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Multimedia files - 5/ 13 Görtler Instability

Multimedia files - 5/ 13 Görtler Instability. Contents : 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.

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Multimedia files - 5/ 13 Görtler Instability

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  1. Multimedia files - 5/13 GörtlerInstability • Contents: • 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

  2. 1. The Eldest Unsolved Linear-Stability Problem

  3. Why Is the Görtler InstabilitySo 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

  4. Görtler Instability on Curved Walls. When Does It Occur? Stable Sketch of SteadyGörtler vortices The necessary and sufficient conditionfor 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)

  5. Why Does Görtler Instability Appear? Governing parameteris 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 R(y<d) Fs d This is similar to unstable stratification (a buoyancy force), which leads to appearance of Görtler instability!

  6. Linear Stability Diagramsand Measurements Neutral curve Neutral curve Representation convenientin experiment: (G,L)-plane Standard representation: (G,b)-plane Floryan & Saric (1982)

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

  8. Amplification of Görtler Vortices Comparison of Experimental Amplification Curvesfor Görtler Vortex Amplitudes with the Linear Stability Theory • 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! • “Theoretical growth rates obtained for the experimental conditions were much higher than the measured growth rates” (Finnis & Brown, 1997)

  9. 2. Modern Approach to Görtler Instability

  10. 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 forthe classic case of Blasius boundary layer! • Whereas other similar problems (like Tollmien-Schlichting instability, cross-flow instability, etc.) have been solved successfully

  11. 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

  12. Steady and Unsteady Vortices • Almost all previous studies were devoted tosteadyGörtler vortices, despite theunsteadyones are often observed in real flows • UnsteadyGörtler vortices seem to dominate at enhanced free-stream turbulence levels, e.g. on turbine blades

  13. Main Fresh Ideas 1. To measure everything accurately How? To tune-off from the zero disturbance frequency and to work withquasi-steadyGörtler vortices instead of exactly steady ones 2. To investigate essentially unsteady Görtler vortices important for practical applications for steady case Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)

  14. 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 What Is Quasi-Steady? Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)

  15. Goals • To develop experimental and theoretical approachesto investigation ofunsteadyGörtler vortices (including quasi-steady ones) • To investigate experimentally and theoretically all main stability characteristics of aboundary layer on a concave surface with respect to such vortices • To perform a detail quantitative comparison ofexperimental and theoretical data onthe boundary-layer instability tounsteady (in general) Görtler vortices Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)

  16. Fan is there Test section Settling chamber Wind-Tunnel T-324 Experimentsare conducted at: Free-stream speedUe = 9.18 m/s and Free-streamturbulence levele = 0.02% Measurements are performed witha hot-wire anemometer Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)

  17. 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)

  18. Experimental Model Traversingmechanism Adjustable Wall Bump Disturbancesource 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)

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

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

  21. Disturbance Source Undisturbed flow to speakers The measurements were performed in 22 main regimes of disturbances excitation in frequency range from 0.5 and 20 Hzfor three values of spanwise wavelength: lz = 8, 12, and 24 mm Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)

  22. 0.6 Exper. Approx. % 0.4 Exper. Approx. 0.2 0.0 Excited Initial Disturbances Spanwise distributions of disturbance amplitude and phase in one of regimeslz=24 mm, f = 11 Hz,x = 400 mm. Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)

  23. Spectra of Eigenmodes of Unsteady Görtler-Instability Problem F = 9.08 F = 22.7 F = 0.57 Amplified modes 1st mode of discretespectrum 2nd mode of discretespectrum Continuous-spectrummodes Attenuated modes Görtler number G = 17.3, spanwise wavelength L = 149 Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)

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

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

  26. 3. Properties of Steady and Unsteady Görtler Vortices

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

  28. Shape of Quasi-Steady Görtler Vortices (f = 2 Hz) Ue Ue Experiment Theory Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)

  29. Shape of Unsteady Görtler Vortices(f = 20 Hz) g Ue Ue Experiment Theory Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)

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

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

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

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

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

  35. Growth of Amplitudes and Phasesof Görtler Modes (lz = 8 mm) Thenon-local, non-parallelstability theory (parabolic stability equations) provides the best agreement with experiment Dependence on frequency Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)

  36. Growth of Amplitudes and Phasesof Görtler Modes (lz = 12 mm) Thenon-local, non-parallelstability theory (parabolic stability equations) provides the best agreement with experiment Dependence on frequency for lz = 12 mm Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)

  37. Frequency Dependence of Increments and Phase Velocities of Görtler Modes lz = 8 mm (b = 0.785 rad/mm) Increments of 1st Görtler modeatG≈ 15 Phase velocities of 1st Görtler modeatG≈ 15 Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)

  38. Frequency Dependence of Increments and Phase Velocities of Görtler Modes lz = 12 mm (b = 0.524 rad/mm) Increments of 1st Görtler modeatG≈ 15 Phase velocities of 1st Görtler modeatG≈ 15 Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)

  39. Growing disturbances (experiment) Attenuating disturbances (experiment) Neutral points (experiment) Contours of increments (LPST) Frequency Evolution of Stability Diagram for Görtler Vortices Tollmien-Schlichting mode First modeof Görtler instability 0.5 Гц 11 Гц 20 Гц 14 Гц 17 Гц 8 Гц 2 Гц 5 Гц Hz Boiko (theory), Ivanov, Kachanov, Mischenko (2005-2007)

  40. Conclusions • Modal approach worksfor Görtler instability problem (steady and unsteady) forat least b ≥ O(1) • Very goodquantitativeagreementbetween experimental and theoretical linear-stability characteristics has bee achieved now forsteadyGörtler vortices (for the most dangerous 1st mode) • Similar,very good agreementis obtained also for unsteady Görtler vortices (again for the 1st, most amplified, mode) • Thenon-local, non-paralleltheorypredicts betterthe most of stability characteristics (to both steady and unsteady Görtler vortices)

  41. Floryan J.M. 1991. On the Görtler instability of boundary layers J. Aerosp. Sci. Vol. 28, pp. 235‒271. Saric W.S. 1994. Görtler vortices. Ann. Rev. Fluid Mech. Vol. 26, p. 379‒409. 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

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