Internal structure of shock waves: Asymptotic behavior in the inviscid limit and features at small Prandtl numbers

Abstract

The internal structure of shock waves is well known for shocks with a Prandtl number of the order of unity, but it is still poorly understood for shocks with extremely large and small Prandtl numbers, and the latter case is of particular interest given the non-existence of smooth solutions for strong inviscid shocks. The literature has suggested unequal contributions of viscosity and heat conduction to shock compression, specifically that viscosity is of more significance, but the observations therein do not provide further insight into the exact role of viscosity in shock waves, i.e., how viscosity affects the shock structure such that a smooth transition is maintained. To elucidate it, this research numerically investigates the internal structure of shock waves, with emphasis on the asymptotic behavior and features of shock profiles in the inviscid (zero-Prandtl-number) limit. The asymptotic analysis demonstrates a curious fact: the distinction between strong shock solutions with positive and zero Prandtl numbers does not vanish as the former's Prandtl number tends to zero. The only way to avoid this gap is to allow the emergence of an unphysical discontinuity (e.g., an isothermal jump) in inviscid solutions, which clearly indicates the essentiality of viscosity. For a better understanding of the underlying mechanisms, features of small-Prandtl-number shocks are further studied and compared with those of ordinary shocks. The result shows a peak in viscous local entropy generation rates within small-Prandtl-number shocks, and reducing the viscosity paradoxically makes it higher. Through this peak, the effect of lowered viscosity is neutralized, and the isothermal jump is smoothed, hence the critical role of viscosity in shock transition.