A STABILIZED CRANK-NICOLSON MIXED FINITE VOLUME ELEMENT FORMULATION FOR THE NON-STATIONARY PARABOLIZED NAVIER-STOKES EQUATIONS

A time semi-discrete Crank-Nicolson （CN） formulation with second-order time accuracy for the non-stationary parabolized Navier-Stokes equations is firstly established. And then, a fully discrete stabilized CN mixed finite volume element （SCNMFVE） formu- lation based on two local Gaussian integrals and parameter-free with the second-order time accuracy is established directly from the time semi-discrete CN formulation so that it could avoid the discussion for semi-discrete SCNMFVE formulation with respect to spatial wriables and its theoretical analysis becomes very simple. Finally, the error estimates of SCNMFVE solutions are provided.

A Stabilized Crank-Nicolson Mixed Finite Element Method for Non-stationary Parabolized Navier-Stokes Equations

In this study, a time semi-discrete Crank-Nicolson （CN） formulation with second-order time accu- racy for the non-stationary parabolized Navier-Stokes equations is firstly established. And then, a fully discrete stabilized CN mixed finite element （SCNMFE） formulation based on two local Gauss integrals and parameter- free with the second-order time accuracy is established directly from the time semi-discrete CN formulation. Thus, it could avoid the discussion for semi-discrete SCNMFE formulation with respect to spatial variables and its theoretical analysis becomes very simple. Finaly, the error estimates of SCNMFE solutions are provided.

AN IMPLICIT-EXPLICIT UPWIND ALGORITHM FOR THE PARABOLIZED NAVIER-STOKES EQUATIONS

In this paper, a second-order implicit-explicit upwind algorithm has been developed for three-dimensional Parabolized Navier-Stokes(PNS) equations. The agreement between the results of the new upwind algorithm and those of the im- plicit upwind algorithm and its ability in marching a long distance along the stream- wise direction have been shown for the supersonic viscous flow past a sphere-cone body. The CPU time is greatly reduced.

An implicit upwind parabolized Navier-Stokes code for chemically nonequilibrium flows

The previously developed single-sweep parabolized Navier-Stokes （SSPNS） space marching code for ideal gas flows has been extended to compute chemically nonequilibrium flows. In the code, the strongly coupled set of gas dynamics, species conservation, and turbulence equations is integrated with the implicit lower-upper symmetric GaussSeidel （LU-SGS） method in the streamwise direction in a space marching manner. The AUSMPW＋ scheme is used to calculate the inviscid fluxes in the crossflow direction, while the conventional central scheme for the viscous fluxes. The k-g two-equation turbulence model is used. The revised SSPNS code is validated by computing the Burrows-Kurkov non-premixed H2/air supersonic combustion flows, premixed H2/air hypersonic combustion flows in a three-dimensional duct with a 15° compression ramp, as well as the hypersonic laminar chemically nonequilibrium air flows around two 10° half-angle cones. The results of these calculations are in good agreement with those of experiments, NASA UPS or Prabhu＇s PNS codes. It can be concluded that the SSPNS code is highly efficient for steady supersonic/ hypersonic chemically reaction flows when there is no large streamwise separation.

A New Finite Volume Element Formulation for the Non-Stationary Navier-Stokes Equations

A semi-discrete scheme about time for the non-stationary Navier-Stokes equations is presented firstly,then a new fully discrete finite volume element(FVE)formulation based on macroelement is directly established from the semi-discrete scheme about time.And the error estimates for the fully discrete FVE solutions are derived by means of the technique of the standard finite element method.It is shown by numerical experiments that the numerical results are consistent with theoretical conclusions.Moreover,it is shown that the FVE method is feasible and efficient for finding the numerical solutions of the non-stationary Navier-Stokes equations and it is one of the most effective numerical methods among the FVE formulation,the finite element formulation,and the finite difference scheme.