Abstract
In many problems encountered in modern gasdynamics, the boundary layer approximations are inadequate to account for the dissipative factors-viscosity and thermal conductivity of the gas-and the solution of the complete system of Navier-Stokes equations is required. This includes, for example, flows with large longitudinal pressure gradients, which in order of magnitude are comparable with or exceed the transverse gradients (temperature jumps, sharp flow rotations, compression shocks, etc.). In many cases, for example in flows with low density, the scale of action of the longitudinal gradients becomes significant, which leads to the need for considering the flow structure in the vicinity of the large gradients. The formulation of certain problems of this type leads to a system of one-dimensional Navier-Stokes equations.
We present a difference scheme for the solution of the system of one-dimensional stationary and nonstationary Navier-Stokes equations and give examples of the calculation of the structure of the stationary shock wave front, unsteady gas flow under the influence of sudden heating of one of the boundaries, and unsteady gas flow in the vicinity of the decay of an initial discontinuity. The solution of the stationary problems is accomplished as a result of stabilization as t → ∞.
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References
N. E. Kochin, I. A. Kibel, and V. N. Roze, Theoretical Hydromechanics [in Russian], Part 2, Fizmatgiz, 1963.
G. Serrin, Mathematical Fundamentals of Classical Fluid Mechanics [Russian translation], Izd. inostr. lit., 1963.
J. Gary, “On certain finite difference schemes for hyperbolic systems”, Math, of Comput., vol. 18, no. 85, 1964.
L. Filler and H. Ludloff, “Stability analysis and integration of the viscous equations of motion”, Math. Comput., 15, no. 75, 1961.
I. Yu. Brailovskaya, “A difference scheme for the numerical solution of the two-dimensional Navier-Stokes equations for a compressible gas”, DAN SSSR, vol. 160, no. 5, 1965.
L. Crocco, “A suggestion for numerical solution of the steady Navier-Stokes equations”, AIAA Journal, vol. 3, no. 10, 1965.
S. K. Godunov and V. S. Ryaben'kii, Introduction to the Theory of Difference Schemes [in Russian], Fizmatgiz, 1962.
D. Gilbarg and D. Paoluccio, “The structure of shock wave in continuum theory of fluids”, J. Rational Mech. and Analysis, vol. 2, 617, 1953.
R. Von Mises, Mathematical Theory of Compressible Fluid Flows [Russian translation], Izd. inostr. lit., 1961.
M. Morduchow and P. Libby, “On a complete solution of the one-dimensional flow equations of a viscous, heat conducting, compressible gas”, J. Aero. Sci., vol. 16, no. 11, 1949.
L. I. Sedov, Similarity and Dimensional Methods in Mechanics [in Russian], Fizmatgiz, 1965.
Ya. B. Zel'dovich and Yu. P. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena [in Russian], Fizmatgiz, 1965.
A. A. Smarskii and I. M. Sobol, “Examples of the numerical calculation of temperature waves”, Zh. vychislit. matem. i matem. fiz., vol. 3, no. 4, 1963.
F. A. Goldsworthy, “The structure of a contact region with application to the reflection of a chock from a heat conducting wall”, J. Fluid Mech., vol. 5, no. 1, 1959.
Yu. A. Dem'yanov and V. T. Kireev, “On the analysis of one-dimensional unsteady gas flows with account for heat conduction and viscosity”, Izv. AN SSSR, Mekhanika, no. 2, 1965.
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The author wishes to thank V. Ya. Likhushin and V. S. Avduevskii for interest in the study and for their valuable counsel during the investigation.
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Polezhaev, V.I. Numerical solution of the system of one-dimensional unsteady Navier-Stokes equations for a compressible gas. Fluid Dyn 1, 21–27 (1966). https://doi.org/10.1007/BF01022272
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DOI: https://doi.org/10.1007/BF01022272