A new pressure-based lattice-Boltzmann method (HRR-p) is proposed for the simulation of flows for Mach numbers ranging from 0 to 1.5. Compatible with nearest-neighbor lattices (e.g., D3Q19), the model consists of a predictor step comparable to classical athermal lattice-Boltzmann methods, appended with a fully local and explicit correction step for the pressure. Energy conservation—for which the Hermitian quadrature is not accurate enough on such a lattice—is solved via a classical finite volume MUSCL-Hancock scheme based on the entropy equation. The Euler part of the model is then validated for the transport of three canonical modes (vortex, entropy, and acoustic propagation), while its diffusive/viscous properties are assessed via thermal Couette flow simulations. All results match the analytical solutions with very limited dissipation. Last, the robustness of the method is tested in a one-dimensional shock tube and a two-dimensional shock–vortex interaction.

International audienceAn original penalization method is applied to model the interaction of magnetically confined plasma with limiter in the frame of minimal transport model for ionic density and parallel momentum. The limiter is considered as a pure particle sink for the plasma and consequently the density and the momentum are enforced to be zero inside. Comparisons of the numerical results with one dimensional analytical solutions show a very good agreement. In particular, presented method provides a plasma velocity which is almost sonic at the boundaries obstacles as expected from the sheath conditions through the Bohm criterion. The new system being solved in an obstacle free domain, an efficient pseudo-spectral algorithm based on a Fast Fourier transform is also proposed, and associated with an exponential filtering of the unphysical oscillations due to Gibbs phenomenon. Finally, the efficiency of the method is illustrated by investigating the flow spreading from the plasma core to the Scrape Off Layer at the wall in a two-dimensional system with one then two limiters neighboring

The numerical simulation of physical problems modeled by systems of conservation laws is difficult due to the presence of discontinuities in the solution. High-order shock capturing schemes combine sharp numerical profiles at discontinuities with a highly accurate approximation in smooth regions, but usually their computational cost is quite large.Following the idea of A. Harten [Comm. we present in this paper a method to reduce the execution time of such simulations. It is based on a point value multiresolution transform that is used to detect regions with singularities. In these regions, an expensive high-resolution shock capturing scheme is applied to compute the numerical flux at cell interfaces. In smooth regions a cheap polynomial interpolation is used to deduce the value of the numerical divergence from values previously obtained on lower resolution scales.This method is applied to solve the two-dimensional compressible Euler equations for two classical configurations. The results are analyzed in terms of quality and efficiency.are routinely used to investigate the behavior of the different HRSC schemes in use, and also their limitations. It is known that some HRSC schemes can produce an anomalous behavior in certain situations; a catalog of numerical pathologies encountered in gas dynamics simulations can be found in [20], where it is observed that some of these pathologies appear only when very fine meshes are used.When using very fine uniform grids, in which the basic code structure of an HRSC scheme is relatively simple, we find that the computational time becomes the main drawback in the numerical simulation. For some HRSC schemes, fine mesh simulations in two dimensions are out of reach simply because they cost too much. The numerical flux evaluations are too expensive, and the computational time is measured by days or months on a personal computer. As an example we notice that a typical computation of a two-dimensional (2D) jet configuration in [19] is 10 to 50 days on an HP710 or 1 to 5 days on an Origin 2000 with 64 processors.It is well known, however, that the heavy-duty flux computations are needed only because nonsmooth structures may develop spontaneously in the solution of a hyperbolic system of conservation laws and evolve in time, and this basic observation has lead researchers to the development of a number of techniques that aim at reducing the computational effort associated to these simulations. Among these, shock tracking and adaptive mesh refinement (AMR) techniques (often combined with one another) are very effective at obtaining high-resolution numerical approximations, but the computational effort is transferred to the programming and the data structure of the code.Starting with the pioneering work of Harten [14], a different multilevel strategy aiming to reduce the computational effort associated to high-cost HRSC methods entered the scene. The key observation is that the information contained in a multiscale decomposition of the numerical approximation can be used to determine i...

This paper deals with the numerical modeling of wave propagation in porous media described by Biot's theory. The viscous efforts between the fluid and the elastic skeleton are assumed to be a linear function of the relative velocity, which is valid in the low-frequency range. The coexistence of propagating fast compressional wave and shear wave, and of a diffusive slow compressional wave, makes numerical modeling tricky. To avoid restrictions on the time step, the Biot's system is splitted into two parts: the propagative part is discretized by a fourth-order ADER scheme, while the diffusive part is solved analytically. Near the material interfaces, a space-time mesh refinement is implemented to capture the small spatial scales related to the slow compressional wave. The jump conditions along the interfaces are discretized by an immersed interface method. Numerical experiments and comparisons with exact solutions confirm the accuracy of the numerical modeling. The efficiency of the approach is illustrated by simulations of multiple scattering.

A penalization method is applied to model the interaction of large Mach number compressible flows with obstacles. A supplementary term is added to the compressible Navier-Stokes system, seeking to simulate the effect of the Brinkmanpenalization technique used in incompressible flow simulations including obstacles. We present a computational study comparing numerical results obtained with this method to theoretical results and to simulations with Fluent software. Our work indicates that this technique can be very promising in applications to complex flows.

In heterogeneous solids such as rocks and concrete, the speed of sound diminishes with the strain amplitude of a dynamic loading (softening). This decrease known as "slow dynamics" occurs at time scales larger than the period of the forcing. Also, hysteresis is observed in the steady-state response. The phenomenological model by Vakhnenko et al. is based on a variable that describes the softening of the material [Phys. Rev. E 70-1, 2004]. However, this model is 1D and it is not thermodynamically admissible. In the present article, a 3D model is derived in the framework of the finite strain theory. An internal variable that describes the softening of the material is introduced, as well as an expression of the specific internal energy. A mechanical constitutive law is deduced from the Clausius-Duhem inequality. Moreover, a family of evolution equations for the internal variable is proposed. Here, an evolution equation with one relaxation time is chosen. By construction, this new model of continuum is thermodynamically admissible and dissipative (inelastic). In the case of small uniaxial deformations, it is shown analytically that the model reproduces qualitatively the main features of real experiments.

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With a proper choice of a single dimensionless control parameter one describes the transition between subsonic and supersonic flows as a bifurcation. The bifurcation point is characterized by specific properties of the control parameter: the control parameter has a vanishing derivative in space and takes the maximum possible value equal to 1. This method is then applied to the sheath plasma with constant temperatures, allowing one to recover the Bohm boundary condition as well as the location of the point where the bifurcation takes place. This analysis is extended to fronts, rarefaction waves and divertor plasmas. Two cases are found, those where departure from quasineutrality is mandatory to generate a maximum in the variation of the control parameter (sheath and fronts) and those where the physics of the quasineutral plasma can generate such a maximum (rarefaction waves and supersonic flow in divertors). The conditions that are required to recover the Bohm condition, when modelling the wall using the penalization technique, are also addressed and generalized.