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DSMC Calculations of Shock Structure with Various Viscosity Laws

By Charles R. Lilley and Michael N. Macrossan

Abstract

It has long been known that: the thickness [Delta] of a plane 1D shock, expressed in terms of the mean free path [lambda_1] in the upstream (pre-shock) flow, is a strong function of shock Mach number; and that the form of this function is sensitive to the form of the viscosity law [mu = mu(T)] of the gas. On the other hand, the approximate kinetic theory method of Mott-Smith [Phys Rev v82, 885-92 (1951)] shows that for many different assumed molecular models and viscosity laws, the average number of collisions suffered by a typical molecule as it traverses the shock quickly approaches a limit as the Mach number increases [Muckenfuss, Phys Fluids v5, 1325-36 (1962)]. This suggests that a mean free path based on a collision cross-section characteristic of the high speed collisions between up and downstream molecules is the appropriate length scale for normalising shock thickness results. One such length scale is a kinetic length scale [L*] defined for the sonic conditions of the flow [Robben and Talbot, Phys Fluids v9, 633-43 (1966)], and another is the mean free path in the downstream flow [lambda_2], where the average collision energy approaches the stagnation energy. Here we use DSMC results, for many different collision models, (and hence viscosity laws), to show that [Delta/lambda_2] is virtually constant for shock Mach numbers M > 3, with little variation shown between different forms of the viscosity law. We also show that [Delta/L*] displays similar behaviour for shock M > 10. Experimental measurements of shock thickness in argon [Alsmeyer, J Fluid Mech v74, 497-513 (1976)], up to downstream temperatures of 2,000 K, for which the viscosity is known from experiments, have been used to estimate a limiting value of [Delta/lambda_2] for argon. Similarly, a limiting value of [Delta/L*] has been estimated using results for which the sonic temperature is < 2,000 K. These limiting values are then used to deduce the viscosity of argon at temperatures far exceeding 2,000 K, for which there appears to be no reliable experimental data

Topics: DSMC, shock thickness, viscosity of argon, shock length scale, 240502 Fluid Physics, 240503 Thermodynamics and Statistical Physics, 290299 Aerospace Engineering not elsewhere classified
Year: 2003
OAI identifier: oai:espace.library.uq.edu.au:UQ:9812

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