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The behaviour of Tollmien–Schlichting waves undergoing small-scale localised distortions

Published online by Cambridge University Press:  03 March 2016

Hui Xu ,
Spencer J. Sherwin ,
Philip Hall  and
Xuesong Wu
Hui Xu
Affiliation:
Department of Aeronautics, Imperial College, London SW7 2AZ, UK Department of Mathematics, Imperial College, London SW7 2AZ, UK
Spencer J. Sherwin*
Affiliation:
Department of Aeronautics, Imperial College, London SW7 2AZ, UK
Philip Hall
Affiliation:
Department of Mathematics, Imperial College, London SW7 2AZ, UK
Xuesong Wu
Affiliation:
Department of Mathematics, Imperial College, London SW7 2AZ, UK
*
†Email address for correspondence: jfluidmech@imperial.ac.uk
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Abstract

This paper is concerned with the behaviour of Tollmien–Schlichting (TS) waves experiencing small localised distortions within an incompressible boundary layer developing over a flat plate. In particular, the distortion is produced by an isolated roughness element located at $\mathit{Re}_{x_{c}}=440\,000$. We considered the amplification of an incoming TS wave governed by the two-dimensional linearised Navier–Stokes equations, where the base flow is obtained from the two-dimensional nonlinear Navier–Stokes equations. We compare these solutions with asymptotic analyses which assume a linearised triple-deck theory for the base flow and determine the validity of this theory in terms of the height of the small-scale humps/indentations taken into account. The height of the humps/indentations is denoted by $h$, which is considered to be less than or equal to $x_{c}\mathit{Re}_{x_{c}}^{-5/8}$ (corresponding to $h/{\it\delta}_{99}<6\,\%$ for our choice of $\mathit{Re}_{x_{c}}$). The rescaled width $\hat{d}~(\equiv d/(x_{c}\mathit{Re}_{x_{c}}^{-3/8}))$ of the distortion is of order $\mathit{O}(1)$ and the width $d$ is shorter than the TS wavelength (${\it\lambda}_{TS}=11.3{\it\delta}_{99}$). We observe that, for distortions which are smaller than 0.1 of the inner deck height ($h/{\it\delta}_{99}<0.4\,\%$), the numerical simulations confirm the asymptotic theory in the vicinity of the distortion. For larger distortions which are still within the inner deck ($0.4\,\%<h/{\it\delta}_{99}<5.5\,\%$) and where the flow is still attached, the numerical solutions show that both humps and indentations are destabilising and deviate from the linear theory even in the vicinity of the distortion. We numerically determine the transmission coefficient which provides the relative amplification of the TS wave over the distortion as compared to the flat plate. We observe that for small distortions, $h/{\it\delta}_{99}<5.5\,\%$, where the width of the distortion is of the order of the boundary layer, a maximum amplification of only 2 % is achieved. This amplification can however be increased as the width of the distortion is increased or if multiple distortions are present. Increasing the height of the distortion so that the flow separates ($7.2\,\%<h/{\it\delta}_{99}<12.8\,\%$) leads to a substantial increase in the transmission coefficient of the hump up to 350 %.

JFM classification

Boundary Layers: Boundary Layers Instability: Instability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 792 , 10 April 2016 , pp. 499 - 525
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© 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
Choudhari, M. & Streett, C. L.1994 Theoretical prediction of boundary-layer receptivity. AIAA Paper 94-2223.CrossRefGoogle Scholar
Corke, T. C., Sever, A. B. & Morkovin, M. V. 1986 Experiments on transition enhancements by distributed roughness. Phys. Fluids 29, 3199–3213.CrossRefGoogle Scholar
Crouch, J. D.1994 Theoretical studies on the receptivity of boundary layers. AIAA Paper 94-2224.CrossRefGoogle Scholar
Dietz, A. J. 1999 Local boundary-layer receptivity to a convected free-stream disturbance. J. Fluid Mech. 378, 291–317.CrossRefGoogle Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Duck, P. W., Ruban, A. I. & Zhikharev, C. N. 1996 Generation of Tollmien–Schlichting waves by free-stream turbulence. J. Fluid Mech. 312, 341–371.CrossRefGoogle Scholar
Fischer, P. & Choudhari, M.2004 Numerical simulation of roughness-induced transient growth in a laminar boundary layer. AIAA Paper 2004-2539.CrossRefGoogle Scholar
Fransson, J. H. M., Talamelli, A., Brandt, L. & Cossu, C. 2006 Delaying transition to turbulence by a passive mechanism. Phys. Rev. Lett. 96, 064501.CrossRefGoogle ScholarPubMed
Gaster, M. 1965 On the generation of spatially growing waves in a boundary layer. J. Fluid Mech. 22, 433–441.CrossRefGoogle Scholar
Gaster, M. 1974 On the effects of boundary-layer growth on flow stability. J. Fluid Mech. 66 (3), 465–480.CrossRefGoogle Scholar
Goldstein, M. E. 1983 The evolution of Tollmien–Schlichting waves near a leading edge. J. Fluid Mech. 127, 59–81.CrossRefGoogle Scholar
Goldstein, M. E. 1985 Scattering of acoustic waves into Tollmien–Schlichting waves by small streamwise variations in surface geometry. J. Fluid Mech. 154, 509–529.CrossRefGoogle Scholar
Goldstein, M. E. & Hultgren, L. S. 1989 Boundary-layer receptivity to long-wave free-stream disturbances. Annu. Rev. Fluid Mech. 21, 137–166.CrossRefGoogle Scholar
Goldstein, M. E., Sockol, P. M. & Sanz, J. 1983 The evolution of Tollmien–Schlichting waves near a leading edge. Part 2. Numerical determination of amplitudes. J. Fluid Mech. 129, 443–453.CrossRefGoogle Scholar
Hall, P. & Smith, F. T. 1984 On the effects of nonparallelism, three-dimensionality, and mode interaction in nonlinear boundary-layer stability. Stud. Appl. Maths 70, 91–120.CrossRefGoogle Scholar
Heinrich, R. A., Choudhari, M. & Kerschen, E. J.1988 A comparison of boundary-layer receptivity mechanisms. AIAA Paper 88-3758.CrossRefGoogle Scholar
Israeli, M. & Orszag, S. A. 1981 Approximation of radiation boundary conditions. J. Comput. Phys. 41, 115–135.CrossRefGoogle Scholar
Kachanov, Y. S. 1994 Physical mechanisms of laminar-boundary-layer transition. Annu. Rev. Fluid Mech. 26, 4110482.CrossRefGoogle Scholar
Kachanov, Y. S. 2000 Three-dimensional receptivity of boundary layers. J. Mech. (B/Fluids) 19, 723–744.CrossRefGoogle Scholar
Karniadaks, G. E. & Sherwin, S. J. 2005 Spectral/HP Element for Computational Fluid Dynamics. Springer.CrossRefGoogle Scholar
Kerschen, E. J.1989 Boundary layer receptivity. AIAA Paper 89-1109.CrossRefGoogle Scholar
Kerschen, E. J. 1990 Boundary layer receptivity theory. Appl. Mech. Rev. 43, S152–S157.CrossRefGoogle Scholar
Kozlov, V. V. & Ryzhov, O. S. 1990 Receptivity of boundary layers: asymptotic theory and experiment. Proc. R. Soc. Lond. A 429, 341–373.Google Scholar
Lin, C. C. 1966 The Theory of Hydrodynamic Stability. Cambridge University Press.Google Scholar
Messiter, A. F. 1970 Boundary-layer flow near the trailing edge of a flat plate. SIAM J. Appl. Math. 18, 241–257.CrossRefGoogle Scholar
Morkovin, M. V. 1969 The many faces of transition. In Viscous Drag Reduction (ed. Wells, C. S.), pp. 1–31. Plenum.Google Scholar
Moston, J., Stewart, P. A. & Cowley, S. J. 2000 On the nonlinear growth of two-dimensional Tollmien–Schlichting waves in a flat-plate boundary layer. J. Fluid Mech. 425, 259–300.CrossRefGoogle Scholar
Murdock, J. W. 1980 The generation of a Tollmien–Schlichting wave by a sound wave. Proc. R. Soc. Lond. A 372, 517–534.Google Scholar
Napolitano, N., Davis, R. T. & Werle, M. J. 1979 A numerical technique for the triple-deck problem. AIAA 17 (7), 78–1133.CrossRefGoogle Scholar
Neiland, V. Ya. 1969 Theory of laminar boundary layer separation in supersonic flow. Mekh. Zhid. Gaz. 4, 53–57.Google Scholar
Nishioka, M. & Morkovin, M. V. 1986 Boundary-layer receptivity to unsteady pressure gradients: experiments and overview. J. Fluid Mech. 171, 219–261.CrossRefGoogle Scholar
Ruban, A. I. 1984 On Tollmien–Schlichting wave generation by sound. Izv. Akad. Nauk SSSR Mekh. Zhidk. Gaza 5, 44–52.Google Scholar
Ruban, A. I. 1985 On Tollmien–Schlichting wave generation by sound. In Laminar–Turbulent Transition (ed. Kozlov, V. V.), pp. 313–320. Springer.CrossRefGoogle Scholar
Saric, W. S., Reed, H. L. & Kerschen, E. J. 2002 Boundary-layer receptivity to free-stream disturbances. Annu. Rev. Fluid Mech. 34, 251–276.CrossRefGoogle Scholar
Savin, D. J., Smith, F. T. & Allen, T. 1999 Transition of free disturbances in inflectional flow over an isolated surface roughness. Proc. R. Soc. Lond. A 455, 491–541.CrossRefGoogle Scholar
Schlichting, H. 1968 Boundary-Layer Theory. McGraw-Hill.Google Scholar
Schubauer, G. B. & Skramstad, H. K. 1948 Laminar-boundary-layer oscillations and transition on a flat plat. NASA TR-909.Google Scholar
Smith, F. T. 1973 Laminar flow over a small hump on a flat plate. J. Fluid Mech. 57, 803–824.CrossRefGoogle Scholar
Smith, F. T. 1979a Nonlinear stability of boundary layers for disturbances of various sizes. Proc. R. Soc. Lond. A 368, 573–589.Google Scholar
Smith, F. T. 1979b On the non-parallel flow stability of the Blasius boundary layer. Proc. R. Soc. Lond. A 366, 91–109.Google Scholar
Smith, F. T. 1981 On boundary-layer flow past two-dimensional obstacles. J. Fluid Mech. 113, 123–152.CrossRefGoogle Scholar
Smith, F. T. & Burggraf, O. R. 1985 On the development of large-sized short-scaled disturbances in boundary layers. Proc. R. Soc. Lond. A 399, 25–55.Google Scholar
Smith, F. T. & Merkin, J. H. 1982 Triple-deck solutions for subsonic flow past humps, steps, concave or convex corners and wedged trailing edges. Comput. Fluids 10 (1), 7–25.CrossRefGoogle Scholar
Stewartson, K. & Williams, P. G. 1969 Self-induced separation. Proc R. Soc. Lond. A 312 (312), 181–206.Google Scholar
Stuart, J. T. 1963 Hydrodynamic stability. In Laminar Boundary Layer (ed. Rosenhead, L.), pp. 492–579. Oxford University Press.Google Scholar
Sykes, R. I. 1978 Stratification effects in boundary-layer flow over hills. Proc. R. Soc. Lond. A 361 (1705), 225–243.Google Scholar
White, E. B. 2002 Transient growth of stationary disturbances in a flat plate boundary layer. Phys. Fluids 14 (12), 4429–4439.CrossRefGoogle Scholar
White, E. B. & Ergin, F. G. 2003 Receptivity and transient growth of roughness-induced disturbances. In 33rd AIAA Fluid Dynamics Conference and Exhibit, Orlando, FL. AIAA 2003-4243.Google Scholar
Wlezien, R. W.1994 Measurement of acoustic receptivity. AIAA Paper 94-2221.CrossRefGoogle Scholar
Wörner, A., Rist, U. & Wagner, S. 2003 Humps/steps influence on stability characteristics of two-dimensional laminar boundary layer. AIAA 41 (2), 192–197.CrossRefGoogle Scholar
Wu, X. S. 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. S. & Hogg, L. W. 2006 Acoustic radiation of Tollmien–Schlichting waves as they undergo rapid distortion. J. Fluid Mech. 550, 307–347.CrossRefGoogle Scholar
Zavol’skii, N. A., Reutov, V. P. & Ryboushkina, G. V. 1983 Excitation of Tollmien–Schlichting waves by acoustic and vortex disturbance scattering in boundary layer on a wavy surface. J. Appl. Mech. Tech. Phys. 24 (3), 355–361.CrossRefGoogle Scholar
Zhuk, V. I. & Ryzhov, O. S. 1982 On locally inviscid perturbations in a boundary layer with self-induced pressure. Dokl. Akad. Nauk SSSR 263 (1), 56–59.Google Scholar