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 $M_1$. These formulas are further reduced to simpler exact asymptotic expressions in the limits of weak and strong shock waves $\left(M_1-1⪡1,M_1⪢1\right)$ and high shock compressibility of the gas $\left(\gamma \to 1\right)$.

Ignition in a supersonic mixing layer interacting with an oblique shock wave is investigated analytically and numerically under conditions such that the post-shock flow remains supersonic. The study requires consideration of the structure of the post-shock ignition kernel that is found to exist around the point of maximum temperature, which may be located either near the edge of the mixing layer or in its interior, depending on the profiles of the fuel concentration, temperature and Mach number across the mixing layer. The ignition kernel displays a balance between the rates of chemical reaction and of post-shock flow expansion, including the acoustic interactions of the chemical heat release with the shock wave, leading to increased front curvature. The analysis, which adopts a one-step chemistry model with large activation energy, indicates that ignition develops as a fold bifurcation, the turning point in the diagram of the peak perturbation induced by the chemical reaction as a function of the Damköhler number providing the critical conditions for ignition. While an explicit formula for the critical Damköhler number for ignition is derived when ignition occurs in the interior of the mixing layer, under which condition the ignition kernel is narrow in the streamwise direction, numerical integration is required for determining ignition when it occurs at the edge, under which condition the kernel is no longer slender. Subsequent to ignition, for the Arrhenius chemistry addressed, the lead shock will rapidly be transformed into a thin detonation on the fuel side of the ignition kernel, and, under suitable conditions, a deflagration may extend far downstream, along with the diffusion flame that must separate the rich and lean reaction products. The results can be helpful in describing supersonic combustion for high-speed propulsion.

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 $\gamma${} and shock strength M1) 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.

We present an analytical model that describes the linear interaction of a planar shock wave with an isotropic random sonic field. First, we study the interaction with a single-mode acoustic field. We present the exact evolution for the pressure, velocity, vorticity, and density field generated behind the shock wave, and we also calculate exact and closed analytical expressions for the asymptotic behavior of these modes. Applying superposition, we use the results obtained from the single-mode analysis in order to compute the interaction with 2D/3D isotropic random acoustic fields. We present analytical expressions for the average turbulent kinetic energy generated behind the shock, as well as the averaged vorticity and the density perturbations as a function of the shock strength M1 and the gas compressibility $\gamma${}. We also study the acoustic energy flux emitted by the shock front. Exact asymptotic analytical scaling laws are given for all the 3D averages downstream. A detailed comparison with previous works is shown.

The response of a shock front to small preshock nonuniformities of density, pressure, and velocity is studied theoretically and numerically. These preshock nonuniformities emulate imperfections of a laser target, due either to its manufacturing, like joints or feeding tubes, or to preshock perturbation seeding/growth, as well as density fluctuations in foam targets, “thermal layers” near heated surfaces, etc. Similarly to the shock-wave interaction with a small nonuniformity localized at a material interface, which triggers a classical Richtmyer-Meshkov (RM) instability, interaction of a shock wave with periodic or localized preshock perturbations distributed in the volume distorts the shape of the shock front and can cause a RM-type instability growth. Explicit asymptotic formulas describing distortion of the shock front and the rate of RM-type growth are presented. These formulas are favorably compared both to the exact solutions of the corresponding initial-boundary-value problem and to numerical simulations. It is demonstrated that a small density modulation localized sufficiently close to a flat target surface produces the same perturbation growth as an “equivalent” ripple on the surface of a uniform target, characterized by the same initial areal mass modulation amplitude.

Convective perturbations arising from nuclear shell burning can play an important role in propelling neutrino-driven core-collapse supernova explosions. In this work, we analyse the impact of vorticity waves on the shock dynamics, and subsequently on the post-shock flow, using the solution of the linear hydrodynamics equations. As a result of the interaction with the shock wave, vorticity waves increase their kinetic energy, and a new set of entropic and acoustic waves is deposited in the post-shock region. These perturbations interact with the neutrino-driven turbulent convection that develops in that region. Although both vorticity and acoustic waves inject non-radial motion into the gain region, the contribution of the acoustic waves is found to be negligibly small in comparison to that of the vorticity waves. On the other hand, entropy waves become buoyant and trigger more convection. Using the concept of critical neutrino luminosity, we assess the impact of these modes on the explosion conditions. While the direct injection of non-radial motion reduces the critical neutrino luminosity by ∼ 12 per cent for typical problem parameters, the buoyancy-driven convection triggered by entropy waves reduces the critical luminosity by ∼ 17–24 per cent, which approximately agrees with the results of three-dimensional neutrino-hydrodynamics simulations. Finally, we discuss the limits of validity of the assumptions employed.