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Three-dimensional spatial normal modes in compressible boundary layers

Published online by Cambridge University Press:  14 August 2007

ANATOLI TUMIN
ANATOLI TUMIN*
Affiliation:
Department of Aerospace and Mechanical Engineering, The University of Arizona, Tucson, AZ 85721, USA
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Abstract

Three-dimensional spatially growing perturbations in a two-dimensional compressible boundary layer are considered within the scope of linearized Navier–Stokes equations. The Cauchy problem is solved under the assumption of a finite growth rate of the disturbances. It is shown that the solution can be presented as an expansion into a biorthogonal eigenfunction system. The result can be used in a decomposition of flow fields derived from computational studies when pressure, temperature, and all the velocity components, together with some of their derivatives, are available. The method can also be used if partial data are available when a priori information may be utilized in the decomposition algorithm.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 586 , 10 September 2007 , pp. 295 - 322
Copyright
Copyright © Cambridge University Press 2007

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References

Ashpis, D. & Reshotko, E. 1990 The vibrating ribbon problem revisited. J. Fluid Mech. 213, 531–547.CrossRefGoogle Scholar
Balakumar, P. & Malik, M. R. 1992 Discrete modes and continuous spectra in supersonic boundary layer. J. Fluid Mech. 239, 631–656.CrossRefGoogle Scholar
Egorov, I. V., Fedorov, A. V. & Nechaev, A. V. 2004 Receptivity of supersonic boundary layer on a blunt plate to acoustic disturbances. AIAA Paper 2004-0249.CrossRefGoogle Scholar
Egorov, I. V., Fedorov, A. V. & Soudakov, V. G. 2005 Direct numerical simulation of supersonic boundary layer receptivity to acoustic disturbances. AIAA Paper 2005-0097.CrossRefGoogle Scholar
Eissler, W. & Bestek, H. 1993 Spatial numerical simulations of nonlinear transition phenomena in supersonic boundary layers. In Transitional and Turbulent Compressible Flows (ed. L. D. Kral & T. A. Zang). FED 151. ASME.Google Scholar
Fedorov, A. V. 1982 Generation and development of instability waves in a boundary layer of a compressible gas. PhD thesis, Moscow Institute of Physics and Technology (in Russian).Google Scholar
Fedorov, A. V. 1984 Excitation of Tollmien-Schlichting waves in a boundary layer by periodic external source located on the body surface. Fluid Dyn. 19, 888–893.CrossRefGoogle Scholar
Fedorov, A. V. 1988 Excitation of waves of instability of the secondary flow in the boundary layer on a swept wing. J. Appl. Mech. Tech. Phys. 29, 643–648.CrossRefGoogle Scholar
Fedorov, A. V. 2003 a Receptivity of a high-speed boundary layer to acoustic disturbances J. Fluid Mech. 491, 101–129.CrossRefGoogle Scholar
Fedorov, A. V. 2003 b Receptivity of hypersonic boundary layer to acoustic disturbances scattered by surface roughness. AIAA Paper 2003-3731.CrossRefGoogle Scholar
Fedorov, A. V. & Khokhlov, A. P. 1991 Mode switching in a supersonic boundary layer. J. Appl. Mech. Tech. Phys. 32, 831–836.Google Scholar
Fedorov, A. V. & Khokhlov, A. P. 2001 Prehistory of instability in a hypersonic boundary layer. Theor. Comp. Fluid Dyn. 14, 359–375.CrossRefGoogle Scholar
Fedorov, A. V. & Khokhlov, A. P. 2002 Receptivity of hypersonic boundary layer to wall disturbances. Theor. Comp. Fluid Dyn. 15, 231–254.CrossRefGoogle Scholar
Fedorov, A. & Tumin, A. 2003 Initial-value problem for hypersonic boundary layer flows. AIAA J. 41, 379–389.CrossRefGoogle Scholar
Forgoston, E. & Tumin, A. 2005 Initial-value problem for three-dimensional disturbances in a hypersonic boundary layer Phys. Fluids 17, 084106.CrossRefGoogle Scholar
Grosch, C. E. & Salwen, H. 1978 The continuous spectrum of the Orr–Sommerfeld equation. Part 1. the spectrum and eigenfunctions. J. Fluid Mech. 87, 33–54.CrossRefGoogle Scholar
Gushchin, V. R. & Fedorov, A. V. 1989 Asymptotic analysis of inviscid perturbations in a supersonic boundary layer. J. Appl. Mech. Tech. Phys. 30, 64–70.CrossRefGoogle Scholar
Gustavsson, L. H. 1979 Initial-value problem for boundary layer flows. Phys. Fluids 22, 1602–1605.CrossRefGoogle Scholar
Guydos, P. 2004 Analysis of small perturbations in compressible boundary layers. MS Thesis, The University of Arizona.Google Scholar
Guydos, P. & Tumin, A. 2004 Multimode decomposition in compressible boundary layers. AIAA J. 42, 1115–1121.CrossRefGoogle Scholar
Hanifi, A., Schmid, P. J. & Henningson, D. S. 1996 Transient growth in compressible boundary layer flow. Phys. Fluids 8, 826–837.CrossRefGoogle Scholar
Hill, D. C. 1995 Adjoint systems and their role in the receptivity problem for boundary-layers. J. Fluid Mech. 292, 183–204.CrossRefGoogle Scholar
Kamke, E. 1959 Differentialgleichungen. Lösungsmethoden und Lösungen. Leipzig: Akademische Verlagsgesellschaft Geest & Portig.Google Scholar
Ma, Y. & Zhong, X. 2001 Numerical simulation of receptivity and stability of nonequilibrium reacting hypersonic boundary layers. AIAA Paper 2001-0892.CrossRefGoogle Scholar
Ma, Y. & Zhong, X. 2003 a Receptivity of a supersonic boundary layer over a flat plate. Part 1: Wave structures and interactions J. Fluid Mech. 488, 31–78.CrossRefGoogle Scholar
Ma, Y. & Zhong, X. 2003 b Receptivity of a supersonic boundary layer over a flat plate. Part 2: Receptivity to freestream sound J. Fluid Mech. 488, 79–121.CrossRefGoogle Scholar
Ma, Y. & Zhong, X. 2005 Receptivity of a supersonic boundary layer over a flat plate. Part 3: Effects of different types of free-stream disturbances. J. Fluid Mech. 532, 63–109.CrossRefGoogle Scholar
Mack, L. M. 1969 Boundary layer stability theory. JPL Report 900-277. Jet Propulsion Lab., California Institute of Technology, Pasadena, CA, USA.Google Scholar
Malik, M. R. 1990 Numerical methods for hypersonic boundary layer stability. J. Comput. Phys. 86, 376–413.CrossRefGoogle Scholar
Nayfeh, A. H. 1980 Stability of three-dimensional boundary layers. AIAA J. 18, 406–416.CrossRefGoogle Scholar
Salwen, H. & Grosch, C. E. 1981 The continuous spectrum of the Orr–Sommerfeld equation. Part 2. Eigenfunction expansion. J. Fluid Mech. 104, 445–465.CrossRefGoogle Scholar
Smith, F. T., Sykes, R. I. & Brighton, P. W. 1977 A two-dimensional boundary layer encountering a three-dimensional hump. J. Fluid Mech. 83, 163–176.CrossRefGoogle Scholar
Terent'ev, E. D. 1981 The linear problem of a vibrator in a subsonic boundary layer. J. Appl. Math. Mech. 45, 791–795.CrossRefGoogle Scholar
Tikhonov, A. N. & Arsenin, V. Y. 1977 Solutions of Ill-Posed Problems. Wiley.Google Scholar
Tikhonov, A. N., Goncharski, A. V., Stepanov, V. V. & Yagola, A. G. 1995 Numerical Methods for the Solution of Ill-Posed Problems. Kluwer.CrossRefGoogle Scholar
Tumin, A. 1983 Excitation of Tollmien–Schlichting waves in a boundary layer on a vibrating surface of an infinitely swept wing. J. Appl. Mech. Tech. Phys. 24, 670–674.CrossRefGoogle Scholar
Tumin, A. 1996 Receptivity of pipe Poiseuille flow. J. Fluid Mech. 315, 119–137.CrossRefGoogle Scholar
Tumin, A. 1998 Subharmonic resonance in a laminar wall jet. Phys. Fluids 10, 1769–1771.CrossRefGoogle Scholar
Tumin, A. 2003 Multimode decomposition of spatially growing perturbations in a two-dimensional boundary layer. Phys. Fluids 15, 2525–2540.CrossRefGoogle Scholar
Tumin, A. 2006 a Biorthogonal eigenfunction system in the triple-deck limit Stud. Appl. Math. 117, 165–190.CrossRefGoogle Scholar
Tumin, A. 2006 b Receptivity of compressible boundary layers to three-dimensional wall perturbations. AIAA Paper 2006-1110.CrossRefGoogle Scholar
Tumin, A. & Aizatulin, L. 1997 Instability and receptivity of laminar wall jets. Theor. Comp. Fluid Dyn. 9, 33–45.CrossRefGoogle Scholar
Tumin, A., Amitay, M., Cohen, J. & Zhou, M. 1996 A normal multi-mode decomposition method for stability experiments. Phys. Fluids 8, 2777–2779.CrossRefGoogle Scholar
Tumin, A. M. & Fedorov, A. V. 1983 a Excitation of instability waves in a boundary layer on a vibrating surface J. Appl. Mech. Tech. Phys. 24, 670–674.CrossRefGoogle Scholar
Tumin, A. M. & Fedorov, A. V. 1983 b Spatial growth of disturbances in a compressible boundary layer J. Appl. Mech. Tech. Phys. 24, 548–554.CrossRefGoogle Scholar
Tumin, A. M. & Fedorov, A. V. 1984 Excitation of instability waves by a vibrator localized in the boundary layer. J. Appl. Mech. Tech. Phys. 25, 867–873.CrossRefGoogle Scholar
Tumin, A. & Reshotko, E. 2003 Optimal disturbances in compressible boundary layers. AIAA J. 42, 2357–2363.CrossRefGoogle Scholar
Tumin, A. & Reshotko, E. 2004 The problem of boundary-layer flow encountering a three-dimensional hump revisited. AIAA Paper 2004-0101.CrossRefGoogle Scholar
Tumin, A. & Reshotko, E. 2005 Receptivity of a boundary-layer flow to a three-dimensional hump at finite Reynolds numbers Phys. Fluids 17, 094101.CrossRefGoogle Scholar
Tumin, A., Wang, X. & Zhong, X. 2007 Direct numerical simulation and the theory of receptivity in a hypersonic boundary layer Phys. Fluids 19, 014101.CrossRefGoogle Scholar
Wang, X. & Zhong, X. 2005 Receptivity of a Mach 8.0 flow over a sharp wedge with half-angle 5.3° to wall blowing-suction. AIAA Paper 2005-5025.CrossRefGoogle Scholar
Wang, X. & Zhong, X. 2007 Numerical simulation of hypersonic boundary-layer receptivity to two and three-dimensional wall perturbations. AIAA Paper 2007-946.CrossRefGoogle Scholar
Wolfram, S. 1999 The Mathematica Book, 4th edn. Cambridge: Wolfram Media and Cambridge University Press.Google Scholar
Zhigulev, V. N. & Fedorov, A. V. 1987 Boundary layer receptivity to acoustic disturbances. J. Appl. Mech. Tech. Phys. 28, 28–34.CrossRefGoogle Scholar
Zhigulev, V. N., Sidorenko, N. V. & Tumin, A. M. 1980 Generation of instability waves in a boundary layer by external turbulence. J. Appl. Mech. Tech. Phys. 21, 774–778.CrossRefGoogle Scholar
Zhigulev, V. N. & Tumin, A. M. 1987 Origin of Turbulence. Novosibirsk: Nauka (in Russian) [transl. NASA TT-20340, October 1988].Google Scholar
Zhong, X. & Ma, Y. 2002 Receptivity and linear stability of Stetson's Mach 8 blunt cone stability experiments. AIAA Paper 2002-2849.Google Scholar