Skip to main content Accessibility help

On the transition between regular and irregular shock patterns of shock-wave/boundary-layer interactions

  • Jan Matheis (a1) and Stefan Hickel (a1) (a2)


The reflection of strong oblique shock waves at turbulent boundary layers is studied numerically and analytically. A particular emphasis is put on the transition between regular shock-wave/boundary-layer interaction (SWBLI) and Mach reflection (irregular SWBLI). The classical two- and three-shock theory and a generalised form of the free interaction theory are used for the analysis of well-resolved large-eddy simulations (LES) and for the derivation of stability criteria. We found that at a critical deflection angle across the incident shock wave, the perturbations related to the turbulent boundary layer cause bi-directional transition processes between regular and irregular shock patterns for a free-stream Mach number of $\mathit{Ma}_{0}=2$ . Computational results show that the mean deflection angle across the separation shock is decoupled from the incident shock wave and can be accurately modelled by the generalised free interaction theory. On the basis of these observations, and the von Neumann and detachment criteria for the asymmetric intersection of shock waves, we derive the critical incident shock deflection angles at which the shock pattern may/must become irregular. Numerical data for a free-stream Mach number of $\mathit{Ma}_{0}=3$ confirm the existence of the dual-solution domain predicted by theory.


Corresponding author

Email address for correspondence:


Hide All
Anderson, J. D. 2001 Fundamentals of Aerodynamic. McGraw-Hill Science/Engineering/Math.
Babinsky, H. & Harvey, J. K. 2011 Shock Wave-Boundary-Layer Interactions. Cambridge University Press.
Bardsley, O. & Mair, W. A. 1950 III. The interaction between an oblique shock-wave and a turbulent boundary-layer. Phil. Mag. 7 42 (324), 2936.
Ben-Dor, G. 2010 Shock Wave Reflection Phenomena. Springer.
Bermejo-Moreno, I., Campo, L., Larsson, J., Bodart, J., Helmer, D. & Eaton, J. K. 2014 Confinement effects in shock wave/turbulent boundary layer interactions through wall-modelled large-eddy simulations. J. Fluid Mech. 758, 562.
Bogdonoff, S. M. 1955 Separation of a supersonic turbulent boundary layer. J. Aeronaut. Sci. (Inst. Aeronaut. Sci.) 22 (6), 414430.
Campo, L. M. & Eaton, J. K. 2015 Shock boundary layer interactions in a low aspect ratio duct. Intl J. Heat Fluid Flow 51, 353371.
Carrière, P., Sirieix, M. & Solignac, J. L.1969 Similarity properties of the laminar or turbulent separation phenomena in a non-uniform supersonic flow. In Applied Mechanics—Proceedings of the Twelfth International Congress of Applied Mechanics, Stanford University, August 26–31, 1968 pp. 145–157.
Chapman, D. R., Kuehn, D. M. & Larson, H. K.1958 Investigation of separated flows in supersonic and subsonic streams with emphasis on the effect of transition. NACA Tech. Rep. 1356.
Charwat, A. F. 1970 Supersonic flows with imbedded separated regions. Adv. Heat Transfer 6, 1131.
Chpoun, A., Passerel, D., Li, H. & Ben-Dor, G. 1995 Reconsideration of oblique shock wave reflections in steady flows. Part 1. Experimental investigation. J. Fluid Mech. 301, 1935.
Coles, D. E.1953 Measurements in the boundary layer on a smooth flat plate in supersonic flow. PhD thesis, California Institute of Technology.
Delery, J. & Dussauge, J. P. 2009 Some physical aspects of shock wave/boundary layer interactions. Shock Waves 19 (6), 453468.
Delery, J. & Marvin, J. G.1986 Shock-wave boundary layer interactions. Tech. Rep. AGARD-AG-280.
Dolling, D. S. 2001 Fifty years of shock-wave/boundary-layer interaction research: what next? AIAA J. 39 (8), 15171531.
Dussauge, J.-P., Dupont, P. & Debieve, J.-F. 2006 Unsteadiness in shock wave boundary layer interactions with separation. Aerosp. Sci. Technol. 10 (2), 8591.
Edney, B. E.1968 Anomalous heat transfer and pressure distributions on blunt bodies at hypersonic speeds in the presence of an impinging shock. Tech. Rep. FFA Report 115, Stockholm.
Erdos, J. & Pallone, A.1963 Shock-boundary layer interaction and flow separations. In Proceedings of the 1962 Heat Transfer and Fluid Mechanics Institute.
Fernholz, H. H. & Finley, P. J.1977 A critical compilation of compressible turbulent boundary layer data. Tech. Rep. AGARD-AG-223.
Foysi, H., Sarkar, S. & Friedrich, R. 2004 Compressibility effects and turbulence scalings in supersonic channel flow. J. Fluid Mech. 509, 207216.
Gadd, G. E., Holder, D. W. & Regan, J. D. 1954 An experimental investigation of the interaction between shock waves and boundary layers. Proc. R. Soc. Lond. A 226 (1165), 227253.
Gottlieb, S. & Shu, C. W. 1998 Total variation diminishing Runge–Kutta schemes. Maths Comput. 67 (221), 7385.
Green, J. E. 1970 Reflexion of an oblique shock wave by a turbulent boundary layer. J. Fluid Mech. 40 (1), 8195.
Grilli, M., Hickel, S. & Adams, N. A. 2013 Large-eddy simulation of a supersonic turbulent boundary layer over a compression–expansion ramp. Intl J. Heat Fluid Flow 42, 7993.
Grilli, M., Schmid, P. J., Hickel, S. & Adams, N. A. 2012 Analysis of unsteady behaviour in shockwave turbulent boundary layer interaction. J. Fluid Mech. 700, 1628.
Guarini, S. E., Moser, R. D., Shariff, K. & Wray, A. 2000 Direct numerical simulation of a supersonic turbulent boundary layer at Mach 2.5. J. Fluid Mech. 414, 133.
Hankey, W. L. Jr. & HJrolden, M. S. 1975 Two-dimensional shock wave-boundary layer interactions in high speed flows. Tech. Rep. AGARD-AG-203.
Henderson, L. F. 1967 The reflexion of a shock wave at a rigid wall in the presence of a boundary layer. J. Fluid Mech. 30 (04), 699722.
Hickel, S., Adams, N. A. & Domaradzki, J. A. 2006 An adaptive local deconvolution method for implicit LES. J. Comput. Phys. 213 (1), 413436.
Hickel, S., Egerer, C. P. & Larsson, J. 2014 Subgrid-scale modeling for implicit large eddy simulation of compressible flows and shock-turbulence interaction. Phys. Fluids 26, 106101.
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 (6), 111.
Hornung, H. G. 1982 Transition from regular to Mach reflection of shock waves Part 2. The steady-flow criterion. J. Fluid Mech. 123, 155164.
Hornung, H. G. 1986 Regular and Mach reflection of shock waves. Annu. Rev. Fluid Mech. 18, 3358.
Hornung, H. G., Oertel, H. & Sandeman, R. J. 1979 Transition to Mach reflexion of shock waves in steady and pseudosteady flow with and without relaxation. J. Fluid Mech. 90 (3), 541560.
Hu, Z. M., Myong, R. S. & Kim, M. S. 2009 Downstream flow condition effects on the RR - MR transition of asymmetric shock waves in steady flows. J. Fluid Mech. 620, 4362.
Humble, R. A., Scarano, F. & van Oudheusden, B. W. 2009 Unsteady aspects of an incident shock wave/turbulent boundary layer interaction. J. Fluid Mech. 635, 4774.
Ivanov, M. S., Ben-Dor, G., Elperin, T., Kudryavtsev, A. N. & Khotyanovsky, D. V. 2002 The reflection of asymmetric shock waves in steady flows: a numerical investigation. J. Fluid Mech. 469, 7187.
Ivanov, M. S., Gimelshein, S. F. & Beylich, A. E. 1995 Hysteresis effect in stationary reflection of shock waves. Phys. Fluids 7 (4), 685687.
Ivanov, M. S., Gimelshein, S. F. & Markelov, G. N. 1998 Statistical simulation of the transition between regular and mach reflection in steady flows. Comput. Maths Applics. 35 (1–2), 113125.
Ivanov, M. S., Kudryavtsev, A. N., Nikiforov, S. B., Pavlov, A. A. & Shiplyuk, A. N.2003 Study of transition between regular and Mach reflections in various wind tunnels. In 41st Aerospace Sciences Meeting and Exhibit.
Ivanov, M. S., Vandromme, D., Fomin, V. M., Kudryavtsev, A. N., Hadjadj, A. & Khotyanovsky, D. V. 2001 Transition between regular and Mach reflection of shock waves: new numerical and experimental results. Shock Waves 11 (3), 199207.
Klein, M., Sadiki, A. & Janicka, J. 2003 A digital filter based generation of inflow data for spatially developing direct numerical or large eddy simulations. J. Comput. Phys. 186 (2), 652665.
Komminaho, J. & Skote, M. 2002 Reynolds stress budgets in Couette and boundary layer flows. Flow Turbul. Combust. 68 (2), 167192.
Krehl, P. & van der Geest, M. 1991 The discovery of the Mach reflection effect and its demonstration in an auditorium. Shock Waves 1 (1), 315.
Kudryavtsev, A. N., Khotyanovsky, D. V., Ivanov, M. S., Hadjadj, A. & Vandromme, D. 2002 Numerical investigations of transition between regular and Mach reflections caused by free-stream disturbances. Shock Waves 12 (2), 157165.
Li, H. & Ben-Dor, G. 1997 A parametric study of mach reflection in steady flows. J. Fluid Mech. 341, 101125.
Li, H., Chpoun, A. & Ben-Dor, G. 1999 Analytical and experimental investigations of the reflection of asymmetric shock waves in steady flows. J. Fluid Mech. 390, 43.
Liepmann, H. W., Roshko, A. & Dhawan, S.1951 On reflection of shock waves from boundary layers. Tech. Rep. 1100 ADA382023.
Maeder, T., Adams, N. A. & Kleiser, L. 2001 Direct simulation of turbulent supersonic boundary layers by an extended temporal approach. J. Fluid Mech. 429, 187216.
Mouton, C. A. & Hornung, H. G. 2007 Mach stem height and growth rate predictions. AIAA J. 45 (8), 19771987.
Mouton, C. A. & Hornung, H. G. 2008 Experiments on the mechanism of inducing transition between regular and Mach reflection. Phys. Fluids 20 (12), 126103.
Östlund, J.2002 Flow processes in rocket engine nozzles with focus on flow separation and side-loads. PhD thesis, KTH, Stockholm.
Pasquariello, V., Grilli, M., Hickel, S. & Adams, N. A. 2014 Large-eddy simulation of passive shock-wave/boundary-layer interaction control. Intl J. Heat Fluid Flow 49, 116127.
Piponniau, S., Dussauge, J. P., Debiêve, J. F. & Dupont, P. 2009 A simple model for low-frequency unsteadiness in shock-induced separation. J. Fluid Mech. 629, 87108.
Pirozzoli, S. & Bernardini, M. 2011 Turbulence in supersonic boundary layers at moderate Reynolds number. J. Fluid Mech. 688 (1), 120168.
Pirozzoli, S. & Grasso, F. 2006 Direct numerical simulation of impinging shock wave/turbulent boundary layer interaction at $M=2.25$ . Phys. Fluids 18 (6), 065113.
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.
Quaatz, J. F., Giglmaier, M., Hickel, S. & Adams, N. A. 2014 Large-eddy simulation of a pseudo-shock system in a Laval nozzle. Intl J. Heat Fluid Flow 49, 108115.
Reijasse, P.2005 Aérodynamique des tuyères propulsives en sur-détente: décollement libre et charges latérales en régime stabilisé (Aerodynamics of overexpanded propulsive nozzles: free separation and side loads in stabilized regime). PhD thesis, Université Pierre-et-Marie-Curie, Paris.
Reijasse, P. & Birkemeyer, J.2002 Semi-empirical flow separation model for subscale nozzles. In Fourth Symposium on Aerothermodynamics for Space Vehicles, p. 407. Capua (IT).
Schlatter, P. & Örlü, R. 2010 Assessment of direct numerical simulation data of turbulent boundary layers. J. Fluid Mech. 659, 116126.
Settles, G. S., Bogdonoff, S. M. & Vas, I. E. 1976 Incipient separation of a supersonic turbulent boundary layer at high Reynolds numbers. AIAA J 14 (1), 5056.
Shang, J. S., Hankey, W. L. Jr. & Law, C. 1976 Numerical simulation of shock wave-turbulent boundary-layer interaction. AIAA J. 14 (10), 14511457.
Simens, M. P., Jiménez, J., Hoyas, S. & Mizuno, Y. 2009 A high-resolution code for turbulent boundary layers. J. Comput. Phys. 228 (1), 42184231.
Smits, A. J. & Dussauge, J. P. 2006 Turbulent Shear Layers in Supersonic Flow. Springer.
Smits, A. J., Matheson, N. & Joubert, P. N. 1983 Low Reynolds number turbulent boundary layers in zero favourable pressure gradients. J. Ship Res. 27, 147157.
Souverein, L. J., Bakker, P. G. & Dupont, P. 2013 A scaling analysis for turbulent shock-wave/boundary-layer interactions. J. Fluid Mech. 714, 505535.
Souverein, L. J., Dupont, P., Debiêve, J. F., Dussauge, J. P., van Oudheusden, B. W. & Scarano, F. 2010 Effect of interaction strength on unsteadiness in turbulent shock-wave-induced separations. AIAA J. 48 (7), 14801493.
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 (2), 79107.
Touber, E. & Sandham, N. D. 2011 Low-order stochastic modelling of low-frequency motions in reflected shock-wave/boundary-layer interactions. J. Fluid Mech. 671, 417465.
van Driest, E. R. 1956 The problem of aerodynamic heating. Aeron. Engng Rev. 15 (10), 2641.
Vuillon, J., Zeitoun, D. & Ben-Dor, G. 1995 Reconsideration of oblique shock wave reflections in steady flows. Part 2. Numerical investigation. J. Fluid Mech. 301, 3750.
Zheltovodov, A. A.1996 Shock waves/turbulent boundary-layer interactions – fundamental studies and applications. AIAA Paper 96-1977, 1–27.
Zheltovodov, A. A. & Yakovlev, V. N.1986 Stages of development, gas dynamic structure and turbulence characteristics of turbulent compressible separated flows in the vicinity of 2-D obstacles. Preprint No. 27–86. Inst. Theor. Appl. Mech. (ITAM), Novosibirsk.
Zukoski, E. E. 1967 Turbulent boundary-layer separation in front of a forward-facing step. AIAA J. 5 (10), 17461753.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification

On the transition between regular and irregular shock patterns of shock-wave/boundary-layer interactions

  • Jan Matheis (a1) and Stefan Hickel (a1) (a2)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed