Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-12-01T04:31:51.389Z Has data issue: false hasContentIssue false
  • English
  • Français

On continuous spectra of the Orr–Sommerfeld/Squire equations and entrainment of free-stream vortical disturbances

Published online by Cambridge University Press:  12 September 2013

Ming Dong  and
Xuesong Wu
Ming Dong
Affiliation:
Department of Mechanics, Tianjin University, Tianjin 300072, China Tianjin Key Laboratory of Modern Engineering Mechanics, Tianjin 300072, China
Xuesong Wu*
Affiliation:
Department of Mechanics, Tianjin University, Tianjin 300072, China Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, UK
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Small-amplitude perturbations are governed by the linearized Navier–Stokes equations, which are, for a parallel or nearly parallel shear flow, customarily reduced to the Orr–Sommerfeld (O-S) and Squire equations. In this paper, we consider continuous spectra (CS) of the O-S and Squire operators for the Blasius and asymptotic suction boundary layers, and address the issue of whether and when continuous modes can represent free-stream vortical disturbances and their entrainment into the shear layer. For the Blasius boundary layer, we highlight two particular properties of the CS: (i) the eigenfunction of a continuous mode simultaneously consists of two components with wall-normal wavenumbers $\pm {k}_{2} $, a phenomenon which we refer to as ‘entanglement of Fourier components’; and (ii) for low-frequency disturbances the presence of the boundary layer forces the streamwise velocity in the free stream to take a much larger amplitude than those of the transverse velocities. Both features appear to be non-physical, and cast some doubt about the appropriateness of using CS to characterize free-stream vortical disturbances and their entrainment into the boundary layer, a practice that has been adopted in some recent studies of bypass transition. A high-Reynolds-number asymptotic description of continuous modes and entrainment is present, and it shows that the entanglement is a result of neglecting non-parallelism, which has a leading-order effect on the entrainment. When this effect is included, entanglement disappears, and moreover the streamwise velocity is significantly amplified in the edge layer when ${R}^{- 1} \ll \omega \ll 1$, where $R$ is the Reynolds number based on the local boundary-layer thickness. For the asymptotic suction boundary layer, which is an exactly parallel flow, both temporal and spatial CS may be defined mathematically. However, at a finite $R$ neither of them represents the physical process of free-stream vortical disturbances penetrating into the boundary layer. The latter must instead be characterized by a peculiar type of continuous modes whose eigenfunctions increase exponentially with the distance from the wall. In the limit $R\gg 1$, all three types of CS are identical at leading order, and hence can be used to represent free-stream vortical disturbances and their entrainment. Low-frequency disturbances are found to generate a large-amplitude streamwise velocity in the boundary layer, which is reminiscent of longitudinal streaks.

JFM classification

Boundary Layers: Boundary Layers Instability: Instability Instability: Transition to turbulence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 732 , 10 October 2013 , pp. 616 - 659
Copyright
©2013 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramowitz, M. & Stegun, I. A. 1964 Handbook of Mathematical Functions, 10th edn. Dover.Google Scholar
Bertolotti, F. P., Herbert, T. & Spalart, P. R. 1992 Linear and nonlinear stability of the Blasius boundary layer. J. Fluid Mech. 242, 441–474.CrossRefGoogle Scholar
Bodonyi, R. J. & Smith, F. T. 1981 The upper-branch stability of the Blasius boundary layer, including non-parallel flow effects. Proc. R. Soc. Lond. A 375, 65–92.Google 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
Craik, A. D. D. 1991 The continuous spectrum of the Orr–Sommerfeld equation: note on a paper of Grosch & Salwen. J. Fluid Mech. 226, 565–571.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
Durbin, P. A. & Wu, X. 2007 Transition beneath vortical disturbances. Annu. Rev. Fluid Mech. 39, 107–128.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
Fransson, J. H. M. & Alfredsson, P. H. 2003 On the disturbance growth in an asymptotic suction boundary layer. J. Fluid Mech. 482, 51–90.CrossRefGoogle Scholar
Fransson, J. H. M., Matsubara, M. & Alfredsson, P. H. 2005 Transition induced by free-stream turbulence. J. Fluid Mech. 527, 1–25.CrossRefGoogle Scholar
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 1. 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
Healey, J. J. 2006 A new convective instability of the rotating-disk boundary layer with growth normal to the disk. J. Fluid Mech. 560, 279–310.CrossRefGoogle Scholar
Hocking, L. M. 1975 Non-linear instability of the asymptotic suction velocity profile. Q. J. Mech. Appl. Maths 28, 341–353.CrossRefGoogle Scholar
Hughes, T. H. & Reid, W. H. 1965 On the stability of the asymptotic suction boundary-layer profile. J. Fluid Mech. 23, 715–735.CrossRefGoogle 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 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
Klebanoff, P. S. 1971 Effect of free stream turbulence on a laminar boundary layer. Bull. Am. Phys. Soc. 16.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. In NASA CP-2386 Transition in Turbines, pp. 161–204.Google 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
Salwen, H. & Grosch, C. E. 1981 The continuous spectrum of the Orr–Sommerfeld equation. Part 2. Eigenfunction expansions. J. Fluid Mech. 104, 445–465.CrossRefGoogle Scholar
Saric, W. S., Reed, H. L. & Kerschen, E. J. 2002 Boundary-layer receptivity to free stream disturbances. Annu. Rev. Fluid Mech. 34, 291–319.CrossRefGoogle Scholar
Smith, F. T. 1979 On the nonparallel flow stability of the blasius boundary layer. Proc. R. Soc. Lond. A 366, 91–109.Google 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
Wu, X., Zhao, D. & Luo, J. 2011 Excitation of steady and unsteady Görtler vortices by free-stream vortical disturbances. J. Fluid Mech. 682, 66–100.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
Yoshioka, S., Fransson, J. H. M. & Alfredsson, P. H. 2004 Free stream turbulence induced disturbances in boundary layers with wall suction. Phys. Fluids 16 (10), 3530–3539.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, 116–147.CrossRefGoogle Scholar