Analytical linear theory for the interaction of a planar shock wave with a two- or three-dimensional random isotropic density field

Abstract

We present an analytical linear model describing the interaction of a planar shock wave with an isotropic random pattern of density nonuniformities. This kind of interaction is important in inertial confinement fusion where shocks travel into weakly inhomogeneous cryogenic deuterium-wicked foams, and also in astrophysics, where shocks interact with interstellar density clumps. The model presented here is based on the exact theory of space and time evolution of the perturbed quantities generated by a corrugated shock wave traveling into a small-amplitude single-mode density field. Corresponding averages in both two and three dimensions are obtained as closed analytical expressions for the turbulent kinetic energy, acoustic energy flux, density amplification, and vorticity generation downstream. They are given as explicit functions of the two parameters (adiabatic exponent γ and shock strength M(1)) that govern the dynamics of the problem. In addition, these explicit formulas are simplified in the important asymptotic limits of weak and strong shocks and highly compressible fluids.

Interaction of a planar shock with an isotropic field of sound waves

“…At the shock surface we require that the fluid perturbations satisfy the linearized Rankine-Hugoniot (RH) conservation equations [16,14,15], and that no sound waves are reflected back to the shock at the weak surface discontinuity (x = 0). The method used to solve Eq.…”

“…The method used to solve Eq. (1) is the same as the one explained in [14,15]. We do not go into the details here for lack of space.…”

“…The periodic forcing of the shock ripple due to the upstream sound waves induces oscillations in the shock shape, which in turn generate evanescent or traveling pressure fluctuations downstream, and density and velocity perturbations. Besides, the weak discontinuity at x = 0 develops Richtmyer-Meshkov instability growth and its ripple evolves in time with amplitude ψ i (t) [14,15]. The shock front ripple oscillations also generate vorticity and thus, the velocity field downstream can be decomposed as the sum of a rotational field associated to the shock generated vorticity and an irrotational field associated to the sound waves that escape behind the shock.…”

“…Therefore, the probability of a particular orientation is equal to sin θ dθ dφ /(4π). Hence, assuming isotropy of rotation on the shock surface, the average of the kinetic energy downstream is [14,15]:…”

“…Experiments were reported for the shock compression of a vorticity field by Barre et al [13]. Quite recently, linear theory analytical models were developed to deal with the interaction of planar shock waves with an isotropic vorticity field [14] and an isotropic density field [15]. In these works, the statistical averages of the interesting quantities downstream (kinetic energy, density, vorticity and acoustic energy flux) were obtained analytically, in terms of elementary functions which could be further reduced to simplified expressions in the important physical limits of weak shocks, strong shocks, and highly compressible fluids.…”