Multimedia files - 5/13
Görtler Instability
Contents:
1. The eldest unsolved linear-stability problem2. Modern approach to Görtler instability3. Properties of steady and unsteady Görtler vortices
Shorten variant of an original lecture by Shorten variant of an original lecture by Yury S. KachanovYury S. Kachanov
1. The Eldest Unsolved Linear-Stability Problem
• 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
Why Is the Görtler InstabilitySo Important?
Görtler Instability on Curved Walls. When Does It Occur?
The necessary and sufficient conditionThe necessary and sufficient conditionfor the flow to be for the flow to be stablestable is: is:(i) d((i) d(UU22)/d)/dyy < 0 for concave wall < 0 for concave walloror(ii) d((ii) d(UU22)/d)/dyy > 0 for convex wall. > 0 for convex wall.
Floryan (1986)Floryan (1986)
Otherwise the instability may occurOtherwise the instability may occur
Stable Sketch of SteadySketch of SteadyGörtler vorticesGörtler vortices
Floryan (1991)Floryan (1991) GGörtlerörtler (1956) (1956)
Why Does Görtler Instability Appear?
As far asAs far as
thenthen
That is why curvature of streamlines is That is why curvature of streamlines is always greater inside boundary layer always greater inside boundary layer
than outside of itthan outside of it
This is similar to unstable stratification This is similar to unstable stratification (a buoyancy force), which leads to (a buoyancy force), which leads to appearance of Gappearance of Görtler instabilityörtler instability!!
R(y≥)
R(y<) Fs
Governing parameterGoverning parameteris Gis Göörtler numberrtler number
Linear Stability Diagramsand Measurements
Floryan & Saric (1982)Floryan & Saric (1982)
Neutral curveNeutral curve
Standard representation: (Standard representation: (GG,,)-plane)-plane Representation convenientRepresentation convenientin experiment: (in experiment: (GG,,)-plane)-plane
Linear Stability Diagramsand Measurements
Experiments by Bippes (1972)Experiments by Bippes (1972)
Experimental check of right branchExperimental check of right branchof the neutral stability curveof the neutral stability curve
GrowingGrowingvorticesvortices
DecayingDecayingvorticesvortices
Left branch of the neutral curve Left branch of the neutral curve obtained from different versions of obtained from different versions of
linear stability theory linear stability theory
After Herbert (1976) and Floryan & Saric (1982)After Herbert (1976) and Floryan & Saric (1982)
Görtler (1941)
Hämmerlin (1955a)
Hämmerlin (1955b)
Smith (1955)
Hämmerlin (1961)
Schultz-Grunow (1973)
Kabawita & Meroney (1973-77)
Kabawita & Meroney (1973-77)
Floryan & Saric (1982)
Hall (1984) has made conclusion that neutral curve does not exist for Hall (1984) has made conclusion that neutral curve does not exist for ≤≤ O(1) O(1)In other words, Hall (1984) conclude that modal approach in invalid for these In other words, Hall (1984) conclude that modal approach in invalid for these
• Any attempts (until recently) to find Any attempts (until recently) to find at least one at least one figurefigure showing direct comparison of measured showing direct comparison of measured amplification curves with amplification curves with linear theorylinear theory of Görtler of Görtler instability failed!!!instability failed!!!
• No quantitative agreementNo quantitative agreement between experiment and between experiment and linear stability theory was obtained for disturbance linear stability theory was obtained for disturbance growth rates!growth rates!
• ““Theoretical growth rates obtained for the Theoretical growth rates obtained for the experimental conditions were experimental conditions were much highermuch higher than the than the measured growth rates” measured growth rates” (Finnis & Brown, 1997)(Finnis & Brown, 1997)
Amplification of Görtler Vortices
Comparison of Experimental Amplification Curves
for Görtler Vortex Amplitudes
with the Linear Stability Theory
2. Modern Approach to Görtler Instability
• Thus, by the beginning of the present century the problem of linear Görtler instability remained unsolved (after almost 70 years of studies) even forthe classic case of Blasius boundary layer!
• Whereas other similar problems (like Tollmien-Schlichting instability, cross-flow instability, etc.) have been solved successfully
Amplification of Görtler Vortices
• 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 r = 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
Why Does This Problem Occur?
• 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
Steady and Unsteady Vortices
Main Fresh Ideas
Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)
1. To measure everything accurately
How?
To tune-off from the zero disturbance frequency and to work with quasi-steady Görtler vortices instead of exactly steady ones
2. To investigate essentially unsteady Görtler vortices important for practical applications
for steady case
What Is Quasi-Steady?
Periodof vortex oscillation >> Timeof flow over model
orX-wavelengthof vortex >> X-sizeof exper. model
E.g. for f = 0.5 Hz, U = 10 m/s, L = 1 m
Periodof vortex oscillation = 2 secTimeof flow over model = 0.1 sec
Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)
• 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
Goals
Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)
Wind-Tunnel T-324
Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)
Experimentsare conducted at:
Free-stream speedUe = 9.18 m/s
and
Free-streamturbulence level
= 0.02%
Measurements are performed with
a hot-wire anemometer
Settlingchamber
Testsection
Fan is there
Experimental Model
((1)1) –– wind-tunnel test-section wallwind-tunnel test-section wall, (2), (2) –– plateplate, (3), (3) –– peace of concave surface with radius peace of concave surface with radius
of curvature ofof curvature of 8 8..37 м, (4)37 м, (4) –– wall bumpwall bump, (5), (5) –– traversetraverse, (6), (6) –– flapflap, (7), (7) –– disturbance source.disturbance source.
Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)
Experimental Model
Test-plate with the concave insert, adjustable wall bump, and Test-plate with the concave insert, adjustable wall bump, and traversetraverse
installed in the wind-tunnel test sectioninstalled in the wind-tunnel test section
Disturbancesource
Traversingmechanism
AdjustableWall Bump
Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)
Boundary Layer
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5 6
y/1
U/U
e
x = 700 мм
x = 900 мм
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5 6
y/1
U/U
e x = 700 мм
x = 900 мм
Блазиус
Measured mean velocity profilesand comparison with theoretical one
Blasius
Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)
Ranges of Measurementson Stability Diagrams
Boiko et al. (2005-2007)Boiko et al. (2005-2007)
f = 0 Hz f = 20 Hz
First modeof Görtler instability
Tollmien-Schlichting mode
Floryan and Saric (1982)Floryan and Saric (1982)
Disturbance Source
к динамикам
U0
к динамикам
U0
к динамикам
U0
to speakersto speakers
The measurements were performed in The measurements were performed in 2222 main regimes main regimes of of disturbances excitation in frequency range from disturbances excitation in frequency range from 00..55 and and 20 20 HzHz
for three values of spanwise wavelengthfor three values of spanwise wavelength: : zz = 8, 12 = 8, 12, and, and 24 24 mmmm
Undisturbed flow
Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)
Excited Initial Disturbances
Spanwise distributions of disturbance amplitude and phase in one of regimesSpanwise distributions of disturbance amplitude and phase in one of regimeszz == 24 24 mmmm, , ff = 11 = 11 HzHz,, xx = 400 = 400 mmmm. .
0
90
180
270
360
440 444 448 452 456 460 464 468 472 476 480 484
z, mm
j, d
eg
Fi1_corr, deg
Series2 Exper. Approx.
0
0.02
0.04
0.06
440 444 448 452 456 460 464 468 472 476 480 484
z, mm
A1, м/c A1_norm, V
Series2
Exper. Approx.
%
0.6
0.4
0.2
0.0
Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)
Spectra of Eigenmodes of Unsteady Görtler-Instability Problem
Görtler number G = 17.3, spanwise wavelength = 149
FF = 0.57 = 0.57 FF = 9.08 = 9.08 FF = 22.7 = 22.7
Continuous-spectrumContinuous-spectrummodesmodes
11stst mode of discrete mode of discretespectrumspectrum
22ndnd mode of discrete mode of discretespectrumspectrum
Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)
Wall-Normal Profiles for Different Spectral Modes
Calculations based on the locally-parallel linear stabilitytheory performed for G = 17.3, F = 0.57, = 149
11stst mode mode
22ndnd mode mode
Mean velocity
U∂U/∂y(non-modal)
11stst mode mode
22ndnd mode mode
11stst--modemode critical layercritical layer
Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)
Disturbance-Source Near-Field.Transient (Non-Modal) Growth
Separation of 1st unsteady Görtler mode due to mode competition
Source near-fieldSource near-fieldTransient (non-modal) behaviorTransient (non-modal) behavior
TransientTransientgrowth in theorygrowth in theory
TransientTransientdecay in theorydecay in theory Modal behavior:Modal behavior:
11stst discrete-spectrum discrete-spectrumGGörtler modeörtler mode
DisturbanceDisturbancesourcesourceTransient decayTransient decay
in experimentin experiment
Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)
3. Properties of Steady and Unsteady Görtler Vortices
Evolution of Quasi-Steady and Unsteady Görtler Vortices
FrequencyFrequency ff = 0,5 = 0,5 HzHz((a a quasi-steadyquasi-steady case case) )
FrequencyFrequency ff = 14 = 14 HzHz((an an essentially unsteadyessentially unsteady case case) )
Streamwise component of velocity disturbance inStreamwise component of velocity disturbance in ( (x,y,tx,y,t)-space)-space((zz = 12 mm = 12 mm))
Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)
Shape of Quasi-Steady Görtler Vortices (f = 2 Hz)
UUee
ExperimentExperiment TheoryTheory
Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)
UUee
Shape of Unsteady Görtler Vortices(f = 20 Hz)
UUee
UUee
ExperimentExperiment TheoryTheory
Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)
Check of Linearity of the Problem
0
0.02
0.04
0.06
0.08
0.1
400 500 600 700 800 900x, mm
A1/A1o
Ao
Ao/2
-180
-90
0
90
180
400 500 600 700 800 900x, mm
j1, град
Ao
Ao/2
Streamwise evolution of Görtler-vortex amplitudes and phasesStreamwise evolution of Görtler-vortex amplitudes and phasesfor two different amplitudes of excitationfor two different amplitudes of excitation
deg
Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)
Wall-Normal Disturbance Profiles
Dependence on streamwiseDependence on streamwisecoordinate, coordinate, zz = = 88 mmmm
z = 8 mm, f = 5 Hz
x = 400 mm
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5 6y/
A/AmaxExperimentPSELST
z = 8 mm, f = 5 Hz
x = 400 mm
-180
-90
0
90
180
0 1 2 3 4 5 6y/
f-f
o, d
eg
ExperimentPSELST
z = 8 mm, f = 5 Hz
x = 500 mm
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5 6y/
A/AmaxExperimentPSELST
z = 8 mm, f = 5 Hz
x = 500 mm
-180
-90
0
90
180
0 1 2 3 4 5 6y/
f-f
o, d
eg
ExperimentPSELST
z = 8 mm, f = 5 Hz
x = 600 mm
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5 6y/
A/AmaxExperimentPSELST
z = 8 mm, f = 5 Hz
x = 600 mm
-180
-90
0
90
180
0 1 2 3 4 5 6y/
f-f
o, d
eg
ExperimentPSELST
z = 8 mm, f = 5 Hz
x = 700 mm
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5 6y/
A/AmaxExperimentPSELST
z = 8 mm, f = 5 Hz
x = 700 mm
-180
-90
0
90
180
0 1 2 3 4 5 6y/
f-f
o, d
eg
ExperimentPSELST
z = 8 mm, f = 5 Hz
x = 800 mm
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5 6y/
A/AmaxExperimentPSELST
z = 8 mm, f = 5 Hz
x = 800 mm
-180
-90
0
90
180
0 1 2 3 4 5 6y/
f-f
o, d
eg
ExperimentPSELST
z = 8 mm, f = 5 Hz
x = 900 mm
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5 6y/
A/AmaxExperimentPSELST
z = 8 mm, f = 5 Hz
x = 900 mm
-180
-90
0
90
180
0 1 2 3 4 5 6y/
f-f
o, d
eg
ExperimentPSELST
First mode of unsteadyFirst mode of unsteadyGGöörtler instability in LSTrtler instability in LST
Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)
Eigenfunctions of Görtler Vortices
Dependence on frequency Dependence on frequency for for zz = 12 = 12 mmmm, , GG = = 17.217.2
z = 12 mm, x = 900 mm
f = 0.5 Hz
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5 6y/
A/AmaxExperimentPSELST
z = 12 mm, x = 900 mmf = 0.5 Hz
-360
-270
-180
-90
0
90
0 1 2 3 4 5 6y/
f-f
o, d
eg
ExperimentPSELST
z = 12 mm, x = 900 mm
f = 2.0 Hz
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5 6y/
A/AmaxExperimentPSELST
z = 12 mm, x = 900 mmf = 2.0 Hz
-360
-270
-180
-90
0
90
0 1 2 3 4 5 6y/
f-f
o, d
eg
ExperimentPSELST
z = 12 mm, x = 900 mm
f = 5.0 Hz
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5 6y/
A/AmaxExperimentPSELST
z = 12 mm, x = 900 mmf = 5.0 Hz
-360
-270
-180
-90
0
90
0 1 2 3 4 5 6y/
f-f
o, d
eg
ExperimentPSELST
z = 12 mm, x = 900 mm
f = 8.0 Hz
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5 6y/
A/AmaxExperimentPSELST
z = 12 mm, x = 900 mmf = 8.0 Hz
-360
-270
-180
-90
0
90
0 1 2 3 4 5 6y/
f-f
o, d
eg
ExperimentPSELST
z = 12 mm, x = 900 mm
f = 11.0 Hz
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5 6y/
A/AmaxExperimentPSELST
z = 12 mm, x = 900 mmf = 11.0 Hz
-360
-270
-180
-90
0
90
0 1 2 3 4 5 6y/
f-f
o, d
eg
ExperimentPSELST
z = 12 mm, x = 900 mm
f = 14.0 Hz
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5 6y/
A/AmaxExperimentPSELST
z = 12 mm, x = 900 mmf = 14.0 Hz
-360
-270
-180
-90
0
90
0 1 2 3 4 5 6y/
f-f
o, d
eg
ExperimentPSELST
z = 12 mm, x = 900 mm
f = 17.0 Hz
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5 6y/
A/AmaxExperimentPSELST
z = 12 mm, x = 900 mmf = 17.0 Hz
-360
-270
-180
-90
0
90
0 1 2 3 4 5 6y/
f-f
o, d
eg
ExperimentPSELST
z = 12 mm, x = 900 mm
f = 20.0 Hz
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5 6y/
A/AmaxExperimentPSELST
z = 12 mm, x = 900 mmf = 20.0 Hz
-360
-270
-180
-90
0
90
0 1 2 3 4 5 6y/
f-f
o, d
eg
ExperimentPSELST
Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)
Eigenfunctions of Görtler Vortices
Dependence on spanwiseDependence on spanwisewavelength, x = 900 mm,wavelength, x = 900 mm,
G = 17.2, G = 17.2, ff = = 55 HzHzf = 5 Hz, x = 900 mm
= 8.0 mm
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5 6y/
A/AmaxExperimentPSELST
z
f = 5 Hz, x = 900 mm
= 8.0 mm
-180
-90
0
90
180
0 1 2 3 4 5 6y/
f-f
o, d
eg
ExperimentPSELST
z
f = 5 Hz, x = 900 mm
= 12.0 mm
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5 6y/
A/AmaxExperimentPSELST
z
f = 5 Hz, x = 900 mm
= 12.0 mm
-180
-90
0
90
180
0 1 2 3 4 5 6y/
f-f
o, d
eg
ExperimentPSELST
z
f = 5 Hz, x = 900 mm
= 24.0 mm
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5 6y/
A/AmaxExperimentPSELST
z
f = 5 Hz, x = 900 mm
= 24.0 mm
-180
-90
0
90
180
0 1 2 3 4 5 6y/
f-f
o, d
eg
ExperimentPSELST
z
First mode of unsteadyFirst mode of unsteadyGGöörtler instability in LSTrtler instability in LST
Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)
Growth of Amplitudes and Phasesof Görtler Modes (f = 2 Hz)
Phase amplification is almost Phase amplification is almost independent of the spanwise wavelength independent of the spanwise wavelength
z = 8 mm
11 13 15 17
f = 2.0 Hz
0.1
1
10
390 490 590 690 790 890x, mm
A/A1
1st Goertler mode in far fieldMixture of modes in near fieldLSTPSE
G
z = 12 mm
11 13 15 17
f = 2.0 Hz
0.1
1
10
390 490 590 690 790 890x, mm
A/A1
1st Goertler mode in far fieldMixture of modes in near fieldLSTPSE
G
z = 24 mm
11 13 15 17
f = 2.0 Hz
0.1
1
10
390 490 590 690 790 890x, mm
A/A1
1st Goertler mode in far fieldMixture of modes in near fieldLSTPSE
G
z = 8 mm
f = 2.0 Hz
0
0.2
0.4
0.6
390 490 590 690 790 890x, mm
fn/
1st Goertler mode in far fieldMixture of modes in near fieldLSTPSE
z = 12 mm
f = 2.0 Hz
0
0.2
0.4
0.6
390 490 590 690 790 890x, mm
fn/
1st Goertler mode in far fieldMixture of modes in near fieldLSTPSE
z = 24 mm
f = 2.0 Hz
0
0.2
0.4
0.6
390 490 590 690 790 890x, mm
fn/
1st Goertler mode in far fieldMixture of modes in near fieldLSTPSE
Dependence onDependence onspanwise wavelengthspanwise wavelength
Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)
Growth of Amplitudes and Phasesof Görtler Modes (z = 8 mm)
TheThe non-local, non-parallelnon-local, non-parallel stability stability theory theory ((parabolic stability equationsparabolic stability equations)) provides the best agreement with provides the best agreement with experimentexperiment
z = 8 mm
11 13 15 17
f = 2.0 Hz
0.1
1
10
390 490 590 690 790 890x, mm
A/A1
1st Goertler mode in far fieldMixture of modes in near fieldLSTPSE
G
z = 8 mm
11 13 15 17
f = 2.0 Hz
0
1
2
3
390 490 590 690 790 890x, mm
fn/
1st Goertler mode in far fieldMixture of modes in near fieldLSTPSE
G
z = 8 mm
11 13 15 17
f = 5.0 Hz
0.1
1
10
390 490 590 690 790 890x, mm
A/A1
1st Goertler mode in far fieldMixture of modes in near fieldLSTPSE
G
z = 8 mm
11 13 15 17
f = 5.0 Hz
0
1
2
3
390 490 590 690 790 890x, mm
fn/
1st Goertler mode in far fieldMixture of modes in near fieldLSTPSE
G
z = 8 mm
11 13 15 17
f = 8.0 Hz
0.1
1
10
390 490 590 690 790 890x, mm
A/A1
1st Goertler mode in far fieldMixture of modes in near fieldLSTPSE
G
z = 8 mm
11 13 15 17
f = 8.0 Hz
0
1
2
3
390 490 590 690 790 890x, mm
fn/
1st Goertler mode in far fieldMixture of modes in near fieldLSTPSE
G
z = 8 mm
11 13 15 17
f = 11.0 Hz
0.1
1
10
390 490 590 690 790 890x, mm
A/A1
1st Goertler mode in far fieldMixture of modes in near fieldLSTPSE
G
z = 8 mm
11 13 15 17
f = 11.0 Hz
0
1
2
3
390 490 590 690 790 890x, mm
fn/
1st Goertler mode in far fieldMixture of modes in near fieldLSTPSE
G
z = 8 mm
11 13 15 17
f = 14.0 Hz
0.1
1
10
390 490 590 690 790 890x, mm
A/A1
1st Goertler mode in far fieldMixture of modes in near fieldLSTPSE
G
z = 8 mm
11 13 15 17
f = 14.0 Hz
0
1
2
3
390 490 590 690 790 890x, mm
fn/
1st Goertler mode in far fieldMixture of modes in near fieldLSTPSE
G
z = 8 mm
11 13 15 17
f = 17.0 Hz
0.1
1
10
390 490 590 690 790 890x, mm
A/A1
1st Goertler mode in far fieldMixture of modes in near fieldLSTPSE
G
z = 8 mm
11 13 15 17
f = 17.0 Hz
0
1
2
3
390 490 590 690 790 890x, mm
fn/
1st Goertler mode in far fieldMixture of modes in near fieldLSTPSE
G
Dependence on frequencyDependence on frequency
Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)
Growth of Amplitudes and Phasesof Görtler Modes (z = 12 mm)
Dependence on frequency Dependence on frequency for for zz = 12 = 12 mmmm
z = 12 mm
11 13 15 17
f = 0.5 Hz
0.1
1
10
390 490 590 690 790 890x, mm
A/A1
1st Goertler mode in far fieldMixture of modes in near fieldLSTPSE
G
z = 12 mm
11 13 15 17
f = 0.5 Hz
0
1
2
3
390 490 590 690 790 890x, mm
fn/
1st Goertler mode in far fieldMixture of modes in near fieldLSTPSE
G
z = 12 mm
11 13 15 17
f = 2.0 Hz
0.1
1
10
390 490 590 690 790 890x, mm
A/A1
1st Goertler mode in far fieldMixture of modes in near fieldLSTPSE
G
z = 12 mm
11 13 15 17
f = 2.0 Hz
0
1
2
3
390 490 590 690 790 890x, mm
fn/
1st Goertler mode in far fieldMixture of modes in near fieldLSTPSE
G
z = 12 mm
11 13 15 17
f = 5.0 Hz
0.1
1
10
390 490 590 690 790 890x, mm
A/A1
1st Goertler mode in far fieldMixture of modes in near fieldLSTPSE
G
z = 12 mm
11 13 15 17
f = 5.0 Hz
0
1
2
3
390 490 590 690 790 890x, mm
fn/
1st Goertler mode in far fieldMixture of modes in near fieldLSTPSE
G
z = 12 mm
11 13 15 17
f = 8.0 Hz
0.1
1
10
390 490 590 690 790 890x, mm
A/A1
1st Goertler mode in far fieldMixture of modes in near fieldLSTPSE
G
z = 12 mm
11 13 15 17
f = 8.0 Hz
0
1
2
3
390 490 590 690 790 890x, mm
fn/
1st Goertler mode in far fieldMixture of modes in near fieldLSTPSE
G
z = 12 mm
11 13 15 17
f = 11.0 Hz
0.1
1
10
390 490 590 690 790 890x, mm
A/A1
1st Goertler mode in far fieldMixture of modes in near fieldLSTPSE
G
z = 12 mm
11 13 15 17
f = 11.0 Hz
0
1
2
3
390 490 590 690 790 890x, mm
fn/
1st Goertler mode in far fieldMixture of modes in near fieldLSTPSE
G
z = 12 mm
11 13 15 17
f = 14.0 Hz
0.1
1
10
390 490 590 690 790 890x, mm
A/A1
1st Goertler mode in far fieldMixture of modes in near fieldLSTPSE
G
z = 12 mm
11 13 15 17
f = 14.0 Hz
0
1
2
3
390 490 590 690 790 890x, mm
fn/
1st Goertler mode in far fieldMixture of modes in near fieldLSTPSE
G
TheThe non-local, non-parallelnon-local, non-parallel stability stability theory theory ((parabolic stability equationsparabolic stability equations)) provides the best agreement with provides the best agreement with experimentexperiment
Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)
Frequency Dependence of Increments and Phase Velocities of Görtler Modes
Increments of 1Increments of 1stst G Görtler modeörtler modeatat GG ≈ 15≈ 15
Phase velocities of 1Phase velocities of 1stst G Görtler modeörtler modeatat GG ≈ 15≈ 15
-0.001
0
0.001
0.002
0.003
0.004
0.005
0 5 10 15 20f , Hz
- i, mm-1
Experiment
LST
PSE
z = 8 mm
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20f , Hz
Сx/Ue
Experiment
LST
PSE
z = 8 mm
zz = 8 mm ( = 8 mm ( = 0.785 rad/mm) = 0.785 rad/mm)
Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)
Frequency Dependence of Increments and Phase Velocities of Görtler Modes
-0.001
0
0.001
0.002
0.003
0.004
0.005
0 5 10 15 20f , Hz
- i, mm-1
Experiment
LST
PSE
z = 12 mm
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20f , Hz
Сx/Ue
Experiment
LST
PSE
z = 12 mm
Increments of 1Increments of 1stst G Görtler modeörtler modeatat GG ≈ 15≈ 15
Phase velocities of 1Phase velocities of 1stst G Görtler modeörtler modeatat GG ≈ 15≈ 15
zz = 12 mm ( = 12 mm ( = 0.524 rad/mm) = 0.524 rad/mm)
Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)
Frequency Evolution of Stability Diagram for Görtler Vortices
Growing disturbances (experiment) Attenuating disturbances (experiment)Neutral points (experiment) Contours of increments (LPST)
0.5 Гц2 Гц5 Гц8 Гц11 Гц14 Гц17 Гц20 Гц HzFirst modeFirst mode
of Gof Göörtler instabilityrtler instability
Tollmien-SchlichtingTollmien-Schlichting mode mode
Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)
• Modal approach worksModal approach works for Gfor Görtler instability örtler instability problem (steady and unsteady) forproblem (steady and unsteady) for at least at least ≥ O(1) ≥ O(1)
• Very goodVery good quantitativequantitative agreementagreement between between experimental and theoretical linear-stability experimental and theoretical linear-stability characteristics has bee achieved now forcharacteristics has bee achieved now for steadysteady Görtler vortices (for the most dangerous 1Görtler vortices (for the most dangerous 1stst mode) mode)
• Similar,Similar, very good agreementvery good agreement is obtained also for is obtained also for unsteady Görtler vortices (again for the 1unsteady Görtler vortices (again for the 1stst, most , most amplified, mode)amplified, mode)
• TheThe non-local, non-parallelnon-local, non-parallel theorytheory predicts betterpredicts better the the most of stability characteristics (to both steady and most of stability characteristics (to both steady and unsteady Görtler vortices)unsteady Görtler vortices)
Conclusions
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.
Recommended Literature