Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-12-01T04:04:36.142Z Has data issue: false hasContentIssue false
  • English
  • Français

Stationary crossflow vortices near the leading edge of three-dimensional boundary layers: the role of non-parallelism and excitation by surface roughness

Published online by Cambridge University Press:  20 April 2018

Adam Butler Open the ORCID record for Adam Butler [Opens in a new window]  and
Xuesong Wu Open the ORCID record for Xuesong Wu [Opens in a new window]
Adam Butler*
Affiliation:
Department of Mathematics, Imperial College London, 180 Queens Gate, London SW7 2AZ, UK
Xuesong Wu
Affiliation:
Department of Mathematics, Imperial College London, 180 Queens Gate, London SW7 2AZ, UK
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Non-parallelism, i.e. the effect of the slow variation of the boundary-layer flow in the chordwise and spanwise directions, in general produces a higher-order correction to the growth rate of instability modes. Here we investigate stationary crossflow vortices, which arise due to the instability of the three-dimensional boundary layer over a swept wing, focusing on a region near the leading edge where non-parallelism plays a leading-order role in their development. In this regime, the vortices align themselves with the local wall shear at leading order, and so have a marginally separated triple-deck structure, consisting of the inviscid main boundary layer, an upper deck and a viscous sublayer. We find that the streamwise (and spanwise) variations of both the base flow and the modal shape must be accounted for. An explicit expression for the growth rate is derived that shows a neutral point occurs in this regime, downstream of which non-parallelism has a stabilising effect. Stationary crossflow vortices thus have a viscous and non-parallel genesis near the leading edge. If an ‘effective pressure minimum’ occurs within this region then the growth rate becomes unbounded, and so the previous analysis is regularised within a localised region around it. A new instability is identified. The mode maintains its three-tiered structure, but the pressure perturbation now plays a passive role. Downstream, the instability evolves into a Cowley, Hocking & Tutty (Phys. Fluids, vol. 28, 1985, pp. 441–443) instability associated with a critical layer located in the lower deck. Finally, we consider the receptivity of the flow in the non-parallel regime: generation of stationary crossflow modes by arrays of chordwise-localised, spanwise-periodic surface roughness elements. The flow responds differently to different Fourier spectral content of the roughness, giving the lower deck a two-part structure. We find that roughness elements with sharper edges generate stronger modes. For roughness elements of fairly moderate height, the resulting nonlinear forcing leads to the so-called super-linearity of receptivity, namely, the amplitude of the generated crossflow mode deviates from the linear dependence on the roughness height even though the perturbation in the boundary layer remains linear.

JFM classification

Aerodynamics: Aerodynamics Boundary Layers: Boundary layer receptivity Boundary Layers: Boundary layer stability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 845 , 25 June 2018 , pp. 93 - 140
Check for updates
Copyright
© 2018 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

Atkin, C., Sunderland, R. & Mughal, S. 2006 Parametric PSE studies on distributed roughness laminar flow control. AIAA Paper 2006–3694.Google Scholar
Bertolotti, F. P. 2000a On the connection between cross-flow vortices and attachment-line instabilities. In Laminar-Turbulent Transition, pp. 625–630. Springer.CrossRefGoogle Scholar
Bertolotti, F. P. 2000b Receptivity of three-dimensional boundary-layers to localized roughness and suction. Phys. Fluids 12, 1799–1809.Google Scholar
Bippes, H. 1999 Basic experiments on transition in three-dimensional boundary layers dominated by crossflow instability. Prog. Aerosp. Sci. 35, 363–412.CrossRefGoogle Scholar
Bippes, H., Müller, B. & Wagner, M. 1991 Measurements and stability calculations of the disturbance growth in an unstable three-dimensional boundary layer. Phys. Fluids A 3 (10), 2371–2377.Google Scholar
Bonfigli, G. & Kloker, M. 2007 Secondary instability of crossflow vortices: validation of the stability theory by direct numerical simulation. J. Fluid Mech. 583, 229–272.CrossRefGoogle Scholar
Butler, A. & Wu, X. 2015 Non-parallel-flow effects on stationary crossflow vortices at their genesis. Procedia IUTAM 14, 311–320.CrossRefGoogle Scholar
Cebeci, T. & Cousteix, J. 2005 Modeling and Computation of Boundary-Layer Flows: Laminar, Turbulent and Transitional Boundary Layers in Incompressible and Compressible Flows. Horizons Pub.Google Scholar
Chernoray, V. G., Dovgal, A. V., Kozlov, V. V. & Löfdahl, L. 2005 Experiments on secondary instability of streamwise vortices in a swept-wing boundary layer. J. Fluid Mech. 534, 295–325.Google Scholar
Choudhari, M.1994a On long-wavelength asymptote of neutral stability curve for stationary crossflow vortices. High Tech. Rep. No. HTC-9405.Google Scholar
Choudhari, M. 1994b Roughness-induced generation of crossflow vortices in three-dimensional boundary layers. Theor. Comput. Fluid Dyn. 6, 1–30.Google Scholar
Choudhari, M. 1995 Long-wavelength asymptotics of unstable crossflow modes, including the effect of surface curvature. Proc. R. Soc. Lond. A 451, 515–541.Google Scholar
Choudhari, M. & Duck, P. W. 1996 Nonlinear excitation of inviscid stationary vortex instabilities in a boundary-layer flow. In IUTAM Symposium on Nonlinear Instability and Transition in Three-Dimensional. Boundary Layers, pp. 409–422. Springer.CrossRefGoogle Scholar
Choudhari, M. & Streett, C. L. 1990 Boundary layer receptivity phenomena in three-dimensional and high-speed boundary layers. AIAA Paper 1990–5258.Google Scholar
Collis, S. S. & Lele, S. K. 1999 Receptivity to surface roughness near a swept leading edge. J. Fluid Mech. 280, 141–168.Google Scholar
Cowley, S. J., Hocking, L. M. & Tutty, D. R. 1985 The stability of solutions of the classical unsteady boundary-layer equation. Phys. Fluids 28, 441–443.CrossRefGoogle Scholar
Dehyle, H. & Bippes, H. 1996 Disturbance growth in an unstable three-dimensional boundary layer and its dependence on environmental conditions. J. Fluid Mech. 316, 73–113.CrossRefGoogle Scholar
Downs, R. S. & White, E. B. 2013 Free-stream turbulence and the development of cross-flow disturbances. J. Fluid Mech. 735, 347–380.Google Scholar
Duan, L., Choudhari, M. & Li, F. 2013 Direct numerical simulation of transition in a swept-wing boundary layer. AIAA Paper 2013–2617.Google 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.Google Scholar
Fischer, T. M. & Dallmann, U. 1991 Primary and secondary instability analysis of a three-dimensional boundary-layer flow. Phys. Fluids A 3 (10), 2378–2391.Google Scholar
Gajjar, J. S. B. 1996 On the nonlinear evolution of a stationary cross-flow vortex in a fully three-dimensional boundary layer flow. In IUTAM Symposium on Nonlinear Instability and Transition in Three-Dimensional Boundary Layers, pp. 317–326. Springer.Google Scholar
Gaponenko, V. R., Ivanov, A. V., Kachanov, Y. S. & Crouch, J. D. 2002 Swept-wing boundary-layer receptivity to surface non-uniformities. J. Fluid Mech. 461, 93–126.CrossRefGoogle Scholar
Goldstein, M. E. 1983 The evolution of Tollmien–Schlichting waves near a leading edge. J. Fluid Mech. 127, 59–81.Google 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.Google Scholar
Goldstein, M. E., Leib, S. J. & Cowley, S. J. 1987 Generation of Tollmien–Schlichting waves on interactive marginally separated flows. J. Fluid Mech. 181, 485–517.Google Scholar
Gregory, N., Stuart, J. T. & Walker, W. S. 1955 On the stability of three-dimensional boundary layers with application to the flow due to a rotating disk. Phil. Trans. R. Soc. Lond. A 155–199.Google Scholar
Hall, P. 1986 An asymptotic investigation of the stationary modes of instability of the boundary layer on a rotating disc. Proc. R. Soc. Lond. A 406, 93–106.Google Scholar
Harris, C. D.1990 NASA supercritical airfoils: a matrix of family-related airfoils. Tech. Rep. NASA.Google Scholar
Haynes, T. S. & Reed, H. L. 2000 Simulation of swept-wing vortices using nonlinear parabolized stability equations. J. Fluid Mech. 405, 325–349.Google Scholar
Högberg, M. & Henningson, D. 1998 Secondary instability of cross-flow vortices in Falkner–Skan--Cooke boundary layers. J. Fluid Mech. 368, 339–357.Google Scholar
Hosseini, S. M., Tempelmann, D., Hanifi, A. & Henningson, D. S. 2013 Stabilization of a swept-wing boundary-layer by distributed roughness elements. J. Fluid Mech. 718, R1.Google Scholar
Hunt, L. E. & Saric, W. S. 2011 Boundary-layer receptivity to three-dimensional roughness arrays on a swept-wing. AIAA Paper 2011–3881.Google Scholar
Joslin, R. D.1998 Overview of laminar flow control. NASA Tech. Rep. 1998-208705.Google Scholar
Kurz, H. B. E. & Kloker, M. 2014 Receptivity of a swept-wing boundary layer to micron-sized discrete roughness elments. J. Fluid Mech. 755, 62–82.Google Scholar
Li, F., Choudhari, M., Carpenter, M., Malik, M., Chang, C.-L. & Streett, C. 2016 Control of crossflow transition at high Reynolds numbers using discrete roughness elements. AIAA J. 54 (1), 39–52.CrossRefGoogle Scholar
Li, F., Choudhari, M., Chang, C.-L., Streett, C. & Carpenter, M. 2009 Roughness based crossflow transition control: a computational assessment. AIAA Paper 2009–4105.Google Scholar
Li, F., Choudhari, M., Chang, C.-L., Streett, C. & Carpenter, M. 2011 Computational modeling of roughness-based laminar flow control on a subsonic swept wing. AIAA J. 49 (3), 520–529.Google Scholar
Li, F., Choudhari, M., Duan, L. & Chang, C.-L. 2014 Nonlinear development and secondary instability of traveling crossflow vortices. Phys. Fluids 26, 064104–1–19.Google Scholar
Malik, M. R., Li, F. & Chang, C.-L. 1994 Crossflow disturbances in three-dimensional boundary layers: nonlinear development, wave interaction and secondary instability. J. Fluid Mech. 268, 1–36.CrossRefGoogle Scholar
Malik, M. R., Li, F., Choudhari, M. & Chang, C.-L. 1999 Secondary instability of crossflow vortices and swept-wing boundary-layer transition. J. Fluid Mech. 399, 85–115.Google Scholar
Morkovin, M. V., Reshotko, E. & Herbert, T. 1994 Transition in open flow systems – a reassessment. Bull. Am. Phys. Soc. 39, 1882.Google Scholar
Müller, B. 1990 Experimental study of the travelling waves in a three-dimensional boundary layer. In Laminar-Turbulent Transition, pp. 489–498. Springer.CrossRefGoogle Scholar
Müller, B. & Bippes, H. 1988 Experimental study of instability modes in a three-dimensional boundary layer. In Fluid Dyn. Three-Dimens. Turbul. Shear Flows Transit., AGARD CP, vol. 438, p. 13.1–13.15. North Atlantic Treaty Organization, Advisory Group for Aerospace Research and Development.Google Scholar
Ng, L. L. & Crouch, J. D. 1999 Roughness-induced receptivity to crossflow vortices on a swept wing. Phys. Fluids 11, 432–438.Google Scholar
Radeztsky, R. H., Reibert, M. S. & Saric, W. S. 1999 Effect of isolated micron-sized roughness on transition in swept-wing flows. AIAA J. 37, 1370–1377.Google Scholar
Reibert, M., Saric, W. S., Carrillo, R. Jr & Chapman, K. 1996 Experiments in nonlinear saturation of stationary crossflow vortices in a swept-wing boundary layer. AIAA Paper 1996–184.Google Scholar
Rizzetta, D. P., Visbal, M. R., Reed, H. L. & Saric, W. S. 2010 Direct numerical simulation of discrete roughness on a swept-wing leading edge. AIAA J. 48, 2260–2273.Google Scholar
Ruban, A. I. 1984 On the generation of Tollmien–Schlichting waves by sound. Izv. Akad. Nauk SSSR Mekh. Zhidk. Gaza 19, 709–717.Google Scholar
Saric, W. S., Carrillo, R. Jr & Reibert, M. S. 1998 Leading-edge roughness as a transition control mechanism. AIAA Paper 1998–781.Google Scholar
Saric, W. S. & Reed, H. L. 2003 Crossflow instabilities – theory & technology. AIAA Paper 2003–771.Google Scholar
Saric, W. S., Reed, H. L. & White, E. B. 2003 Stability and transition of three-dimensional boundary layers. Annu. Rev. Fluid Mech. 35, 413–440.CrossRefGoogle Scholar
Saric, W. S., West, D. E., Tufts, M. W. & Reed, H. L. 2015 Flight test experiments on discrete roughness element technology for laminar flow control. AIAA Paper 2015–539.Google Scholar
Serpieri, J. & Kotsonis, M. 2016 Three-dimensional organization of primary and secondary crossflow instability. J. Fluid Mech. 799, 200–245.Google Scholar
Tempelmann, D., Schrader, L. U., Hanifi, A., Brandt, L. & Henningson, D. S. 2012 Swept wing boundary-layer receptivity to localized surface roughness. J. Fluid Mech. 711, 516–544.CrossRefGoogle Scholar
Theofilis, V., Fedorov, A., Obrist, D. & Dallmann, U. C. 2003 The extended Görtler–Hämmerlin model for linear instability of three-dimensional incompressible swept attachment-line boundary layer flow. J. Fluid Mech. 487, 271–313.Google Scholar
Trefethen, L. N. 2000 Spectral Methods in MATLAB, vol. 10. SIAM.CrossRefGoogle Scholar
Wassermann, P. & Kloker, M. 2002 Mechanisms and passive control of crossflow-vortex-induced transition in a three-dimensional boundary layer. J. Fluid Mech. 456, 49–84.Google Scholar
Wassermann, P. & Kloker, M. 2003 Transition mechanisms induced by travelling crossflow vortices in a three-dimensional boundary layer. J. Fluid Mech. 483, 67–89.Google Scholar
White, E. B. & Saric, W. S. 2005 Secondary instability of crossflow vortices. J. Fluid Mech. 525, 275–308.Google Scholar
Woodruff, M. J., Saric, W. S. & Reed, H. L. 2011 Receptivity measurements on a swept-wing model. AIAA Paper 2011–3882.Google Scholar
Wu, X. 1999 Generation of Tollmien–Schlichting waves by convecting gusts interacting with sound. J. Fluid Mech. 397, 285–316.Google Scholar
Wu, X. 2001a On local boundary-layer receptivity to vortical disturbances in the free stream. J. Fluid Mech. 431, 91–133.Google Scholar
Wu, X. 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, 91–133.Google Scholar
Zhao, D.2011 Instability and receptivity of boundary layers on concave surfaces and swept wings. PhD thesis, Imperial College London.Google Scholar