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Spatial optimal growth in three-dimensional compressible boundary layers

  • David Tempelmann (a1), Ardeshir Hanifi (a1) (a2) and Dan S. Henningson (a1)


This paper represents a continuation of the work by Tempelmann et al. (J. Fluid Mech., vol. 646, 2010b, pp. 5–37) on spatial optimal growth in incompressible boundary layers over swept flat plates. We present an extension of the methodology to compressible flow. Also, we account for curvature effects. Spatial optimal growth is studied for boundary layers over both flat and curved swept plates with adiabatic and cooled walls. We find that optimal growth increases for higher Mach numbers. In general, extensive non-modal growth is observed for all boundary layer cases even in subcritical regions, i.e. where the flow is stable with respect to modal crossflow disturbances. Wall cooling, despite stabilizing crossflow modes, destabilizes disturbances of non-modal nature. Curvature acts similarly on modal as well as non-modal disturbances. Convex walls have a stabilizing effect on the boundary layer whereas concave walls have a destabilizing effect. The physical mechanisms of optimal growth in all studied boundary layers are found to be similar to those identified for incompressible flat-plate boundary layers.


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1. Anderson, J. D. 2006 Hypersonic and High-Temperature Gas Dynamics, 2nd edn. AIAA.
2. Andersson, P., Berggren, M. B. & Henningson, D. S. 1999 Optimal disturbances and bypass transition in boundary layers. Phys. Fluids 11, 134150.
3. Andersson, P., Henningson, D. S. & Hanifi, A. 1998 On a stabilization procedure for the parabolic stability equations. J. Engng Math. 33 (3), 311332.
4. Bertolotti, F. P. & Herbert, T. 1991 Analysis of the linear stability of compressible boundary layers using the PSE. Theor. Comput. Fluid Dyn. 3, 117124.
5. Bertolotti, F. P., Herbert, T. & Spalart, P. R. 1992 Linear and nonlinear stability of the Blasius boundary layer. J. Fluid Mech. 242, 441474.
6. Bippes, H. 1999 Basic experiments on transition in three-dimensional boundary layers dominated by cross-flow instability. Prog. Aerosp. Sci. 35, 363412.
7. Bottaro, A. 2010 A receptive boundary layer. J. Fluid Mech. 646, 14.
8. Breuer, K. S. & Kuraishi, T. 1994 Transient growth in two- and three-dimensional boundary layers. Phys. Fluids 6 (6), 19831993.
9. Butler, K. M. & Farrell, B. F. 1992 Three-dimensional optimal perturbations in viscous shear flow. Phys. Fluids 4, 16371650.
10. Cooke, J. C. 1950 The boundary layer of a class of infinite yawed cylinders. Proc. Camb. Phil. Soc. 46, 645648.
11. Corbett, P. & Bottaro, A. 2001 Optimal linear growth in swept boundary layers. J. Fluid Mech. 435, 123.
12. Cossu, C., Chomaz, J.-M., Huerre, P. & Costa, M. 2000 Maximum spatial growth of Görtler vortices. Flow Turbul. Combust. 65, 369392.
13. Ellingsen, T. & Palm, E. 1975 Stability of linear flow. Phys. Fluids 18 (4), 487488.
14. Fischer, P. F., Lottes, J. W. & Kerkemeier, S. G. 2008 Nek5000 web page.
15. Flügge, S. & Tresdell, C. 1959 Handbuch der Physik, vol. VIII/1, Strömungsmechanik I. Springer.
16. Gaponov, S. A. & Smorodskii, B. V. 2008 Linear stability of three-dimensional boundary layers. J. Appl. Mech. Tech. Phys. 49 (2), 157166.
17. Graves, R. E. & Arrow, B. M. 1999 Bulk viscosity: past to present. J. Thermodyn. Heat Transfer 13 (3), 337342.
18. Haj-Hariri, H. 1994 Characteristics analysis of the parabolized stability equations. Stud. Appl. Math. 92, 4153.
19. Hanifi, A. & Henningson, D. S. 1998 The compressible inviscid algebraic instability for streamwise independent disturbances. Phys. Fluids 10 (8), 17841786.
20. Hanifi, A., Henningson, D. S., Hein, S., Bertolotti, F. P & Simen, M. 1994 Linear non-local instability analysis: the linear NOLOT code. FFA TN 1994-54.
21. Hanifi, A., Schmid, P. J. & Henningson, D. S. 1996 Transient growth in compressible boundary layer flow. Phys. Fluids 8 (3), 826837.
22. Kufner, E. 1995 Numerische Untersuchungen der Strömungsinstabilitäten an spitzen und stumpfen Kegeln bei hypersonischen Machzahlen. PhD thesis, DLR Göttingen.
23. Kurian, T., Fransson, J. H. M. & Alfredsson, P. H. 2011 Boundary layer receptivity to free stream turbulence and surface roughness over a swept flat plate. Phys. Fluids 23, 034107.
24. Landahl, M. T. 1980 A note on algebraic instability of inviscid parallel shear flows. J. Fluid Mech. 98, 243251.
25. Levin, O. & Henningson, D. S. 2003 Exponential vs algebraic growth and transition prediction in boundary layer flow. Flow Turbul. Combust. 70, 183210.
26. Luchini, P. 2000 Reynolds-number-independent instability of the boundary layer over a flat surface: optimal perturbations. J. Fluid Mech. 404, 289309.
27. Mack, L. M. 1965 Computation of the stability of the laminar compressible boundary layer. Meth. Comput. Phys. 4, 247.
28. Mack, L. M. 1969 Boundary layer stability theory. JPL Rep. 900-277. Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA, USA.
29. Mack, L. M. 1984 Boundary-layer linear stability theory. AGARD Rep. 709.
30. Mack, L. M. 1986 Boundary layer stability analysis for sharp cones at zero angle-of-attack. AFWAL-TR-86-3022, Flight Dynamics Laboratory, Air Force Wright Aeronautical Laboratories, Wright-Patterson AFB, Ohio, USA.
31. Pralits, J. O., Airiau, C., Hanifi, A. & Henningson, D. S. 2000 Sensitivity analysis using adjoint parabolized stability equations for compressible flows. Flow Turbul. Combust. 65, 321346.
32. Reddy, S. C. & Henningson, D. S. 1993 Energy growth in viscous channel flow. J. Fluid Mech. 252, 209238.
33. Reibert, M. S., Saric, W. S., Carillo, R. B. & Chapman, K. L. 1996 Experiments in nonlinear saturation of stationary cross-flow vortices in a swept-wing boundary layer. AIAA Paper 96-0184.
34. Reshotko, E. & Tumin, A. 2004 a Optimal disturbances in the boundary layer over a sphere. AIAA Paper 2004-2241.
35. Reshotko, E. & Tumin, A. 2004b Role of transient growth in roughness-induced transition. AIAA J. 42 (4), 766770.
36. Schrader, L. U., Brandt, L. & Henningson, D. S. 2009 Receptivity mechanisms in three-dimensional boundary layer flows. J. Fluid Mech. 618, 209241.
37. Simen, M. 1992 Local and non-local stability theory of spatially varying flows. In Instability, Transitions, and Turbulence, pp. 181195. Springer.
38. Tempelmann, D. 2009 Stability and receptivity of three-dimensional boundary layers. TRITA-MEK 2009:19, licentiate thesis, KTH Stockholm.
39. Tempelmann, D., Hanifi, A. & Henningson, D. S. 2010 a Optimal disturbances and receptivity in three-dimensional boundary layers. In Proc. 5th European Conference on Computational Fluid Dynamics ECCOMAS CFD (ed. J. C. F. Pereira & A. Sequeira) Lisbon, Portugal, 14–17 June 2010.
40. Tempelmann, D., Hanifi, A. & Henningson, D. S. 2010b Spatial optimal growth in three-dimensional boundary layers. J. Fluid Mech. 646, 537.
41. Ting, L. 1965 On the initial conditions for boundary layer equations. J. Math. Phys. 44, 353367.
42. Trefethen, L. N., Trefethen, A. E., Reddy, S. C. & Driscoll, T. A. 1993 Hydrodynamic stability without eigenvalues. Science 261, 578584.
43. Tumin, A. & Reshotko, E. 2001 Spatial theory of optimal disturbances in boundary layers. Phys. Fluids 13 (7).
44. Tumin, A. & Reshotko, E. 2003 Optimal disturbances in compressible boundary layers. AIAA Paper 2003-0792.
45. Weideman, J. A. & Reddy, S. C. 2000 A MATLAB differentiation matrix suite. ACM Trans. Math. Softw. (TOMS) 26 (4).
46. Zuccher, S., Tumin, A. & Reshotko, E. 2006 Parabolic approach to optimal perturbations in compressible boundary layers. J. Fluid Mech. 556, 189216.
47. Zurigat, Y. H., Nayfeh, A. H. & Masad, J. A. 1990 Effect of pressure gradient on the stability of compressible boundary layers. AIAA Paper 90-1451.
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Spatial optimal growth in three-dimensional compressible boundary layers

  • David Tempelmann (a1), Ardeshir Hanifi (a1) (a2) and Dan S. Henningson (a1)


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