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Entrainment of short-wavelength free-stream vortical disturbances in compressible and incompressible boundary layers

Published online by Cambridge University Press:  24 May 2016

Xuesong Wu  and
Ming Dong
Xuesong Wu*
Affiliation:
Department of Mechanics, Tianjin University, Tianjin 300072, PR China Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, UK
Ming Dong
Affiliation:
Department of Mechanics, Tianjin University, Tianjin 300072, PR China
*
†Email address for correspondence: [email protected]
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Abstract

The fundamental difference between continuous modes of the Orr–Sommerfeld/Squire equations and the entrainment of free-stream vortical disturbances (FSVD) into the boundary layer has been investigated in a recent paper (Dong & Wu, J. Fluid Mech., vol. 732, 2013, pp. 616–659). It was shown there that the non-parallel-flow effect plays a leading-order role in the entrainment, and neglecting it at the outset, as is done in the continuous-mode formulation, leads to non-physical features of ‘Fourier entanglement’ and abnormal anisotropy. The analysis, which was for incompressible boundary layers and for FSVD with a characteristic wavelength of the order of the local boundary-layer thickness, is extended in this paper to compressible boundary layers and FSVD with even shorter wavelengths, which are comparable with the width of the so-called edge layer. Non-parallelism remains a leading-order effect in the present scaling, which turns out to be more general in that the equations and solutions in the previous paper are recovered in the appropriate limit. Appropriate asymptotic solutions in the main and edge layers are obtained to characterize the entrainment. It is found that when the Prandtl number $\mathit{Pr}<1$, free-stream vortical disturbances of relatively low frequency generate very strong temperature fluctuations within the edge layer, leading to formation of thermal streaks. A composite solution, uniformly valid across the entire boundary layer, is constructed, and it can be used in receptivity studies and as inlet conditions for direct numerical simulations of bypass transition. For compressible boundary layers, continuous spectra of the disturbance equations linearized about a parallel base flow exhibit entanglement between vortical and entropy modes, namely, a vortical mode necessarily induces an entropy disturbance in the free stream and vice versa, and this amounts to a further non-physical behaviour. High Reynolds number asymptotic analysis yields the relations between the amplitudes of entangled modes.

JFM classification

Boundary Layers: Boundary layer receptivity Boundary Layers: Boundary layer stability Instability: Transition to turbulence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 797 , 25 June 2016 , pp. 683 - 728
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Copyright
© 2016 Cambridge University Press 

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References

Abramowitz, M. & Stegun, I. A. 1964 Handbook of Mathematical Functions. National Bureau of Standards.Google Scholar
Balakumar, P. & Malik, M. R. 1992 Discrete modes and continuous spectra in supersonic boundary layers. J. Fluid Mech. 239, 631–656.CrossRefGoogle Scholar
Bertolotti, F. P., Herbert, Th. & Spalart, P. R. 1992 Linear and nonlinear stability of the Blasius boundary layer. J. Fluid Mech. 242, 441–474.CrossRefGoogle Scholar
Brandt, L., Schlatter, P. & Henningson, D. S. 2004 Transition in boundary layers subject to free-stream turbulence. J. Fluid Mech. 517, 167–198.CrossRefGoogle Scholar
Crow, S. C. 1966 The spanwise perturbation of two-dimensional boundary-layers. J. Fluid Mech. 24, 153–164.CrossRefGoogle Scholar
Deguchi, K. & Hall, P. 2015 Free-stream coherent structures in growing boundary layers: a link to near-wall streaks. J. Fluid Mech. 778, 451–484.CrossRefGoogle Scholar
Dong, M. & Wu, X. 2013 On continuous spectra of the Orr–Sommerfeld/Squire equations and entrainment of free-stream vortical disturbances. J. Fluid Mech. 732, 616–659.CrossRefGoogle Scholar
Dong, M. & Wu, X. 2015 Entrainment of short-wavelength free-stream vortical disturbances into boundary layers. Procedia IUTAM 14, 96–104.CrossRefGoogle Scholar
Duck, P. W., Ruban, A. I. & Zhikharev, C. N. 1996 The generation of Tollmien–Schlichting waves by free-stream turbulence. J. Fluid Mech. 312, 341–371.CrossRefGoogle Scholar
Fasel, H. & Konzelmann, U. 1990 Non-parallel stability of a flat-plate boundary layer using the complete Navier–Stokes equations. J. Fluid Mech. 221, 311–347.CrossRefGoogle Scholar
Goldstein, M. E. 2014 Effect of free-stream turbulence on boundary layer transition. Phil. Trans R. Soc. A 372; doi:10.1098/rsta.2013.0354.Google ScholarPubMed
Goldstein, M. E. & Leib, S. J. 1993 Three-dimensional boundary layer instability and separation induced by small-amplitude streamwise vorticity in the upstream flow. J. Fluid Mech. 246, 21–41.CrossRefGoogle Scholar
Goldstein, M. E., Leib, S. J. & Cowley, S. J. 1992 Distortion of a flat plate boundary layer by free stream vorticity normal to the plate. J. Fluid Mech. 237, 231–260.CrossRefGoogle Scholar
Grosch, C. E. & Salwen, H. 1978 The continuous spectrum of the Orr–Sommerfeld equation. Part I. The spectrum and the eigenfunctions. J. Fluid Mech. 87, 33–54.CrossRefGoogle Scholar
Gulyaev, A. N., Kozlov, V. E., Kuznetsov, V. R., Mineev, B. I. & Sekundov, A. N. 1989 Interaction of laminar boundary layer with external turbulence. Izv. Akad. Nauk SSSR Mekh. Zhidk. Gaza 6, 700–710.Google Scholar
Jacobs, R. & Durbin, P. 1998 Shear sheltering and the continuous spectrum of the Orr–Sommerfeld equation. Phys. Fluids 10, 2006–2011.CrossRefGoogle Scholar
Jacobs, R. & Durbin, P. 2001 Simulations of bypass transition. J. Fluid Mech. 428, 185–212.CrossRefGoogle Scholar
Joo, J. & Durbin, P. A.2010 Approximate decoupling of continuous modes and mode interaction in high-speed boundary-layer transition. AIAA Paper 2010-4283.CrossRefGoogle Scholar
Joo, J. & Durbin, P. A. 2012 Continuous mode transition in high-speed boundary-layers. Flow Turbul. Combust. 88, 407–430.CrossRefGoogle Scholar
Kachanov, Y. S 1994 Physical mechanisms of the laminar-boundary-layer transition. Annu. Rev. Fluid Mech. 26, 411–482.CrossRefGoogle Scholar
Kemp, N.1951 The laminar three-dimensional boundary layer and a study of the flow past a side edge. MSc thesis, Cornell University.Google Scholar
Kendall, J. M.1990 Boundary layer receptivity to freestream turbulence. AIAA Paper 85-1695.CrossRefGoogle Scholar
Klebanoff, P. S. 1971 Effect of free-stream turbulence on a laminar boundary layer. Bull. Am. Phys. Soc. 10, 1323–1334.Google Scholar
Kovasznay, L. S. 1953 Turbulence in supersonic flow. J. Aero. Sci. 20, 657–682.Google Scholar
Leib, S. J., Wundrow, D. W. & Goldstein, M. E. 1999 Effect of free-stream turbulence and other vortical disturbances on a laminar boundary layer. J. Fluid Mech. 380, 169–203.CrossRefGoogle Scholar
Liu, Y., Zaki, T. & Durbin, P. 2008 Boundary-layer transition by interaction of discrete and continuous modes. J. Fluid Mech. 605, 199–233.CrossRefGoogle Scholar
Malik, M.1990 Finite difference solution of the compressible stability eigenvalue problem. NASA Tech. Rep. 16572.Google Scholar
Matsubara, M. & Alfredsson, P. H. 2001 Disturbance growth in boundary layers subjected to free-stream turbulence. J. Fluid Mech. 430, 149–168.CrossRefGoogle Scholar
Morkovin, M. V.1984 Bypass transition to turbulence and research desiderata. NASA CP-2386 Transition in Turbines, pp. 161–204.Google Scholar
Nagarajan, S., Lele, S. K. & Ferziger, J. H. 2007 Leading-edge effects in bypass transition. J. Fluid Mech. 572, 471–504.CrossRefGoogle Scholar
Ovchinnikov, V., Piomelli, U. & Choudhari, M. M. 2006 Numerical simulations of boundary-layer transition induced by a cylinder wake. J. Fluid Mech. 547, 413–441.CrossRefGoogle Scholar
Ricco, P., Luo, J. & Wu, X. 2011 Evolution and instability of unsteady nonlinear streaks generated by free-stream vortical disturbances. J. Fluid Mech. 677, 1–38.CrossRefGoogle Scholar
Ricco, P. & Wu, X. 2007 Response of a compressible laminar boundary layer to free-stream vortical disturbances. J. Fluid Mech. 587, 97–138.CrossRefGoogle Scholar
Saric, W. S., Reed, H. L. & Kerschen, E. J. 2002 Boundary-layer receptivity to freestream disturbances. Annu. Rev. Fluid Mech. 34, 291–319.CrossRefGoogle Scholar
Schrader, L.-U., Brandt, L., Mavriplis, C. & Henningson, D. S. 2010 Receptivity to free-stream vorticity of the flow past a flat plate with elliptic leading edge. J. Fluid Mech. 653, 245–271.CrossRefGoogle Scholar
Stewartson, K. 1964 The Theory of Laminar Boundary Layers in Compressible Fluids. Claredon Press.CrossRefGoogle Scholar
Westin, K. J. A., Boiko, A. V., Klingmann, B. G. B., Kozlov, V. V. & Alfredsson, P. H. 1994 Experiments in a boundary layer subjected to free stream turbulence. Part 1. Boundary layer structure and receptivity. J. Fluid Mech. 281, 193–218.CrossRefGoogle Scholar
Wu, X. 2001 Receptivity of boundary layers with distributed roughness to vortical and acoustic disturbances: a second-order asymptotic theory and comparison with experiments. J. Fluid Mech. 431, 91–133.CrossRefGoogle Scholar
Wundrow, D. W. & Goldstein, M. E. 2001 Effect on a laminar boundary layer of small-amplitude streamwise vorticity in the upstream flow. J. Fluid Mech. 426, 229–262.CrossRefGoogle Scholar
Zaki, T. A. & Durbin, P. 2005 Mode interaction and the bypass route to transition. J. Fluid Mech. 531, 85–811.CrossRefGoogle Scholar
Zaki, T. A. & Saha, S. 2009 On shear sheltering and the structure of vortical modes in single- and two-fluid boundary layers. J. Fluid Mech. 626, 111–147.CrossRefGoogle Scholar