Turbulence generation by a shock wave interacting with a random density inhomogeneity field

TL;DR

This paper develops an exact linear theory to describe how shock waves interacting with random density inhomogeneities generate turbulence, providing analytical expressions for turbulence characteristics relevant in fusion and astrophysics.

Contribution

It introduces a novel analytical framework for shock-density inhomogeneity interactions, including explicit formulas for turbulence metrics in various regimes.

Findings

Derived closed-form expressions for turbulent kinetic energy and anisotropy.

Quantified shock amplification of density non-uniformities.

Analyzed turbulence generation in weak and strong shock regimes.

Abstract

When a planar shock wave interacts with a random pattern of pre-shock density non-uniformities, it generates an anisotropic turbulent velocity/vorticity field. This turbulence plays an important role at the early stages of the mixing process in the compressed fluid. This situation emerges naturally in shock interaction with weakly inhomogeneous deuterium-wicked foam targets in Inertial Confinement Fusion (ICF) and with density clumps/clouds in astrophysics. We present an exact small-amplitude linear theory describing such interaction. It is based on the exact theory of time and space evolution of the perturbed quantities behind a corrugated shock front for a single-mode pre-shock non-uniformity. Appropriate mode averaging in 2D results in closed analytical expressions for the turbulent kinetic energy, degree of anisotropy of velocity and vorticity fields in the shocked fluid, shock amplification of the density non-uniformity, and sonic energy flux radiated downstream. These explicit formulas are further simplified in the important asymptotic limits of weak/strong shocks and highly compressible fluids. A comparison with the related problem of a shock interacting with a pre-shock isotropic vorticity field is also presented.