Analytic solution for the zero-order postshock profiles when an incident planar shock hits a planar contact surface

An explicit analytical solution to calculate the profiles after the shock collision with a planar contact surface is presented. The case when a shock is reflected after the incident shock refraction is considered. The goal of this work is to present explicit formulas to obtain the quantities behind the transmitted and reflected shocks valid for arbitrary initial preshock parameters.

Analytical scalings of the linear Richtmyer-Meshkov instability when a rarefaction is reflected

The Richtmyer-Meshkov instability for the case of a reflected rarefaction is studied in detail following the growth of the contact surface in the linear regime and providing explicit analytical expressions for the asymptotic velocities in different physical limits. This work is a continuation of the similar problem when a shock is reflected [Phys. Rev. E 93, 053111 (2016)1539-375510.1103/PhysRevE.93.053111]. Explicit analytical expressions for the asymptotic normal velocity of the rippled surface (δv_{i}^{∞}) are shown. The known analytical solution of the perturbations growing inside the rarefaction fan is coupled to the pressure perturbations between the transmitted shock front and the rarefaction trailing edge. The surface ripple growth (ψ_{i}) is followed from t=0+ up to the asymptotic stage inside the linear regime. As in the shock reflected case, an asymptotic behavior of the form ψ_{i}(t)≅ψ_{∞}+δv_{i}^{∞}t is observed, where ψ_{∞} is an asymptotic ordinate to the origin. Approximate expressions for the asymptotic velocities are given for arbitrary values of the shock Mach number. The asymptotic velocity field is calculated at both sides of the contact surface. The kinetic energy content of the velocity field is explicitly calculated. It is seen that a significant part of the motion occurs inside a fluid layer very near the material surface in good qualitative agreement with recent simulations. The important physical limits of weak and strong shocks and high and low preshock density ratio are also discussed and exact Taylor expansions are given. The results of the linear theory are compared to simulations and experimental work [R. L. Holmes et al., J. Fluid Mech. 389, 55 (1999)JFLSA70022-112010.1017/S0022112099004838; C. Mariani et al., Phys. Rev. Lett. 100, 254503 (2008)PRLTAO0031-900710.1103/PhysRevLett.100.254503]. The theoretical predictions of δv_{i}^{∞} and ψ_{∞} show good agreement with the experimental and numerical reported values.

Analytical scalings of the linear Richtmyer-Meshkov instability when a shock is reflected

When a planar shock hits a corrugated contact surface between two fluids, hydrodynamic perturbations are generated in both fluids that result in asymptotic normal and tangential velocity perturbations in the linear stage, the so called Richtmyer-Meshkov instability. In this work, explicit and exact analytical expansions of the asymptotic normal velocity (δv_{i}^{∞}) are presented for the general case in which a shock is reflected back. The expansions are derived from the conservation equations and take into account the whole perturbation history between the transmitted and reflected fronts. The important physical limits of weak and strong shocks and the high/low preshock density ratio at the contact surface are shown. An approximate expression for the normal velocity, valid even for high compression regimes, is given. A comparison with recent experimental data is done. The contact surface ripple growth is studied during the linear phase showing good agreement between theory and experiments done in a wide range of incident shock Mach numbers and preshock density ratios, for the cases in which the initial ripple amplitude is small enough. In particular, it is shown that in the linear asymptotic phase, the contact surface ripple (ψ_{i}) grows as ψ_{∞}+δv_{i}^{∞}t, where ψ_{∞} is an asymptotic ordinate different from the postshock ripple amplitude at t=0+. This work is a continuation of the calculations of F. Cobos Campos and J. G. Wouchuk, [Phys. Rev. E 90, 053007 (2014)PLEEE81539-375510.1103/PhysRevE.90.053007] for a single shock moving into one fluid.

Normal velocity freeze-out of the Richtmyer-Meshkov instability when a rarefaction is reflected

The Richtmyer-Meshkov instability (RMI) develops when a shock front hits a rippled contact surface separating two different fluids. After the incident shock refraction, a transmitted shock is always formed and another shock or a rarefaction is reflected back. The pressure-entropy-vorticity fields generated by the rippled wave fronts are responsible for the generation of hydrodynamic perturbations in both fluids. In linear theory, the contact surface ripple reaches an asymptotic normal velocity which is dependent on the incident shock Mach number, fluids density ratio, and compressibilities. It was speculated in the past about the possibility of getting a zero value for the asymptotic normal velocity, a phenomenon that was called "freeze-out" [G. Fraley, Phys. Fluids 29, 376 (1986); K. Mikaelian, Phys. Fluids 6, 356 (1994), A. L. Velikovich et al., Phys. Plasmas 8, 592 (2001)]. In a previous paper, freeze-out was studied for the case when a shock is reflected at the contact surface [J. G. Wouchuk and K. Nishihara, Phys. Rev. E 70, 026305 (2004)]. In this work the freeze-out of the RMI is studied for the case in which a rarefaction is reflected back. Two different regimes are found: nearly equal preshock densities at the interface at any shock intensity, and very large density difference for strong shocks. The contour curves that relate shock Mach number and preshock density ratio are obtained in both regimes for fluids with equal and different compressibilities. An analysis of the temporal evolution of different cases of freeze-out is shown. It is seen that the freeze-out is the result of the interaction between the unstable interface and the rippled wave fronts. As a general and qualitative criterion to look for freeze-out situations, it is seen that a necessary condition for freeze-out is the same orientation for the tangential velocities generated at each side of the contact surface at t=0+. A comparison with the results of previous works is also shown.

Analytical asymptotic velocities in linear Richtmyer-Meshkov-like flows

An analytical model to study the perturbation flow that evolves between a rippled piston and a shock is presented. Two boundary conditions are considered: rigid and free surface. Any time a corrugated shock is launched inside a fluid, pressure, velocity, density, and vorticity perturbations are generated downstream. As the shock separates, the pressure field decays in time and a quiescent velocity field emerges in the space in front of the piston. Depending on the boundary conditions imposed at the driving piston, either tangential or normal velocity perturbations evolve asymptotically on its surface. The goal of this work is to present explicit analytical formulas to calculate the asymptotic velocities at the piston. This is done in the important physical limits of weak and strong shocks. An approximate formula for any shock strength is also discussed for both boundary conditions.

Effect of shock-generated turbulence on the Hugoniot jump conditions

Interaction of a shock wave with preshock random density nonuniformities is known to generate turbulence in the postshock flow. The turbulent motion, in turn, modifies the shock jump conditions. As first detected in the simulations by Hazak et al. [Phys. Plasmas 5, 4357 (1998)], shock compression of a deuterium-filled foam is less than that predicted for a uniform medium of the same average density. Exact analytical small-amplitude theory of this shock undercompression effect is reported, and explicit formulas for the turbulent corrections to the strong-shock Hugoniot jump conditions are presented.

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

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 M(1) and the gas compressibility γ. 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.

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

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.

Richtmyer-Meshkov instability: theory of linear and nonlinear evolution

A theoretical framework to study linear and nonlinear Richtmyer-Meshkov instability (RMI) is presented. This instability typically develops when an incident shock crosses a corrugated material interface separating two fluids with different thermodynamic properties. Because the contact surface is rippled, the transmitted and reflected wavefronts are also corrugated, and some circulation is generated at the material boundary. The velocity circulation is progressively modified by the sound wave field radiated by the wavefronts, and ripple growth at the contact surface reaches a constant asymptotic normal velocity when the shocks/rarefactions are distant enough. The instability growth is driven by two effects: an initial deposition of velocity circulation at the material interface by the corrugated shock fronts and its subsequent variation in time due to the sonic field of pressure perturbations radiated by the deformed shocks. First, an exact analytical model to determine the asymptotic linear growth rate is presented and its dependence on the governing parameters is briefly discussed. Instabilities referred to as RM-like, driven by localized non-uniform vorticity, also exist; they are either initially deposited or supplied by external sources. Ablative RMI and its stabilization mechanisms are discussed as an example. When the ripple amplitude increases and becomes comparable to the perturbation wavelength, the instability enters the nonlinear phase and the perturbation velocity starts to decrease. An analytical model to describe this second stage of instability evolution is presented within the limit of incompressible and irrotational fluids, based on the dynamics of the contact surface circulation. RMI in solids and liquids is also presented via molecular dynamics simulations for planar and cylindrical geometries, where we show the generation of vorticity even in viscid materials.

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

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) .

Instability of a planar expansion wave

An expansion wave is produced when an incident shock wave interacts with a surface separating a fluid from a vacuum. Such an interaction starts the feedout process that transfers perturbations from the rippled inner (rear) to the outer (front) surface of a target in inertial confinement fusion. Being essentially a standing sonic wave superimposed on a centered expansion wave, a rippled expansion wave in an ideal gas, like a rippled shock wave, typically produces decaying oscillations of all fluid variables. Its behavior, however, is different at large and small values of the adiabatic exponent gamma. At gamma > 3, the mass modulation amplitude delta(m) in a rippled expansion wave exhibits a power-law growth with time alpha(t)beta, where beta = (gamma - 3)/(gamma - 1). This is the only example of a hydrodynamic instability whose law of growth, dependent on the equation of state, is expressed in a closed analytical form. The growth is shown to be driven by a physical mechanism similar to that of a classical Richtmyer-Meshkov instability. In the opposite extreme gamma - 1 << 1, delta(m) exhibits oscillatory growth, approximately linear with time, until it reaches its peak value approximately (gamma - 1)(-1/2), and then starts to decrease. The mechanism driving the growth is the same as that of Vishniac's instability of a blast wave in a gas with low . Exact analytical expressions for the growth rates are derived for both cases and favorably compared to hydrodynamic simulation results.

Spontaneous acoustic emission of a corrugated shock wave in the presence of a reflecting surface

An analytic model to study perturbation evolution in the space between a corrugated shock and a piston surface is presented. The conditions for stable oscillation patterns are obtained by looking at the poles of the exact Laplace transform. It is seen that besides the standard D'yakov-Kontorovich (DK) mode of oscillation, the shock surface can exhibit an additional finite set of discrete frequencies, due to the interaction with the piston which reflects sound waves from behind. The additional eigenmodes are excited when the shock is launched at t= 0(+) . The first eigenmode (the DK mode) is always present, if the Hugoniot curve has the correct slope in the V-p plane. However, the additional frequencies could be excited for strong enough shocks. The predictions of the model are verified for particular cases by studying a van der Waals gas, as in the work of Phys. Fluids 11, 462 (1999)]; Phys. Rev. Lett. 84, 1180 (2000)]. Only acoustic emission modes are considered.

Normal velocity freeze-out of the Richtmyer-Meshkov instability when a shock is reflected

It is known that for some values of the initial parameters that define the Richtmyer-Meshkov instability, the normal velocity at the contact surface vanishes asymptotically in time. This phenomenon, called freeze-out, is studied here with an exact analytic model. The instability freeze-out, already considered by previous authors [K.O. Mikaelian, Phys. Fluids 6, 356 (1994), Y. Yang, Q. Zhang, and D.H. Sharp, Phys. Fluids 6, 1856 (1994), and A.L. Velikovich, Phys. Fluids 8, 1666 (1996)], is the result of a subtle interaction between the unstable surface and the corrugated shock fronts. In particular, it is seen that the transmitted shock at the contact surface plays a key role in determining the asymptotic behavior of the normal velocity at the contact surface. By properly tuning the fluids compressibilities, the density jump, and the incident shock Mach number, the value of the initial circulation deposited by the reflected and transmitted shocks at the material interface can be adjusted in such a way that the normal growth at the contact surface will vanish for large times. The conditions for this to happen are calculated exactly, by expressing the initial density ratio as a function of the other parameters of the problem: fluids compressibilities and incident shock Mach number. This is done by means of a linear theory model developed in a previous work [J.G. Wouchuk, Phys. Rev. E. 63, 056303 (2001)]. A general and qualitative criterion to decide the conditions for freezing-out is derived, and the evolution of different cases (freeze-out and non-freeze-out) are studied with some detail. A comparison with previous works is also presented.

Growth rate of the linear Richtmyer-Meshkov instability when a shock is reflected

An analytic model is presented to calculate the growth rate of the linear Richtmyer-Meshkov instability in the shock-reflected case. The model allows us to calculate the asymptotic contact surface perturbation velocity for any value of the incident shock intensity, arbitrary fluids compressibilities, and for any density ratio at the interface. The growth rate comes out as the solution of a system of two coupled functional equations and is expressed formally as an infinite series. The distinguishing feature of the procedure shown here is the high speed of convergence of the intermediate calculations. There is excellent agreement with previous linear simulations and experiments done in shock tubes.