Cooling Effect of the Richtmyer-Meshkov Instability

ESAIM: PROCEEDINGS AND SURVEYS, December 2015, Vol. 52, p. 66-75 S. Dellacherie, F. Dubois, S. Fauve, R. Gatignol, Editors COOLING EFFECT OF THE RICHTMYER-MESHKOV INSTABILITY Farhang Mohseni 1 , Miller Mendoza 1 , Sauro Succi 2 and Hans J. Herrmann 1, 3 Abstract. We provide numerical evidence that the Richtmyer-Meshkov (RM) instability contributes to the cooling of a relativistic fluid. Due to the presence of jet particles traveling throughout the medium, shock waves are generated in the form of Mach cones. The interaction of multiple shock waves can trigger the RM instability, and we have found that this process leads to a down-cooling of the relativistic fluid. To confirm the cooling effect of the instability, shock tube Richtmyer-Meshkov instability simulations are performed. Additionally, in order to provide an experimental observable of the RM instability resulting from the Mach cone interaction, we measure the two particle correlation function and highlight the effects of the interaction. The simulations have been performed with an improved version of the relativistic lattice Boltzmann model, including general equations of state and external forces. 1. Introduction Particles traveling through a compressible fluid generate waves moving at the speed of sound. Moreover, if the particles travel faster than the speed of sound of the medium, the disturbances in the fluid are confined to the so-called Mach cone. This phenomenon is very common in many natural systems, including astrophysics and high energy physics [1–5], where relativistic fluid effects are important. The existence of relativistic shock-waves in the presence of density variations, leads to the appearance of the RM instability, one of the fundamental fluid instabilities, which occurs whenever a shock wave passes through an interface between regions at different densities. This instability was theoretically predicted by Richtmyer [6] and experimentally detected by Meshkov [7], in the non-relativistic context. The study of the RM instability is of major importance in several fields, ranging from high energy physics [2–5] to astrophysics [8] and plasma physics [9]. Density variations can appear in the relativistic fluid whenever particles travel through the medium, due to the sweeping effect of the shock waves [10], as well as due to external mechanisms. In this work, we show numerically that the RM instability may reduce the average temperature of the relativistic fluid. In particular, we investigate the interaction of two relativistic Mach shocks, and show that the RM instability arises due to this interaction (see Fig. 1). Furthermore, we find that the appearance of this hydrodynamic instability leads to a decrease in the average temperature of the medium. To justify this finding and to single out the effect of the instability, shock tube RM instability simulations are carried out. The effect of initial domain temperature and density ratio on the cooling effect of the instability is also investigated. Since the growth rate of the instability depends explicitly on its form, the study of the instability can provide information on the equation of state (EoS), the same way shock waves can offer insights on the EoS [11, 12]. 1 ETH Zürich, Computational Physics for Engineering Materials, Institute for Building Materials, Wolfgang-Pauli-Strasse 27, HIT, CH-8093 Zürich (Switzerland) e-mail: & 2 Istituto per le Applicazioni del Calcolo C.N.R., Via dei Taurini, 19 00185, Rome (Italy) e-mail: 3 Departamento de Fı́sica, Universidade Federal do Ceará, 60455-760 Fortaleza, Ceará, (Brazil) e-mail: c EDP Sciences, SMAI 2015 Article published online by EDP Sciences and available at http://www.esaim-proc.org or http://dx.doi.org/10.1051/proc/201552004 67 ESAIM: PROCEEDINGS AND SURVEYS Thus, gaining information about the RM instability may provide a new means of studying the thermodynamic properties of relativistic fluids. Finally, we propose a way to detect the interaction between Mach cones from an experimental observable, namely the two-point correlation function. For the numerical simulations, the recently developed relativistic Lattice Boltzmann (LB) model [13] is extended to deal with the ideal gas equation of state and external forces. This paper is organised as follows. In Sec. 2, we explain the numerical model and the extensions needed to describe a relativistic fluid with an arbitrary equation of state. The results of our numerical simulations of the RM instability are presented in Sec. 3, and finally, in Sec. 4 we discuss our results and comment on future works. 2. Numerical model Let us start the description of the numerical model by presenting the conservation equations for relativistic fluid dynamics, namely ∂α T αβ = 0, αβ ∂α N α = 0, α (1) β 2 αβ αβ where the energy-momentum tensor is defined as T = (ǫ + p)U U /c − pη + π , and the current density as N α = nU α [14]. Here, n is the number density, p the hydrostatic pressure, ǫ the energy density, c the speed of light, π αβ the shear-stress tensor, and η αβ the Minkowski metric tensor with the signature (+, −,p −, −). The macroscopic four-velocity is (U α ) = (c, ~u)γ(u), ~u being the three-dimensional velocity and γ(u) = 1/ 1 − u2 /c2 the Lorentz’s factor. The Einstein summation convention and natural units i.e., c = kB = ~ = 1, are assumed here and throughout this paper. All our numerical simulations are performed using the extended version of the relativistic LB model recently proposed in Ref. [13]. This method is a numerical approach based on a minimal lattice version of the relativistic Boltzmann equation, which can be solved to find the probability distribution function in phase space [13,15]. The equilibrium distribution f eq for the relativistic Boltzmann equation, in the single-relaxation time approximation for the collision operator, is the Maxwell-Jüttner distribution function [16]. An extension to simulate high velocities was also proposed in Ref. [13], based on the ultra-relativistic equation of state, i.e., ǫ = 3p. However, for this case, the equation for the conservation of energy and momentum is not affected by the density field [13], and the two conservation equations become decoupled. This effect suppresses the RM instability, where both equations must be coupled. Therefore, we are interested in a more general ideal gas equation of state, of the form [17]: p = (Γ − 1)(ǫ − n), (2) where Γ = cp /cv , with cp and cv being the specific heats at constant pressure and volume, respectively. For low temperatures, i.e. mc2 /kB T ≪ 1 , Γ = 5/3, while for high temperatures, i.e. mc2 /kB T ≫ 1 , Γ = 4/3. In the ultra-relativistic limit, by replacing Γ = 4/3 and considering the condition n ≪ ǫ, the ultra-relativistic equation of state is recovered. In the relativistic lattice Boltzmann methods based on the model of Marle [18] for the collision operator, the macroscopic variables can be calculated by solving a system of equati (...truncated)