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Görtler Instability
Contents:
1.
2.
3.
The eldest unsolved linear-stability problem
Modern approach to Görtler instability
Properties of steady and unsteady Görtler vortices
Shorten variant of an original lecture by Yury S. Kachanov
1. The Eldest Unsolved LinearStability Problem
Why Is the Görtler Instability
So Important?
•
Görtler instability may occur in flows near curved walls and
lead to amplification of streamwise vortices, which are able to
result in:
•
(i) the laminar-turbulent transition,
•
(ii) the enhancement of heat and mass fluxes,
•
(iii) strong change of viscous drag
•
(iii) other changes important for aerodynamics
Görtler Instability on Curved Walls.
When Does It Occur?
Stable
Sketch of Steady
TheGörtler
necessary
and
vortices
sufficient condition
for the flow to be stable is:
(i) d(U2)/dy < 0 for concave wall
or
(ii) d(U2)/dy > 0 for convex wall.
Floryan (1986)
Otherwise the instability may occur
Floryan (1991)
Görtler (1956)
Why Does Görtler Instability Appear?
Governing parameter
is Görtler number
As far as
then
R(y≥d)
That is why curvature of streamlines is
always greater inside boundary layer
than outside of it
This is similar to unstable stratification
(a buoyancy force), which leads to
appearance of Görtler instability!
d
R(y<d)
Fs
Linear Stability Diagrams
and Measurements
Neutral curve
Neutral curve
Standard representation: (G,b)-plane
Representation convenient
in experiment: (G,L)-plane
Floryan & Saric (1982)
Linear Stability Diagrams
and Measurements
In
other
words,
(1984)
conclude
modal
approach
in invalid
Hall
(1984)
hasHall
made
conclusion
thatthat
neutral
curve
does not
exist forfor
b ≤these
O(1)b
Görtler (1941)
Hämmerlin (1955b)
Growing
vortices
Hämmerlin (1961)
Schultz-Grunow (1973)
Decaying
vortices
Kabawita & Meroney (1973-77)
Smith (1955)
Hämmerlin (1955a)
Floryan & Saric (1982)
b
Experimental check of right branch
of the neutral stability curve
Experiments by Bippes (1972)
Kabawita & Meroney (1973-77)
b
Left branch of the neutral curve
obtained from different versions of
linear stability theory
After Herbert (1976) and Floryan & Saric (1982)
Amplification of Görtler Vortices
• Any attempts (until recently) to find at least one
figure showing direct comparison of measured
Comparison
of Experimental
Amplification
amplification
curves
with linear
theory Curves
of Görtler
instability failed!!!
for Görtler Vortex Amplitudes
• No quantitative agreement between experiment and
with the Linear Stability Theory
linear stability theory was obtained for disturbance
growth rates!
• “Theoretical growth rates obtained for the
experimental conditions were much higher than the
measured growth rates” (Finnis & Brown, 1997)
2. Modern Approach to Görtler
Instability
Amplification of Görtler Vortices
• Thus, by the beginning of the present century the problem
of linear Görtler instability remained unsolved (after
almost 70 years of studies) even for
the classic case of Blasius boundary layer!
• Whereas other similar problems (like TollmienSchlichting instability, cross-flow instability, etc.) have
been solved successfully
Why Does This Problem Occur?
• Very poor accuracy of measurements at zero frequency of
perturbations (perhaps ±several%)
• Researchers were forced to work at very large amplitudes (10%
and more) resulted in nonlinearities
• Near-field effects of disturbance source (transient growth, etc.)
were not taken into account properly in the most of cases
• Meanwhile, there effects (i.e. the influence of initial spectrum, or
shape of disturbances) are very important for Görtler instability
(because ar = 0 for steady vortices)
• Range of validity of Hall’s conclusion on non-applicability of the
eigenvalue problem (i.e. on infinite length of the disturbance source
near-field) remained unclear
Steady and Unsteady Vortices
• Almost all previous studies were devoted to
steady Görtler vortices, despite the unsteady ones
are often observed in real flows
• Unsteady Görtler vortices seem to dominate at
enhanced free-stream turbulence levels, e.g. on
turbine blades
Main Fresh Ideas
1.
2. To
To measure
investigate
everything
essentially
accurately
unsteady Görtler
vortices
for steady
case
important for practical
How?
applications
To tune-off from the zero disturbance frequency
and to work with quasi-steady Görtler vortices
instead of exactly steady ones
Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)
What Is Quasi-Steady?
Periodof vortex oscillation >> Timeof flow over model
or
X-wavelengthof vortex >> X-sizeof exper. model
E.g. for f = 0.5 Hz, U = 10 m/s, L = 1 m
Periodof vortex oscillation = 2 sec
Timeof flow over model = 0.1 sec
Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)
Goals
• To develop experimental and theoretical approaches to
investigation of unsteady Görtler vortices (including
quasi-steady ones)
• To investigate experimentally and theoretically all main
stability characteristics of a boundary layer on a concave
surface with respect to such vortices
• To perform a detail quantitative comparison of
experimental and theoretical data on the boundary-layer
instability to unsteady (in general) Görtler vortices
Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)
Wind-Tunnel T-324
Experiments
are conducted at:
Fan is there
Free-stream speed
Ue = 9.18 m/s
and
Free-stream
turbulence level
e = 0.02%
Measurements are
performed with
a hot-wire
anemometer
Test
section
Settling
chamber
Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)
Experimental Model
(1) – wind-tunnel test-section wall, (2) – plate, (3) – peace of concave surface with radius
of curvature of 8.37 м, (4) – wall bump, (5) – traverse, (6) – flap, (7) – disturbance source.
Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)
Experimental Model
Adjustable
Wall Bump
Traversing
mechanism
Disturbance
source
Test-plate with the concave insert, adjustable wall bump, and traverse
installed in the wind-tunnel test section
Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)
Boundary Layer
1.2
1
U/Ue
0.8
x = 700 мм
x = 700 мм
x = 900 мм
x = 900 мм
Блазиус
Blasius
0.6
0.4
0.2
0
0
1
2
3
4
5
6
y/d 1
Measured mean velocity profiles
and comparison with theoretical one
Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)
Ranges of Measurements
on Stability Diagrams
Tollmien-Schlichting
mode
First mode
of Görtler instability
f = 0 Hz
Floryan and Saric (1982)
f = 20 Hz
Boiko et al. (2005-2007)
Disturbance Source
U0
Undisturbed flow
кtoдинамикам
speakers
The measurements were performed in 22 main regimes of disturbances
excitation in frequency range from 0.5 and 20 Hz for three values of
spanwise wavelength: lz = 8, 12, and 24 mm
Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)
Excited Initial Disturbances
0.06
0.6
%
A1, м/c
360
A1_norm,
Exper. V
Approx.
Series2
270
j , deg
0.04
0.4
180
0.02
0.2
90
0.0
0
Fi1_corr,
Exper.deg
Approx.
Series2
0
440 444 448 452 456 460 464 468 472 476 480 484
z, mm
440 444 448 452 456 460 464 468 472 476 480 484
z, mm
Spanwise distributions of disturbance amplitude and phase in one of regimes
lz = 24 mm, f = 11 Hz, x = 400 mm.
Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)
Spectra of Eigenmodes of Unsteady
Görtler-Instability Problem
F = 0.57
F = 9.08
F = 22.7
1st mode of discrete
spectrum
2nd mode of discrete
spectrum
Continuous-spectrum
modes
Görtler number G = 17.3, spanwise wavelength L = 149
Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)
Wall-Normal Profiles for Different
Spectral Modes
Mean velocity
1st mode
2nd mode
U∂U/∂y
(non-modal)
1st mode
2nd mode
1st-mode critical layer
Calculations based on the locally-parallel linear stability
theory performed for G = 17.3, F = 0.57, L = 149
Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)
Disturbance-Source Near-Field.
Transient (Non-Modal) Growth
Transient (non-modal) behavior
Transient decay
in experiment
Source near-field
Disturbance
source
Transient
decay in theory
1st
Modal behavior:
discrete-spectrum
Görtler mode
Transient
growth in theory
Separation of 1st unsteady Görtler mode due to mode competition
Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)
3. Properties of Steady and
Unsteady Görtler Vortices
Evolution of Quasi-Steady and
Unsteady Görtler Vortices
Streamwise component of velocity disturbance in (x,y,t)-space
(lz = 12 mm)
Frequency f = 0,5 Hz
(a quasi-steady case)
Frequency f = 14 Hz
(an essentially unsteady case)
Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)
Shape of Quasi-Steady Görtler Vortices
(f = 2 Hz)
Ue
Ue
Experiment
Theory
Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)
Shape of Unsteady Görtler Vortices
(f = 20 Hz)
g
Ue
Ue
Experiment
Theory
Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)
Check of Linearity of the Problem
180
0.1
jdeg
1, град
A1/A1o
0.08
Ao
Ao/2
Ao
Ao/2
90
0.06
0
0.04
-90
0.02
-180
0
400
500
600
700
800 x, mm 900
400
500
600
700
800 x, mm 900
Streamwise evolution of Görtler-vortex amplitudes and phases
for two different amplitudes of excitation
Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)
Wall-Normal Disturbance Profiles
A/A1.2
max
Dependence on streamwise
coordinate, lz = 8 mm
180
90
f-f o, deg
l z = 8 mm, f = 5 Hz
Experiment
PSE
LST
1
Experiment
PSE
LST
0.8
l z = 8 mm, f = 5 Hz
0.6
0.4
x = 900
400 mm
500
600
700
800
0.2
x = 900
400 mm
500
600
700
800
0
0
0
1
2
3
4
5
-90
First mode of unsteady
Görtler instability in LST
-180
0
1
2
3
4
5
y/d 6
Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)
y/d 6
Eigenfunctions of Görtler Vortices
Dependence on frequency
for lz = 12 mm, G = 17.2
A/A1.2
max
Experiment
PSE
LST
1
0.8
l z = 12 mm, x = 900 mm
90
l z = 12 mm, x = 900 mm
f = 20.0
0.5 Hz
2.0
5.0
8.0
11.0
14.0
17.0
Hz
f-f o, deg
0
0.6
0.4
-90
0.2
-180
0
Hz
0.5 Hz
2.0
5.0
8.0
11.0
14.0
17.0
f = 20.0
0
1
2
3
4
Experiment
PSE
LST
-270
-360
0
1
2
3
4
5
y/d 6
Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)
5
y/d 6
Eigenfunctions of Görtler Vortices
Dependence on spanwise
wavelength, x = 900 mm,
G = 17.2, f = 5 Hz
A/A1.2
max
1
0.8
180
f = 5 Hz, x = 900 mm
0.6
f = 5 Hz, x = 900 mm
l z = 24.0
8.0 mm
12.0
mm
0.4
90
f-f o, deg
Experiment
PSE
LST
l z = 24.0
8.0 mm
12.0
mm
0.2
0
0
0
-90
Experiment
PSE
LST
1
2
2
3
4
5
First mode of unsteady
Görtler instability in LST
-180
0
1
3
4
5
y/d 6
Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)
y/d 6
Growth of Amplitudes and Phases
of Görtler Modes (f = 2 Hz)
10
11
15
13
1st Goertler mode in far field
Mixture of modes in near field
LST
PSE
A/A1
G
17
mm
8 mm
12
llzz==24
Phase amplification is almost
independent of the spanwise wavelength
f n/
1
1st Goertler mode in far field
Mixture of modes in near field
LST
PSE
0.6
0.4
f = 2.0 Hz
llzz==24
12
8 mm
mm
f = 2.0 Hz
0.1
390
490
590
690
790
Dependence on
spanwise wavelength
mm
x,890
0.2
0
390
490
590
690
Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)
790
x,890
mm
Growth of Amplitudes and Phases
of Görtler Modes (lz = 8 mm)
10
11
15
13
1st Goertler mode in far field
Mixture of modes in near field
LST
PSE
A/A1
G
17
l z = 8 mm
The non-local, non-parallel stability
theory (parabolic stability equations)
provides the best agreement with
experiment
f n/
1
15
13
G
17
1st Goertler mode in far field
Mixture of modes in near field
LST
PSE
3
f = 17.0
2.0 Hz
5.0
8.0
11.0
14.0
Hz
11
2
l z = 8 mm
f = 17.0
2.0 Hz
5.0
8.0
11.0
14.0
Hz
0.1
390
490
590
690
790
Dependence on frequency
x,890
mm
1
0
390
490
590
690
Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)
790
x,890
mm
Growth of Amplitudes and Phases
of Görtler Modes (lz = 12 mm)
10
11
15
13
1st Goertler mode in far field
Mixture of modes in near field
LST
PSE
A/A1
G
17
l z = 12 mm
The non-local, non-parallel stability
theory (parabolic stability equations)
provides the best agreement with
experiment
f n/
1
15
13
G
17
1st Goertler mode in far field
Mixture of modes in near field
LST
PSE
3
Hz
0.5 Hz
2.0
5.0
8.0
11.0
f = 14.0
11
2
l z = 12 mm
f = 14.0
0.5 Hz
2.0
5.0
8.0
11.0
Hz
0.1
390
490
590
690
790
Dependence on frequency
for lz = 12 mm
mm
x,890
1
0
390
490
590
690
Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)
790
x,890
mm
Frequency Dependence of Increments
and Phase Velocities of Görtler Modes
lz = 8 mm (b = 0.785 rad/mm)
1
-a i0.005
, mm-1
Сx/Ue
lz = 8 mm
0.004
l z = 8 mm
0.8
0.003
0.6
0.002
0.4
Experiment
LST
PSE
0.001
0
Experiment
LST
PSE
0.2
-0.001
0
0
5
10
15
Increments of 1st Görtler mode
at G ≈ 15
f , Hz
20
0
5
10
15
f , Hz
20
Phase velocities of 1st Görtler mode
at G ≈ 15
Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)
Frequency Dependence of Increments
and Phase Velocities of Görtler Modes
lz = 12 mm (b = 0.524 rad/mm)
1
-a i0.005
, mm-1
Сx/Ue
lz = 12 mm
0.004
l z = 12 mm
0.8
0.003
0.6
0.002
0.4
Experiment
LST
PSE
0.001
0
Experiment
LST
PSE
0.2
-0.001
0
0
5
10
15
Increments of 1st Görtler mode
at G ≈ 15
f , Hz
20
0
5
10
15
f , Hz
20
Phase velocities of 1st Görtler mode
at G ≈ 15
Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)
Frequency Evolution of Stability
Diagram for Görtler Vortices
Tollmien-Schlichting
mode
Growing disturbances (experiment)
Attenuating disturbances (experiment)
Neutral points (experiment)
Contours of increments (LPST)
Hz
0.5
14
17
20
11
2
8
5 Гц
First mode
of Görtler instability
Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)
Conclusions
• Modal approach works for Görtler instability
problem (steady and unsteady) for at least b ≥ O(1)
• Very good quantitative agreement between
experimental and theoretical linear-stability
characteristics has bee achieved now for steady
Görtler vortices (for the most dangerous 1st mode)
• Similar, very good agreement is obtained also for
unsteady Görtler vortices (again for the 1st, most
amplified, mode)
• The non-local, non-parallel theory predicts better the
most of stability characteristics (to both steady and
unsteady Görtler vortices)
Recommended Literature
1. Floryan J.M. 1991. On the Görtler instability of
boundary layers J. Aerosp. Sci. Vol. 28, pp. 235‒271.
2. Saric W.S. 1994. Görtler vortices. Ann. Rev. Fluid
Mech. Vol. 26, p. 379‒409.
3. A.V. Boiko, A.V. Ivanov, Y.S. Kachanov, D.A.
Mischenko (2010) Steady and unsteady Görtler
boundary-layer instability on concave wall. Eur. J.
Mech./B Fluids, Vol. 29, pp. 61‒83.