Improved gas-kinetic unified algorithm for high rarefied to continuum flows by computable modeling of the Boltzmann equation

In this paper, we present an improved gas-kinetic unified algorithm (IGKUA) for high rarefied transition to continuum flows by computable modeling of Boltzmann equation. Compared with the original algorithm, the new method utilizes less needed discrete velocity ordinate points to obtain accurate results and removes the dependency of flow regime on computational time step, which can speed up the convergence in continuum flows. One of the novel strategies adopted in IGKUA is to develop a type of gas-kinetic quadrature rule that can exactly preserve conservation constraint of the model by adjusting the integral weights, increasing efficiency, and reducing nonphysical sources. Another key innovation is to introduce the analytical solutions of colliding-relaxation equation by considering the evolutions of associated macroscopic flow variables first, leading to no limit on the permissible time step. Numerical explicit and implicit schemes for unsteady flows are constructed to solve the physical convective equation, and Fourier spectral method is applied for the molecular-velocity convective movement equation analytically when the flows are under external-force fields. The IGKUA is tested using some numerical examples, including the shock-tube problems, Rayleigh flow, Couette flow, lid-driven cavity, external force-driven Poiseuille flow, and hypersonic flow past an infinite flat plate. Simulation results are in high resolution of the flow fields and match well with the results of the analytical, direct simulation Monte Carlo, Navier-Stokes solvers, and other reference methods. In addition, the new algorithm is better than the original one in the aspects of computational amount and time, which are more obvious when simulating the continuum flows.