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Discrete Boltzmann model of compressible flows with spherical or cylindrical symmetry

To study simultaneously the hydrodynamic and thermodynamic behaviors in compressible flow systems with spherical or cylindrical symmetry, we present a theoretical framework for constructing Discrete Boltzmann Model(DBM) with spherical or cylindrical symmetry in spherical or cylindrical coordinates. To this aim, a key technique is to use \emph{local} Cartesian coordinates to describe the particle velocity in the kinetic model. Thus, the geometric effects, like the divergence and convergence, are described as a \textquotedblleft force term\textquotedblright . Even though the hydrodynamic models are one- or two-dimensional, the DBM needs a Discrete Velocity Model(DVM) with 3 dimensions. We use a DVM with 26 velocities to formulate the DBM which recovers the Navier-Stokes equations with spherical or cylindrical symmetry in the hydrodynamic limit. For the system with \emph{global} cylindrical symmetry, we formulated also a DBM based on a DVM with 2 dimensions and 16 velocities. In terms of the nonconserved moments, we define two sets of measures for the deviations of the system from its thermodynamic equilibrium state. The extension of current model to the multiple-relaxation-time version is straightforward.