Hostname: page-component-848d4c4894-75dct Total loading time: 0 Render date: 2024-05-17T23:48:25.027Z Has data issue: false hasContentIssue false

Direct numerical simulation of transonic shock/boundary layer interaction under conditions of incipient separation

Published online by Cambridge University Press:  24 June 2010

Dipartimento di Meccanica e Aeronautica, Università di Roma ‘La Sapienza’, Via Eudossiana 18, 00184 Rome, Italy
Dipartimento di Meccanica e Aeronautica, Università di Roma ‘La Sapienza’, Via Eudossiana 18, 00184 Rome, Italy
Dipartimento di Meccanica e Aeronautica, Università di Roma ‘La Sapienza’, Via Eudossiana 18, 00184 Rome, Italy
Email address for correspondence:


The interaction of a normal shock wave with a turbulent boundary layer developing over a flat plate at free-stream Mach number M = 1.3 and Reynolds number Reθ ≈ 1200 (based on the momentum thickness of the upstream boundary layer) is analysed by means of direct numerical simulation of the compressible Navier–Stokes equations. The computational methodology is based on a hybrid linear/weighted essentially non-oscillatory conservative finite-difference approach, whereby the switch is controlled by the local regularity of the solution, so as to minimize numerical dissipation. As found in experiments, the mean flow pattern consists of an upstream fan of compression waves associated with the thickening of the boundary layer, and the supersonic region is terminated by a nearly normal shock, with substantial bending of the interacting shock. At the selected conditions the flow does not exhibit separation in the mean. However, the interaction region is characterized by ‘intermittent transitory detachment’ with scattered spots of instantaneous flow reversal throughout the interaction zone, and by the formation of a turbulent mixing layer, with associated unsteady release of vortical structures. As found in supersonic impinging shock interactions, we observe a different amplification of the longitudinal Reynolds stress component with respect to the others. Indeed, the effect of the adverse pressure gradient is to reduce the mean shear, with subsequent suppression of the near-wall streaks, and isotropization of turbulence. The recovery of the boundary layer past the interaction zone follows a quasi-equilibrium process, characterized by a self-similar distribution of the mean flow properties.

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.)



Adams, N. A. 2000 Direct simulation of the turbulent boundary layer along a compressible ramp at M = 3 and Re θ = 1685. J. Fluid Mech. 420, 4783.CrossRefGoogle Scholar
Atkin, C. J. & Squire, L. C. 1992 A study on the interaction of a normal shock wave with a turbulent boundary layer at Mach numbers between 1.30 and 1.55. Eur. J. Mech. B/Fluids 11, 93118.Google Scholar
Barakos, G. & Drikakis, D. 2000 Investigation of nonlinear eddy-viscosity turbulence models in shock/boundary layer interaction. AIAA J. 38, 461469.CrossRefGoogle Scholar
Batten, P., Craft, T. J., Leschziner, M. A. & Loyau, H. 1999 Reynolds-stress-transport modeling for compressible aerodynamics applications. AIAA J. 37, 785797.CrossRefGoogle Scholar
Bruce, P. J. K. 2008 Transonic shock/boundary layer interactions subject to downstream pressure perturbations. PhD thesis, Magdalene College, Department of Engineering, University of Cambridge.Google Scholar
Bruce, P. J. K. & Babinsky, H. 2008 Unsteady shock wave dynamics. J. Fluid Mech. 603, 463473.CrossRefGoogle Scholar
Bruce, P. J. K. & Babinsky, H. 2009 Behaviour of unsteady transonic shock/boundary layer interactions with three-dimensional effects. AIAA Paper 2009-1590.CrossRefGoogle Scholar
Bull, M. K. 1967 Wall pressure fluctuations associated with subsonic turbulent boundary layer flow. J. Fluid Mech. 28, 719754.CrossRefGoogle Scholar
Castillo, L. & George, W. K. 2001 Similarity analysis for turbulent boundary layer with pressure gradient: outer flow. AIAA J. 39, 4147.CrossRefGoogle Scholar
Clauser, F. H. 1954 Turbulent boundary layers in adverse pressure gradients. J. Aerosp. Sci. 21, 91108.Google Scholar
Davidson, L. 1995 Reynolds stress transport modelling of shock-induced separated flow. Comput. Fluids 24, 253268.CrossRefGoogle Scholar
Délery, J. 1983 Experimental investigation of turbulence properties in transonic shock/boundary layer interactions. AIAA J. 21, 180185.CrossRefGoogle Scholar
Délery, J. & Marvin, J. G. 1986 Shock-wave boundary layer interactions. AGARDograph 280.Google Scholar
Dolling, D. S. 2001 Fifty years of shock-wave/boundary layer interaction research: what next? AIAA J. 39, 15171531.CrossRefGoogle Scholar
East, L. F. 1976 The application of a laser anemometer to the investigation of shock wave boundary layer interactions. AGARD-CPP 193.Google Scholar
Erm, L. P. & Joubert, J. 1991 Low Reynolds number turbulent boundary layers. J. Fluid Mech. 230, 144.CrossRefGoogle Scholar
Fernholz, H. H. & Finley, P. J. 1980 A critical commentary on mean flow data for two-dimensional compressible turbulent boundary layers. AGARDograph 253.Google Scholar
Garnier, E., Sagaut, P. & Deville, M. 2002 Large-eddy simulation of shock/boundary-layer interaction. AIAA J. 40 (10), 19351944.CrossRefGoogle Scholar
Gerolymos, G. A. & Vallet, I. 1997 Near-wall Reynolds-stress three-dimensional transonic flow computation. AIAA J. 35, 228236.CrossRefGoogle Scholar
Guarini, S. E., Moser, R. D., Shariff, K. & Wray, A. 2000 Direct numerical simulation of a supersonic boundary layer at Mach 2.5. J. Fluid Mech. 414, 133.CrossRefGoogle Scholar
Hopkins, E. J. & Inouye, M. 1971 An evaluation of theories for predicting turbulent skin friction and heat transfer on flat plates at supersonic and hypersonic Mach numbers. AIAA J. 9, 9931003.CrossRefGoogle Scholar
Jiang, G. S. & Shu, C. W. 1996 Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202.CrossRefGoogle Scholar
Jiménez, J. & Wray, A. A. 1998 On the characteristics of vortex filaments in isotropic turbulence. J. Fluid Mech. 373, 255285.CrossRefGoogle Scholar
Knight, D., Yana, H., Panaras, A. G. & Zheltovodov, A. 2003 Advances in CFD prediction of shock wave turbulent boundary layer interactions. Prog. Aerosp. Sci. 39, 121184.CrossRefGoogle Scholar
Lam, K. & Banerjee, S. 1992 On the condition of streak formation in a bounded turbulent flow. Phys. Fluids 4 (2), 306320.CrossRefGoogle Scholar
Leschziner, M. A., Batten, P. & Loyau, H. 2000 Modelling shock-affected near-wall flows with anisotropy-resolving turbulence closures. Intl J. Heat Fluid Flow 21, 239251.CrossRefGoogle Scholar
Leschziner, M. A. & Drikakis, D. 2002 Turbulence modelling and turbulent-flow computation in aeronautics. Aeronaut. J. 106, 349384.CrossRefGoogle Scholar
Liu, X. & Squire, L. C. 1988 An investigation of shock/boundary-layer interactions on curved surfaces at transonic speeds. J. Fluid Mech. 187, 467486.CrossRefGoogle Scholar
Lumley, J. L. 1978 Computational modeling of turbulent flows. Adv. Appl. Mech. 18, 123176.CrossRefGoogle Scholar
Lund, T. S., Wu, X. & Squires, K. D. 1998 Generation of turbulent inflow data for spatially-developing boundary layer simulations. J. Fluid Mech. 140, 233258.Google Scholar
Morkovin, M. V. 1961 Effects of compressibility on turbulent flows. In Mécanique de la Turbulence (ed. Favre, A.), p. 367. CNRS.Google Scholar
Na, Y. & Moin, P. 1998 a Direct numerical simulation of a separated turbulent boundary layer. J. Fluid Mech. 374, 379404.CrossRefGoogle Scholar
Na, Y. & Moin, P. 1998 b The structure of wall-pressure fluctuations in turbulent boundary layers with adverse pressure gradient and separation. J. Fluid Mech. 377, 347373.CrossRefGoogle Scholar
Pirozzoli, S. 2002 Conservative hybrid compact-WENO schemes for shock–turbulence interaction. J. Comput. Phys. 178, 81117.CrossRefGoogle Scholar
Pirozzoli, S., Bernardini, M. & Grasso, F. 2007 Aeroacoustics of transonic shock–boundary layer interactions. AIAA Paper 2007-3416.CrossRefGoogle Scholar
Pirozzoli, S., Bernardini, M. & Grasso, F. 2008 Characterization of coherent vortical structures in a supersonic turbulent boundary layer. J. Fluid Mech. 613, 205231.CrossRefGoogle Scholar
Pirozzoli, S. & Grasso, F. 2004 Direct numerical simulations of isotropic compressible turbulence: influence of compressibility on dynamics and structures. Phys. Fluids 16 (12), 43864407.CrossRefGoogle Scholar
Pirozzoli, S. & Grasso, F. 2006 Direct numerical simulation of impinging shock wave turbulent boundary layer interaction at M = 2.25. Phys. Fluids 18, 065113.CrossRefGoogle Scholar
Pirozzoli, S., Grasso, F. & Gatski, T. B. 2004 Direct numerical simulation and analysis of a spatially evolving supersonic turbulent boundary layer at M = 2.25. Phys. Fluids 16 (3), 530545.CrossRefGoogle Scholar
Poinsot, T. J. & Lele, S. K. 1992 Boundary conditions for direct simulations of compressible viscous flows. J. Comput. Phys. 101, 104129.CrossRefGoogle Scholar
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.CrossRefGoogle Scholar
Rogers, M. & Moser, R. 1993 Direct numerical simulation of a self-similar turbulent mixing layer. Phys. Fluids A 6, 903923.CrossRefGoogle Scholar
Sagaut, P., Garnier, E., Tromeur, E., Larcheveque, L. & Labourasse, E. 2004 Turbulent inflow conditions for large eddy simulation of compressible wall-bounded flows. AIAA J. 42, 469477.CrossRefGoogle Scholar
Sandham, N. D., Yao, Y. F. & Lawal, A. A. 2003 Large-eddy simulation of transonic flow over a bump. Intl J. Heat Fluid Flow 24, 584595.CrossRefGoogle Scholar
Seddon, J. 1960 The flow produced by interaction of a turbulent boundary layer with a normal shock wave of strength sufficient to cause separation. Tech. Rep. 3502. ARC R & M.Google Scholar
Simpson, R. L. 1989 Turbulent boundary layer separation. Annu. Rev. Fluid Mech. 21, 205234.CrossRefGoogle Scholar
Smits, A. J. & Dussauge, J. P. 2006 Turbulent Shear Layers in Supersonic Flow. American Institute of Physics.Google Scholar
Spalart, P., Strelets, M. & Travin, A. 2006 Direct numerical simulation of large-eddy-break-up devices in a boundary layer. Intl J. Heat Fluid Flow 27, 902910.CrossRefGoogle Scholar
Spalart, P. R. 1988 Direct numerical simulation of a turbulent boundary layer up to Re θ = 1410. J. Fluid Mech. 187, 6198.CrossRefGoogle Scholar
Stolz, S. & Adams, N. A. 2003 Large-eddy simulation of high-Reynolds-number supersonic boundary layers using the approximate deconvolution model and a rescaling and recycling technique. Phys. Fluids 15, 23982412.CrossRefGoogle Scholar
Touber, E. & Sandham, N. D. 2009 Large-eddy simulation of low-frequency unsteadiness in a turbulent shock-induced separation bubble. Theor. Comput. Fluid Dyn. 23, 79107.CrossRefGoogle Scholar
White, F. M. 1974 Viscous Fluid Flow. McGraw-Hill.Google Scholar
Wu, M. & Martin, M. P. 2007 Direct numerical simulation of supersonic turbulent boundary layer over a compression ramp. AIAA J. 45, 879889.CrossRefGoogle Scholar
Wu, X. & Moin, P. 2009 Direct numerical simulation of turbulence in a nominally zero-pressure-gradient flat-plate boundary layer. J. Fluid Mech. 630, 541.CrossRefGoogle Scholar
Xu, S. & Martin, M. P. 2004 Assessment of inflow boundary conditions for compressible turbulent boundary layers. Phys. Fluids 16, 26232639.CrossRefGoogle Scholar
Zagarola, M. & Smits, A. J. 1998 Mean-flow scaling of turbulent pipe flow. J. Fluid Mech. 373, 3379.CrossRefGoogle Scholar

Pirozzoli et al. supplementary movie

Transonic shock wave/boundary layer interaction. Animation of the pressure field in streamwise, wall-normal plane in the proximity of the interaction zone. Pressure is normalized by its free-stream value, and 32 contour levels are shown, from 0.77 to 1.53.

Download Pirozzoli et al. supplementary movie(Video)
Video 10.5 MB