The Construction And Application Of The Implicit Algorithm Of Unified Gas Dynamic Theory Format

With the further exploration of aerospace,the flow problems that would confront aircrafts are getting more and more complex.For instance,the launching and reentry of spacecrafts and the jet satellites attitude control are closely related to the solution of multiscale flow problems.Thus,it is necessary to develop an efficient numerical method,which is capable of solving problems in all flow regimes covering from continuum flow to free molecule flow and at all speeds from subsonic flow to hypersonic flow,and which has great potential in engineering practices.The unified gas kinetic scheme(UGKS)is a direct modeling finite-volume method based on the scale of numerical discretization.The application of the analytic solution of BGK model to the construction of flux passing through cell interface makes it possible to capture the physics in all flow regimes.In order to capture the nonequilibrium effects,the discretized distribution functions in velocity space are used.The UGKS is a promising tool to solve multiscale problems because it can avoid some troubles caused by utilization of hybrid method in traditional CFD method,such as partition of flow regions and coupling of the macroscopic method and the particle-based method.However,due to discretized distribution function,it requires more memory and computational resources than macroscopic method in CFD,which makes it of low efficiency in solving problems in continuum flow regime.In order to promote the development of UGKS in engineering application,this paper presents an implicit unified gas kinetic scheme which is capable and efficient to solve steady state problems in all flow regimes.The physical time step to determine the physical flow scale and the numerical time step to control the marching on in time are used in this implicit scheme to maintain the capability of UGKS for all flow regimes and to take the advantages of implicit algorithms for large time step.A robust and efficient full implicit unified scheme is obtained from coupling of the implicit macroscopic governing equations and the implicit microscopic one,which makes it still of high convergence efficiency in continuum flow.The utilization of the LU-SGS iteration technique which can solve the huge sparse matrix by twice symmetric Gauss-Seidel iterations can much reduce the computation of the implicit governing equations and improve the computational efficiency.Several numerical test cases,such as Couette flow,lid-driven cavity flow and supersonic flow passing over a circle cylinder are computed to demonstrate the validity,robustness,accuracy,efficiency and computational cost of the implicit unified scheme.It turns out that the implicit UGKS is a numerical method of multiscale solution with second order of accuracy and its convergence efficiency for steady state solution in all flow regimes at all speeds can be improved by one or two orders of magnitude.