Analytical linear theory for the interaction of a planar shock wave with an isotropic turbulent vorticity field

Abstract

An exact analytical model for the interaction between an isolated shock wave and an isotropic turbulent vorticity field is presented. The interaction with a single-mode two-dimensional (2D) divergence-free vorticity field is analyzed in detail, giving the time and space evolutions of the perturbed quantities downstream. The results are generalized to study the interaction of a planar shock wave with an isotropic three-dimensional (3D) or 2D preshock vorticity field. This field is decomposed into Fourier modes, and each mode is assumed to interact independently with the shock front. Averages of the downstream quantities are made by integrating over the angles that define the orientation of the upstream velocity field. The ratio of downstream/upstream kinetic energies is in good agreement with existing numerical and experimental results for both 3D and 2D preshock vorticity fields. The generation of sound and the sonic energy flux radiated downstream from the shock front is also discussed in detail, as well as the amplification of transverse vorticity across the shock front. The anisotropy is calculated for the far downstream fields of both velocity and vorticity. All the quantities characteristic of the shock-turbulence interaction are reduced to closed-form exact analytical expressions. They are presented as explicit functions of the two parameters that govern the dynamics of the interaction: the adiabatic exponent gamma and the shock Mach number M1 . These formulas are further reduced to simpler exact asymptotic expressions in the limits of weak and strong shock waves (M_{1}-11, M_{1}1) and high shock compressibility of the gas (gamma-->1) .

Richtmyer–Meshkov instability: theory of linear and nonlinear evolution

“…It should also be mentioned that it is possible to extend the theory in §3c,d for a case with three-dimensional interface and two-dimensional sheet strength. Although transition to a turbulent regime is not well understood yet, there has been an attempt to solve shock interaction with an isotropic turbulent vorticity field (Wouchuk et al 2009). …”

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.…”

“…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.…”