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Instabilities in oblique shock wave/laminar boundary-layer interactions

Published online by Cambridge University Press:  15 January 2016

F. Guiho ,
F. Alizard  and
J.-Ch. Robinet
F. Guiho
Affiliation:
DynFluid Lab., Arts and Métiers ParisTech, 151, Bd. de l’Hôpital, 75013, Paris, France CNES, Direction des lanceurs, 52, rue Jacques Hillairet, 75012, Paris, France
F. Alizard
Affiliation:
DynFluid Lab., CNAM, 151, Bd. de l’Hôpital, 75013, Paris, France
J.-Ch. Robinet*
Affiliation:
DynFluid Lab., Arts and Métiers ParisTech, 151, Bd. de l’Hôpital, 75013, Paris, France
*
†Email address for correspondence: [email protected]
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Abstract

The interaction of an oblique shock wave and a laminar boundary layer developing over a flat plate is investigated by means of numerical simulation and global linear-stability analysis. Under the selected flow conditions (free-stream Mach numbers, Reynolds numbers and shock-wave angles), the incoming boundary layer undergoes separation due to the adverse pressure gradient. For a wide range of flow parameters, the oblique shock wave/boundary-layer interaction (OSWBLI) is seen to be globally stable. We show that the onset of two-dimensional large-scale structures is generated by selective noise amplification that is described for each frequency, in a linear framework, by wave-packet trains composed of several global modes. A detailed analysis of both the eigenspectrum and eigenfunctions gives some insight into the relationship between spatial scales (shape and localization) and frequencies. In particular, OSWBLI exhibits a universal behaviour. The lowest frequencies correspond to structures mainly located near the separated shock that emit radiation in the form of Mach waves and are scaled by the interaction length. The medium frequencies are associated with structures mainly localized in the shear layer and are scaled by the displacement thickness at the impact. The linear process by which OSWBLI selects frequencies is analysed by means of the global resolvent. It shows that unsteadiness are mainly associated with instabilities arising from the shear layer. For the lower frequency range, there is no particular selectivity in a linear framework. Two-dimensional numerical simulations show that the linear behaviour is modified for moderate forcing amplitudes by nonlinear mechanisms leading to a significant amplification of low frequencies. Finally, based on the present results, we draw some hypotheses concerning the onset of unsteadiness observed in shock wave/turbulent boundary-layer interactions.

JFM classification

Compressible Flows: Compressible boundary layers Instability: Instability Wakes/Jets: Separated flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 789 , 25 February 2016 , pp. 1 - 35
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© 2016 Cambridge University Press 

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References

Ackeret, J., Feldmann, F. & Rott, N.1947 Investigation of compression shocks and boundary layers in gases moving at high speed. NACA TM 1113.Google Scholar
Arnoldi, W. E. 1951 The principle of minimized iterations in the solution of the matrix eigenvalue problem. Q. Appl. Maths 9, 17–29.CrossRefGoogle Scholar
Aubard, G., Gloerfelt, X. & Robinet, J.-C. 2013 Large-eddy simulation of broadband unsteadiness in a shock/boundary-layer interaction. AIAA J. 51, 2395–2409.Google Scholar
Bagheri, S., Akervik, E., Brandt, L. & Henningson, D. S. 2009 Matrix-free methods for the stability and control of boundary layers. AIAA J. 47 (5), 1057–1068.Google Scholar
Balakumar, P., Zhao, H. & Atkins, H. 2005 Stability of hypersonic boundary layers over a compression corner. AIAA J. 43 (4), 760–767.Google Scholar
Barkley, D., Blackburn, H. M. & Sherwin, S. J. 2008 Direct optimal growth analysis for timesteppers. Intl J. Numer. Meth. Fluids 57, 1435–1458.Google Scholar
Bedarev, I. A., Maslov, A., Sidorenko, A. A., Fedorova, N. N. & Shiplyuk, A. 2002 Experimental and numerical study of hypersonic separated flow in the vicinity of a cone-flare model. J. Appl. Mech. Tech. Phys. 43 (6), 867–876.Google Scholar
Beneddine, S., Mettot, C. & Sipp, D. 2015 Global stability analysis of underexpanded screeching jets. Eur. J. Mech. (B/Fluids) 49, 392–399.Google Scholar
Boin, J. P., Robinet, J.-Ch., Corre, Ch. & Deniau, H. 2006 3D steady and unsteady bifurcations in a shock-wave/laminar boundary layer interaction; a numerical study. Theor. Comput. Fluid Dyn. 20 (3), 163–180.Google Scholar
Bres, G. & Colonius, T. 2008 Three-dimensional instabilities in compressible flow over open cavities. J. Fluid Mech. 599, 309–339.Google Scholar
Cheung, L. C. & Lele, S. K. 2009 Linear and nonlinear processes in two-dimensional mixing layer dynamics and sound generation. J. Fluid Mech. 625, 321–351.Google Scholar
Chomaz, J. M. 2005 Global instabilities in spatially developing flows: non-normality and nonlinearity. Annu. Rev. Fluid Mech. 37, 357–392.Google Scholar
Clemens, N. T. & Narayanaswamy, V. 2014 Low-frequency unsteadiness of shock wave/turbulent boundary layer interactions. Annu. Rev. Fluid Mech. 46, 469–492.Google Scholar
Cossu, C. & Chomaz, J.-M. 1997 Global measures of local convective instability. Phys. Rev. Lett. 78, 4387–4390.Google Scholar
Crouch, J. D., Garbaruk, A. & Magidov, D. 2007 Predicting the onset of flow unsteadiness based on global instability. J. Comput. Phys. 224 (2), 924–940.Google Scholar
Degrez, G., Boccadoro, C. H. & Wendt, J. F. 1987 The interaction of an oblique shock wave with a laminar boundary layer revisited. An experimental and numerical study. J. Fluid Mech. 177, 247–263.Google Scholar
Délery, J. & Marvin, J. G.1986 Shock-wave boundary layer interactions. Tech. Rep. AGARDograph.Google Scholar
Deleuze, J.1995 Structure d’une couche limite turbulente soumise à une onde de choc incidente. PhD thesis. University Aix-Marseille II.Google Scholar
Dolling, D. S. 2001 Fifty years of shock-wave/boundary-layer interaction research: what next? AIAA J. 39 (8), 1517–1531.Google Scholar
Dovgal, A. V., Kozlov, V. V. & Michalke, A. 1994 Laminar boundary-layer separation: instability and associated phenomena. Prog. Aerosp. Sci. 30, 61–94.CrossRefGoogle Scholar
Dupont, P., Haddad, C., Ardissone, J. P. & Debiève, J.-F. 2005 Space and time organisation of a shock wave/turbulent boundary layer interaction. Aerosp. Sci. Technol. 9 (7), 561–572.Google Scholar
Dupont, P., Haddad, C. & Debiève, J.-F. 2006 Space and time organization in a shock induced boundary layer. J. Fluid Mech. 559, 255–277.Google Scholar
Dupont, P., Piponniau, S., Sidorenko, A. & Debieve, J. F. 2008 Investigation by particle image velocimetry measurements of oblique shock reflection with separation. AIAA J. 46 (6), 1365–1370.Google Scholar
Dussauge, J.-P., Dupont, P. & Debiève, J.-F. 2006 Unsteadiness in shock wave boundary layer interaction with separation. Aerosp. Sci. Technol. 10 (2), 85–91.Google Scholar
Dussauge, J. P. & Piponniau, S. 2008 Shock/boundary-layer interactions: possible sources of unsteadiness. J. Fluids Struct. 24 (8), 1166–1175.Google Scholar
Edwards, W. S., Tuckerman, L. S., Friesner, R. A. & Sorensen, D. 1994 Krylov methods for the incompressible Navier–Stokes equations. J. Comput. Phys. 110, 82–101.Google Scholar
Ehrenstein, U. & Gallaire, F. 2008 Two-dimensional global low-frequency oscillations in a separating boundary-layer flow. J. Fluid Mech. 614, 315–327.Google Scholar
Ganapathisubramani, B., Clemens, N. T. & Dolling, D. S. 2007 Effects of upstream boundary layer on the unsteadiness of shock-induced separation. J. Fluid Mech. 585, 369–394.Google Scholar
Garnaud, X., Lesshafft, L., Schmid, P. J. & Huerre, P. 2013 The preferred mode of incompressible jets: linear frequency response analysis. J. Fluid Mech. 716, 189–202.Google Scholar
Garnier, E., Sagaut, P. & Deville, M. 2002 Large-eddy simulation of shock/boundary-layer interaction. AIAA J. 40 (10), 1935–1944.Google Scholar
Grilli, M., Schmid, P. J., Hickel, S. & Adams, N. A. 2011 Analysis of unsteady behaviour in shockwave turbulent boundary layer interaction. J. Fluid Mech. 700, 16–28.Google Scholar
Hammond, D. A. & Redekopp, L. G. 1998 Local and global instability properties of separation bubbles. Eur. J. Mech. (B/Fluids) 17 (2), 145–164.Google Scholar
Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22, 473.Google Scholar
Humble, R. A., Sacarano, F. & van Oudheusden, B. W. 2009 Unsteady aspects of an incident shock wave/turbulent boundary layer interaction. J. Fluid Mech. 635, 47–74.Google Scholar
Jackson, C. P. 1987 A finite-element study of the onset of vortex shedding in flow past variously shaped bodies. J. Fluid Mech. 182, 23–45.Google Scholar
Jameson, A.1991 Time-dependent calculations using multigrid with applications to unsteady flows past airfoils and wings. AIAA Paper 91-1596.Google Scholar
Jaunet, V., Debieve, J. F. & Dupont, P. 2014 Length scales and time scales of a heated shock-wave/boundary-layer interaction. AIAA J. 52 (6), 2524–2532.Google Scholar
Laurent, H.1996 Turbulence d’une interaction onde de choc/couche limite sur une paroi adiabatique ou chauffée. PhD thesis, University Aix-Marseille II.Google Scholar
Loiseau, J.-C., Robinet, J.-C., Cherubini, S. & Leriche, E. 2014 Investigation of the roughness-induced transition: global stability analyses and direct numerical simulations. J. Fluid Mech. 760, 175–211.Google Scholar
Mack, C. J. & Schmid, P. J. 2011a Global stability of swept flow around a parabolic body: features of the global spectrum. J. Fluid Mech. 669, 375–396.Google Scholar
Mack, C. J. & Schmid, P. J. 2011b Global stability of swept flow around a parabolic body: the neutral curve. J. Fluid Mech. 678, 589–599.CrossRefGoogle Scholar
Mack, C. J., Schmid, P. J. & Sesterhenn, J. L. 2008 Global stability of swept flow around a parabolic body: connecting attachment-line and crossflow modes. J. Fluid Mech. 611, 205–214.Google Scholar
Mack, L.1969 Boundary layer stability theory. Tech. Rep. 900-277, Jet Propulsion Laboratory, Pasadena.Google Scholar
Méliga, Ph., Sipp, D. & Chomaz, J.-M. 2008 Absolute instability in axisymmetric wakes: compressible and density variation effects. J. Fluid Mech. 600, 373–401.CrossRefGoogle Scholar
Méliga, P., Sipp, D. & Chomaz, J.-M. 2010 Effect of compressibility on the global stability of axisymmetric wake flows. J. Fluid Mech. 660, 499–526.Google Scholar
Monkewitz, P. A., Huerre, P. & Chomaz, J.-M. 1993 Global linear stability analysis of weakly non-parallel shear flows. J. Fluid Mech. 251, 1–20.Google Scholar
Morgan, B., Duraisamy, K., Nguyen, N., Kawai, S. & Lele, S. K. 2013 Flow physics and RANS modelling of oblique shock/turbulent boundary layer interaction. J. Fluid Mech. 729, 231–284.Google Scholar
Nichols, J. W. & Lele, S. K. 2011 Global modes and transient response of a cold supersonic jet. J. Fluid Mech. 669, 225–241.Google Scholar
Pagella, A., Rist, U. & Wagner, S. 2000 Numerical investigations of small-amplitude disturbances in a laminar boundary layer with impinging shock waves. Proceedings of the 12th DGLR Fach-Symposium der AG Stab, Stuttgart.Google Scholar
Pagella, A., Rist, U. & Wagner, S. 2002 Numerical investigations of small-amplitude disturbances in a boundary layer with impinging shock wave at $Ma=4.8$ . Phys. Fluids 14 (7), 2088–2101.Google Scholar
Piponniau, S., Dussauge, J. P., Debieve, J. F. & Dupont, P. 2009 A simple model for low-frequency unsteadiness in shock-induced separation. J. Fluid Mech. 629, 87–108.CrossRefGoogle Scholar
Pirozzoli, S. & Bernardini, M. 2011 Direct numerical simulation database for impinging shock wave/turbulent boundary-layer interaction. AIAA J. 49 (6), 1307–1312.Google 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.Google Scholar
Pirozzoli, S., Larsson, J., Nichols, J. W., Bernardini, M., Morgan, B. E. & Lele, S. K. 2010 Analysis of unsteady effects in shock/boundary-layer interactions. In Center for Turbulence Research, Proceedings of the Summer Program 2010.Google Scholar
Priebe, S. & Martin, M. P. 2012 Low-frequency unsteadiness in shock wave/turbulent boundary layer interaction. J. Fluid Mech. 699, 1–49.Google Scholar
Priebe, S., Wu, M. & Martin, M. P. 2009 Direct numerical simulation of a reflected-shock-wave/turbulent-boundary-layer interaction. AIAA J. 47 (5), 1173–1185.Google Scholar
Lehoucq, R. B., Sorensen, D. C. & Yang, C.1997 ARPACK User’s guide: solution of large scale eigenvalue problems with implicitly restarted Arnoldi methods. Technical Note.Google Scholar
Ribner, H. S.1953 Convection of pattern of vorticity through a shock wave. NASA Tech. Rep.Google Scholar
Robinet, J.-Ch. 2007 Bifurcations in shock wave/laminar boundary layer interaction: global instability approach. J. Fluid Mech. 578, 67–94.Google Scholar
Robinet, J.-C. & Casalis, G. 2001 Critical interaction of a shock wave with an acoustic wave. Phys. Fluids 13 (4), 1047–1059.CrossRefGoogle Scholar
Rodriguez, D. & Theofilis, V. 2010 Structural changes of laminar separation bubbles induced by global linear instability. J. Fluid Mech. 655, 280–305.Google Scholar
Roe, P. 1981 Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43 (2), 357–372.Google Scholar
Sansica, A., Sandham, N. D. & Hu, Z. 2014 Forced response of a laminar shock-induced separation bubble. Phys. Fluids 26, 093601.CrossRefGoogle Scholar
Sartor, F.2014 Unsteadiness in transonic shock-wave/boundary-layer interactions: experimental investigation and global stability analysis. PhD thesis, Aix-Marseille University.Google Scholar
Sartor, F., Mettot, C., Bur, R. & Sipp, D. 2015 Unsteadiness in transonic shock-wave/boundary-layer interactions: experimental investigation and global stability analysis. J. Fluid Mech. 781, 550–577.Google Scholar
Sartor, F., Mettot, C., Sipp, D. & Bur, R.2013 Dynamics of a shock-induced separation in a transonic flow: a linearized approach. AIAA Paper 2013-2735.Google Scholar
Schmid, P. J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129–162.Google Scholar
Sipp, D., Marquet, O., Méliga, P. & Barbagallo, A. 2010 Dynamics and control of global instabilities in open-flows: a linearized approach. Appl. Mech. Rev. 63, 030801.Google Scholar
Song, G., Alizard, F., Robinet, J.-C. & Gloerfelt, X. 2013 Global and Koopman modes of sound generation in mixing layers. Phys. Fluids 25, 124101.Google Scholar
Souverein, L. J., Bakker, P. J. & Dupont, P. 2013 A scaling analysis for turbulent shock-wave/boundary-layer interactions. J. Fluid Mech. 714, 505–535.Google Scholar
Souverein, L. J., Dupont, P., Debieve, 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), 1480–1493.Google Scholar
Tam, C. W. & Burton, D. E. 1984 Sound generated by instability waves of supersonic flows. Part 1. Two-dimensional mixing layers. J. Fluid Mech. 138, 249–271.Google Scholar
Theofilis, V. 2003 Advances in global linear instability analysis of nonparallel and three-dimensional flows. Prog. Aerospace Sci. 39, 249–315.Google Scholar
Theofilis, V. 2011 Global linear instability. Annu. Rev. Fluid Mech. 43, 319–352.Google Scholar
Theofilis, V. & Colonius, T.2003 An algorithm for the recovery of 2- and 3-D BiGlobal instabilities of compressible flow over 2-D open cavities. AIAA Paper 2003-4143.Google Scholar
Theofilis, V. & Colonius, T.2004 Three-dimensional instabilities of compressible flow over open cavities: direct solution of the BiGlobal eigenvalue problem. AIAA Paper 2004-2544.CrossRefGoogle Scholar
Theofilis, V., Hein, S. & Dallmann, U. 2000 On the origin of unsteadiness and three-dimensionality in a laminar separation bubble. Phil. Trans. R. Soc. Lond. A 358, 3229–3246.Google Scholar
Toh, K.-C. & Trefethen, L. N. 1996 Calculation of pseudospectra by the Arnoldi iteration. SIAM J. Sci. Comput. 17, 1–15.Google Scholar
Touber, E. & Sandham, N. D. 2009a Comparison of three large-eddy simulations of shock-induced turbulent separation bubbles. Shock Waves 19, 469–478.Google Scholar
Touber, E. & Sandham, N. D. 2009b Large-eddy simulation of low-frequency unsteadiness in a turbulent shock-induced separation bubble. Theor. Comput. Fluid Dyn. 23, 79–107.Google Scholar
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, 417–465.Google Scholar
Trefethen, L. N. & Embree, M. 2005 Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators. Princeton University Press.Google Scholar
Weiss, P.-E., Deck, S., Robinet, J.-C. & Sagaut, P. 2009 On the dynamics of axisymmetric turbulent separating/reattaching flows. Phys. Fluids 21, 075103.Google Scholar
Wu, M. & Martin, M. P. 2008 Analysis of shock motion in shockwave and turbulent boundary layer interaction using direct numerical simulation data. J. Fluid Mech. 594, 71–83.Google Scholar
Yamouni, S., Sipp, D. & Jacquin, L. 2013 Interaction between feedback aeroacoustic and acoustic resonance mechanisms in a cavity flow: a global stability analysis. J. Fluid Mech. 717, 134–165.Google Scholar
Yao, Y., Krishnan, L., Sandham, N. D. & Roberts, G. T. 2007 The effect of Mach number on unstable disturbances in shock/boundary-layer interactions. Phys. Fluids 19, 054104.Google Scholar
Zebib, A. 1987 Stability of viscous flow past a circular cylinder. J. Engng Maths 21 (2), 155–165.Google Scholar