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Higher Order Conditions for Weak Shocks: Modified Prandtl Relation
Author(s) -
Blackmore D.,
Ting L.
Publication year - 2002
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/1617-7061(200203)1:1<397::aid-pamm397>3.0.co;2-g
Subject(s) - prandtl number , shock (circulatory) , inviscid flow , curvature , turbulent prandtl number , shock wave , tangent , physics , oblique shock , moving shock , classical mechanics , mechanics , plane (geometry) , mathematical analysis , mathematics , reynolds number , geometry , turbulence , heat transfer , nusselt number , medicine
The classical equations for the jumps in the state variables across a planar shock wave in an inviscid flow ‐ which can be solved by starting with the Prandtl relation for the jump in the normal velocity ‐ have been applied locally in gas dynamics to a tangent plane of a curved shock. This tangent plane approximation has been validated for real fluids using matched asymptotics with a small parameter ϵ ≔ δ/l, where δ is the effective shock thickness and l is a typical radius of curvature of the shock; ϵ is of order 1/Re where Re is the Reynolds number based on l. The leading order inner solution in the scale δ yields the classical shock structure resolving the discontinuities across the shock in the larger scale l, while the O(ϵ) corrections to the classical shock conditions account for the shock structure, shock curvature and the flow field gradients behind and ahead of the shock. We prove that there is an analog (correction) of the classical Prandtl relation that shows that the first order correction to the tangent plane approximation is O(ϵ/Δ), where Δ denotes the scaled shock strength. Hence, the correction to the Prandtl relation is of paramount importance in the analysis of weak shocks.