is described by Greitzer et al.
141
and
Smith [17]. The focus of the this thesis is reversible wake attenuation by unsteady
mechanisms and differential work transfer.
20
Reversible attenuation results in reduced mixing losses and improved component
performance.
Figure 1-1 shows rotor wake depth plotted versus distance through
the compressor stator at a peak pressure rise condition [201.
The "viscous" only
curve indicates viscous mixing (irreversible attenuation) at uniform pressure. The
two lines labeled "stretching only" and "viscous + stretching" represent reversible
wake attenuation alone and reversible and irreversible wake attenuation combined.
Laser anemometer measurements are shown by the crosses.
The curve including
both viscous and reversible attenuation has a smaller viscous component than the
"viscous only" line, because the smaller viscous mixing is reduced by the reversible
wake attenuation.
Figure 1-1 illustrates the means by which a compressor wake may be attenuated.
Initially the wake fluid has a lower stagnation pressure than the freestream. If the
wake fluid receives more work than the freestream as it travels through the passage,
the stagnation pressure defect decreases. Similarly, for a turbine, if less work were
extracted from the wake fluid than the freestream fluid, the wake defect would be
decreased.
vlscous only
Si
NS
hs n -stretching only
-
0.2
viscous+stretchingW
0
20
40
so
so
100
2io
% Stator Axial Chard
Figure 1-1: Compressor rotor wake evolution through a stator passage at peak pressure rise condition; lines present analysis, symbols present data [201.
Smith has explained this effect using Kelvin's theorem' [17] for an inviscid wake.
'Kelvin's theorem states that for an inviscid and barotropic flow with conservative body forces,
the circulation around a fluid contour is constant in time.
21
This analysis captures the effect well for a compressor, and Van Zante et al. identify it as the dominant mechanism in reversible wake attenuation [20]. As a wake
segment passes through a blade row, the wake segment changes through end-to-end
stretching of the wake segments. If we assume the wake segments remains linear and
are uniformly stretched, Kelvin's theorem implies that stretching of the wake along
the centerline direction means a consequent decrease in the wake velocity defect and
thus the mixing losses. This is seen in Figure 1-2 where segment AB is stretched so
that CD > AB as it passes through the compressor rotor.
Fixed Incoming Wake
B
\U
C
D
Moving Rotor
Streamlines\
'
\
Fixed "Avenue"
Along Which
Wakes Proceed
Figure 1-2: Chopping and stretching of an incoming wake
1.1.2
[181
Hot Streak Processing
Hot streak processing in high pressure turbines is an active research area and a summary of the findings and related fluid mechanics are presented below.
A hot streak is defined, for purposes of this thesis, as a temperature non-uniformity
at the combustor-turbine interface caused by discrete fuel injectors in the combustor.
It has been observed that there is a decrease in stage efficiency in high pressure
turbines when subjected to a hot streak, but at the present the cause is still not clear.
Several mechanisms for migration of turbine hot streaks have been presented.
These include radial migration of the hot streak, baroclinic torque, inlet swirl, and tip
flow ingestion [16, 111. Circumferential migration mechanisms include what is known
22
as the segregation effect or the Kerrebrock-Mikolajczak effect [101, which has been
used to explain the increased heat load seen on the pressure surface. The increased
heat load at the tip has been ascribed to radial migration, however, the link with
aerodynamic performance is not yet clear. The dominant effect isolated in this thesis
is the relative flow angle variation and subsequent blade loading imposed by the hot
streak at the rotor.
Research Questions
1.2
The aim of this work is to quantify the impact of turbine inlet non-uniformities
on stage performance. The specific topics are stagnation pressure defects from upstream stator wakes and stagnation temperature non-uniformities from combustor
hot streaks. The research questions are:
e Are stator wakes attenuated through a turbine rotor row?
e How does wake evolution differ between turbine and compressor blade rows?
e How does a hot streak at the combustor-turbine interface change the high pressure turbine stage efficiency with and without the presence of a tip gap?
9 Are mixing plane calculations sufficient to predict the variation in stage efficiency with inlet hot streak strength?
1.3
1.3.1
Methodology
Wake Attenuation
To answer the first two research questions, two types of two-dimensional calculations
have been used.
The first utilizes RANS CFD for an isolated rotor to illustrate
the convection of tracer particles, a surrogate for the vorticity in the wake which is
convected by a background steady flow. The second is an unsteady simulation of a
moving blade row with a stagnation pressure non-uniformity at the inlet. This set of
23
calculations captures the same effect, but at a higher fidelity, to give insight into the
mechanisms of reversible wake attenuation.
Results from the steady calculations show reversible attenuation is possible, despite compression of the wake segment, and quantify the magnitude for a representative turbine geometry. A reversible attenuation of 46% is found from the steady
background flow calculations.
The unsteady calculations show reversible wake at-
tenuation of 74% one quarter chord downstream of the trailing edge for the same
geometry.
1.3.2
Hot Streak Processing
To answer the third and fourth research questions, we have analyzed two geometries
to determine the dependence of turbine efficiency sensitivity to hot streaks for a
stage with a tip clearance and a stage without tip gap. Both RANS and URANS
calculations for a vane-rotor first stage high pressure turbine geometry were used to
characterize the sensitivity of the stage efficiency to hot streaks for the two geometries.
A comparison of efficiency definitions is also presented to show the importance
of selecting a proper efficiency metric. Viscous dissipation in the rotor is used to
quantify loss sources and highlight tip leakage flow mixing as a dominant effect in
setting stage efficiency sensitivity to inlet hot streak.
1.4
Contributions
The contributions of this thesis are:
9 It is shown that there can be reversible wake attenuation in a turbine. The
process is different than for a compressor because, velocity gradients normal
to the wake orientation deform the wake. An inviscid attenuation of 74% has
been demonstrated for a representative geometry.
It is also shown that the
wake can increase in stagnation pressure relative to the freestream near the
pressure surface if the wake is turned such that the velocity defect vector points
24
downstream.
" Wake segments in a turbine are not purely stretched or compressed as can
be approximated for a compressor.
Bowing and turning of the segments can
reorient the velocity defect from pointing upstream to downstream, decreasing
or increasing the wake flow-through time compared to the freestream.
The
unsteady pressure field in the relative coordinate system can also provide work
extraction from the wake relative to the freestream in the passage.
" The calculated efficiency sensitivity to inlet hot streak for a rotor with a tip
clearance is similar for steady CFD with a mixing plane and unsteady CFD
with phase-lag boundary conditions.
" For the turbine geometry examined, a turbine rotor with a 2% clearance is
roughly twice as sensitive to hot streaks than the same rotor with no tip gap.
The difference in hot streak sensitivity is attributed to a reduction in tip leakage
mass flow and thus tip leakage mixing losses, compared to the rotor operating
without a hot streak.
" A turbine performance efficiency definition is proposed for evaluating turbine
efficiency sensitivity to hot streaks. The efficiency definition utilizes a workaveraged inlet condition and mixed-out exit condition to conserve enthalpy flux
and work output between the original three-dimensional unsteady flow field and
the averages.
1.5
Organization of Thesis
The remainder of this thesis is separated into two primary sections to address the
preceding research questions; Chapters 2 through 4 address wake attenuation and
Chapters 5 and 6 address hot streak processing. Chapter 2 describes a model used
to highlight the change in wake flow-through time relative to freestream.
Chapter
3 shows, with unsteady CFD, that reversible attenuation is possible by two inviscid
25
mechanisms. Chapter 4 shows where each mechanism contributes to wake attenuation. Chapter 5 describes the approach and design of computations for the hot streak
research questions. Chapter 6 evaluates the effect of inlet hot streak on the turbine
stage and highlights tip leakage flow as the cause of the difference in stage performance sensitivity to inlet hot streak for the two geometries. Finally, Chapter 7 gives
a summary, conclusions, and recommendations for future work.
26
Chapter 2
Wake Kinematics: Kelvin's Theorem
and Wake Convection
2.1
Introduction
In this chapter, wake convection by a steady background flow is used as a qualitative
tool to interpret wake attenuation and the differences between turbine and compressor
wake evolution. We extend work by Joslyn, Caspar, and Dring f8J to determination of
the downstream stagnation pressure distribution in the wake, using Kelvin's theorem
and the wake geometry. The results show a reversible attenuation of the wake by 46%
which is associated with the first of two mechanisms to be described.
The analysis in this chapter is based on thin wakes with small velocity defect which
are convected by a steady background flow. In contrast to previous approximations
of the wake as a single linear segment, we use a distribution of infinitesimal control
volumes along the wake to calculate the distribution of wake quantities downstream
of the turbine. The analysis is useful to explain results in subsequent chapters and
to identify the importance of wake segment turning in the wake attenuation process.
27
2.2
Background
For a compressor blade row, a streamtube expands as it passes through the row. If
we approximate the wake as a linear segment with uniform velocity defect, Kelvin's
theorem can be stated as in Equation 2.1 where L denotes the segment length and AV
denotes the wake velocity defect. In a compressor, wake stretching typically occurs
because of both streamtube expansion and blade circulation, leading to end-to-end
stretching of the wake, described as
LcO
=o A
<1
AV
(2.1)
where 0 denotes a station far upstream, and oc denotes far downstream, of the rotor.
Figure 2-1 presents tracer particles convected by an inviscid, steady background
flow in a compressor. Upstream wake segments are absolute streamlines and each
line segment represents a snapshot of the wake as it convects through the blade
passages. Figure 2-1 illustrates wake segment stretching through the turbine blade
row as described in Equation 2.1.
Suppose now that we apply these arguments to a turbine blade row. Assuming
the wake remains straight with uniform stretching as before, we would conclude that
the wake segment is compressed, the wake defect is amplified, and that wake mixing
losses are increased. Figure 2-2 however shows that this viewpoint is too naive; the
wake segment end-to-end length decreases through the turbine blade row, but the
arc length of the wake segment increases. The explanation for wake attenuation in
a compressor thus cannot be directly extended to turbine blade rows because the
assumptions are not valid.
2.3
Kelvin's Theorem
Kelvin's theorem states that the circulation, F, of a material contour remains constant
in time for an inviscid, barotropic flow with conservative body forces. The circulation
around a closed material contour, N, is defined as
28
V1
UM
Um
8
Ci
5
C1
6
V
60
V22
WAKE
1 -0
Figure 2-2: Computed turbine
transport, 0 = 0.34 [8].
Figure 2-1: Computed compressor wake
transport, 0 = 0.85 [81.
F = Ju
-d,
vake
(2.2)
N
where u is velocity and df is an element of fluid contour N. The theorem is expressed
as
DF
Dt
0
(2.3)
If it is assumed a material contour enclosing a segment of the wake and the
neighboring freestream fluid is convected by the freestream, and the wake velocity
defect does not deform the material contour, the circulation is defined as AV df
where AV is the upstream defect velocity and df is the length of the differential
wake segment. Kelvin's theorem states that the circulation of this material contour
is constant. At two time instants, 1 and 2, the differential wake segment length and
velocity defect are related by
A V2
df
= A V, d .(2.4)
Equation 2.4 implies that if a differential wake segment is stretched (d1/df2 < 1)
29
as it passes through a geometry, the local wake defect decreases (AV/A V2 > 1). If the
wake defect is decreased, wake mixing losses are also decreased because mixing losses
scale with (AV) 2 for incompressible flow. Wake stretching is thus analogous to wake
attenuation and reduced mixing losses. The wake mixing entropy production scales
with (AV) 2 but is also a function of the defect vector orientation. The stagnation
pressure defect is used here as a metric for potential loss generation as it incorporates
both velocity defect and orientation.
The velocity defect used in Equation 2.4 is a scalar and represents the wake velocity
defect (the freestream velocity minus the wake centerline velocity).
The velocity
component in the wake normal to the wake centerline, must be equal to the freestream
normal component because the normal velocity flux to the material contour must be
equal to zero. The velocity defect vector thus must have a magnitude equal to AV
and direction parallel to the local orientation of the wake segment, de.
For a wake with small thickness and wake defect, the wake may be viewed as
an unsteady perturbation on a steady background flow. Using Kelvin's theorem,
the wake defect evolution can be analyzed by tracking the geometry of the wake as
it deforms in the passage. This is the methodology used in the remainder of this
chapter.
2.4
Computational Methodology
The calculations described in this chapter use an inlet velocity boundary condition
that models the stagnation pressure defect from an upstream stator. There is no
stator-rotor interaction and the background flow field is steady in the rotor frame.
ANSYS@ Fluent v14.5 was used as the flow solver for this work. The inviscid
solver was employed to separate inviscid reversible wake attenuation from viscous
attenuation. The flow is incompressible.
30
2.4.1
Blade Design Specification
The turbine airfoil is designed using the 11 parameter model developed by Pritchard
115] and selected by Mazur [12] for his work. The relative and absolute flow angles
are set by the definition of the work coefficient (0), the flow coefficient (0), and the
stage reaction (R). Table 2.1 shows the turbine stage parameters.
Table 2.1: Turbine stage parameters
Parameter
Assigned Value
Work Coefficient (/)
Flow Coefficient (#)
Stage Reaction (R)
Axial Chord
Tangential Chord
Unguided Turning
Inlet Half Wedge Angle
Number of Blades
Number of Stators
1.60
0.75
0.50
Specified so that Z, = 0.8
Specified by Kacker and Okapuu Correlation 191
2.4.2
150
140
58
58
Boundary Conditions and Mesh
Two-dimensional linear cascade meshes of approximately forty thousand cells were
created using the Rolls-Royce code PADRAM. The computational domain extends
one chord upstream to provide space to seed the wake particles, and two chords
downstream to let the wake develop after exiting the blade row. At the inlet, the
velocity and flow-angle are specified. At the outlet, o( )/&x is set to zero. Finally,
the flow is circumferentially periodic.
2.5
Results
The geometry in Table 2.1 which is representative of turbine stage parameters, exhibits a reduction of the end-to-end length of the wake segments. However, wake
attenuation occurs, in conflict with the analysis described in Equation 2.1.
31
2.5.1
Wake Attenuation Metric
A metric for wake attenuation is defined in Equation 2.5, referred to as the stagnation
pressure defect ratio. Subscripts fs and w denote freestream and wake respectively.
All values of stagnation pressure are defined in the absolute reference frame. Spatial
coordinates are defined in the relative coordinate system.
In Equation 2.5, fs is
defined as the steady background flow and w is defined as the conditions in the
convected infinitesimal wake. Equation 2.5 represents the stagnation pressure defect
at station x, non-dimensionalized by the stagnation pressure defect upstream of the
blade row. A value less than one indicates reversible wake attenuation (and less
potential for wake mixing losses). Similarly, values greater than one represent wake
growth and increased potential for mixing losses.
(Pt,!s - PtW)X
(Pt,fs - Pt'W)O
(2.5)
In later sections, the stagnation pressure defect ratio is evaluated with time averaging to construct representative quantities for fs and w. Details on the averaging
technique will be given in Section 3.4.
2.5.2
Attenuation Quantification
The turbine geometry is shown in Figure 2-3 with the centerlines of the convected wake
segments marked at uniformly distributed time steps. Upstream of the blade row,
the wake centerlines are straight and represent absolute streamlines. The contours
identify pressure coefficient using the upstream static pressure and relative dynamic
pressure as reference. The contour intervals are 0.25.
As discussed in Section 2.2, the wake velocity defect and orientation may be
inferred from the wake centerline geometry, with magnitude defined by the localized
stretching of the wake and orientation determined by the orientation of the local wake
segment. Figure 2-4 shows velocity vectors, in the absolute reference frame, at the
inlet station and at a downstream station.
The wake geometries in Figure 2-4 were extracted (far upstream and far down32
[-
1.0
0.0
-1.0
-2.0
-3.0
0.0
1.0
0.5
x/cx
Figure 2-3: Convected wake geometry evolution through passage, 2 (p - po)/p u4
Downstream
Upstream
1.0
1.0
-
---
0.5
0.5
0.0
0.0
free-streaim
defect
wake
0.5
1.0
0.0
0.0
0.5
1.0
x/cx
x/cx
Figure 2-4: Upstream and downstream wake velocity triangles
stream) from the solution shown in Figure 2-3. The velocity triangles are not indicated on the upstream wake segment because in the absolute reference frame they are
collinear with the wake segment (the upstream wake segment is an absolute streamline). The downstream segment is taken far enough downstream to be outside the
pressure field of the rotor in Figure 2-3. The wake segment is (i) compressed, (ii)
bowed, and (iii) turned with respect to the segment end points. In the figure, the
freestream vector denotes the velocity of the background flow in the absolute reference
frame. The defect is a vector of magnitude inversely proportional to the local wake
stretching and with direction of the local wake angle. The wake velocity is then the
33
vector summation of the freestream velocity and the wake defect velocity. Although
the initial velocity defect is equal to 0+ for this analysis, the velocity triangle is scaled
up for visualization.
For an infinitesimal initial wake velocity defect, we can calculate the stagnation
pressure defect ratio explicitly from the wake geometry evolution. Figure 2-5 displays
the downstream stagnation pressure defect ratio for this limiting case (i.e. Figure 2-5
is the stagnation pressure derived from the velocity triangles in Figure 2-4 taken at
the limit as the wake thickness and upstream defect vector goes to zero). In Figure
2-4, y/W = 0.00 denotes the suction surface stagnation streamline and y/W
1.00
denotes the pressure surface stagnation streamline.
1.5
-
1.0
-
0.5
-
0.0
-
-0.5
3 -1.0
-1.5
0.00
0.25
0.50
0.75
1.00
y/W
Figure 2-5: Stagnation pressure defect far downstream for a convected wake segment
Figure 2-5 shows that the wake is attenuated for all pitch-wise locations, except
for y/W from 0.24 to 0.43. An average stagnation pressure defect (average wake
attenuation) can be calculated by using the stretching to determine local wake width.
For a differential wake segment, the area of the material contour (product of w and
df) is constant in an incompressible flow, as in Equation 2.6,
dA = w df = constant.
(2.6)
An area average (denoted by superscript A) of some quantity F is calculated
34
using Equation 2.7. The variable is weighted by the local differential area, dA. In
the integral bounds, 0 and L represent the endpoints of the wake.
L
F
=_
f dA
A
Fwdd
(2.7)
0
Regions where the wake is compressed are weighted larger in an area averaged
stagnation pressure defect due to increased wake area per differential length. The
calculated average stagnation pressure defect is equal to 0.54 for Figure 2-5, or 46%
reversible attenuation of the wake.
This wake kinematics are one of two mechanisms identified as cause for reversible
wake attenuation. As to be shown in Equation 3.3 in Chapter 3, the work extraction
is a function of axial velocity and the unsteady pressure. If the wake fluid has a higher
axial velocity, it can pass through the blade row with less work extracted, as on the
pressure surface of the turbine in Figure 2-4 and Figure 2-5. The increase in axial
velocity is related not only to stretching of the wake segment, but also its turning.
The combination of stretching and turning results in attenuation and can cause an
increase in wake stagnation pressure above freestream (pt, > pt,fs) near the pressure
surface.
The stagnation pressure defect distribution can be used to further interpret Figures
2-1 and 2-2. For a compressor, the wake segment is stretched such that its velocity
defect decreases. In the turbine however, the velocity defect vector is turned in the
wake near the pressure surface so the velocity defect is directed downstream (i.e. wake
fluid has higher axial velocity than freestream fluid), and local stagnation pressure
in the wake is increased relative to the freestream. The increase in wake stagnation
pressure above the freestream stagnation pressure is consistent with the geometric
results in Figure 2-4 and the quantitative results shown in Figure 2-5.
35
36
Chapter 3
Assessing Wake Attenuation in a
Turbine Blade Row
3.1
Introduction
In this chapter we identify a second mechanism of inviscid wake attenuation using
unsteady computations of an isolated rotor and a wake with non-zero thickness and
velocity defect. We show that the unsteady pressure field (in the relative reference
frame) increases work extraction in the freestream relative to the wake fluid. While
the analysis in Chapter 2 yielded a wake attenuation of 46%, the results in Chapter 3
predict wake attenuation of 74% when both mechanisms are included. This chapter
establishes an upper bound for wake attenuation for the given geometry in the absence
of viscous attenuation.
3.2
Background
For inviscid incompressible flow, the relation between the temporal variation in static
pressure and the change in stagnation pressure for a fluid particle is given in Equation
3.1.
Dpt
O_
~(3.1)
Ot
Dt
37
Far upstream or downstream of the blade row, &p/&t ~ 0, and the stagnation pressure
is then a convected quantity of a given particle, Dpt/Dt = 0. For the turbine blade
row, each particle must travel the same axial distance from inlet to exit. It is therefore
useful to examine the rate of change of a particle's stagnation pressure with respect
to axial position, x, rather than time. In Equation 3.2, the left hand side, Dpt/Dx,
is the rate of change of a particle's stagnation pressure defined with respect to axial
distance traveled over time increment Dt; Dx = u Dt
Dpt
Dx
Dpt
-1
(3.2)
ux Dt
Substitution of Equation 3.2 into Equation 3.1 yields Equation 3.3 which shows
the two effects that contribute to stagnation pressure changes in the passage. The
first is the axial velocity, ux. If the value for Op/lt is negative (work extraction) as
it is in a turbine, a wake particle that spends less time in the passage will have less
work extraction. This is referred to as flow-through time, as described by Smith for a
compressor [17]. If &p/&tvaries such that the unsteady pressure magnitude is smaller
for a wake particle, then reversible wake attenuation may also be achieved even if the
flow-through time is the same for wake and freestream.
Dp-= lp
Dx
3.3
ux Ot
(3.3)
Computational Methodology
As mentioned in Chapter 2, ANSYS® Fluent v14.5 was used as the flow solver. The
calculations described in Chapter 3 and 4 are time-accurate with the flow modeled
as inviscid and incompressible.
3.3.1
Boundary Conditions and Mesh
It is assumed that the flow far upstream is steady in the stationary reference frame
and there is no upstream influence at the wake injection point. The wake profile is
38
parameterized by depth (e), profile geometry, and wake thickness (8). The parameter,
e is a measure of wake strength, as in Equation 3.4, based on the upstream freestream
velocity, uma, and the velocity at the wake centerline, Umin.
Umax
Umin
-
(3.4)
Umax
To mitigate artificial dissipation, a wake profile was selected to minimize the spatial second derivative. This results in a piecewise quadratic profile with a continuous
first derivative and discontinuous second derivative as in Equation 3.6.
1
Uxmax
0
Ux~~m{x
UI
(e /2W)
-
if y ;> 6/2
1(Yif (2 8/46
2
if 6/4 < jyj <6 /2
6
+1
(3.5)
(3.6)
62y<;/
All cases presented in this document utilize 8 = 0.3W where W is the pitch. This
value is a compromise between reducing velocity gradients and simulating vane wake
profiles in detail, it is representative for the effects of interest here. A time trace of
the upstream wake in the relative frame is shown over one stator passing in Figure
3-1 for e = 0.25 and 6 = 0.3 W.
The isolated rotor is circumferentially periodic. The outlet pressure is specified
as a gauge pressure (absolute pressure is not relevant in incompressible flow), and
the outlet boundary condition is applied two chords downstream, out of the region
where upstream influence is important. The mesh interface between stationary and
translating zones is one-quarter chord upstream of the leading edge. The stationary
stator zone extends one and one-half chord further upstream. The pitch to axial chord
ratio is approximately 3 : 4. The blade geometry utilizes a slip wall condition.
39
1.2
1.0
-
0.8
0.6
free-stream
wake
free-stream
0.25
0.50
t/T
0.75
0.4
0.2
0.0
0.00
Figure 3-1: Inlet boundary condition, c
3.3.2
1.00
0.25
Blade Parameters and Mesh
The rotor geometry described in 2.1 is used in the unsteady finite wake propagation
calculations. The unsteady wake processing requires a higher fidelity mesh to accommodate the larger spatial gradients in the wake regions. The baseline two-dimensional
mesh contains approximately sixty-five thousand cells (compared to forty thousand
in Chapter 2). A sliding mesh interface is used between stationary and translating
zones of the computational domain.
The temporal solver uses a second order implicit scheme. The spatial solver is
a third-order MUSCL momentum scheme and a second order pressure solver. The
temporal resolution has 720 time steps per blade passing.
3.4
Averaging
The stagnation pressure defect in Equation 2.5 was utilized as the metric for reversible
wake attenuation. The subscripts fs and w are more complex than the definition
presented in Chapter 2 because pt,f,(X, y) - pt,w(x, y) is defined at a given coordinate
and thus requires a time averaged definition, i.e. the presence of a freestream and
wake particle at a specific spatial coordinate is mutually exclusive. The freestream
40
and wake stagnation pressure values are thus defined at each point as weighted timeaverage values of the fluid stagnation pressure passing over a location during one blade
passing. These values are substituted into the stagnation pressure defect in Equation
2.5 to evaluate wake attenuation in the passage.
Upstream of the blade row, any
fluid within 6/2 of the wake centerline is specified as wake fluid ,with everything else
defined as freestream fluid. In the computations, the upstream fluid has a numerical
marker, P, (1 for freestream fluid and 0 for wake fluid) that convects with the fluid
particles to segregate wake from freestream fluid.
The averaging scheme used to define fs and w is presented in Equation 3.7 for
an intrinsic property of the fluid, F. Subscript fs denotes the average value of F for
fluid marked as freestream that has traveled through coordinate (x, y) over one blade
passing. The example shown in Equation 3.7 is for averaging over the freestream
fluid, and thus uses the subscript fs, but the definition is applicable to both wake
and freestream fluid.
Ff s (x, y)
=F(x,
K(x, y)
y, t) dt
(3.7)
The two unknowns in Equation 3.7 are K and C. C is defined in Equation 3.8 as
the set of times during one blade passing when freestream fluid is located at coordinate
(x, y). The flow is periodic in time T, so F(x, y, t) = F(x, y, t + a T) where a is any
integer.
C(x, y) = {t I(a E IR, t c [a, a + T]) n (P(x, y, t) = 1)}
(3.8)
The variable K, defined in Equation 3.9, is the cumulative residency time of
freestream particles, i.e. the total time that freestream (or wake as appropriate)
particles are present at coordinate (x, y) over one blade passing, T,
K
J dt.
(3.9)
C
The averaging in Equation 3.7, which is implied for any variable with subscripts,
41
fs or w, converts unsteady quantities, f(x, y, t), into steady quantities, f(x, y), to
permit evaluation of the differences in freestream and wake quantities at a specified
x, y location.
Results
3.5
The results below are for the geometry of Table 2.1 with c = 0.1. The stagnation
pressure defect defined in Equation 2.5 is used to quantify attenuation. If this ratio
is greater than one, the local stagnation pressure defect is greater than the initial
upstream condition.
3.5.1
Reversible Wake Attenuation
Figure 3-2 shows contours of the time-averaged absolute stagnation pressure ratio
(Equation 2.5) in the blade passage. Values greater than one represent wake amplification and values less than one represent attenuation.
1.5
1.0
0.5
0.0
-0.5
0.0
0.5
1.0
-1.0
x/cx
Figure 3-2: Stagnation pressure defect ratio contour, c = 0.1
Downstream of the trailing edge, the stagnation pressure ratio, defined locally
using the scheme established in Section 3.4, is less than one for all pitchwise coordinates. Although attenuation is achieved, localized amplification can be seen within
42
the passage. The wake amplification achieves a maximum of 1.36 at an axial location
of x/cx = 0.23 and a pitch of y/W = 0.44. The region of amplification is small,
however, and occurs in the center of the passage where freestream shear stresses, in
a physical turbine stage with viscous stresses, are small.
Therefore, the localized
wake amplification expected to be of less importance than the attenuation occurring
elsewhere in the passage. This is observed downstream where the attenuation has a
larger magnitude than the amplification in the passage.
1.5
1.0
0.5 --
0.0
-0.5
-
--- -
1.
.o
1.5
0.00
-
-
C
-
/C = 1.00
x/c
= 1.00
x/c = 1.25
'
.0
0.25
0.50
0.75
1.00
y/W
Figure 3-3: Stagnation pressure defect ratio at axial locations, 6 = 0.1
The stagnation pressure defect ratio at three different x/cx values is shown in
Figure 3-3 as a function of y/W. At the leading edge, there is localized amplification
but by the trailing edge of the rotor, the wake is attenuated at all circumferential locations. The location, x/cx = 1.25 represents an axial station approximately halfway
between the rotor and downstream stator. At this location, the wake is attenuated
by 74%.
Attenuation is most prominent on the pressure surface, consistent with the estimates in Chapter 2, suggesting wake attenuation is a result of reduced flow-through
time for wake particles near the pressure surface.
Figure 3-4 displays the difference in axial velocity between the freestream and
wake normalized by the value at the inlet. The axial velocity of the wake is larger
than the freestream fluid on the pressure surface, resulting in a lower flow-through
43
time and reduction in the work extraction from the wake fluid. This may be seen
in Equation 3.3 where the change in stagnation pressure is a function of both axial
velocity and the unsteady pressure.
3
-
2
0
\f
\
---x/c
-3
0.00
/cX
=
1
0.00
x/c = 1.00
= 1.25
0.25
0.50
0.75
1.00
y/W
Figure 3-4: Axial velocity defect in wake, c
0.1
The difference between Figure 3-2 and Figure 2-5 suggests there is a mechanism
captured by the unsteady calculations that is not included in the wake tracing analysis
of Chapter 2. To show this, the stagnation pressure ratio for the convected wake
(Chapter 2) and the unsteady finite wake (Chapter 3) is presented in Figure 3-5.
The solutions agree well on the pressure surface but on the suction surface there is a
significant discrepancy.
In Chapter 2, we addressed the mechanism of wake stretching and retention time
only. This is equivalent to saying that
p/Ot does not depend on the presence of
the wake and the primary mechanism is changes in the axial velocity, ux. Figure 3-6
displays the unsteady pressure for several axial locations in the passage and indicates
Op/&t does vary between freestream and wake fluid. More work is extracted from
the freestream than wake fluid (wake attenuation) for y-axis values less than zero in
Figure 3-6. Near the suction surface, Op/&t decreases in magnitude when wake fluid
is present compared to freestream fluid, which reduces the difference between wake
and freestream stagnation pressure. A description of this effect is given in Chapter 4.
44
1.5
1.0-
<0.0
-
-
0.5
-0.5
-1.5
0.00
--
/Cx = 00, E= 0.0
x/cx = 1.25, e = 0.1
'
0.25
'
0.50
y/W
-
'
1.0 .
0.75
1.00
Figure 3-5: Difference between convected and finite wake analyses
In summary, pressure changes due to the presence of the wake have an influence
comparable to that of flow-through time in reversible wake attenuation in a turbine.
For the geometry investigated, the computed value for wake attenuation increases
from 46% to 74% when the effect of unsteady pressure is included.
45
5.0
-
2.5
-
----
x/cx 0.25
x/c= 0.50
x
=0.75
0.0 I-
;31
-2.5
CZ)
-
-5.0
0.00
0.25
0.50
0.75
1.00
y/W
Figure 3-6: Difference in Op/&t between wake and freestream fluid at axial slices,
C = 0.1
46
Chapter 4
Separation of the Wake Attenuation
Mechanisms
In this chapter, we present an analysis to illustrate the unsteady pressure (Op/&t)
and axial velocity (uX) mechanisms described in Chapter 3. We develop a linearized
form of the convective derivative, Dpt/Dx, to obtain an expression for D(Apt)/Dx,
separate the two mechanisms, and quantify the rate of change of stagnation pressure
defect through the blade row.
4.1
Introduction
We use the stagnation pressure defect ratio of Equation 2.5 and analyze the same
isolated rotor. The computations are carried out with, c = 0.025, to approximate a
linearized expression. Figure 4-1 is a visualization of the stagnation pressure defect
for c = 0.025.
The similarity with Figure 3-2 indicates that non-linear terms are
small in both E = 0.025 and E = 0.1 because of the similar appearance (Figure 3-2
and Figure 4-1).
The contours in Figures 3-2 and 4-1 agree more near the inlet than at the trailing
edge because of the increased amount of fluid traveling along the wake at larger c
(i.e. the so-called negative jet [6]). However, at one quarter chord downstream of the
trailing edge, cases with c = {0.025, 0.050, 0.100, 0.200} all show the wake attenuation
47
1.5
1.0
0.5
0.0
-0.5
-1.0
0.0
1.0
0.5
X/cX
Figure 4-1: Stagnation pressure defect, c = 0.025
to be between 74% and 81%. Figure 4-2 shows the stagnation pressure defect for
the four e values at one quarter chord downstream. All four values of wake depth
show nearly the same attenuation and show the good agreement also indicates the
linearization is appropriate for the chosen values of c.
4.2
Governing Equation
The equation for changes in stagnation pressure was presented as Equation 3.1, but
it is given again in Equation 4.1 for reference. The mechanisms that drive the evolution of stagnation pressure are nonlinear for large amplitude perturbations, but for
small perturbations, Section 4.1 shows a linearized form of Equation 4.1 can be used.
The left hand side of Equation 4.1 describes the rate of change of a particle's stagnation pressure with respect to axial position, which is a useful metric for evaluating
stagnation pressure changes across a blade row and can be computed directly.
Dpt
1
lOp
Ut at
Dx
48
(4.1)
1.5
1.0
0.5
- 0.0
= 0.025
-0.5 -
e=0.050
-1.0 .=0.100
c=0.200
-1.5
0.75
0.50
y/w
0.25
0.00
1.00
Figure 4-2: Wake attenuation for four wake depths, x/cx = 1.25
4.3
Linearization
The variables on the right hand side of Equation 4.1, au and Dp/&t, are defined as the
the summation of a time-averaged flow field and an unsteady perturbation. The flow
is periodic and ix is the time average of ux over one blade passing. Primes denote
the perturbations.
ux(x, y, t) =x(x, y) + , (x, y, t)
(X, y, t)
at
=9
at
(X, y) +
at
(x, y, t)
(4.2)
(4.3)
Figures 4-3 and 4-4 present the two components of Equation 4.2. Figure 4-3 is
the time-averaged background flow and Figure 4-4 is a snapshot in time showing the
axial velocity perturbations from the background time-averaged flow field. The same
process is used for
p/&t but
is not shown.
If the definitions in Equations 4.2 and 4.3 are substituted into Equation 3.3 we
obtain Equation 4.4 which describes the stagnation pressure evolution of a material
49
2.5
3.0
2.0
2.0
1.0
1.5
0.0
1.0
-1.0
0.5
-2.0
0.0
0.5
x/c,
0.0
Figure 4-3:
0.0
1.0
Figure
Time averaged velocity,
0.5
/Cx
4-4:
1.0
Velocity
-3.0
perturbation,
U' /(C e)
X/UX'O
element,
FOp1 + (4.4)
9P'
x + '4 Lat ot
Dpt
-Dx
I
We now linearize Equation 4.4 to obtain Equation 4.5.
1 0p'
-(-5
1 Op
--
_
-
ix at
ait
U'
_p
at
(
Dpt
Dx
Equation 4.5 also implies a summation of a time-averaged component and a perturbation component for Dp/Dt and Dp'/Dt respectively.
4.4
Results
Equation 4.5 is the linearized form of the governing equation for changes in stagnation
pressure. The stagnation pressure defect of interest is a time-averaged quantity, therefore Equation 4.5 must be time averaged. The quantities (Dpt/Dx)f, and (Dpt/Dx).,
are the time-averaged rate of change for all freestream and wake particles respectively
passing through an area element centered on a point x, y with respect to their axial position.
The difference in the freestream and wake derivatives then yield an
expression for the derivative of the stagnation pressure defect, Apt,
50
D(Apt)
Dx
DpK
Dx
Dpt(
k,Dx
(46
Equation 4.6 combined with Equation 4.5, yields an expression for the rate of
change of the stagnation pressure defect, given in Equation 4.7. The time averaged
term, (Dp/&t)/i, which appears in both the freestream and wake components cancels
out leaving only the perturbed quantities.
Equation 4.7 gives the difference in the time average rate of change of the stagnation pressure defect with respect to axial position,
D(Apt)
Dx
1
_
Dp>'
Kk at(")fs
(4U-,7) 1&p
' pl
a
iix
ft a;p
2
UJI
at
(47
at
We non-dimensionalize this expression to yield Equation 4.8. In Equation 4.8,
there are four distinct terms which affect the attenuation of the stagnation pressure
defect in the rotor passage.
2c D(Apt)
2_
EpU2
_
Dx
_2
2 c[
EpU2
1
&X
Op'
at f
1
iiX
p'
p '
"
at
+___
OPXf
_-2
at ii
(4.8)
(4j
at
ii2
The terms in Equation 4.8 may be grouped into the two mechanisms described
previously. The first two terms on the right hand side have a time-averaged axial
velocity field and unsteady perturbations in the static pressure. This represents the
differential work extraction through the unsteady pressure variations. The last two
terms have a time-averaged background pressure derivative and perturbations in the
axial velocity and represent the flow-through time effect.
The values for both of the terms previously described are shown in Figures 4-5
and 4-6. Interpretation of these two figures will be expanded on in Section 4.4.1.
The summation of the two mechanisms in Figures 4-5 and 4-6 is equivalent to a
linearized approximation for (2 c;)/(EpU2) (D(Apt)/Dx) and can be compared to the
computed value from the (nonlinear) CFD. The derivative, Dpt/Dx, is a modification
of the material derivative definition, as in Equation 4.9, and is evaluated for all spatial
51
0.0
0.5
x/cx
5.0
5.0 5.0
2.5
2.5
0.0
0.0
-2.5
-2.5
-5.0
-5.0
1.0
0.0
0.5
1.0
/c,
Figure 4-6: Non-dimensionalized axial
velocity perturbation
Figure 4-5: Non-dimensionalized unsteady pressure perturbation
coordinates and time in the computations.
Equation 4.9 represents the material
derivative definition, Dpt/Dt, divided by ux.
Dpt
Dx
-
1 apt
_
-
-+
Ot
Opt +Upt
+
Ux OY
&x
(4.9)
49
The material derivative is averaged using the technique to determine fs and w, and
is combined with Equation 4.6 to calculate the CFD based stagnation pressure defect
derivative, D(Apt)/Dx. A comparison between the CFD results and the linearized
results is shown in Figures 4-7 and 4-8. The similarity between Figures 4-7 and 4-8
supports the linear assumption for small c.
Figures 4-5 and 4-6 demonstrate the two mechanisms that cause reversible wake
attenuation in a turbine. The flow-through time mechanism (Figure 4-6) gives a region
of attenuation near the pressure surface and amplification near the suction surface.
The unsteady mechanism in Figure 4-5 is more difficult to characterize. The primary
effect is that there exists a region in the passage where the unsteady pressure (as seen
in the relative coordinate system) removes energy from the freestream, counteracting
the amplification due to the first mechanism. A more detailed discussion is given in
Section 4.4.1.
52
- 5.0
7].
2.5
2.5
0.0
0.0
-2.5
-2.5
-5.0
-5.0
0.0
0.5
xIC.x
0.0
1.0
0.5
1.0
/c,
Figure 4-8: Computed D(Apt)/Dx from
linearized analysis
Figure 4-7: Computed D(Apt)/Dx from
unsteady CFD
4.4.1
5.0
Attenuation Mechanism Interpretation
Figures 4-5 and 4-6 display the spatial dependence of the two mechanisms responsible
for reversible wake attenuation. The mechanism in Figure 4-5 (pressure fluctuations)
is primarily caused by blade loading changes from impingement of the wake on the
rotor.
This is most apparent in the upstream section of the passage where wake
impingement on the suction surface occurs. A cartoon is shown in Figure 4-9 where
the wake impingement near the leading edge imposes a localized high pressure region.
The high pressure region convects downstream along the suction surface such that
fluid directly downstream of the impingement point experiences a ap/&t larger than
when the wake is not present. The wake is in front of the impingement point so less
work is extracted from the wake than the surrounding freestream fluid. Although
first identified by Meyer [131, this effect is described in-depth by Hodson and Dawes
16] for a wake in a turbine passage.
The second mechanism is variation in the axial velocity and hence in particle flowthrough time between the wake and freestream. This mechanism includes a weighting
factor of &p/&t which determines whether work extraction or addition occurs locally in
the passage. Figure 4-6 shows that this mechanism imposes attenuation upstream of
the pressure surface stagnation point and downstream of the trailing edge stagnation
point. Wake attenuation occurs in these regions due to the work addition occurring
53
UX
+1
Phigh
act~
H
+
X "
0.0
0.5
0.0
1.0
1.0
0.5
x/cX
x/cX,
Figure 4-9: Description of unsteady
pressure mechanism
Figure 4-10: Description of flow-through
time mechanism
by
p/&t
and the longer residency time of wake particles in this region. Conversely,
work extraction occurs in the remainder of the passage. The weighting term, Op/Dt
causes work extraction in the main passage but, the shorter flow-through time of wake
fluid leads to a reduction in work extraction in the wake. This is shown in Figure
4-10 where the positive and negative regions denote work addition and extraction
respectively. The positive region of the wake segment, u ,g
traveling faster than freestream fluid and the contrary for u', 8
[+],
[-].
denotes wake fluid
These two regions
describe wake attenuation and amplification respectively in agreement with the result
shown in Figure 4-6.
54
Chapter 5
Effects of Hot Streaks on High
Pressure Turbine Efficiency
5.1
Introduction
In Chapters 5 and 6, we address a second subject: the effect of hot streaks on high
pressure turbine efficiency. Three topics are addressed. First, a method of timeaveraging is outlined that reduces flow time requirements by use of a finite impulse
response filter. Second, a generalized efficiency definition for unsteady non-uniform
flow is introduced. Third, the results are interrogated and show the efficiency of a
turbine with a tip clearance is 2.5 times less sensitive to inlet hot streak than the
same turbine with no tip gap.
5.2
Computational Methodology
The sensitivity of turbine efficiency to inlet hot streak magnitude is shortened herein
to hot streak sensitivity. Hot streak sensitivity is defined as 07r/(OTDF)'. It is
reported here for a given geometry with varying inlet hot streak magnitude.
To quantify hot streak sensitivity, two separate turbine configurations are ana'OTDF = max(T, 4 (r, 0) - Tk1)/(T/ - Ttj') where Al denotes mass average, station 3 is the
compressor-combustor interface, and station 4 is the combustor-turbine interface.
55
(a) Meridional view of vane and rotor mesh
(b) Rotor tip mesh
Figure 5-1: Vane and rotor surface mesh, TG-2%
lyzed: one with a finite tip gap and another without a tip gap. The first, with no tip
gap, is referred to as the 0%
(TG-0%) clearance case. The second geometry has a tip
clearance of 2% of the rotor span (TG-2%). Both turbine geometries share a common
vane geometry and a common rotor geometry with a different casing endwall. Steady
calculations include vane and rotor and utilize a mixing plane at an interstage station
between them. The mixing plane circumferentially mixes the flow exiting the vane
and permits coupled steady CFD calculations of the two computational domains. The
unsteady calculations utilize a phase-lag boundary condition and are initialized from
the converged mixing-plane calculations.
All results related to hot streak processing utilize the Rolls-Royce proprietary CFD
code HYDRA, which solves the Reynolds-Averaged Navier-Stokes equations (RANS)
and unsteady RANS (URANS) equations. All geometries use a structured mesh and
the HYDRA solver utilizes an unstructured node based solver.
An example of the
mesh used for the TG-2% case is shown in Figure 5-1. Both meshes used in this study
have an average y+ of 1 to resolve the boundary layers.
The Spalart-Allmaras (SA) one equation turbulence model is used and initialized
with a turbulent to laminar viscosity ratio of 100 at the inlet to account for high
levels of turbulence of the fluid exiting the combustor
[1]. The hot streak sensitivity
is found to be weakly dependent on the turbulent viscosity ratio; the sensitivity
56
reduced by 11% when the viscosity ratio was lowered from 100 to 10, much smaller
than the variations caused by geometry alterations (250%). It is shown that the
change in tip leakage flow mixing with the main flow is the dominant mechanism
for turbine sensitivity reduction with hot streak strength. Huang also found that tip
leakage losses are insensitive to the inlet turbulent viscosity ratio [71, supporting the
preceding comments about turbulent viscosity ratio.
5.2.1
Turbine Geometry and Mesh Specification
The turbine simulations in this thesis utilized a mixing plane for the RANS solver and
a phase-lag boundary condition for the URANS solver to preserve full annulus blade
counts. All calculations described utilize thirty-six vanes and fifty-eight blades per
revolution. The combustor contains eighteen burners and the computational domain
includes two vanes to match the burner count, yielding a flow periodicity of onehalf revolution. A method is presented in Section 5.2.3 to reduce the computational
complexity of time-averaging efficiency and eliminate the need for the full period.
A meridional view of the turbine with the mixing plane included is shown in Figure
5-2. The black line denotes the TG-0% and red denotes TG-2%. The black vertical
line is the mixing plane and the gray lines are the vane and rotor. The rotor and vane
geometry was held constant between tip gap configurations and the casing endwall is
shifted as in Figure 5-2. No cooling flow is included and all surfaces are modeled as
adiabatic. The fluid is an ideal gas with constant specific heats, and -y = 1.3.
x
Figure 5-2: First stage meridional view comparison for TG-0% and TG-2%
57
The stage parameters associated with the turbine rotor and two cases used are
shown in Table 5.1.
Tip Clearance
Work Coefficient
Flow Coefficient
Stagnation-to-Static Pressure Ratio
Solidity
Aspect Ratio
0.00% 2.00%
2.03
1.97
0.44
0.46
1.94
1.96
0.97
1.25
Table 5.1: Turbine stage parameters, OTDF = 0.0
5.2.2
Boundary Conditions
The inlet boundary condition is held constant for the simulations with the TG-0% and
TG-2% configurations for a given hot streak strength. A two-dimensional distribution
of stagnation pressure, stagnation temperature, turbulent to laminar viscosity ratio,
and flow angle is specified at the turbine inlet in the r, 0 plane. At the outlet, the
static pressure at the hub is specified and the radial distribution of static pressure
that satisfies radial equilibrium is established by the solver. All walls have no slip
boundary conditions.
The measure used for hot streak strength in this thesis is the overall temperature distribution function (OTDF). The OTDF represents the difference between the
maximum temperature and the mass average temperature, non-dimensionalized by
the temperature rise across the combustor. The radial temperature distribution function (RTDF) is a non-dimensional circumferentially averaged temperature.
and RTDF are defined in Equations 5.1 and 5.2 respectively.
OTDF
The superscript M
denotes a mass average quantity, station 4 is the combustor-turbine interface and
station 3 is the compressor-combustor interface.
OTDF = max
r,o0
Tt, 4 (r, 0) - TA'
'
Lt,4
58
-L t,3
(5.1)
27r
RTDF
1 f Tt, 4 (r, 0)- T^'d(
d
270
T4 -T,3
(5.2)
The cases analyzed with CFD are shown in Table 5.2 denoted by x's.
The
OTDF denotes the inlet boundary condition as defined in Equation 5.1. The case of
OTDF = 0.0 can be regarded as a baseline case with uniform inlet stagnation temperature and is sometimes referred to as the reference state denoted by the subscript
0.
OTDF
RANS
URANS
0.0
x
x
0.2
X
X
0.4
x
X
0.6
x
Table 5.2: CFD inlet boundary conditions and solver
For this study, a single temperature profile was used, scaled about T
by varying
OTDF. For all cases it was assumed that the OTDF is twice the maximum RTDF
value, a representative approximation at the combustor-turbine interface. Figure 5-3
shows the radial temperature profile with RTDF and min/max temperature distributions overlaid for the OTDF = 0.6 case. The peak value of temperature occurs
at approximately mid-span and corresponds to OTDF = 0.6. The peak RTDF also
occurs at mid-span and corresponds to RTDF
-
0.3.
Figure 5-4 shows the non-dimensional stagnation temperature profile at the combustorturbine interface over the domain of two vanes (10 degrees). This temperature profile,
scaled about Td by OTDF, is utilized for all cases in the following study.
Figure 5-5 displays the circumferential clocking2 of the hot streak at mid-span.
The hot streak impinges the leading edge and the majority of the hot streak travels
on the pressure surface to permit effective cooling. The figure also identifies the vane
to hot streak ratio as 2:1. The clocking configuration shown is used for all the cases
explored here.
The inlet stagnation pressure is taken as uniform. Heat addition may be approximated as a steady frictionless flow with the changes in stagnation pressure given
2Circumferential position of hot streak relative to vane leading edge
59
I
0.8
-eQ
-
/
-/
0.6
N
-
110
0.4
0
0.21-
'
1
RTDF
-- Min and Max
/
z
-1.2
-0.9
0
-0.3
-0.6
0.3
0.6
0.9
)
(T - T, 3)/(T, 4 - T, 3
Figure 5-3: Circumferentially averaged inlet temperature distribution, OTDF = 0.6
1
-eQt 0.75
P4
0
0-0.50
-0.25
0.00
0.50
0.5
0.25
- 0 .0 00
01
5
.25-
0 .25
2.5
0
-2.5
Circumferential Position [deg]
5
Figure 5-4: Inlet temperature profile, OTDF = 0.6
by
dpt
Pt
_y
M2
2
dT
T)
(5.3)
If M 2 < 1, the stagnation pressure changes are small and pt can be approximated
as uniform.
Steady CFD calculations were conducted where the hub and casing
boundary layers each extended into 40% of the flow and it was found that the hot
streak sensitivity only increased by 7%. The weak dependence combined with the
low Mach numbers in the combustor support the use of a uniform inlet stagnation
60
-0.3
Figure 5-5: OTDF contour, circumferential clocking of the hot streak
pressure.
5.2.3
Unsteady Phase Averaging
The unsteady calculations utilize phase-lag boundary conditions at the circumferential
bounds of the computational domain and retain the first seven temporal harmonics of
the solution. The flow is periodic over one-half a revolution and, due to computational
restrictions, a method to reduce the time period used in the time-averaging process
is necessary.
The isentropic efficiency for a turbine is defined as
ht, 1 - ht,2(54
ht, 1 - ht,2sl
,(5.4) r/
=
where ht,2 , is the ideal exit enthalpy for an isentropic process occurring between the
inlet stagnation pressure and temperature and exit stagnation pressure of the turbine.
Station 1 is the inlet and station 2 is the exit. Figure 5-6 shows an evaluation of
Equation 5.4 in the time domain for a turbine where OTDF = 0.4. Figure 5-6 shows
61
a full revolution of the turbine with red and blue lines representing separate half
revolutions to convey the temporal convergence. The unsteady efficiency is compared
to the mixing plane efficiency (77steady) for the same turbine.
32-
0-
-2-3
0
0.1
0.2
0.3
Revolutions
0.4
0.5
Figure 5-6: Instantaneous efficiency over two cycles, OTDF = 0.4
The unsteady efficiency in Figure 5-6 has a dominant mode with frequency, 18/rev,
that scales with OTDF. This oscillation is caused by unsteady chopping of the hot
streak by the blade and a subsequent oscillation in T, 2 as the hot streak exits the
computational domain. The black line in Figure 5-6 is a moving average with an
averaging period of 27r/18Q where 27r/Q is the period of revolution and Q is the
rotational frequency. The moving average removes the hot streak induced amplitude
but retains the time averaged mean. The moving average (black line in Figure 5-6)
has a low frequency oscillation (27r/4 Q) and an amplitude of 0.15%.
A power spectrum of an instantaneous evaluation of Equation 5.4 is given in Figure
5-7. The peaks describe the amplitude of the unsteady modes in Figure 5-6. Figure
5-7 shows that the dominant frequencies in the efficiency are multiples of the vane
count, the hot streak count, and at a lower magnitude, a low frequency harmonic
with four cycles per revolution.
The time averaging methodology is now described. Time averaging of quantity
62
1.5
RTDF = 0.2
= 0.1
RTDF = 0.0
_RTDF
0.5
0
0
0.5
3
2.5
2
1.5
1
N,
2irf
/Q
Vane Frequency
3.5
4
Figure 5-7: Power spectrum of instantaneous efficiency
A (t) where A(t) is periodic over time interval T is defined as
(5.5
)
A (t) dt.
A' (t ) dt' =
A= lin
0
0
The proposed averaging method reduces T from one half of a revolution to one sixth
of a revolution. As shown in Figure 5-6 the quantity is filtered with a moving average
over period 27r/18 Q , the hot streak period. The resulting moving averaged efficiency
has an amplitude of 0.15% over a periodicity of 2 7r/4 Q. Finally, a sinusoidal regression of the moving averaged efficiency yields the time-averaged value of the original
quantity.
5.3
Generalized Efficiency Definition for Unsteady
Non-Uniform Flows
At a snapshot in an unsteady flow, enthalpy is not conserved between the inlet and
exit; there are changes in energy (unsteady energy equation has dE/dt) within the
control volume. Over time period, T, the unsteady term time averages to zero and
63
the power and ideal power can be expressed as a function of the inlet and exit fluxes.
The time averaged power extracted by the turbine over the period is defined by
Equation 5.6.
Power =
(Th TM - T2 rM) dt
(5.6)
If the exit pressure is approximately constant, the ideal power can be defined by
an isentropic expansion from the inlet stagnation temperature and pressure to the exit
stagnation pressure. This is shown in Equation 5.7 where wa denotes work averaging
and X denotes mixed out averaging.
T
Cp
Ideal Power
M
(
T
T T,
1
X
t,2
dt
(5.7)
Work average stagnation pressure, pfa, has been presented by Cumptsy and Horlock [2] who defined it as
p
f
[
a
A(T
_d.
(5.8)
The work averaged stagnation pressure conserves work output across the stage
between the non-uniform flow and the average. The mixed out average represents
the state of the flow after fully mixing out to a uniform state in a constant area,
adiabatic, and frictionless duct.
The time averaged efficiency definition based on Equations 5.6 and 5.7 is then the
power divided by the ideal power.
f
1h
T
rh TAI TMr,)
-1 2T
T
wa
dt(59
dtdt
The two conclusions from Equation 5.9 are the use of the work average and the
mixed out average in the stagnation pressure ratio. The definition in this equation is
64
referred to as the mixed-work efficiency. The definition is essentially a work-averaged
efficiency with the exit state defined as a mixed out condition so that exit length of the
domain is removed from the sensitivity, i.e. the turbine is penalized for downstream
mixing.
5.4
Effect of Tip Gap and Hot Streak on Efficiency
The drop in efficiency of a turbine when subjected to a hot streak is quantified by
rq(OTDF) - 7, where q, is the efficiency with uniform inlet stagnation temperature
(OTDF = 0.0). Figure 5-8, which utilizes the efficiency definition in Equation 5.9,
shows a comparison of the efficiency for all the cases listed in Table 5.2. The y-axis
represents the change in efficiency from the value at OTDF = 0.0. In the range of
OTDF = 0.2 to OTDF = 0.6, the turbine with no tip gap is 2.5 times more sensitive
to inlet hot streak strength and decreases by approximately 3% per OTDF.
0.5
0
-0.5
IN
1.5 --
TG-O% Steady
TG-2% Steady
-
-
-
TG-2% Unsteady
-2
0
0.4
0.2
0.6
OTDF
Figure 5-8: Hot streak sensitivity for TG-0% and TG-2%
Figure 5-8 indicates that there is only a small difference in the hot streak sensitivity
between unsteady CFD and steady CFD with a mixing plane. Additionally, the stage
efficiency increases by approximately 0.3 points.
65
Figure 5-8 confirms that stage efficiency decreases with increasing hot streak
strength and it also presents a new finding; the turbine with no tip clearance has
a higher hot streak sensitivity than the turbine with a 2% tip clearance.
Chapter
6 examines this result and highlights tip leakage mixing losses as the cause for the
difference in hot streak sensitivity.
66
Chapter 6
Hot Streak Loss Mechanisms and
Utility of Entropy as a Turbine Loss
Metric
6.1
Introduction
This chapter discusses two mechanisms that create the different hot streak sensitivities
for the turbines with and without tip clearance. The topic of entropy generation as
a metric for lost work and the choice of an appropriate efficiency is also addressed.
" The choice of a mixed-work efficiency is compared to various other definitions.
* The difference in hot streak sensitivity is linked to a decrease in tip leakage
mass flow and associated tip leakage mixing losses. This is due to reduced
blade loading near the tip.
" An efficient turbine will decrease in efficiency from an entropy generation source
more than an inefficient turbine because of the nonlinear relationship between
efficiency and loss generation. This effect accounts for 15% of the hot streak
sensitivity difference between geometries.
67
6.2
Spatial Averaging and Efficiency
In this section we describe the effect of efficiency definition on computed efficiency
and the use of viscous mixing losses as a measure for lost work.
For a one-dimensional, adiabatic, steady control volume, the second law of thermodynamics takes the form
r si - ms
2
+
(6.1)
irr = 0.
The term, Sirr, is the control volume integral of local volumetric entropy generation
and represents the overall entropy generation within control volume V.
J
Sirr =
(6.2)
dV
V
In Equation 6.1 stations 1 and 2 here denote the inlet and exit of the turbine
respectively. For an ideal gas, the entropy at the exit may be related to the stagnation
temperature and stagnation pressure at the outlet and a reference state by
= cln
\
t,ef
R In
/
(
\
2
.
(6.3)
Pt,ref
/
2
When Sirr = 0, s1 = S2. For a given exit and reference stagnation pressure, the
inlet entropy may be expressed as a function of the exit pressure and an isentropic
exit temperature, T, 2,.
Si
= cP ln
(
Tt,2)
R In (Pt'2
(6.4)
Combining Equations 6.1, 6.3, and 6.4 the reference state cancels out as shown in
Equation 6.5 which is an expression for the isentropic exit temperature of the control
volume. Equation 6.5 expresses the minimum stagnation temperature at the outlet
of the control volume for an isentropic process given the calculated exit temperature
from CFD and the total entropy generation in the control volume. The expression
relates the non-isentropic process with nonzero
68
Sirr
and exit temperature, T, 2 , to a
representative isentropic process with zero
Tt,2= T,
2
Sirr
and exit temperature, Tt,2,-
- .
exp
(6.5)
-
m cp
Equations 6.4 and 6.3 may also be equated
(Si = S2)
to express the isentropic exit
total temperature as an ideal expansion from the inlet stagnation temperature and
pressure to the outlet stagnation pressure.
T
= T,
-,2,
(6.6)
1
\Pt'iJ
In conclusion, the isentropic exit temperature may be represented as a function
of either entropy generation in the domain or the expansion ratio and inlet temperature.
These two definitions are equivalent for the one-dimensional case.
For
multi-dimensional flow, proper averaging of the inlet and outlet must be employed.
The following section addresses averaging methods, conserved quantities, and the
effect of efficiency definition on the calculated hot streak sensitivity.
6.2.1
Averaging Non-Uniform Flow
We now address different averaging methods for use in Equation 6.6 and quantify the
effect of averaging definition on calculated hot streak sensitivity.
Mass Averaging
Although often used in averaging stagnation pressure, this use has no physical basis
for flows where stagnation temperature is not uniform.
Mass averaging stagnation
temperature flux conserves stagnation enthalpy and is defined as
Tm =-J
T di
A
denoted by a M superscript.
69
(6.7)
Mixed Out Averaging
Mixed out averaging describes a process where the flow is regarded as undergoing a
mixing process to a uniform state. In this thesis, the term mixed out denotes the
process of mixing out in a constant area duct with no wall shear stress and no heat
addition. Conservation of mass, momentum, and energy on a control volume basis
uniquely define the exit mixed out conditions. This averaging process penalizes the
turbine for all downstream entropy generation despite the potential for attenuation
by downstream components. Although mixed-out averaging at the exit provides a
lower bound on efficiency magnitude, hot streak sensitivity is a measure of changes
in a efficiency and not efficiency magnitude.
It is sometimes the practice to use a mixed out average for both inlet and outlet
in the efficiency definition in Equation 6.6. If the inlet has uniform stagnation pressure and non-uniform stagnation temperature, the mixed out inlet will have a lower
stagnation pressure than the actual uniform value. Using a mixed out average at the
inlet implies the turbine cannot take advantage of work lost through the inlet mixing
process. This lost work, scales with inlet OTDF and reduces the calculated hot streak
sensitivity.
Work Averaging
Work averaging defines an average stagnation pressure and stagnation temperature
that provide the same work output as a non-uniform flow exiting at a specified uniform exit stagnation pressure. The appropriate average stagnation temperature is
the mass-average temperature to conserve enthalpy. The work averaged stagnation
pressure is specified in Equation 6.8
Pt
[2].
L
(6.8)
fATtdm
-- 1
fATtIpC dm _
Conceptually, work-averaging denotes a process where the flow is separated into
differential streams that are isentropically expanded to a pressure,
70
pa,
such that
the mass average total temperature of the two states are equal. In other words the
work output from isentropic expansion of the high pressure differential streams is
equal in magnitude to the work input from isentropic compression of the low pressure
differential streams.
Availability Averaging
Availability averaging (denoted by aa) conserves both availability and enthalpy flux
of the non-uniform flow, and hence conserves entropy flux. The definition is given in
Equation 6.9, in which T" is the mass average total temperature.
Ptp" = exp
--
11
In
ln pt
Pt-dm-drIii
T
In (h)
dh
(6.9)
.
_ A
7y
For a uniform stagnation pressure, the availability averaged stagnation pressure
may be larger than the uniform value if the stagnation temperature is non-uniform.
The additional stagnation pressure represents the work potential for a reversible
Carnot cycle that operates between the temperature difference at the inlet.
6.2.2
Effect of Efficiency Definition on Computed Turbine Stage
Sensitivity
The following addresses the choice of definition for T2,,, the enthalpy associated with
an isentropic process occurring through the turbine. Several definitions of Tt, 2, are
used to evaluate the variation of efficiency with respect to the ideal exit temperature
definition. The importance of conserved quantities and appropriate efficiency definition is highlighted and the work average efficiency definition leads to examination of
viscous dissipation in the turbine blade row.
Table 6.1 presents four definitions using the preceding averages. The inlet stagnation pressure is uniform for the problem addressed and thus ptj= p'. The entropy
efficiency is defined in Equation 6.5. The mixed-work as defined in Equation 5.9 is
the recommended metric.
71
IT,2s
Mass
Availability
Entropy
Mixed-Work
T, 1 (P" /Pt,1)
T, I(pt"/Pa)
T, 2 exp[-S/Th cp]
)- 1/T, 1 (p/p
Table 6.1: Isentropic expansion temperature definitions
Evaluation of the efficiency for the two geometries is shown in Figure 6-1. The
figure shows the effect of T, 2, definition as given in Table 6.1 on the computed hot
streak sensitivity for both turbine configurations.
0
-
-2
-2
-3 -
Mass
_4 .Entropy
Availability
Mixed-Work
-5 -
TGO%
-6
--
0
-
TG2%
0.4
0.2
0.6
OTDF
Figure 6-1: Hot streak sensitivity as a function of efficiency definition
In Figure 6-1, the x-axis represents hot streak strength and the y-axis represents
changes in efficiency with respect to the baseline case with uniform inlet stagnation
conditions. Solid lines are for TG-0% and dashed lines are for TG-2% with color
denoting the averaging used.
Figure 6-1 shows that the availability and entropy
definitions are approximately equivalent and yield the highest hot streak sensitivities.
The availability definition preserves entropy flux and for the case of adiabatic walls,
is equal to the entropy definition. The entropy definition includes both the viscous
entropy generation and thermal entropy generation components.
72
It will be seen that only viscous mixing losses should be analyzed in turbine
component design. The entropy and availability definitions thus have a disconnect
with effects a designer has control over. The mass average efficiency under predicts
the sensitivity by approximately 32%.
The mixed-work efficiency is advantageous as it utilizes both a work averaged inlet
and a mixed out average at the exit and thus conserves work extraction across the
stage. This definition follows on from the definition developed by Miller [14] but is
altered where the exit plane is defined as fully mixed out.
6.3
6.3.1
Results
Entropy Generation Rate
The entropy production rate can be decomposed into two separate parts, thermal
dissipation (i.e.
heat transfer across a finite temperature difference) and viscous
dissipation. Both represent lost work, with the thermal mixing representing work that
could be achieved by Carnot cycles operating between a finite temperature difference.
The two parts are given in Equations 6.10 and 6.11 expressed as entropy generation
rates per unit volume. In Equation 6.10, v denotes a process related to viscosity and
pteff is the effective viscosity from the SA turbulence model. In Equation 6.11, th
denotes thermal, k is the thermal conductivity, and pt is the turbulent viscosity from
the SA turbulence model.
F
2
/
8W
Ou
+
OZ
OX
th
k+cp40.9
T2
2
\2
+
+
82
09z
W
+
2
Oy
OTT2 + OT T2 T 2
Ox
73
\2
/
Oy
2
Ou
3
Ox
Ow
Ov
+
OZ
Oy
2
(6.10)
+
Oz)
The entropy production rates are locally defined but Equation 6.2 is used to relate
the rate of local entropy generation within the control volume to the total rate of
entropy generation.
The total entropy production can be further separated into control volumes for
vane and rotor domains. The dividing line is defined as the inter-stage plane halfway
between the vane trailing edge and the rotor leading edge. The total entropy production in each control volume is shown as a function of OTDF in Figure 6-2 and
6-3. The y-axis represents changes in the entropy generation rate with respect to the
baseline OTDF = 0.0 case.
-
0.06
0.05
-
,1 0.04-
-
TG-0% Vane
-
TG-2% Vane
TG-0% Rotor
TG-2% Rotor
-
0.03
U
0.02-
0.01
0
-0.01
0
0.4
0.2
0.6
OTDF
Figure 6-2: Viscous entropy generation rate within control volume
The work averaged efficiency definition may be written as a function of the viscous
dissipation in the control volume as shown in Equation 6.12
[14].
There exists a
thermal creation term in the denominator of Equation 6.12 that has been identified
by Miller [141 but it is shown that this term is of negligible importance for the problem
of interest here.
Power
Power + fff vdV
74
(6.12)
(6.12
0.060.05
-
-
-
-
0.04 .
TG-0% Vane
TG-2% Vane
TG-0% Rotor
TG-2% Rotor
/
0.03
-
0.02
-
0.01
E4
0
-0.01
0
0.4
0.2
0.6
OTDF
Figure 6-3: Thermal entropy generation rate within control volume
The thermal mixing term, whose magnitude is given by Figure 6-3, does not appear in
the denominator of Equation 6.12. When performing component design, the viscous
mixing losses can be altered by changes in the design whereas the thermal mixing of
the inlet temperature gradients is seen as an inevitable loss. This topic is still being
debated in the literature
119] and requires further inquiring, but that is beyond the
scope of this thesis. When examining turbine efficiency in this thesis we thus use the
definition of Equation 6.12 and only viscous mixing losses are included.
The y-axis in Figure 6-2 thus shows the changes in the denominator of Equation
6.12, representing either increases or decreases in lost work. The increase seen in the
viscous terms as a function of OTDF is directly linked to the hot streak sensitivity.
Using Equation 6.12, the hot streak sensitivity can be evaluated by examining
changes in S,. Figure 6-2 shows that generation of viscous losses in the turbine rotor
decrease in magnitude for TG-2% relative to TG-0%.
As the hot streak strength
increases, associated viscous mixing losses increase in the passage but there is reduced viscous mixing losses for the case with a finite tip gap.
This reduction in
viscous mixing losses describes the difference in hot streak sensitivity between the
two geometries.
75
6.3.2
Blade Incidence Angle
At the inlet to the computational domain, the endwalls are parallel (see Figure 5-2)
and the inlet boundary condition specifies parallel flow. For a constant stagnation
pressure inlet with parallel streamlines, the Mach number will also be uniform. The
axial velocity is thus
Ux = M V R T = .1 V/' R T f (- , M/).
Ux cX/
(6.13)
(6.14)
At the inlet, the hot streak has a higher axial velocity than the surrounding cold
fluid. At the vane exit, the Mach number is approximately the same for the hot and
cold streams so the velocity difference will increase. At the interstage plane there
is thus a velocity non-uniformity with approximately aligned flow in the stationary
reference frame. However, in the relative reference frame, the hot streak has increased
incidence seen by the rotor. The loading will be reduced for regions where the fluid
is colder than the mass averaged value. Figure 6-4 shows the rotor incidence angle
for OTDF cases of 0.0 and 0.6 for the 2% configuration.
At 95% span, where the fluid is colder than the mass average, the incidence angle
is reduced by 140 for OTDF = 0.6. This reduces loading of the rotor at the hub and
tip. The static pressure distribution for TG-2% at 95% span is shown in Figure 6-5
for the same two OTDF values as in Figure 6-4. A reduction in the loading of the
blade tip region can be seen which is due to the hot streak.
6.3.3
Tip Leakage Massflow
We now examine the tip leakage flow in the TG-2% geometry. A reduction in tip
loading (pressure difference) implies a reduction in tip leakage mass flow rate and tip
leakage losses. Figure 6-6 shows the tip leakage flow as a fraction of the main stream
flow rate versus OTDF. There is a reduction in physical tip leakage mass flow by
76
I
-
0.8
-
0.6
-
0.4
0.2
-
20
OTDF = 0.0, TG-2%
OTDF = 0.6, TG-2%
-
25
30
35
40
45
50
55
Relative Flow Angle [deg]
Figure 6-4: Interstage rotor relative incidence angle
10.5% as OTDF goes from 0.0 to 0.6.
6.3.4
Tip Gap Loss Scaling
We now examine the different loss generation terms and their scaling with OTDF
and tip gap. Tip gap loss (also referred to as Gap in Figure 6-7) in this context is
defined as the entropy generation that occurs within the tip gap. The tip leakage flow
mixing loss is defined as the loss incurred by mixing of the tip leakage flow with the
mainstream flow.
Figure 6-7 from Huang
I7]
presents a decomposition of the entropy generation
terms for a similar turbine design with tip gaps of 0% and 2%.
y-axis,
In Figure 6-7 the
, is defined as
Tt,2 As
,(6.15)
Aht (
=
'
a non-dimensional loss metric. The rotor geometry in this thesis and Huang's is the
same from 0% to 50%. Figure 6-7 indicates the order of magnitude of the different
loss components for a similar turbine.
The primary differences between the two
77
0.7
0.75
0.5
-
-
OTDF =0.0
OTDF= 0.6
0.25
0
0.2
0.4
0.6
0.8
1
X/cX
Figure 6-5: Pressure contour at 95% span, TG-2%
geometries in Figure 6-7 is the existence of a tip gap loss and a more importantly,
a loss associated with the tip leakage flow mixing with the mainstream in the upper
half of the span. The latter effect accounts for the majority of the difference seen in
the total viscous loss.
Figure 6-8 shows the scaling of losses within the tip gap with respect to OTDF
for the TG-2% geometry defined in this thesis. Figure 6-8 shows that losses in the
tip gap reduce by 11.5% at an OTDF of 0.6, a result of the 10.5% reduction in tip
mass flow in Figure 6-6. More importantly, Figure 6-7, shows that the contribution
of tip gap losses is small compared to mixing losses in the upper half span. In the
TG-2% configuration, tip gap losses contribute just 12% of the overall loss. Our focus
is thus on the tip flow leakage mixing loss which represents a larger influence on stage
efficiency and hot streak sensitivity.
Denton describes a control volume model for mixing out of tip leakage flow in a
mainstream flow
[3] which is applicable when pressure gradients in the mainstream are
small. A scaling conclusion from the control volume analysis is that the mixing losses
scale with the non-dimensionalized mass flow rate of the tip leakage flow, ritip/rmain.
As shown in Figure 6-7, the difference in entropy generation between TG-0% and
78
5.4
5.2-
5 --
4.8
4.6
0
0.2
0.6
0.4
OTDF
Figure 6-6: Tip leakage mass flow rate reduction with OTDF
TG-2% is attributed to losses imposed by the finite tip gap.
We can scale the tip
leakage flow mixing losses by mhtip/mmrnain as an estimate of change in tip leakage
mixing losses.
Figure 6-9 shows the variation in efficiency with OTDF using the
scaling for tip leakage flow mixing losses and the calculated tip gap losses. Figure 6-9
shows modifications to the TG-2% efficiency curve to approximate the TG-0% curve.
Correction 1 denotes the removal of the tip leakage mixing loss calculated using the
Denton scaling argument. Correction 2 denotes the removal of both the tip leakage
mixing loss and the calculated tip gap losses. Figure 6-9 confirms the hypothesis that
losses in the tip gap have a small contribution.
The tip gap losses and tip leakage
mixing losses captures 82% of the hot streak sensitivity difference seen between the
two geometries, a 205% increase in hot streak sensitivity from TG-2% to TG-0%.
6.3.5
Efficiency Sensitivity to Inlet Hot Streak
The hot streak sensitivity is different for the two turbine geometries due to different
amounts of viscous entropy production in the two geometries, but the efficiency is
also non-linearly related to entropy generation and thus the hot streak sensitivity
79
0.12
0.1
0.08
4 0.06
00% clearance
M2% clearance
0.04
--
0.02
0
Total Viscous
Gap
Lower Half
Upper Half
Freestream
Freestream
Blade + Hub
Loss
Figure 6-7: Turbine stage loss components
varies with the reference efficiency, r,
of the turbine. The efficiency is defined as a
function of non-dimensional entropy generation, T, 2 S/rhAht, in Equation 6.16. We
wish to quantify the sensitivity of rT to entropy production at different values of S
and describe how the efficiency will change with an additional loss source, A5..
1
1
r7'
(6.16)
1+ Tt,2 sv
h Aht
If we assume that the viscous loss scaling is independent of tip clearance, we
can determine a relationship between the efficiency and the entropy generation when
OTDF = 0.0.
In other words, we can view the inlet hot streak as a perturbation
from the baseline flow field and quantify the stage sensitivity for perturbations with
both TG-0% and TG-2%. Equation 6.17 decomposes the entropy generation into an
So term that represents the entropy generation for a given geometry at OTDF = 0.0
and a perturbation in viscous mixing losses due to by the inlet hot streak, Alc.
=
1 + STv
0 +ASv
Aht
(6.17)
Taking the first derivative with respect to Al5, evaluated at OTDF = 0.0, we find
80
U
-5
-10
-15
0.6
0.4
0.2
0
OTDF
Figure 6-8: Entropy generation in the tip gap, TG-2%
1cd
(A,) OTDF=O
(
m
O
2.(6.18)
1)
Equation 6.18 states that the sensitivity of turbine efficiency to a loss perturbation
scales with 1/502, so a turbine stage with a higher efficiency is more sensitive to
increases in viscous dissipation than a stage with a lower efficiency.
Figure 6-10 shows a curve of efficiency,
ij,
T4Aht
eration, T, 2 S/rhAht, based on Equation 6.16.
versus non-dimensional entropy genThe two curves include the linear
assumption for high efficiency turbines and the efficiency definition in Equation 6.16.
The slope of the curve in Figure 6-10 shows the change in efficiency for a given change
in entropy generation. Figure 6-10 shows that a more efficient stage will change efficiency more than an inefficient stage.
For the two turbines in this study, the difference in ro and thus efficiency sensitivity
to perturbations in viscous dissipation accounts for 15% of the difference in hot streak
sensitivity between TG-0% and TG-2%. This effect is relatively small and thus the
reduction in tip leakage mixing losses associated with blade loading is identified as
the dominant mechanism.
81
0
-4
-0.5
-1
N
1.5
-
- -
-
- -
N
N
TG-O%
TG-2%
--
-2
~.
'N
N' 'N
NN.N
N
TG-2% Correction 1
TG-2% Correction 2
0
NN
N
N
N
N
0.6
0.4
0.2
OTDF
Figure 6-9: Efficiency with control volume model correction
-
90
-
1/(1 + T, 2 $/Aht)
1-T, 2 S/Ah
.
100
80
70
60
50
0
0.25
0.75
0.5
-T,
2
S/IT
1
Ah
Figure 6-10: Efficiency sensitivity to entropy production
82
Chapter 7
Summary, Conclusions, and
Recommendations for Future Work
Summary and Conclusions
7.1
Turbine stage performance dependence on inlet non-uniformities has been assessed
through two model problems, upstream wake attenuation and hot streak effects on
turbine efficiency. A combination of two-dimensional and three-dimensional, RANS
and URANS, calculations have been used.
Reversible Wake Attenuation
7.1.1
" An upstream wake was represented as vorticity convected by a steady background flow. The geometric distortion of the wake, Kelvin's theorem, and a ID
distribution of infinitesimal control volumes along the wake was used to estimate the evolution of wake velocity defect in the turbine passage. Reversible
wake attenuation of the 46% was seen.
" The process of wake attenuation in a turbine is different than for a compressor
because, velocity gradients normal to the wake orientation deform the wake,
i.e. wake segments do not remain straight. An inviscid attenuation of 74% was
demonstrated for a representative geometry with unsteady CFD. It was also
83
shown that the wake can increase in stagnation pressure above the freestream
near the pressure surface if the wake is turned such that the velocity defect
vector points downstream.
e Two-dimensional unsteady calculations identify two wake attenuation mechanisms.
Bowing and turning of the segments can reorient the velocity defect
from pointing upstream to downstream, decreasing or increasing the wake flowthrough time compared to freestream.
Wake segments in a turbine are not
purely stretched or compressed as can be approximated for a compressor. The
unsteady pressure field in the relative coordinate system can also provide work
extraction from the wake relative to the freestream in the passage.
7.1.2
Effects of Hot Streaks on Turbine Stage Efficiency
* Steady three-dimensional RANS calculations have shown that blade loading
near the tip and thus tip leakage flow rate, are the primary cause for different
hot streak sensitivity between a turbine geometry with a finite tip gap and one
with no tip gap.
" A numerical example is used to convey the importance of efficiency definition
for both steady and unsteady CFD with non-uniform inlets.
" The calculated efficiency sensitivity to inlet hot streak for a rotor with a tip
clearance is similar for steady CFD with a mixing plane and unsteady CFD
case with phase-lag boundary conditions.
* For the turbine geometry examined, a turbine rotor with a 2% clearance is
roughly twice as sensitive to hot streaks than the same rotor with no tip gap.
The difference in hot streak sensitivity is attributed to a reduction in tip leakage
mass flow and thus tip leakage mixing losses, compared to the rotor operating
without a hot streak.
" A turbine performance efficiency definition has been proposed for evaluating
turbine sensitivity to hot streaks.
The efficiency definition utilizes a work84
averaged inlet conditions and mixed out exit conditions to conserve enthalpy
flux and work output between the original three-dimensional unsteady flow field
and the averages.
7.2
Recommendations for Future Work
(i) A next step in the second research problem could be to augment the blade
count to permit sliding plane unsteady calculations of a similar geometry, i.e.
to change the rotor and vane count from 58 and 36 respectively to 54 and 36.
The geometry change reduces flow periodicity from one-half a revolution to oneeighteenth. The reduced solution time permits time averaging of flow parameters
(e.g entropy generation and inlet flow angles) and further investigation of how
well mixing-plane calculations capture the unsteady three dimensional flow field.
(ii) The effect of an asymmetric vane row may also be evaluated.
A proposed
design could include two vanes, one designed for hot streak impingement and
one designed for the neighbor vane without a hot streak. The spanwise turning
angle distribution could be designed such that the turning of the hot streak
aligns the relative flow angles exiting the vane row for vane with hot streak and
its neighboring vane without a hot streak. This reduces the unsteadiness of
the flow entering the rotor and permits a single point optimization of the rotor
rather than a robust design for the periodic alternating flow angle currently seen
at the rotor inlet.
(iii) The effect of swirl and corresponding radial migration should be analyzed.
Swirling flow at the inlet may be used to cause radial migrations towards the
hub in the vain and a reduction in hot streak ingestion at the rotor tip.
(iv) Finally, the convected wake analysis in a steady flow can be augmented with
a one-dimensional momentum equation to predict unsteady pressure variations
within the wake. Currently the model assumes that the pressure in the wake
is the same as the pressure in the freestream. A one-dimensional momentum
85
equation may be included in the model which uses the velocity defect vector
variation and the freestream pressure to determine &p/&t in the wake. If the
model agrees with the unsteady calculations, the parameterization of reversible
wake attenuation can be used for design.
86
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