A unified gas-kinetic scheme for continuum and rarefied flows IV: Full Boltzmann and model equations

With discretized particle velocity space, a unified gas-kinetic scheme for entire Knudsen number flows is constructed based on the BGK model. In comparison with many existing kinetic schemes for the Boltzmann equation, the current method has no difficulty to get accurate solution in the continuum flow regime, such as the solution of the Navier-Stokes (NS) equations with the time step being much larger than the particle collision time, and the rarefied flow solution, even for the free molecule flow. The unified scheme is an extension of the gas-kinetic BGK-NS scheme from the continuum flow to the rarefied regime with the discretization of particle velocity space. The success of the method is due to the un-splitting treatment for the particle transport and collision in the evaluation of local solution of the gas distribution function. For these methods which use operator splitting technique to solve the transport and collision separately, it is usually required that the time step is less than the particle collision time, which basically makes these methods useless in the continuum flow regime, especially in the high Reynolds cases. Theoretically, once the physical process of particle transport and collision is modeled statistically by the gas-kinetic Boltzmann equation, the transport and collision become continuous operators in space and time, and their numerical discretization should be done consistently. With the use of the integral solution of the BGK, the unified scheme can simulate the flow accurately in the whole flow regime from the continuum Navier-Stokes solutions to the free molecule flow. At the same time, the time step in the high Reynolds number continuum flow region is only determined by the CFL condition of the macroscopic equations, instead of particle collision time.