Hostname: page-component-7d684dbfc8-mqbnt Total loading time: 0 Render date: 2023-09-28T02:25:23.924Z Has data issue: false Feature Flags: { "corePageComponentGetUserInfoFromSharedSession": true, "coreDisableEcommerce": false, "coreDisableSocialShare": false, "coreDisableEcommerceForArticlePurchase": false, "coreDisableEcommerceForBookPurchase": false, "coreDisableEcommerceForElementPurchase": false, "coreUseNewShare": true, "useRatesEcommerce": true } hasContentIssue false

Receptivity to free-stream vorticity of flow past a flat plate with elliptic leading edge

Published online by Cambridge University Press:  27 April 2010

Linné Flow Centre, KTH Mechanics, SE-100 44 Stockholm, Sweden
Linné Flow Centre, KTH Mechanics, SE-100 44 Stockholm, Sweden
Department of Mechanical Engineering, University of Ottawa, Ottawa K1N 6N5Canada
Linné Flow Centre, KTH Mechanics, SE-100 44 Stockholm, Sweden
Email address for correspondence:


Receptivity of the two-dimensional boundary layer on a flat plate with elliptic leading edge is studied by numerical simulation. Vortical perturbations in the oncoming free stream are considered, impinging on two leading edges with different aspect ratio to identify the effect of bluntness. The relevance of the three vorticity components of natural free-stream turbulence is illuminated by considering axial, vertical and spanwise vorticity separately at different angular frequencies. The boundary layer is most receptive to zero-frequency axial vorticity, triggering a streaky pattern of alternating positive and negative streamwise disturbance velocity. This is in line with earlier numerical studies on non-modal growth of elongated structures in the Blasius boundary layer. We find that the effect of leading-edge bluntness is insignificant for axial free-stream vortices alone. On the other hand, vertical free-stream vorticity is also able to excite non-modal instability in particular at zero and low frequencies. This mechanism relies on the generation of streamwise vorticity through stretching and tilting of the vertical vortex columns at the leading edge and is significantly stronger when the leading edge is blunt. It can thus be concluded that the non-modal boundary-layer response to a free-stream turbulence field with three-dimensional vorticity is enhanced in the presence of a blunt leading edge. At high frequencies of the disturbances the boundary layer becomes receptive to spanwise free-stream vorticity, triggering Tollmien–Schlichting (T-S) modes and receptivity increases with leading-edge bluntness. The receptivity coefficients to free-stream vortices are found to be about 15% of those to sound waves reported in the literature. For the boundary layers and free-stream perturbations considered, the amplitude of the T-S waves remains small compared with the low-frequency streak amplitudes.

Copyright © Cambridge University Press 2010

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.)



Bertolotti, F. P. 1997 Response of the Blasius boundary layer to free-stream vorticity. Phys. Fluids 9 (8), 22862299.CrossRefGoogle Scholar
Bertolotti, F. P. & Kendall, J. M. 1997 Response of the Blasius boundary layer to controlled free-stream vortices of axial form. AIAA Paper 97-2018.Google Scholar
Brandt, L., Schlatter, P. & Henningson, D. S. 2004 Transition in boundary layers subject to free-stream turbulence. J. Fluid Mech. 517, 167198.CrossRefGoogle Scholar
Buter, T. A. & Reed, H. L. 1994 Boundary layer receptivity to free-stream vorticity. Phys. Fluids 6 (10), 33683379.CrossRefGoogle Scholar
Choudhari, M. & Streett, C. L. 1992 A finite Reynolds number approach for the prediction of boundary layer receptivity in localized regions. Phys. Fluids A 4, 24952514.CrossRefGoogle Scholar
Collis, S. S. & Lele, S. K. 1999 Receptivity to surface roughness near a swept leading edge. J. Fluid Mech. 380, 141168.CrossRefGoogle Scholar
Crouch, J. D. 1992 Localized receptivity of boundary layers. Phys. Fluids A 4 (7), 14081414.CrossRefGoogle Scholar
Feng, H. & Mavriplis, C. 2002 Adaptive spectral element simulations of thin flame sheet deformation. J. Sci. Comput. 17 (1–3), 385395.CrossRefGoogle Scholar
Fischer, P., Kruse, J., Mullen, J., Tufo, H., Lottes, J. & Kerkemeier, S. 2008 NEK5000 – Open Source Spectral Element CFD solver. Scholar
Fischer, P. F. 1997 An overlapping Schwarz method for spectral element solution of the incompressible Navier–Stokes equations. J. Comput. Phys. 133, 84101.CrossRefGoogle Scholar
Fischer, P. F. & Mullen, J. 2001 Filter-based stabilization of spectral element methods. C. R. Acad. Sci. Paris 332 (Série I), 265270.CrossRefGoogle Scholar
Fuciarelli, D., Reed, H. & Lyttle, I. 2000 Direct numerical simulation of leading-edge receptivity to sound. AIAA J. 38 (7), 11591165.CrossRefGoogle Scholar
Goldstein, M. E. 1983 The evolution of Tollmien–Schlichting waves near a leading edge. J. Fluid Mech. 127, 5981.CrossRefGoogle Scholar
Goldstein, M. E. & Hultgren, L. S. 1987 A note on the generation of Tollmien–Schlichting waves by sudden surface-curvature change. J. Fluid Mech. 181, 519525.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, 2141.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, 231260.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, 443453.CrossRefGoogle Scholar
Goldstein, M. E. & Wundrow, D. W. 1998 On the environmental realizability of algebraically growing disturbances and their relation to Klebanoff modes. Theoret. Comput. Fluid Dyn. 10, 171186.CrossRefGoogle Scholar
Haddad, O. M. & Corke, T. C. 1998 Boundary layer receptivity to free-stream sound on parabolic bodies. J. Fluid Mech. 368, 126.CrossRefGoogle Scholar
Hammerton, P. W. & Kerschen, E. J. 1996 Boundary-layer receptivity for a parabolic leading edge. J. Fluid Mech. 310, 243267.CrossRefGoogle Scholar
Hammerton, P. W. & Kerschen, E. J. 1997 Boundary-layer receptivity for a parabolic leading edge. Part 2. The small-Strouhal-number limit. J. Fluid Mech. 353, 205220.CrossRefGoogle Scholar
Hammerton, P. W. & Kerschen, E. J. 2005 Leading-edge receptivity for bodies with mean aerodynamic loading. J. Fluid Mech. 535, 132.CrossRefGoogle Scholar
Heinrich, R. A. & Kerschen, E. J. 1989 Leading-edge boundary-layer receptivity to free-stream disturbance structures. Z. Angew. Math. Mech. 69 (6), T596T598.Google Scholar
Kendall, J. M. 1985 Experimental study of disturbances produced in a pre-transitional laminar boundary layer by weak free-stream turbulence. AIAA Paper 85-1695, pp. 1–10.Google Scholar
Kendall, J. M. 1998 Experiments on boundary-layer receptivity to free stream turbulence. AIAA Paper 98-0530.Google Scholar
Kerschen, E. J., Choudhari, M. & Heinrich, R. A. 1990 Generation of boundary layer instability waves by acoustic and vortical free-stream disturbances. In Laminar-Turbulent Transition: Proceedings of the IUTAM Symposium, Toulouse, France, pp. 477488. Springer.CrossRefGoogle Scholar
Lin, N., Reed, H. & Saric, W. 1992 Effect of leading edge geometry on boundary-layer receptivity to free stream sound. In Instability, Transition and Turbulence (ed. Hussaini, M., Kumar, A. & Streett, C.), pp. 421440. Springer.CrossRefGoogle Scholar
Maday, Y. & Patera, A. T. 1989 Spectral element methods for the Navier–Stokes equations. In State of the Art Surveys in Computational Mechanics (ed. Noor, A. K.), pp. 71143. ASME.Google Scholar
Maday, Y., Patera, A. T. & Ronquist, E. M. 1990 An operator-integration-factor splitting method for time-dependent problems: application to incompressible fluid flow. J. Sci. Comput. 5 (4), 310337.CrossRefGoogle Scholar
Matsubara, M. & Alfredsson, P. H. 2001 Disturbance growth in boundary layers subjected to free-stream turbulence. J. Fluid Mech. 430, 149168.CrossRefGoogle Scholar
Nagarajan, S., Lele, S. K. & Ferziger, J. H. 2007 Leading-edge effects in bypass transition. J. Fluid Mech. 572, 471504.CrossRefGoogle Scholar
Ohlsson, J., Schlatter, P., Mavriplis, C. & Henningson, D. S. 2009 The spectral-element and pseudo-spectral methods – a comparative study. In ICOSAHOM 09, Lecture Notes in Computational Science and Engineering. Springer, Trondheim Norway.Google Scholar
Patera, A. T. 1984 A spectral element method for fluid dynamics: laminar flow in a channel expansion. J. Comput. Phys. 54, 468488.CrossRefGoogle Scholar
Rogler, H. L. & Reshotko, E. 1976 Spatially decaying array of vortices. Phys. Fluids 19 (12), 18431850.CrossRefGoogle Scholar
Saric, W. S., Reed, H. L. & Kerschen, E. J. 2002 Boundary-layer receptivity to free stream disturbances. Annu. Rev. Fluid Mech. 34, 291319.CrossRefGoogle Scholar
Schrader, L.-U. 2008 Receptivity of boundary layers under pressure gradient. Tech. Rep. TRITA-MEK 2008:08. Department of Mechanics, Royal Institute of Technology (KTH).Google Scholar
Schrader, L.-U., Brandt, L. & Henningson, D. S. 2009 Receptivity mechanisms in three-dimensional boundary-layer flows. J. Fluid Mech. 618, 209241.CrossRefGoogle Scholar
Tufo, H. M. & Fischer, P. F. 1999 Terascale spectral element algorithms and implementations. In Supercomputing, ACM/IEEE 1999 Conference, Portland, OR.Google Scholar
Wanderley, J. B. V. & Corke, T. C. 2001 Boundary layer receptivity to free-stream sound on elliptic leading edges of flat plates. J. Fluid Mech. 429, 121.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, 229262.CrossRefGoogle Scholar
Xiong, Z. & Lele, S. K. 2007 Stagnation-point flow under free-stream turbulence. J. Fluid Mech. 590, 133.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, 111147.CrossRefGoogle Scholar