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Abstract

We simulate transmission of a small-amplitude disturbance wave through a shock wave. Results of our numerical experiments performed with different high-order shock-capturing schemes show that the capability of a scheme to correctly predict the amplification of the disturbances crucially depends on the Riemann solver used in evaluation of numerical fluxes. Incorrectly high amplification rates are produced by the solvers resolving shock waves sharply, with no interior points in numerical profiles of steady shock waves. In particular, both the exact Riemann solver and the popular Roe flux difference splitting demonstrate such unphysical behavior. A possible explanation of such behavior is proposed. More dissipative solvers, such as the global Lax–Friedrichs splitting, produce transmission coefficients close to the predictions of linear theory.

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References

Smits

A.J.

Dussauge

J.-P.

, Turbulent Shear Layers in Supersonic Flow, 2nd edition., Springer, N. Y., 2006.

Gatski

T.B.

and Bonnet

J.-P.

, Compressibility, Turbulence and High-Speed Flow, Elsevier, Oxford, Amsterdam, 2009.

Blokhintsev

D.I.

, Sound receiver in motion, Dokl. Akad. Nauk SSSR, 1945, 47(1), 22–25.

Burgers

J.M.

, On the transmission of sound waves through a shock, Proc. Kon. Ned. Akad. Wet, 1946, 49, 273–281.

Dyakov

S.P.

, Interaction of shock waves with small disturbances, 1, J. Experimental and Theoretical Physics – JETP, 1957, 33(4), 948–961.

Dyakov

S.P.

, Interaction of shock waves with small disturbances, 2, J. Experimental and Theoretical Physics – JETP, 1957, 33 (4), 962–974.

Kontorovitch

V.M.

, Reflection and refraction of the sound on the shock waves, Acoustical Physics (Akusticheskii Zhurnal), 1957, 5(3), 314–323.

McKenzie

J.F.

and Westphal

K.O.

, Interaction of linear waves with oblique shock waves, Physics of Fluids, 1968, 11 (11), 2350–2362.

Shugaev

F.V.

and Shtemenko

L.S.

, Propagation and Reflection of Shock Waves, Series on Advances in Mathematics for Applied Sciences (49), World Scientific, Singapore , 1998.

10.

Yamamoto

and Daiguji

, Higher-order-accurate upwind schemes for solving the compressible Euler and Navier-Stokes equations, Computers and Fluids, 1993, 22 (2–3), 259–270.

11.

Godunov

S.K.

A finite difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics

Mat. Sb., 1959, 47 (3), 271–306.

12.

Roe

P.L.

, Approximate Riemann solvers, parameter Vectors, and difference schemes. J. Comput. Phys., 1981, 43 (2), 357–372.

13.

Harten

, On a class of high resolution total-variation-stable finite difference schemes, SIAM J. Numerical. Analysis, 1981, 43, 357–372.

14.

van Leer

Flux-vector splitting for the Euler equations, in: Lecture Notes in Physics, Springer, Berlin, 1982, 170, 507–512.

15.

Jiang

G.S.

and Shu

C.W.

, Efficient implementation of weighted ENO Schemes, J. Computational Physics, 1996, 126 (1), 202–228.

16.

Zhong

, High-order finite-difference schemes for numerical simulation of hypersonic boundary-layer transition, J. Computational Physics, 1998, 144 (2), 662–709.

17.

Lele

, Compact finite-difference schemes with spectral-like resolution. J. Computational Physics, 1992, 103 (1), 16–42.

18.

Hairer

Nørsett

S.P.

Wanner

, Solving ordinary differential equations I. Nonstiff problems, Springer Series in Computational Mathematics (8), Springer Verlag, 1987.

19.

Hayes

W. D.

, The basic theory of gasdynamic discontinuities, in: Fundamentals of Gas Dynamics, Vol. III, High Speed Aerodynamics and Jet Propulsion, Emmons

H. W.

(Ed.), Princeton University Press, 1958, Chap 4.

20.

Lubchich

A.A.

, and Pudovkin

M.I.

Interaction of small perturbations with shock waves, Physics of Fluids, 2004, 16 (12), 4489–4505.

21.

Landau

L.D.

and Lifshitz

E.M.

, Fluid Mechanics, 2nd edition: Volume 6 of Course of Theoretical Physics, Pergamon Press, 1987.

22.

Roberts

T.W.

, The behavior of flux difference splitting schemes near slowly moving shock waves, J. Computational Physics, 1990, 90 (1), 141–160.

A Comparative Study of Accuracy of Shock Capturing Schemes for Simulation of Shock/Acoustic Wave Interactions

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