A Paradigm for Modeling and Computation of Gas Dynamics
  In the continuum flow regime, the Navier-Stokes equations are usually used for the description of gas dynamics. On the other hand, the Boltzmann equation is applied for the rarefied gas dynamics. Both equations are constructed from modeling flow physics in different scales. Fortunately, due to the distinct separation of scales, i.e., the hydrodynamic and kinetic ones, both Navier-Stokes equations and the Boltzmann equation are valid in their respectable domains. However, in real physical application, there may not have such a distinctive scale separation. For example, around a hypersonic flying vehicle, the flow physics at different regions may correspond to different regimes, where the local Knudsen number can be changed in several order of magnitudes. With a variation of modeling scale, theoretically a continuous governing equation from kinetic Boltzmann equation to the hydrodynamic Navier-Stokes equations should exist. However, due to the difficulties of a direct modeling of flow physics in the scale between the kinetic and hydrodynamic ones, there is basically no reliable theory or valid governing equation to cover the whole transition regime. In fact, it is an unresolved problem about the exact scale for the validity of the NS equations as Reynolds number decreases. The traditional computational fluid dynamics (CFD) is based on the numerical solution of partial differential equations (PDE), and it targets on the recovering of the exact solution of the PDEs as mesh size and time step converging to zero. This methodology can be hardly applied here because there is no such a complete PDE for flow physics in all scales. It may be the time to combine the modeling and computation together without going through the process of constructing PDEs. In other words, the CFD research is not only to obtain the numerical solution of governing equation, but also to construct a valid discrete governing equation to identify the flow physics in the mesh size and time step scales. In this paper, we are going to present the idea for the modeling and computation. This methodology leads to the unified gas-kinetic scheme (UGKS) for flow simulation in all flow regimes. Based on UGKS, the boundary for the validation of the Navier-Stokes equations can be quantitatively evaluated. The combination of modeling and computation provides a paradigm for the description of multiscale transport process. 
Kun Xu and Chang Liu
physics.flu-dyn