Skip to main content
×
×
Home

Influence of localised smooth steps on the instability of a boundary layer

  • Hui Xu (a1), Jean-Eloi W. Lombard (a1) and Spencer J. Sherwin (a1)
Abstract

We consider a smooth, spanwise-uniform forward-facing step defined by a Gauss error function of height 4 %–30 % and four times the width of the local boundary layer thickness $\unicode[STIX]{x1D6FF}_{99}$ . The boundary layer flow over a smooth forward-facing stepped plate is studied with particular emphasis on stabilisation and destabilisation of the two-dimensional Tollmien–Schlichting (TS) waves and subsequently on three-dimensional disturbances at transition. The interaction between TS waves at a range of frequencies and a base flow over a single or two forward-facing smooth steps is conducted by linear analysis. The results indicate that for a TS wave with a frequency ${\mathcal{F}}\in [140,160]$ ( ${\mathcal{F}}=\unicode[STIX]{x1D714}\unicode[STIX]{x1D708}/U_{\infty }^{2}\times 10^{6}$ , where $\unicode[STIX]{x1D714}$ and $U_{\infty }$ denote the perturbation angle frequency and free-stream velocity magnitude, respectively, and $\unicode[STIX]{x1D708}$ denotes kinematic viscosity), the amplitude of the TS wave is attenuated in the unstable regime of the neutral stability curve corresponding to a flat plate boundary layer. Furthermore, it is observed that two smooth forward-facing steps lead to a more acute reduction of the amplitude of the TS wave. When the height of a step is increased to more than 20 % of the local boundary layer thickness for a fixed width parameter, the TS wave is amplified, and thereby a destabilisation effect is introduced. Therefore, the stabilisation or destabilisation effect of a smooth step is typically dependent on its shape parameters. To validate the results of the linear stability analysis, where a TS wave is damped by the forward-facing smooth steps direct numerical simulation (DNS) is performed. The results of the DNS correlate favourably with the linear analysis and show that for the investigated frequency of the TS wave, the K-type transition process is altered whereas the onset of the H-type transition is delayed. The results of the DNS suggest that for the perturbation with the non-dimensional frequency parameter ${\mathcal{F}}=150$ and in the absence of other external perturbations, two forward-facing smooth steps of height 5 % and 12 % of the boundary layer thickness delayed the H-type transition scenario and completely suppressed for the K-type transition. By considering Gaussian white noise with both fixed and random phase shifts, it is demonstrated by DNS that transition is postponed in time and space by two forward-facing smooth steps.

  • View HTML
    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Influence of localised smooth steps on the instability of a boundary layer
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Influence of localised smooth steps on the instability of a boundary layer
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Influence of localised smooth steps on the instability of a boundary layer
      Available formats
      ×
Copyright
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Corresponding author
Email address for correspondence: jfluidmech@imperial.ac.uk
References
Hide All
Bassom, A. P. & Hall, P. 1994 The receptivity problem for O(1) wavelength Görtler vortices. Proc. R. Soc. Lond. A 446, 499516.
Bassom, A. P. & Seddougui, S. O. 1995 The receptivity problem for O(1) wavelength Görtler vortices. Theor. Comput. Fluid Dyn. 7, 317339.
Berlin, S., Wiegel, M. & Henningson, D. S. 1999 Numerical and experimental investigations of oblique boundary layer transition. J. Fluid Mech. 393, 2357.
Bertolotti, F. P. 1996 On the birth and evolution of disturbances in three-dimensional boundary layers. In IUTAM Symposium on Nonlinear Instability and Transition in Three-Dimensional Boundary Layers (ed. Duck, P. W. & Hall, P.), vol. 35, pp. 247256. Springer.
Bertolotti, F. P. 2003 Response of the Blasius boundary layer to free-stream vorticity. Phys. Fluids 9 (8), 22862299.
Bippes, H. 1999 Basic experiments on transition in three-dimensional boundary layers dominated by crossflow instability. Prog. Aerosp. Sci. 35 (4), 363412.
Borodulin, V. I., Ivanov, A. V., Kachanov, Y. S. & Roschektaev, A. P. 2013 Receptivity cocients at excitation of cross-flow waves by free-stream vortices in the presence of surface roughness. J. Fluid Mech. 716, 487527.
Borodulin, V. I., Kachanov, Y. S. & Koptsev, D. B. 2002 Experimental study of resonant interactions of instability waves in self-similar boundary layer with an adverse pressure gradient: part I. Tuned resonances. J. Turbul. 3 (62), 138.
Brehm, C., Koevary, C., Dackermann, T & Fasel, H. F.2011 Numerical investigations of the influence of distributed roughness on Blasius boundary layer stability. AIAA Paper 2011–563.
Cantwell, C. D., Moxey, D., Comerford, A., Bolis, A., Rocco, G., Mengaldo, G., Grazia, D. D. E., Yakovlev, S., Lombard, J. E., Ekelschot, D. et al. 2015 Nektar++: an open-source spectral/hp element framework. Comput. Phys. Commun. 192, 205219.
Choudhari, M.1994 Boundary-layer receptivity to three-dimensional unsteady vortical disturbances in free stream. AIAA Paper 96–0181.
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.
Corke, T. C., Sever, A. B. & Morkovin, M. V. 1986 Experiments on transition enhancements by distributed roughness. Phys. Fluids 29, 31993213.
Cossu, C. & Brandt, L. 2002 Stabilization of Tollmien–Schlichting waves by finite amplitude optimal streaks in the Blasius boundary layer. Phys. Fluids 14 (8), L57L60.
Cossu, C. & Brandt, L. 2004 On Tollmien–Schlichting-like waves in streaky boundary layers. Eur. J. Mech. (B/Fluids) 23, 815833.
Crouch, J. D. 1994 Distributed excitation of Tollmien–Schlichting waves by vortical free stream disturbances. Phys. Fluids 6 (1), 217223.
Crouch, J., Kosorygin, V. & Ng, L. 2006 Modeling the effects of steps on boundary-layer transition. In IUTAM Symposium on Laminar–Turbulent Transition (ed. Govindarajan, R.), Fluid Mechanics and Its Applications, vol. 78, pp. 3744. Springer.
Davies, C. & Carpenter, P. W. 1996 Numerical simulation of the evolution of Tollmien–Schlichting waves over finite compliant panels. J. Fluid Mech. 335, 361392.
Denier, J. P., Hall, P. & Seddougui, S. O. 1991 On the receptivity problem for Görtler vortices: vortex motions induced by wall roughness. Phil. Trans. R. Soc. A 335, 5185.
Dietz, A. J. 1999 Local boundary-layer receptivity to a convected free-stream disturbance. J. Fluid Mech. 378, 291317.
Downs, R. D. & Fransson, J. H. M. 2014 Tollmien–Schlichting wave growth over spanwise-periodic surface patterns. J. Fluid Mech. 754, 3974.
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.
Duck, P. W., Ruban, A. I. & Zhikharev, C. N. 1996 Generation of Tollmien–Schlichting waves by free-stream turbulence. J. Fluid Mech. 312, 341371.
Edelmann, C. & Rist, U. 2013 Impact of forward-facing steps on laminar–turbulent transition in transonic flows without pressure gradient. In 51st AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition.
Edelmann, C. & Rist, U. 2015 Impact of forward-facing steps on laminar–turbulent transition in transonic flows. AIAA J. 53 (9), 25042511.
Fasel, H. & Konzelmann, U. 1990 Non-parallel stability of a flat plate boundary layer using the complete Navier–Stokes equations. J. Fluid Mech. 221, 311347.
Fransson, J. H. M., Brandt, L., Talamelli, A. & Cossu, C. 2005 Experimental study of the stabilization of Tollmien–Schlichting waves by finite amplitude streaks. Phys. Fluids 17 (5), 054110.
Fransson, J. H. M., Talamelli, A., Brandt, L. & Cossu, C. 2006 Delaying transition to turbulence by a passive mechanism. Phys. Rev. Lett. 96, 064501.
Gao, B., Park, D. H. & Park, S. O. 2011 Stability analysis of a boundary layer over a hump using parabolized stability equations. Fluid Dyn. Res. 43 (5), 055503.
Garzon, G. A. & Roberts, M. W.2013 Effect of a small surface wave on boundary-layer transition. AIAA Paper 2013–3110.
Gaster, M. 1965 On the generation of spatially growing waves in a boundary layer. J. Fluid Mech. 22, 433441.
Goldstein, M. E. 1983 The evolution of Tollien–Schlichting waves near a leading edge. J. Fluid Mech. 127, 5981.
Goldstein, M. E. 1985 Scattering of acoustic waves into Tollmien–Schlichting waves by small streamwise variations in surface geometry. J. Fluid Mech. 154, 509529.
Goldstein, M. E. & Hultgren, L. S. 1987 A note on the generation of Tollmien–Schlichting waves by sudden curvature change. J. Fluid Mech. 181, 519525.
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.
Gray, W. E.1952 The effect of wing sweep on laminar flow. Tech. Rep. RAE TM Aero 255. British Royal Aircraft Establishment.
Hall, P. 1990 Görtler vortices in growing boundary layers: the leading edge receptivity problem, linear growth and the nonlinear breakdown stage. Mathematika 37 (02), 151189.
Hammond, D. A. & Redekopp, L. G. 1998 Local and global instability properties of separation bubbles. Eur. J. Mech.—(B/Fluids) 27 (2), 145164.
Herbert, T. 1988 Secondary instability of boundary-layers. Annu. Rev. Fluid Mech. 20, 487526.
Horton, H. P.1968 A semi-empirical theory for the growth and bursting of laminar separation bubbles. PhD dissertation, University of London.
Huai, X., Joslin, R. D. & Piomelli, U. 1997 Large-eddy simulation of trantion to turbulence in boundary layers. Theor. Comput. Fluid Dyn. 9, 149163.
Jones, B. M. 1938 Stalling. J. R. Aero. Soc. 38, 747770.
Kachanov, Y. S. 1994 Physical mechanisms of laminar-boundary-layer transition. Annu. Rev. Fluid Mech. 26, 4110482.
Kachanov, Y. S., Kozlov, V. V. & Levchenko, V. Y. 1979a Origin of Tollmien–Schlichting waves in boundary layer under the influence of external disturbances. Fluid Dyn. 13, 704711.
Kachanov, Y. S., Kozlov, V. V., Levchenko, V. Y. & Maksimov, V. P. 1979b Transformation of external disturbances into the boundary layer waves. In Sixth International Conference on Numerical Methods in Fluid Dyn (ed. Cabannes, H., Holt, M. & Rusanov, V.), pp. 299307. Springer.
Kachanov, Y. S. & Levchenko, V. Y. 1984 The resonant interaction of disturbances at laminar–turbulent transition in a boundary layer. J. Fluid Mech. 138, 209247.
Karniadakis, G., Israeli, M. & Orszag, S. 1991 High-order splitting methods for the incompressible Navier–Stokes equations. J. Comput. Phys. 97 (2), 414443.
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.
Kendall, J. M.1990 Boundary-layer receptivity to free stream turbulence. AIAA Paper 90–1504.
Kendall, J. M. 1991 Studies on laminar boundary-layer receptivity to free stream turbulence near a leading edge. In Boundary Layer Stability and Transition to Turbulence (ed. Reda, D. C., Reed, H. L. & Kobayashi, R. K.), vol. 114, pp. 2330. ASME FED.
Kerschen, E. J.1989 Boundary layer receptivity. AIAA Paper 89–1109.
Kerschen, E. J. 1990 Boundary layer receptivity theory. Appl. Mech. Rev. 43, S152S157.
Klebanoff, P. S., Tidstrom, K. D. & Sargent, L. M. 1962 The three-dimensional nature of boundary-layer instability. J. Fluid Mech. 12 (1), 134.
Kozlov, V. V. & Ryzhov, O. S. 1990 Receptivity of boundary layers: asymptotic theory and experiment. Proc. R. Soc. Lond. A 429, 341373.
Laurien, E. & Kleiser, L. 1989 Numerical simulation of boundary-layer transition and transition control. J. Fluid Mech. 199, 403440.
Morkovin, M. V.1969a Critical evaluation of transition from laminar to turbulent shear layers with emphasis on hypersonically travelling bodies. Tech. Rep. AFFDL-TR 68-149. US Air Force Flight Dynamics Laboratory, Wright Patterson Air Force Base, Ohio.
Morkovin, M. V. 1969b The many faces of transition. In Viscous Drag Reduction (ed. Wells, C. S.), pp. 131. Plenum.
Murdock, J. W. 1980 The generation of a Tollmien–Schlichting wave by a sound wave. Proc. R. Soc. Lond. A 372, 517534.
Nenni, J. P. & Gluyas, G. L. 1966 Aerodynamic design and anlaysis of an LFC surface. Astron. Aeronaut. 4 (7), 131.
Nishioka, M. & Morkovin, M. V. 1986 Boundary-layer receptivity to unsteady pressure gradients: experiments and overview. J. Fluid Mech. 171, 219261.
Park, D. & Park, S. O. 2013 Linear and non-linear stability analysis of incompressible boundary layer over a two-dimensional hump. Comput. Fluids 73, 8096.
Perraud, J., Arnal, D., Seraudie, A. & Tran, D. 2004 Laminar–turbulent transition on aerodynamic surfaces with imperfections. In NATO Research and Technology Organisation Applied Vehicle Technology Panel 111 Symposium, Prague, Czech Republic.
Reed, H. L. & Saric, W. S. 1989 Stability of three-dimensional boundary layers. Annu. Rev. Fluid Mech. 21, 235284.
Reibert, M. S., Saric, W. S., Carrillo, R. B. Jr. & Chapman, K. L. 1996 Experiments in nonlinear saturation of stationary crossflow vortices in a swept-wing boundary layer. AIAA Paper 96–0184.
Rist, U. 1993 Nonlinear effects of two-dimensional and three-dimensional disturbances on laminar separation bubbles. In Nonlinear Instability of Nonparallel Flows (ed. Lin, S. P., Phillips, W. R. C. & Valentine, D. T.), pp. 330339. Springer.
Rogler, H. L. & Reshotko, E. 1975 Disturbances in a boundary layer introduced by a low intensity array of vortices. SIAM J. Appl. Mech. 28 (2), 431462.
Ruban, A. I. 1984 On Tollmien–Schlichting wave generation by sound. Izv. Akad. Nauk SSSR Mekh. Zhidk. Gaza 5, 4452.
Ruban, A. I. 1985 On tollmien–schlichting wave generation by sound. In Laminar–Turbulent Transition (ed. Kozlov, V. V.), pp. 313320. Springer.
Ruban, A. I., Bernots, T. & Pryce, D. 2013 Receptivity of the boundary layer to vibrations of the wing surface. J. Fluid Mech. 723, 480528.
Saric, W. S. 1990 Low-speed experiments: requirements for stability measurements. In Instability and Transition (ed. Hussaini, M. Y. & Voight, R. G.), ICASE/NASA LaRC Series, vol. 1, pp. 162174. Springer.
Saric, W. S. 1994 Görtler vortices. Annu. Rev. Fluid Mech. 26, 379409.
Saric, W. S., Carrillo, R. B. Jr & Reibert, M. S. 1998 Nonlinear stability and transition in 3-D boundary layers. Meccanica 33, 469487.
Saric, W. S., Reed, H. L. & Kerschen, E. J. 2002 Boundary-layer receptivity to free-stream disturbances. Annu. Rev. Fluid Mech. 34, 251276.
Saric, W. S., Reed, H. L. & White, E. B. 2003 Stability and transition of three-dimensional boundary layers. Annu. Rev. Fluid Mech. 35, 413440.
Sayadi, T., Hamman, C. W. & Moin, P. 2013 Direct numerical simulation of complete H-type and K-type transitions with implications for the dynamics of turbulent boundary layers. J. Fluid Mech. 724, 480509.
Schlichting, H. & Gersten, K. 1968 Boundary-Layer Theory. Macgraw-Hill.
Schubauer, G. B. & Skramstad, H. K. 1948 Laminar-boundary-layer oscillations and transition on a flat plat. NASA TR-909.
Shahinfar, S., Sattarzadeh, S. S., Fransson, J. H. M. & Talamelli, A. 2012 Revival of classical vortex generators now for transition delay. Phys. Rev. Lett. 109, 074501.
Stuart, J. T. 1963 Hydrodynamic stability. In Laminar Boundary Layer (ed. Rosenhead, L.), pp. 492579. Oxford University Press.
Templelmann, D. T. 2011 Numerical study of boundary-layer receptivity on a swept wing. AIAA J. 20113294.
Theofilis, V. 2000 Global linear instability in laminar separated boundary layer flow. In Proceedings of the IUTAM Laminar–Turbulent Symposium V (ed. Fasel, H. & Saric, W.), pp. 663668. Springer.
Wang, Y. X. & Gaster, M. 2005 Effect of surface steps on boundary layer transition. Exp. Fluids 39 (4), 679686.
Wörner, A., Rist, U. & Wagner, S. 2003 Humps/steps influence on stability characteristics of two-dimensional laminar boundary layer. AIAA J. 41 (2), 192197.
Wu, X. S. 2001a On local boundary-layer receptivity to vortical disturbances in the free stream. J. Fluid Mech. 449, 373393.
Wu, X. S. 2001b 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, 91133.
Wu, X. S. & Hogg, L. W. 2006 Acoustic radiation of Tollmien–Schlichting waves as they undergo rapid distortion. J. Fluid Mech. 550, 307347.
Xu, H., Sherwin, S., Hall, P. & Wu, X. S. 2016 The behaviour of Tollmien–Schlichting waves undergoing small-scale localised distortions. J. Fluid Mech. 792, 499525.
Young, A. D. & Horton, H. P. 1966 Stalling. AGARD CP 4, 779811.
Zavol’skii, N. A., Reutov, V. P. & Rybushkina, 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, 355361.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

JFM classification

Metrics

Full text views

Total number of HTML views: 25
Total number of PDF views: 602 *
Loading metrics...

Abstract views

Total abstract views: 512 *
Loading metrics...

* Views captured on Cambridge Core between 15th March 2017 - 20th August 2018. This data will be updated every 24 hours.