Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-26T07:22:24.320Z Has data issue: false hasContentIssue false
  • English
  • Français

The propagation of weak shocks in non-uniform flows

Published online by Cambridge University Press:  26 April 2006

N. K.-R. Kevlahan
N. K.-R. Kevlahan
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge, CB3 9EW, UK Present address: LMD-CNRS, 24, rue Lhomond, 75231 Paris cédex 05, France.
Get access Rights & Permissions [Opens in a new window]

Abstract

A new theory of the propagation of weak shocks into non-uniform, two-dimensional flows is introduced. The theory is based on a description of shock propagation in terms of a manifold equation together with compatibility conditions for shock strength and its normal derivatives behind the shock. This approach was developed by Ravindran & Prasad (1993) for shocks of arbitrary strength propagating into a medium at rest and is extended here to non-uniform media and restricted to moderately weak shocks. The theory is tested against known analytical solutions for cylindrical and plane shocks, and against a full direct numerical simulation (DNS) of a shock propagating into a sinusoidal shear flow. The test against DNS shows that the present theory accurately predicts the evolution of a moderately weak shock front, including the formation of shock-shocks due to shock focusing. The theory is then applied to the focusing of an initially parabolic shock, and to the propagation of an initially straight shock into a variety of simple flows (sinusoidal shear, vortex array, point-vortex array) exhibiting some fundamental properties of turbulent flows. A number of relations are deduced for the variation of shock quantities with initial shock strength MS0 and the Mach number of the flow ahead of the shock MU (e.g. separation of shock-shocks and maximum shock strength at a focus). It is found that shock-shocks are likely to form in turbulent flows with Mt/M1N > 0.14–0.25, where Mt is the average Mach number of the turbulence and M1N is the Mach number of the shock in a flow at rest. The shock moves up to 1.5% faster in a two-dimensional vortex array than in uniform flow.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 327 , 25 November 1996 , pp. 161 - 197
Copyright
© 1996 Cambridge University Press

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

Anyiwo, J. C. & Bushnell, D. M. 1982 Turbulence amplification in shock-wave and boundary-layer interaction. AIAA J. 20, 893–899.Google Scholar
Bryson, A. E. & Gross, R. W. F. 1961 Diffraction of strong shocks by cones, cylinders and spheres. J. Fluid Mech. 10, 1–16.Google Scholar
Catherasoo, C. J. & Sturtevant, B. 1983 Shock dynamics in non-uniform media. J. Fluid Mech. 127, 539–561.Google Scholar
Courant, R. & Friedrichs, K. O. 1948 Supersonic Flow and Shock Waves. Interscience Publishers.
Courant, R. & Hilbert, D. 1953 Methods of Mathematical Physics. Interscience.
Durbin, P. A. & Zeman, O. 1992 Rapid distortion theory for homogeneous turbulence with application to modelling. J. Fluid Mech. 242, 349–370.Google Scholar
Germain, P. & Guiraud, J.-P. 1966 Conditions de choc et structure des ondes de choc dans un écoulement non stationnaire de fluide dissipatif. J. Math. Pures Appl. 45, 311–358.Google Scholar
Grinfel'd, I. M. 1978 Ray method for calculating the wavefront intensity in nonlinear elastic material. J. Appl. Math. Mech. 42, 958–977.Google Scholar
Guderley, G. 1942 Starke kugelige und zylindrische Verdichtungsstösse in der Nähe des Kugelmittelpunktes bzw der Zylinderachse. Luftfahrtforschung 19, 302–312.Google Scholar
Hayes, W. D. 1968 Self-similar strong shocks in an exponential medium. J. Fluid Mech. 32, 305–315.Google Scholar
Hunt, J. C. R. & Vassilicos, J. C. 1991 Kolmogorov's contribution to the physical and geometrical understanding of small-scale turbulence and recent developments. Proc. R. Soc. Lond. A 434, 183–210.Google Scholar
Keller, J. B. 1954 Geometrical acoustics. I. The theory of weak shock waves. J. Appl. Phys. 25, 938–947.Google Scholar
Kevlahan, N. 1996 The vorticity jump across a shock. Submitted to J. Fluid Mech.Google Scholar
Kevlahan, N., Krishnan, M. & Lee, S. 1992 Evolution of the shock front and turbulence structures in the shock/turbulence interaction. In Studying Turbulence using Numerical Simulation Databases - IV, Proc. 1992 Summer Program. Stanford: CTR.
Kida, S. & Orszag, S. A. 1990 Enstrophy budget in decaying compressible turbulence. J. Sci. Comput. 5, 1–34.Google Scholar
Kovásznay, L. S. G. 1953 Turbulence in supersonic flow. J. Aero. Sci. 20(10), 657–682.Google Scholar
Kulkarny, V. A. & White, B. S. 1982 Focusing of waves in a turbulent inhomogeneous media. Phys. Fluids 25, 1770–1784.Google Scholar
Landau, L. D. 1945 On shock waves at large distances from the place of their origin. Sov. J. Phys. 9, 496–500.Google Scholar
Lee, S., Lele, S. K. & Moin, P. 1993 Direct numerical simulation of isotropic turbulence interacting with a weak shock wave. J. Fluid Mech. 251, 533–562.Google Scholar
Lele, S. K. 1992 Shock-jump relations in a turbulent flow. Phys. Fluids A 4, 2900–2905.Google Scholar
McKenzie, J. F. & Westphal, K. O. 1968 Interaction of linear waves with oblique shock waves. Phys. Fluids 11, 2350–2362.Google Scholar
Maslov, V. P. 1978 Propagation of shock waves in an isentropic non-viscous gas. J. Sov. Math. 13; English transl. (1980) 119–163.Google Scholar
Moore, F. K. 1953 Unsteady oblique interaction of a shock wave with a plane disturbance. NACA TN-2879.Google Scholar
Obermeier, F. 1983 On the propagation of weak and moderately strong, curved shock waves. J. Fluid Mech. 129, 123–136.Google Scholar
Peters, N. 1992 A spectral closure for premixed turbulent combustion in the flamelet regime. J. Fluid Mech. 242, 611–629.Google Scholar
Prasad, P. 1982 Kinematics of a multi-dimensional shock of arbitrary strength in an ideal gas. Acta Mech. 45, 163–176.Google Scholar
Prasad, P., Ravindran, R. & Sau, A. 1991 The characteristic rule for shocks. Appl. Math. Lett. 4, 5–8.Google Scholar
Ravindran, R. & Prasad, P. 1993 On an infinite system of compatibility conditions along a shock ray. Q. J. Mech. Appl. Maths, 46, 131–140.Google Scholar
Ravindran, R., Suder, S. & Prasad, P. 1994 Long time behaviour of the solution of a system of equations from new theory of shock dynamics. Comput. Math. Appl. 27(12), 91–104.Google Scholar
Ribner, H. S. 1954 Convection of a pattern of vorticity through a shock waveL. NACA TN-2864.Google Scholar
Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.
Sturtevant, B. & Kulkarny, V. A. 1976 The focusing of weak shock waves. J. Fluid Mech. 73, 651–671.Google Scholar
Van Dyke, M. & Guttmann, A. J. 1982 The converging shock wave from a spherical or cylindrical piston. J. Fluid Mech. 120, 451–462Google Scholar
Vassilicos, J. C. & Hunt, J. C. R. 1992 Turbulent flamelet propagation. Combust. Sci. Tech. 87, 291–327.Google Scholar
Whitham, G. B. 1957 A new approach to problems of shock dynamics. Part I. Two-dimensional problems. J. Fluid Mech. 2, 145–171.Google Scholar
Whitham, G. B. 1968 A note on shock dynamics relative to a moving frame. J. Fluid Mech. 31, 449–453.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. John Wiley & Sons.
Williams, F. A. 1985 Turbulent combustion. In The Mathematics of Combustion (ed. J. Buckmaster), pp. 97–131. SIAM.