2. Multimedia files - 5/13 Görtler Instability Contents: 1. The eldest unsolved linear-stability problem 2. Modern approach to Görtler instability 3. Properties of steady and unsteady Görtler vortices Shorten variant of an original lecture by Yury S. Kachanov

3. 1. The Eldest Unsolved Linear- Stability Problem

4. 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 Instability So Important?

5. Görtler Instability on Curved Walls. When Does It Occur? The necessary and sufficient condition for the flow to be stable is: (i) d(U 2 )/dy < 0 for concave wall or (ii) d(U 2 )/dy > 0 for convex wall. Floryan (1986) Otherwise the instability may occur Stable Sketch of Steady Görtler vortices Floryan (1991) Görtler (1956)

6. Why Does Görtler Instability Appear? As far as then 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! R(y≥)R(y≥) R(y<)R(y<) FsFs Governing parameter is Görtler number 

7. Linear Stability Diagrams and Measurements Floryan & Saric (1982) Neutral curve Standard representation: (G,  )-plane Representation convenient in experiment: (G,  )-plane

8. Linear Stability Diagrams and Measurements Experiments by Bippes (1972) Experimental check of right branch of the neutral stability curve  Growing vortices Decaying vortices Left branch of the neutral curve obtained from different versions of linear stability theory After Herbert (1976) and Floryan & Saric (1982)  Görtler (1941) Hämmerlin (1955a) Hämmerlin (1955b) Smith (1955) Hämmerlin (1961) Schultz-Grunow (1973) Kabawita & Meroney (1973-77) Floryan & Saric (1982) Hall (1984) has made conclusion that neutral curve does not exist for  ≤ O(1) In other words, Hall (1984) conclude that modal approach in invalid for these 

9. Any attempts (until recently) to find at least one figure showing direct comparison of measured amplification curves with linear theory of Görtler instability failed!!!Any attempts (until recently) to find at least one figure showing direct comparison of measured amplification curves with linear theory of Görtler instability failed!!! No quantitative agreement between experiment and linear stability theory was obtained for disturbance growth rates!No quantitative agreement between experiment and 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)“Theoretical growth rates obtained for the experimental conditions were much higher than the measured growth rates” (Finnis & Brown, 1997) Amplification of Görtler Vortices Comparison of Experimental Amplification Curves for Görtler Vortex Amplitudes with the Linear Stability Theory

10. 2. Modern Approach to Görtler Instability

11. 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 Tollmien- Schlichting instability, cross-flow instability, etc.) have been solved successfully Amplification of Görtler Vortices

12. 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?

13. 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

14. Main Fresh Ideas 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

15. What Is Quasi-Steady? Period of vortex oscillation >> Time of flow over model or X-wavelength of vortex >> X-size of exper. model E.g. for f = 0.5 Hz, U = 10 m/s, L = 1 m Period of vortex oscillation = 2 sec Time of flow over model = 0.1 sec Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)

16. 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)

17. Wind-Tunnel T-324 Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007) Experiments are conducted at: Free-stream speed U e = 9.18 m/s and Free-stream turbulence level  = 0.02% Measurements are performed with a hot-wire anemometer Settling chamber Test section Fan is there

18. 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. ( 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)

19. Experimental Model Test-plate with the concave insert, adjustable wall bump, and traverse installed in the wind-tunnel test section Disturbance source Traversing mechanism Adjustable Wall Bump Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)

20. Boundary Layer Measured mean velocity profiles and comparison with theoretical one Blasius Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)

21. Ranges of Measurements on Stability Diagrams Boiko et al. (2005-2007) f = 0 Hzf = 20 Hz First mode of Görtler instability Tollmien-Schlichting mode Floryan and Saric (1982)

22. Disturbance Source to 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: z = 8, 12, and 24 mm Undisturbed flow Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)

23. Excited Initial Disturbances Spanwise distributions of disturbance amplitude and phase in one of regimes z = 24 mm, f = 11 Hz, x = 400 mm. Exper. Approx. % 0.6 0.4 0.2 0.0 Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)

24. Spectra of Eigenmodes of Unsteady Görtler-Instability Problem Görtler number G = 17.3, spanwise wavelength  = 149 F = 0.57 F = 9.08 F = 22.7 Continuous-spectrum modes 1 st mode of discrete spectrum 2 nd mode of discrete spectrum Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)

25. Wall-Normal Profiles for Different Spectral Modes Calculations based on the locally-parallel linear stability theory performed for G = 17.3, F = 0.57,  = 149 1 st mode 2 nd mode Mean velocity U∂U/∂y (non-modal) 1 st mode 2 nd mode 1 st -mode critical layer Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)

26. Disturbance-Source Near-Field. Transient (Non-Modal) Growth Separation of 1 st unsteady Görtler mode due to mode competition Source near-field Transient (non-modal) behavior Transient growth in theory Transient decay in theory Modal behavior: 1 st discrete-spectrum Görtler mode Disturbance source Transient decay in experiment Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)

27. 3. Properties of Steady and Unsteady Görtler Vortices

28. Evolution of Quasi-Steady and Unsteady Görtler Vortices Frequency f = 0,5 Hz (a quasi-steady case) Frequency f = 14 Hz (an essentially unsteady case) Streamwise component of velocity disturbance in (x,y,t)-space ( z = 12 mm) Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)

29. Shape of Quasi-Steady Görtler Vortices (f = 2 Hz) UeUeUeUe ExperimentTheory Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007) UeUeUeUe

30. Shape of Unsteady Görtler Vortices (f = 20 Hz) UeUeUeUe UeUeUeUe ExperimentTheory  Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)

31. Check of Linearity of the Problem Streamwise evolution of Görtler-vortex amplitudes and phases for two different amplitudes of excitation deg Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)

32. Wall-Normal Disturbance Profiles Dependence on streamwise coordinate, z = 8 mm First mode of unsteady Görtler instability in LST Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)

33. Eigenfunctions of Görtler Vortices Dependence on frequency for z = 12 mm, G = 17.2 Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)

34. Eigenfunctions of Görtler Vortices Dependence on spanwise wavelength, x = 900 mm, G = 17.2, f = 5 Hz First mode of unsteady Görtler instability in LST Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)

35. Growth of Amplitudes and Phases of Görtler Modes (f = 2 Hz) Phase amplification is almost independent of the spanwise wavelength Dependence on spanwise wavelength Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)

36. Growth of Amplitudes and Phases of Görtler Modes ( z = 8 mm) The non-local, non-parallel stability theory (parabolic stability equations) provides the best agreement with experiment Dependence on frequency Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)

37. Growth of Amplitudes and Phases of Görtler Modes ( z = 12 mm) Dependence on frequency for z = 12 mm The non-local, non-parallel stability theory (parabolic stability equations) provides the best agreement with experiment Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)

38. Frequency Dependence of Increments and Phase Velocities of Görtler Modes Increments of 1 st Görtler mode at G ≈ 15 Phase velocities of 1 st Görtler mode at G ≈ 15 z = 8 mm (  = 0.785 rad/mm) z = 8 mm (  = 0.785 rad/mm) Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)

39. Frequency Dependence of Increments and Phase Velocities of Görtler Modes Increments of 1 st Görtler mode at G ≈ 15 Phase velocities of 1 st Görtler mode at G ≈ 15 z = 12 mm (  = 0.524 rad/mm) z = 12 mm (  = 0.524 rad/mm) Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)

40. 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 Hz First mode of Görtler instability Tollmien-Schlichting mode Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)

41. Modal approach works for Görtler instability problem (steady and unsteady) for at least  ≥ O(1)Modal approach works for Görtler instability problem (steady and unsteady) for at least  ≥ 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 1 st mode)Very good quantitative agreement between experimental and theoretical linear-stability characteristics has bee achieved now for steady Görtler vortices (for the most dangerous 1 st mode) Similar, very good agreement is obtained also for unsteady Görtler vortices (again for the 1 st, most amplified, mode)Similar, very good agreement is obtained also for unsteady Görtler vortices (again for the 1 st, most amplified, mode) The non-local, non-parallel theory predicts better the most of stability characteristics (to both steady and unsteady Görtler vortices)The non-local, non-parallel theory predicts better the most of stability characteristics (to both steady and unsteady Görtler vortices) Conclusions

42. 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