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Initial-value-problem solution for isolated rippled shock fronts in arbitrary fluid media.

Jason W Bates¹

¹Plasma Physics Division, U.S. Naval Research Laboratory, Washington, DC 20375, USA. bates@this.nrl.navy.mil

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|July 13, 2004

Summary

This study analyzes shock front perturbations, revealing stable solutions with oscillations decaying over time. Shorter wavelengths dissipate faster, influenced by shock strength and material properties.

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Area of Science:

Fluid Dynamics
Shock Wave Propagation
Nonlinear Dynamics

Background:

Previous work by Roberts established foundational understanding of shock front behavior.
Investigating perturbations is crucial for predicting shock stability and evolution.
Understanding shock dynamics in inviscid fluids with arbitrary equations of state is a key challenge.

Purpose of the Study:

To analyze the impact of 2D perturbations on planar shock fronts.
To derive analytical expressions for linearized, time-dependent Fourier coefficients.
To characterize the stability and decay of perturbed shock fronts.

Main Methods:

Linearized analysis of an initial-value problem for a planar shock front.
Derivation of analytical expressions for Fourier coefficients describing corrugations.

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Comparison of theoretical predictions with FAST2D numerical simulations.

Main Results:

Explicit analytical expressions for time-dependent Fourier coefficients were derived.
The temporal evolution exhibits damped oscillations, distinct from simple harmonic motion.
Two families of stable solutions were identified, with decay envelopes proportional to t(-3/2).
Shorter wavelengths were observed to decay more rapidly than longer ones.

Conclusions:

The study provides a theoretical framework for understanding perturbed shock front dynamics.
Stability and decay rates are dependent on shock strength and the fluid's equation of state.
Theoretical findings align with numerical simulations, validating the analytical model.