The asymptotic analysis of wave interactions and numerical calculations of transonic nozzle flow. (English) Zbl 0598.76065

Transonic flow through a duct of variable cross section for which nonlinear resonance effects are important is considered. The complicated local interactions of nonlinear waves are resolved through asymptotic analysis and this is then used to construct a random choice method to calculate general unsteady flow fields. The method produces sharp shocks without oscillations, is accurate in smooth regions, and converges to a stable steady flow.


76H05 Transonic flows
76N15 Gas dynamics (general theory)
76M99 Basic methods in fluid mechanics
76E30 Nonlinear effects in hydrodynamic stability
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[1] Chorin, A.J, Random choice solution of hyperbolic systems, J. comp. phys., 22, 517-533, (1976) · Zbl 0354.65047
[2] Colella, P, Glimm’s method for gas dynamics, SIAM J. sci. statist. comp., 3, 76-110, (1982) · Zbl 0502.76073
[3] {\scP. Colella and H. M. Glaz}, Numerical modelling of inviscid shocked flows of real gases, in “Proc. Eighth International Conference on Numerical Methods in Fluid Dynamics” (E. Krause, Ed.), Springer-Verlag Lecture Notes in Physics, Vol. 170.
[4] Courant, R; Friedrichs, K.O, Supersonic flow and shock waves, (1948), Wiley-Interscience New York · Zbl 0041.11302
[5] Fok, S.K, Extension of Glimm’s method to the problem of gas flow in a duct of variable cross-section, Lawrence Berkeley laboratory report LBL-12322, (1980)
[6] Glimm, J, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. pure appl. math., 18, 697-715, (1965) · Zbl 0141.28902
[7] Lax, P.D, Hyperbolic systems of conservation laws, II, Comm. pure appl. math., 10, 537-566, (1957) · Zbl 0081.08803
[8] Li, C.-Z; Liu, T.-P, Asymptotic states for hyperbolic conservation laws with a moving source, Adv. appl. math., 4, 353-379, (1983) · Zbl 0544.76059
[9] Liepmann, H.W; Roshko, A, Elements of gas dynamics, (1956), Wiley New York, Chap. 5 · Zbl 0078.39901
[10] Liu, T.-P, Quasilinear hyperbolic systems, Comm. math. phys., 68, 141-172, (1979) · Zbl 0435.35054
[11] Liu, T.-P, Transonic gas flows along a duct of varying area, Arch. rat. mech. anal., 80, 1-18, (1982) · Zbl 0503.76076
[12] Liu, T.-P, Nonlinear stability and instability of transonic gas flow through a nozzle, Comm. math. phys., 83, 243-260, (1982) · Zbl 0576.76053
[13] Liu, T.-P, Resonance for quasilinear hyperbolic equation, Bull. amer. math. soc., 6, 463-465, (1982) · Zbl 0501.76048
[14] Shubin, G.R; Stephens, A.B; Glaz, H.M, Steady shock tracking and Newton’s method applied to one-dimensional duct flow, J. comp. phys., 39, 364-374, (1981) · Zbl 0468.76061
[15] Sod, G, A numerical study of a converging cylindrical shock, J. fluid mech., 83, 785-794, (1977) · Zbl 0366.76055
[16] van Leer, B, On the relation between the upwind-differencing schemes of Godunov, engquist-osher, and roe, ICASE report no. 81-11, (1981)
[17] {\scJ. Glimm, G. Marshall, and B. Plohr}, A generalized random choice method for gas dynamics, preprint. · Zbl 0566.76056
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