# Shock motion inside a varying cross-section channel and consequences on the downstream flow

Abstract : Shock wave propagation in a variable cross-section channel is a recurrent issue in the literature. Seminal work regarding this flow configuration has been proposed by~\citeapos{whitham_propagation_1958} through the derivation of a one-dimensional approach connecting the shock Mach number and the area channel: the A-M relation. It is based on strong theoretical restrictions: (i) shock equations applied on a $C^+$, (ii) omission of the post-shock influence, and (iii) initial conditions at rest. It has been the focus of many studies aimed at generalizing it. However, very little attention has been paid to the study of the shock motion outside the varying cross-sectional region since~\citeapos{russell_shock-wave_2018}. The objective of the current work is to describe and to explain the shock wave behaviour in a constant area channel behind a convergent or divergent channel. It is found that the shock propagation in the downstream uniform area region is influenced by the post-shock flow unsteadinesses. Thence, the Whitham model is not suited to the shock motion study in a constant area region downstream of a convergent or divergent area region. A detailed flow description is provided and a quasi-steady model for determining the waves intensity at large times is proposed. This model gives accurate results without any assumptions on the shock strength, area variation or the shock upstream state. Finally, this study points out the limits of the use of Whitham's theory in a variable area channel with increasing section variation rate (where $|\mathrm{dA/A(x+dx)}| \geq |\mathrm{dA/A(x)}|$).
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Journal articles

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Submitted on : Tuesday, April 27, 2021 - 10:11:41 AM
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Florian Hermet, Jérémie Gressier, Nicolas Binder. Shock motion inside a varying cross-section channel and consequences on the downstream flow. Physical Review Fluids, American Physical Society, 2021, 6 (4), ⟨10.1103/PhysRevFluids.6.044802⟩. ⟨hal-03209251⟩

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