multimedia files - 5/13 görtler instability contents: 1. the eldest unsolved linear-stability...

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


• 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


Wind-Tunnel T-324


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.


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


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


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


U0


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


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


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


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


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


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))


Shape of Quasi-Steady Görtler Vortices (f = 2 Hz)

UUee

ExperimentExperiment TheoryTheory


UUee

Shape of Unsteady Görtler Vortices(f = 20 Hz)

UUee

UUee

ExperimentExperiment TheoryTheory


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


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/


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/


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/


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/


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/


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


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/


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/


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/


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/


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/


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/


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/


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/


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


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/


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/


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/


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


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


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


G

z = 8 mm

f = 2.0 Hz

0

0.2

0.4

0.6

390 490 590 690 790 890x, mm

fn/


z = 12 mm

f = 2.0 Hz

0

0.2

0.4

0.6

390 490 590 690 790 890x, mm

fn/


z = 24 mm

f = 2.0 Hz

0

0.2

0.4

0.6

390 490 590 690 790 890x, mm

fn/


Dependence onDependence onspanwise wavelengthspanwise wavelength


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


G

z = 8 mm

11 13 15 17

f = 2.0 Hz

0

1

2

3

390 490 590 690 790 890x, mm

fn/


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


G

z = 8 mm

11 13 15 17

f = 5.0 Hz

0

1

2

3

390 490 590 690 790 890x, mm

fn/


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


G

z = 8 mm

11 13 15 17

f = 8.0 Hz

0

1

2

3

390 490 590 690 790 890x, mm

fn/


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


G

z = 8 mm

11 13 15 17

f = 11.0 Hz

0

1

2

3

390 490 590 690 790 890x, mm

fn/


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


G

z = 8 mm

11 13 15 17

f = 14.0 Hz

0

1

2

3

390 490 590 690 790 890x, mm

fn/


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


G

z = 8 mm

11 13 15 17

f = 17.0 Hz

0

1

2

3

390 490 590 690 790 890x, mm

fn/


G

Dependence on frequencyDependence on frequency


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


G

z = 12 mm

11 13 15 17

f = 0.5 Hz

0

1

2

3

390 490 590 690 790 890x, mm

fn/


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


G

z = 12 mm

11 13 15 17

f = 2.0 Hz

0

1

2

3

390 490 590 690 790 890x, mm

fn/


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


G

z = 12 mm

11 13 15 17

f = 5.0 Hz

0

1

2

3

390 490 590 690 790 890x, mm

fn/


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


G

z = 12 mm

11 13 15 17

f = 8.0 Hz

0

1

2

3

390 490 590 690 790 890x, mm

fn/


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


G

z = 12 mm

11 13 15 17

f = 11.0 Hz

0

1

2

3

390 490 590 690 790 890x, mm

fn/


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


G

z = 12 mm

11 13 15 17

f = 14.0 Hz

0

1

2

3

390 490 590 690 790 890x, mm

fn/


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


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)


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)


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


• 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.

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