Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-02T21:46:26.927Z Has data issue: false hasContentIssue false
  • English
  • Français

Low-order stochastic modelling of low-frequency motions in reflected shock-wave/boundary-layer interactions

Published online by Cambridge University Press:  07 March 2011

EMILE TOUBER  and
NEIL D. SANDHAM
EMILE TOUBER*
Affiliation:
Aerodynamics and Flight Mechanics Research Group, School of Engineering Sciences, University of Southampton, University Road, Southampton, SO17 1BJ, UK
NEIL D. SANDHAM
Affiliation:
Aerodynamics and Flight Mechanics Research Group, School of Engineering Sciences, University of Southampton, University Road, Southampton, SO17 1BJ, UK
*
†Present address: Department of Aeronautics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK. Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

A combined numerical and analytical approach is used to study the low-frequency shock motions observed in shock/turbulent-boundary-layer interactions in the particular case of a shock-reflection configuration. Starting from an exact form of the momentum integral equation and guided by data from large-eddy simulations, a stochastic ordinary differential equation for the reflected-shock-foot low-frequency motions is derived. During the derivation a similarity hypothesis is verified for the streamwise evolution of boundary-layer thickness measures in the interaction zone. In its simplest form, the derived governing equation is mathematically equivalent to that postulated without proof by Plotkin (AIAA J., vol. 13, 1975, p. 1036). In the present contribution, all the terms in the equation are modelled, leading to a closed form of the system, which is then applied to a wide range of input parameters. The resulting map of the most energetic low-frequency motions is presented. It is found that while the mean boundary-layer properties are important in controlling the interaction size, they do not contribute significantly to the dynamics. Moreover, the frequency of the most energetic fluctuations is shown to be a robust feature, in agreement with earlier experimental observations. The model is proved capable of reproducing available low-frequency experimental and numerical wall-pressure spectra. The coupling between the shock and the boundary layer is found to be mathematically equivalent to a first-order low-pass filter. It is argued that the observed low-frequency unsteadiness in such interactions is not necessarily a property of the forcing, either from upstream or downstream of the shock, but an intrinsic property of the coupled system, whose response to white-noise forcing is in excellent agreement with actual spectra.

JFM classification

Compressible Flows: Compressible boundary layers Nonlinear Dynamical Systems: Low-dimensional models Compressible Flows: Shock waves
Type
Papers
Information
Journal of Fluid Mechanics , Volume 671 , 25 March 2011 , pp. 417 - 465
Copyright
Copyright © Cambridge University Press 2011

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

REFERENCES

Carpenter, M. H., Nordstrom, J. & Gottlieb, D. 1998 A stable and conservative interface treatment of arbitrary spatial accuracy. Tech. Rep. CR-206921. NASA.Google Scholar
Ducros, F., Ferrand, V., Nicoud, F., Weber, C., Darracq, D., Gacherieu, C. & Poinsot, T. 1999 Large-eddy simulation of the shock/turbulence interaction. J. Comput. Phys. 152, 517–549.CrossRefGoogle Scholar
Dupont, P., Haddad, C. & Debiève, J. F. 2006 Space and time organization in a shock-induced separated boundary layer. J. Fluid Mech. 559, 255–277.CrossRefGoogle Scholar
Dussauge, J.-P., Dupont, P. & Debiève, J.-F. 2006 Unsteadiness in shock wave boundary layer interaction with separation. Aerosp. Sci. Technol. 10, 85–91.CrossRefGoogle Scholar
Dussauge, J.-P. & Piponniau, S. 2008 Shock/boundary-layer interactions: possible sources of unsteadiness. J. Fluids Struct. 24, 1166–1175.CrossRefGoogle 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.CrossRefGoogle Scholar
Ganapathisubramani, B., Clemens, N. T. & Dolling, D. S. 2009 Low-frequency dynamics of shock-induced separation in a compression ramp interaction. J. Fluid Mech. 585, 397–425.CrossRefGoogle Scholar
Garnier, E. 2009 Stimulated detached eddy simulation of three-dimensional shock/boundary layer interaction. Shock Waves 19 (6), 479–486.CrossRefGoogle Scholar
Garnier, E., Sagaut, P. & Deville, M. 2002 Large eddy simulation of shock/boundary-layer interaction. AIAA J. 40 (10), 1935–1944.CrossRefGoogle Scholar
Humble, R. A., Elsinga, G. E., Scarana, F. & van Oudheusden, B. W. 2009 Three-dimensional instantaneous structure of a shock wave/turbulent boundary layer interaction. J. Fluid Mech. 622, 33–62.CrossRefGoogle Scholar
Inagaki, M., Kondoh, T. & Nagano, Y. 2005 A mixed-time-scale SGS model with fixed model-parameters for practical LES. J. Fluids Engng 127, 1–13.CrossRefGoogle Scholar
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, 652–665.CrossRefGoogle Scholar
Moin, P., Squires, K., Cabot, W. & Lee, S. 1991 A dynamic subgrid-scale model for compressible turbulence and scalar transport. Phys. Fluids 3 (11), 2746–2757.CrossRefGoogle Scholar
Piponniau, S., Dussauge, J.-P., Debiève, 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. & Grasso, F. 2006 Direct numerical simulation of impinging shock wave/turbulent boundary layer interaction at M = 2.25. Phys. Fluids 18 (6), 065113.CrossRefGoogle Scholar
Plotkin, K. J. 1975 Shock wave oscillation driven by turbulent boundary-layer fluctuations. AIAA J. 13 (8), 1036–1040.CrossRefGoogle Scholar
Poggie, J. & Smits, A. J. 2001 Shock unsteadiness in a reattaching shear layer. J. Fluid Mech. 429, 155–185.CrossRefGoogle Scholar
Poggie, J. & Smits, A. J. 2005 Experimental evidence for Plotkin model of shock unsteadiness in separated flow. Phys. Fluids 17 (1), 018107.CrossRefGoogle Scholar
Polivanov, P. A., Sidorenko, A. A. & Maslov, A. A. 2009 Experimental study of unsteady effects in shock wave/turbulent boundary layer interaction. In 47th AIAA Aerospace Sciences Meeting, Orlando, Florida, USA.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), 1174–1185.CrossRefGoogle Scholar
Ringuette, M. J., Bookey, P., Wyckham, C. & Smits, A. J. 2009 Experimental study of a Mach 3 compression ramp interaction at Re θ = 2400. AIAA J. 47 (2), 373–385.CrossRefGoogle Scholar
Risken, H. 1989 The Fokker–Plank Equation, 2nd edn. Springer.Google Scholar
Robinet, J.-C. 2007 Bifurcations in shock-wave/laminar-boundary-layer interaction: global instability approach. J. Fluid Mech. 579, 85–112.CrossRefGoogle Scholar
Sagaut, P. 2005 Large Eddy Simulations for Incompressible Flows, 3rd edn. Springer.Google Scholar
Sandham, N. D., Li, Q. & Yee, H. C. 2002 Entropy splitting for high-order numerical simulation of compressible turbulence. J. Comput. Phys. 178, 307–322.CrossRefGoogle Scholar
Sandhu, H. S. & Sandham, N. D. 1994 Boundary conditions for spatially growing compressible shear layers. Tech. Rep. QMW-EP-1100. Queen Mary, University of London.Google Scholar
Thompson, K. W. 1987 Time dependent boundary conditions for hyperbolic systems. J. Comput. Phys. 68 (1), 1–24.CrossRefGoogle Scholar
Touber, E. 2010 Unsteadiness in shock-wave/boundary-layer interactions. PhD thesis, University of Southampton, School of Engineering Sciences.Google Scholar
Touber, E. & Sandham, N. D. 2008 Oblique shock impinging on a turbulent boundary layer: low-frequency mechanisms. In 38th Fluid Dynamics Conference, Seattle, Washington, USA. AIAA Paper 2008-4170.Google Scholar
Touber, E. & Sandham, N. D. 2009 a Comparison of three large-eddy simulations of shock-induced turbulent separation bubbles. Shock Waves 19 (6), 469–478.CrossRefGoogle Scholar
Touber, E. & Sandham, N. D. 2009 b Large-eddy simulation of low-frequency unsteadiness in a turbulent shock-induced separation bubble. Theor. Comput. Fluid Dyn. 23, 79–107.CrossRefGoogle Scholar
Touber, E. & Sandham, N. D. 2009 c Large-eddy simulations of an oblique shock impinging on a turbulent boundary layer: effect of the spanwise confinement on the low-frequency oscillations. In 2nd Intl Conf. on Turbulence and Interactions, Sainte-Luce, Martinique.CrossRefGoogle Scholar
Touber, E. & Sandham, N. D. 2010 Oblique-shock/turbulent-boundary-layer interaction. arXiv:1010.2044v1. Available at: http://arxiv.org/abs/1010.2044v1.Google Scholar
White, F. M. 1991 Viscous Fluid Flow, 2nd edn. McGraw-Hill.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.CrossRefGoogle Scholar
Yee, H. C., Sandham, N. D. & Djomehri, M. J. 1999 Low-dissipative high-order shock-capturing methods using characteristic-based filters. J. Comput. Phys. 150, 199–238.CrossRefGoogle Scholar