drives a spherical shock into the target, with the goal being to compress the fuel to such an extent that the temperature and density at the center are sufficient to initiate a fusion reaction. The RMI promotes mixing between the capsule material and the fuel. This mixing limits the final compression of the fuel and hence the possibility of achieving energy break-even or production (Lindl et al., 1992). The RMI is also important in astrophysical phenomena. It has been used to account for the lack of stratification in the products of supernova 1987A and is required in stellar evolution models (Arnett, 2000). In supersonic and hypersonic air breathing engines, the RMI may be used to enhance the mixing of fuel and air (Yang et al., 1993). The RMI also arises in many combustion systems where shock-flame interactions occur, the resulting **instability** is significant in deflagration-to-detonation transition (Khokhlov et al., 1999). Finally, in reflected shock tunnels, the RMI is a possible mechanism for explaining driver gas contamination in the absence of shock bifurcation due to the wall boundary layer (Stalker and Crane, 1978, Brouillette and Bonazza, 1999).

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in **magnetohydrodynamics** [80]. The new solver was compared to the previous version, which was based on a method of linearized characteristics, for the case of a solid–gas interface with a difference in pressure that replicated a **Richtmyer**–**Meshkov** shock–interface interaction, showing that results were closer to the exact solution when employing HLLD. The study of the multi-material capabili- ties was complemented with tests showing the influence of the AMR algorithm and the level set in mass conservation. This tests revealed extremely small errors due to the restriction–prolongation algorithm of AMR and at most first-order convergence rate in mass errors as the base grid is refined. Chapter 4 explored the problem of cylindrically and spherically symmetric converging shocks in elastic–plastic solids. This interesting problem was judged to be a useful precursor to the study of the cylindrically convergent **Richtmyer**–**Meshkov** **instability**. Whitham’s shock dynamics equations for compressible neo-Hookean elastic–plastic solids were derived obtaining closed expressions for the shock evolution in terms of definite integrals. Comparison with numerical simulations revealed that this approximate theory is extremely accurate, even when shear deformations and plasticity are considered. Strong-shock limits were identified for purely elastic and elastic–perfectly plastic solids, exhibiting a large dependence on the constitutive law used to model the material. This study was closed with the numerical analysis of the transition of a shock from the elastic to the plastic regime due to the effects of a converging geometry. Contrary to the elastic precursor–plastic shock structure observed in planar shocks, the plastic shock travels faster in this case and eventually catches up with the elastic precursor, producing a single imploding plastic shock.

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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.

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Even without the rotation, the relativistic jet poten- tially becomes unstable to the Rayleigh-Taylor **instability** [23]. A radial inertia force, which naturally arises from a pressure mismatch between the jet and cocoon when the jet propagates through the ambient medium, drives the ra- dial oscillating motion of the jet, yielding the reconfine- ment region inside the jet [7, 24]. This radial inertia force triggers the Rayleigh-Taylor **instability** at the jet interface. When considering the non-axisymmetric evolution of the jet, the radial oscillation-induced Rayleigh-Taylor in- stability at the interface of the jet might have a po- tential impact on the deformation and morphology of the relativistic jet. In these proceedings, the nonlin- ear development of the relativistic jet is studied using three-dimensional (3D) special relativistic hydrodynamic (SRHD) simulations.

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are compared with experiments both when the gravitational acceleration is responsible for the mixing and when a suddent acceleration (shock) forces the mixing. The first case constitutes a Rayleigh-Taylor (RT) **instability** driven mixing front, and the second case forces **Richtmyer**-**Meshkov** (RM) **instability** that produces further mixing coupled with heat and mass transport. The **instability** produced, RT in its simplest forms, occurs when a layer of dense fluid is placed on top of a less dense layer in a gravitational field. On the other hand, if a stable two fluid layer configuration accelerating (e.g. falling in a gravitational field) is suddenly decelerated, then RM develops during the short time that the upward acceleration dominates gravitational acceleration, g.

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The results of the single-mode tests were compared to the Omega experiments for single-mode ripples at the six same laser energies. To simulate the different energies (70 J, 100 J, 120 J, 150 J, 200 J, and 250 J), each simulation was subjected to a different velocity profile, which was obtained from the rad-hydro code Hyades. The velocity profile was applied as a boundary condition to a heat shield that sits in front of the Ta sample. A wave travels through the heat shield, passing through the interface of the heat shield and Ta sample, and creating the **Richtmyer**-**Meshkov** **instability**. Following the simulations, growth factors were calculated through visualization and nodal displacement analysis. Each successive simulation was performed at higher drive energy, and it was expected that with increasing energy the growth factor would increase. The results are summarized in Chapter 4 and the data shows that the values for peak growth factor do increase with increasing laser energy. The trends of growth factor versus time were also observed. For laser energies ranging from 70 to 150 J, the growth factor at the end increased. For laser energies 200 and 250 J, the growth factor at the end decreased, but the ripples were compressed substantially after the initial growth. Two different mesh sizes were used during these simulations. The finer mesh provided more accuracy, but it did not affect the values of the peak growth factor or the trends of growth factor over time. Predictions of peak pressure were also estimated in the simulations and agreed reasonably well with the predictions from the rad-hydro code Hyades.

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This paper investigates **Richtmyer**–**Meshkov** **instability** in shock-tubes using a novel algorithm. This is done to val- idate the algorithm used and to test a moving mesh tech- nique which has been implemented into the existing Cran- field in-house code, CHOC (see Section 2 for full details). The moving mesh technique is useful for reshock and conver- gent geometries whereby the region of interest moves con- siderably. Using this method cells can be clustered around regions of interest and track the interfaces without the com- putational penalty of using a highly refined grid throughout the domain. Here the algorithm is validated for a number of cases including the Sod-shock [7] tube case and the single- mode **Richtmyer**–**Meshkov** **instability** (SM-RMI) case com- pared with the results of Collins and Jacobs [6].

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This paper presents a hybrid compressible-incompressible approach for simulating the **Richtmyer**-**Meshkov** in- stability (RMI) and associated mixing. The proposed numerical approach aims to circumvent the numerical defi- ciencies of compressible methods at low Mach numbers, when the flow has become essentially incompressible. A compressible flow solver is used at the initial stage of the interaction of the shock wave with the fluids interface and the development of the **Richtmyer**-**Meshkov** **instability**. When the flow becomes sufficiently incompressible, based on a Mach number prescribed threshold, the simulation is carried out using an incompressible flow solver. Both the compressible and incompressible solvers use Godunov-type methods and high-resolution numerical reconstruction schemes for computing the fluxes at the cell interfaces. The accuracy of the model is assessed by using results for a 2D single-mode RMI.

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The interactions of shock waves with perturbed interfaces separating fluids of different properties are of crucial importance in compressible turbulence, as they occur in a myriad of applications, both natural and man-made. This class of problems is generally referred to as the **Richtmyer**-**Meshkov** **instability** (RMI), after **Richtmyer**, who first rigorously analyzed the growth rate of a perturbation at a plane density inhomogeneity following an impulsive acceleration modeling the passage of a shock parallel to the interface [94], and **Meshkov**, who confirmed, at least qualitatively, Richtmyer’s predictions using shock-tube experiments [77]. In [94] **Richtmyer** also compared his analytical results with numerical simulations of the linearized compressible Euler equations. The RMI is sometimes thought of as an impulsive, or shock-induced, version of the Rayleigh-Taylor **instability** (RTI), where the density interface is submitted to a finite sustained acceleration (e.g., gravitational field) [107, 92]. The dominant fluid dynamics process responsible for the amplification of the interface pertur- bation is local vorticity generation by means of baroclinic torque, due to the misalignment of the pressure gradient across the shock and the local density gradient at the interface during shock pas- sage. Consider the evolution equation of the vorticity field ω = ∇ × u, with u velocity field. In the absence of dissipation terms,

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∇ × ∇ ) at the interface, and induces the turbulent mixing at the late times. This phenomenon exists in a variety of man-made applications and natural phenomena such as the inertial confinement fusion (ICF) [3], deflagration-to-detonation transition (DDT) [4], high-speed combustion [5] and astrophysics (i.e. supernova explosions) [6]. It is so impor- tant that many scientists devote themselves to the study of the **Richtmyer**-**Meshkov** **instability**.

In this paper, based on the multi-viscous-fluid piece- wise parabolic method [10], the Vreman [11] and Sma- gorinsky [12] subgrid eddy viscosity models are employed to solve the Navier-Stokes equations. A three-dimensional large eddy simulation (LES) code MVFT3D (3D multi- viscous-fluid and turbulence) for the multi-viscosity-fluid and turbulence from the fluid interface **instability** is de- veloped. The SGS dissipation and molecular viscosity dis- sipation have been analyzed by the simulation of AWE’s shuck tube [13] RM **instability** experiment. We mainly simulate the experiment of fluid **instability** of shock-ac- celerated interface by Poggi in this paper. Experiment shows that a turbulent mixing zone is generated by the incident shock wave. We can see in numerical simula- tions the decay of the turbulent kinetic energy before the first reflected shock wave-mixing zone interaction and its strong enhancement by re-shocks. The computational mixing zone width under double re-shock agreement well with the experiment, and the decaying law of the turbu- lent kinetic energy is consistent with Mohamed and Larue’s investigation. By the numerical simulations, we compare the factors that affect the mixing zone width, which include the Smagorinsky model and the Vreman SGS model, as well as the three kinds of random wave- length ranges. We focus on the three dimensional simu- lation of shock induced turbulence. The goal of our simulation is to perform highly resolved three-dimen- sional numerical simulations of flows subsequent to the RM **instability**, and study both the transitional and the turbulent regimes. In the last part of the paper, we give some simulation results of the inverse Chevron model from AWE. However, because there is no definite ex- perimental data in the literature [14], we have no quanti- tative comparison. After all, through quantitative and qualitative comparison with experiment, the method and code of MVFT3D is validated and can be used to inves- tigate the problem of RM **instability** induced turbulence.

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In contrast to the enormous advances in the theoretical and numerical analysis of the DGM, the development of a viable, attractive, competitive, and ultimately superior DG method over the more mature and well-established second order methods is relatively an untouched area. This is mainly due to the fact that the DGM have a number of weaknesses that have yet to be addressed, before they can be robustly used for problems of practical interest in **magnetohydrodynamics** in a complex configuration environment. In particular, how to effectively control spurious oscillations in the presence of strong discontinuities, and how to reduce the computing costs for the DGM remain the two most challenging and unresolved issues in the DGM. Indeed, compared to the finite element methods and finite volume methods, the DGM require solutions of systems of equations with more unknowns for the same grids. Consequently, these methods have been recognized as expensive in terms of both computational costs and storage requirements.

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Torres-Rincon 2015; Buividovich & Ulybyshev 2016 ) . This **instability** may be relevant in the physics of the early universe ( Joyce & Shaposhnikov 1997; Fröhlich & Pedrini 2000, 2002; Semikoz & Sokoloff 2004; Semikoz et al. 2009, 2012; Boyarsky et al. 2012a, 2012b; Dvornikov & Semikoz 2012, 2013, 2014, 2017; Tashiro et al. 2012; Manuel & Torres-Rincon 2015; Gorbar et al. 2016; Pavlovi ć et al. 2016, 2017 ) , of the quark – gluon plasmas ( Akamatsu & Yamamoto 2013; Hirono et al. 2015; Taghavi & Wiedemann 2015 ) , or of neutron stars ( Ohnishi & Yamamoto 2014; Dvornikov & Semikoz 2015a, 2015b; Dvornikov 2016; Sigl & Leite 2016; Yamamoto 2016 ) . However, to the best of our knowledge, a systematic analysis of the system of chiral MHD equations, including the back- reaction of the magnetic ﬁ eld on the chiral chemical potential and the coupling to the plasma velocity ﬁ eld, U ( t, x ) , appears to be missing.

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In this paper, we aim to study the effect of plasma cooling on the damping rate of slow standing MHD waves in coronal loops for arbitrary thermal conduction coefficient.. The paper is or[r]

It is now well recognized that magnetic fields play a very important role in many astrophysical phenomena and in particular in those involving relativistic outflows. The magnetic fields are likely to be involved in launching, powering and collimation of such outflows. The dynamics of relativistic magnetized plasma can be studied using diverse mathematical frameworks. The most developed one so far is the single fluid ideal relativistic **magnetohydrodynamics** (RMHD). During the last decade, an impressive progress has been achieved in developing robust and efficient computer codes for integration of its equations. Only within a scope of proper review one can acknowledge the numerous contributions made by many different research groups and individuals. This framework is most suitable for studying large-scale macroscopic motions as it does not require to resolve the microphysical scales for numerical stability. One of its obvious disadvantages is that it allows only numerical dissipation. While at shocks this is not of a problem, the dissipation observed in simulations at other locations may be questionable. This becomes more of an issue as ever growing body of evidence suggests the importance of magnetic dissipation associated with magnetic reconnection in dynamics of highly magnetized relativistic plasma. The framework of resistive RMHD allows us to incorporate this magnetic dissipation, but the inevitably phenomenological nature

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Description Ratio of smoothing length to particle spacing Smoothing length gradient correction term Tolerance in smoothing length-density iterations Scalar part of kernel gradient Deriva[r]

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Abstract. We present a new adaptive multiresoltion method for the numerical simulation of ideal **magnetohydrodynamics**. The governing equations, i.e. , the compressible Euler equations coupled with the Maxwell equations are discretized using a finite volume scheme on a two-dimensional Cartesian mesh. Adaptivity in space is obtained via Harten’s cell average multiresolution analysis, which allows the reliable introduction of a locally refined mesh while controlling the error. The explicit time dis- cretization uses a compact Runge–Kutta method for local time stepping and an embedded Runge-Kutta scheme for automatic time step control. An extended generalized Lagrangian multiplier approach with the mixed hyperbolic-parabolic correction type is used to control the incompressibility of the magnetic field. Applications to a two-dimensional problem illustrate the properties of the method. Memory savings and numerical divergences of magnetic field are reported and the accuracy of the adaptive computations is assessed by comparing with the available exact solution.

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We investigate whether or not the low ionization fractions in molecular cloud cores can solve the ‘magnetic braking catastrophe’, where magnetic fields prevent the formation of circumstellar discs around young stars. We perform three-dimensional smoothed particle non- ideal **magnetohydrodynamics** (MHD) simulations of the gravitational collapse of one solar mass molecular cloud cores, incorporating the effects of ambipolar diffusion, Ohmic resistivity and the Hall effect alongside a self-consistent calculation of the ionization chemistry assuming 0.1 μ m grains. When including only ambipolar diffusion or Ohmic resistivity, discs do not form in the presence of strong magnetic fields, similar to the cases using ideal MHD. With the Hall effect included, disc formation depends on the direction of the magnetic field with respect to the rotation vector of the gas cloud. When the vectors are aligned, strong magnetic braking occurs and no disc is formed. When the vectors are anti-aligned, a disc with radius of 13 au can form even in strong magnetic when all three non-ideal terms are present, and a disc of 38 au can form when only the Hall effect is present; in both cases, a counter-rotating envelope forms around the first hydrostatic core. For weaker, anti-aligned fields, the Hall effect produces massive discs comparable to those produced in the absence of magnetic fields, suggesting that planet formation via gravitational **instability** may depend on the sign of the magnetic field in the precursor molecular cloud core.

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SUMMARY AND CONCLUDING REMARKS I improved on previous simulations of wind-cloud interactions by 1 using higher numerical resolutions to adequately capture small-scale perturbations arisi[r]

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Eringen (1964) earlier developed the fluid mechanics of deformable microelements, which were termed as Simple Microfluids. Eringen defined the simple microfluid as: “A viscous medium whose behavior and properties are affected by the local motion of particles in its microvolume”. These fluids are characterized by 22 viscosity and material constants and when applied to flow problems the result is a system of 19 partial differential equations with 19 unknown that may not be amenable to be solved. Eringen (1966) subsequently introduced a subclass of fluids which he named micropolar fluids that ignores the deformation of the microelements but still allows for the particle micromotion to take place. The theory of micropolar fluids, which consist of rigid, randomly oriented particles suspended in a viscous medium. These special features of micropolar fluids were discussed comprehensively by Ariman et al. (1973). In general, as part of the momentum is lost in rotating of particles, the flow of a micropolar fluid is less prone to **instability** than that of a classical fluid. The stability of micropolar fluids problems have been investigated by Lakshmana Rao (1970) as well as Sastry and Das (1985).

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