On the Thickness of Normal Shock Waves in a Perfect Gas

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Abstract
An approximate method for calculating the thickness of shocks in perfect gases, similar to the integral methods of ordinary boundary-layer theory, is formulated and leads to an explicit formula relating a shock-thickness Reynolds number to the Prandtl number, ratio of specific heats, and Mach number entering the shock. The method takes account of the variation of viscosity with temperature. The results of the approximate theory are in exact accord with the known exact solutions for (a) the case where the Prandtl number is 3/4 and the viscosity and heat conductivity are constant, (b) the case where the shock is very weak, and (c) the case where the gas has finite viscosity but zero conductivity. In addition, the approximate theory is in good agreement with results obtained on a differential analyzer showing the effects of Prandtl number and variation of viscosity with temperature. The results also are expressed in terms of the ratio of shock thickness to mean free molecular path in the inlet gas. This ratio is given in terms of the Prandtl number, ratio of specific heats, and shock strength. In the absence of relaxation effects, the numerical magnitudes suggest that the continuum theory is accurate for inlet Mach numbers below about 1.5, whereas for higher Mach numbers the continuum theory is indicative of general orders of magnitude.

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