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A local scattering approach for the effects of abrupt changes on boundary-layer instability and transition: a finite-Reynolds-number formulation for isolated distortions

Published online by Cambridge University Press:  06 June 2017

Zhangfeng Huang  and
Xuesong Wu Open the ORCID record for Xuesong Wu [Opens in a new window]
Zhangfeng Huang
Affiliation:
Department of Mechanics, Tianjin University, Tianjin 300072, China State Key Laboratory of Aerodynamics, China Aerodynamic Research and Development Center, Mianyang Sichuan, 621000, 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]
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Abstract

We investigate the influence of abrupt changes on boundary-layer instability and transition. Such changes can take different forms including a local porous wall, suction/injection and surface roughness as well as junctions between rigid and porous walls. They may modify the boundary conditions and/or the mean flow, and their effects on transition have usually been assessed by performing stability analysis for the modified base flow and/or boundary conditions. However, such a conventional local linear stability theory (LST) becomes invalid if the change occurs over a relatively short scale comparable with, or even shorter than, the characteristic wavelength of the instability. In this case, the influence on transition is through scattering with the abrupt change acting as a local scatter, that is, an instability mode propagating through the region of abrupt change is scattered by the strong streamwise inhomogeneity to acquire a different amplitude. A local scattering approach (LSA) should be formulated instead, in which a transmission coefficient, defined as the ratio of the amplitude of the instability wave after the scatter to that before, is introduced to characterize the effect on instability and transition. In the present study, we present a finite-Reynolds-number formulation of LSA for isolated changes including a rigid plate interspersed by a local porous panel and a wall suction through a narrow slot. When the weak non-parallelism of the unperturbed base flow is ignored, the local scattering problem can be cast as an eigenvalue problem, in which the transmission coefficient appears as the eigenvalue. We also improved the method to take into account the non-parallelism of the unperturbed base flow, where it is found that the weak non-parallelism has a rather minor effect. The general formulation is specialized to two-dimensional Tollmien–Schlichting (T–S) waves. The resulting eigenvalue problem is solved, and full direct numerical simulations (DNS) are performed to verify some of the predictions by LSA. A parametric study indicates that conventional LST is valid only when the change is sufficiently gradual, and becomes either inaccurate or invalid when the scale of the local distortion is short. A local porous panel enhances T–S waves, while a local suction with a moderate mass flux significantly inhibits T–S waves. In the latter case, a comprehensive comparison is made between the theoretical predictions and experimental data, and a satisfactory quantitative agreement was observed.

JFM classification

Boundary Layers: Boundary layer stability Low-Reynolds-number flows: Porous media Waves/Free-surface Flows: Wave scattering
Type
Papers
Information
Journal of Fluid Mechanics , Volume 822 , 10 July 2017 , pp. 444 - 483
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Copyright
© 2017 Cambridge University Press 

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References

Airiau, C., Bottaro, A., Walther, S. & Legendre, D. 2003 A methodology for optimal laminar flow control: application to the damping of Tollmien–Schlichting waves in a boundary layer. Phys. Fluids 15 (5), 1131–1145.Google Scholar
Babucke, A., Kloker, M. & Rist, U. 2008 DNS of a plane mixing layer for the investigation of sound generation mechanisms. Comput. Fluids 37, 360–368.Google Scholar
Bertolotti, F. P., Herbert, Th. & Spalart, P. R. 1992 Linear and nonlinear stability of the Blasius boundary layer. J. Fluid Mech. 242, 441–474.Google Scholar
Cebeci, T. & Egan, D. A. 1989 Prediction of transition due to isolated roughness. AIAA J. 27 (7), 870–875.Google Scholar
Chang, C.-L. & Malik, M. R. 1994 Oblique-mode breakdown and secondary instability in supersonic boundary layers. J. Fluid Mech. 273, 323–360.Google Scholar
Choudhari, M. & Duck, P. W. 1996 Nonlinear excitation of inviscid stationary vortex instabilities in a boundary-layer flow. In Proceeding of IUTAM Symposium on Nonlinear Instability and Transition in Three-Dimensional Boundary Layers, Manchester, UK, 17–20 July 1995 (ed. Duck, P. W. & Philip, H.), pp. 409–422. Springer.Google Scholar
Colonius, T. 2004 Modelling artificial boundary conditions for compressible flow. Annu. Rev. Fluid Mech. 36 (1), 315–345.Google Scholar
Darcy, H. 1856 Les Fontaines Publiques de la Ville de Dijon. Dalmont.Google Scholar
Edelmann, C. A. & Rist, U.2013 Impact of forward-facing steps on laminar–turbulent transition in transonic flows without pressure gradient. AIAA Paper 2013-0080.Google Scholar
Fasel, H. 1976 Investigation of the stability of boundary layers by a finte-difference model of the Navier–Stokes equations. J. Fluid Mech. 78, 355–383.CrossRefGoogle Scholar
Fedorov, A. V. 2011 Transition and stability of high-speed boundary layers. Annu. Rev. Fluid Mech. 43 (1), 79–95.Google Scholar
Fong, K. D., Wang, X. W. & Zhong, X. L.2013 Stabilization of hypersonic boundary layer by 2-D surface roughness. AIAA Paper 2013-2985.Google Scholar
Fujii, K. 2006 Experiment of the two-dimensional roughness effect on hypersonic boundary-layer transition. AIAA J. 43 (4), 731–738.Google Scholar
Gaster, M. 1974 On the effects of boundary-layer growth on flow stability. J. Fluid Mech. 66 (03), 465–480.Google Scholar
Herbert, T. 1997 Parabolized stability equations. Annu. Rev. Fluid Mech. 29 (1), 245–283.CrossRefGoogle Scholar
Huang, Z. F., Cao, W. & Zhou, H. 2005a The mechanism of breakdown in laminar–turbulent transition of a supersonic boundary layer on a flat plate – temporal mode. Sci. China Ser. G 48, 614–625.Google Scholar
Huang, Z. F. & Wu, X. 2015 A non-perturbative approach to spatial instability of weakly non-parallel shear flows. Phys. Fluids 27, 054102.Google Scholar
Huang, Z. F., Zhou, H. & Luo, J. S. 2005b Direct numerical simulation of a supersonic turbulent boundary layer on a flat plate and its analysis. Sci. China Ser. G 48, 626–640.Google Scholar
Kleiser, L. & Zang, T. A. 1991 Direct numerical simulation of transition in wall-bounded shear flows. Annu. Rev. Fluid Mech. 23 (1), 495–537.Google Scholar
Kloker, M., Konzelmann, U. & Fasel, H. 1993 Outflow boundary conditions for spatial Navier–Stokes simulations of transitional boundary layers. AIAA J. 31 (4), 620–628.Google Scholar
Kurz, H. & Kloker, M. J. 2014 Receptivity of a swept-wing boundary layer to micron-sized discrete roughness elements. J. Fluid Mech. 755, 62–82.Google Scholar
Malik, M. R.1990 Finite difference solution of the compressible stability eigenvalue problem. NASA Tech. Rep. 16572.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.Google Scholar
Marxen, O., Iaccarino, G. & Shaqfeh, E. S. G. 2010 Disturbance evolution in a Mach 4.8 boundary layer with two-dimensional roughness-induced separation and shock. J. Fluid Mech. 648, 436–469.Google Scholar
Masad, J. A. 1995 Transition in flow over heat-transfer strips. Phys. Fluids 7 (9), 2163–2174.Google Scholar
Masad, J. A. & Nayfeh, A. H. 1992 Laminar flow control of subsonic boundary layers by suction and heat-transfer strips. Phys. Fluids 4 (6), 1259–1272.Google Scholar
Meitz, H. L. & Fasel, H. 2000 A compact-difference scheme for the Navier–Stokes equaions in vorticity–velocity formulation. J. Comput. Phys. 157 (1), 371–403.Google Scholar
Nayfeh, A. H. & Abu-Khajeel, H. T. 1996 Effect of a hump on the stability of subsonic boundary layers over an airfoil. Intl J. Engng Sci. 34 (6), 599–628.Google Scholar
Nayfeh, A. H., Ragab, S. A. & Almaaitah, A. A. 1988 Effect of bulges on the stability of boundary layers. Phys. Fluids 31 (4), 795–806.Google Scholar
Nayfeh, A. H. & Reed, H. L. 1985 Stability of flow over axisymmetric bodies with porous suction strips. Phys. Fluids 28 (10), 2990–2998.Google Scholar
Park, D. & Park, S. O. 2013 Linear and non-linear stability analysis of incompressible boundary layer over a two-dimensional hump. Comput. Fluids 73, 80–96.Google Scholar
Pralits, J. O. & Hanifi, A. 2003 Optimization of steady suction for disturbance control on infinite swept wings. Phys. Fluids 15 (9), 2756–2772.Google Scholar
Pralits, J. O., Hanifi, A. & Henningson, D. S. 2002 Adjoint-based optimization of steady suction for disturbance control in incompressible flows. J. Fluid Mech. 467, 129–161.Google Scholar
Reed, H. L. & Nayfeh, A. H. 1986 Numerical-perturbation technique for stability of flat-plate boundary layers with suction. AIAA J. 24 (2), 208–214.Google Scholar
Reed, H. L., Saric, W. S. & Arnal, D. 1996 Linear stability theory applied to boundary layers. Annu. Rev. Fluid Mech. 28 (1), 389–428.Google Scholar
Reshotko, E. 1976 Boundary-layer stability and transition. Annu. Rev. Fluid Mech. 8 (1), 311–349.Google Scholar
Reynolds, G. A. & Saric, W. S. 1986 Experiments on the stability of the flat-plate boundary layer with suction. AIAA J. 24 (2), 202–207.Google Scholar
Rist, U. & Fasel, H. 1995 Direct numerical simulation of controlled transition in flat-plate boundary layer. J. Fluid Mech. 298, 211–248.Google Scholar
Wang, X. W. & Zhong, X. L. 2012 The stabilization of a hypersonic boundary layer using local sections of porous coating. Phys. Fluids 24, 034105.Google Scholar
Wie, Y. & Malik, M. R. 1998 Effect of surface waviness on boundary-layer transition in two-dimensional flow. Comput. Fluids 27 (2), 157–181.Google Scholar
Worner, A., Rist, U. & Wagner, S. 2003 Humps/steps influence on stability characteristics of two-dimensional laminar boundary layer. AIAA J. 41 (2), 192–197.Google Scholar
Wu, X. & Dong, M. 2016 A local scattering theory for the effects of isolated roughness on boundary-layer instability and transition: transmission coefficient as an eigenvalue. J. Fluid Mech. 794, 68–108.Google Scholar
Wu, X. & Hogg, L. W. 2006 Acoustic radiation of Tollmien–Schlichting waves as they undergo rapid distortion. J. Fluid Mech. 550, 307–347.Google Scholar
Xu, H., Sherwin, S. J., Hall, P. & Wu, X. 2016 The behaviour of Tollmien–Schlichting waves undergoing small-scale localised distortions. J. Fluid Mech. 792, 499–525.Google Scholar
Zhong, X. L. & Wang, X. W. 2012 Direct numerical simulation on the receptivity, instability, and transition of hypersonic boundary layers. Annu. Rev. Fluid Mech. 44 (1), 527–561.Google Scholar