Discrete Boltzmann model for implosion- and explosion- related compressible flow with spherical symmetry

To kinetically model implosion- and explosion-related phenomena, we present a theoretical framework for constructing a discrete Boltzmann model （DBM） with spherical symmetry in spherical coordinates. To achieve this goal, a key technique is to use local Cartesian coordinates to describe the particle velocity in the kinetic model. Therefore, geometric effects, such as divergence and convergence, are described as a ＂force term＂. To better access the nonequilibrium behavior, even though the corre- sponding hydrodynamic model is one-dimensional, the DBM uses a discrete velocity model （DVM） with three dimensions. A new scheme is introduced so that the DBM can use the same DVM regard- less of whether or not there are extra degrees of freedom. As an example, a DVM with 26 velocities is formulated to construct the DBM at the Navier-Stokes level. Via the DBM, one can study simulta- neously the hydrodynamic and thermodynamic nonequilibrium behaviors in implosion and explosion processes that are not very close to the spherical center. The extension of the current model to a multiple-relaxation-time version is straightforward.

Improvement on Spherical Symmetry in Two-Dimensional Cylindrical Coordinates for a Class of Control Volume Lagrangian Schemes

In[14],Maire developed a class of cell-centered Lagrangian schemes for solving Euler equations of compressible gas dynamics in cylindrical coordinates.These schemes use a node-based discretization of the numerical fluxes.The control volume version has several distinguished properties,including the conservation of mass,momentum and total energy and compatibility with the geometric conservation law(GCL).However it also has a limitation in that it cannot preserve spherical symmetry for one-dimensional spherical flow.An alternative is also given to use the first order area-weighted approach which can ensure spherical symmetry,at the price of sacrificing conservation of momentum.In this paper,we apply the methodology proposed in our recent work[8]to the first order control volume scheme of Maire in[14]to obtain the spherical symmetry property.The modified scheme can preserve one-dimensional spherical symmetry in a two-dimensional cylindrical geometry when computed on an equal-angle-zoned initial grid,andmeanwhile itmaintains its original good properties such as conservation and GCL.Several two-dimensional numerical examples in cylindrical coordinates are presented to demonstrate the good performance of the scheme in terms of symmetry,non-oscillation and robustness properties.

Global Spherically Symmetric Solutions for a Coupled Compressible Navier–Stokes/Allen–Cahn System

In this paper,we consider the global spherically symmetric solutions for the initial boundary value problem of a coupled compressible Navier–Stokes/Allen–Cahn system which describes the motion of two-phase viscous compressible fluids.We prove the existence and uniqueness of global classical solution,weak solution and strong solution under the assumption of spherically symmetry condition for initial dataρ0 without vacuum state.