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Accepted: 2017-10-05, 資料番号: SA1170187000

Accepted: 2016-03-24, 資料番号: SA1160148000

The newly developed unifying discontinuous formulation named the correction pro- cedure via reconstruction (CPR) for conservation laws is extended to solve the Navier-Stokes equations for 3D mixed grids. In the current development, tetrahedrons and triangular prisms are considered. The CPR method can unify several popular high order methods including the dis- continuous Galerkin and the spectral volume methods into a more efficient differential form. By selecting the solution points to coincide with the flux points, solution reconstruction can be com- pletely avoided. Accuracy studies confirmed that the optimal order of accuracy can be achieved with the method. Several benchmark test cases are computed by solving the Euler and compress- ible Navier-Stokes equations to demonstrate its performance.

An efficient implicit lower-upper symmetric Gauss-Seidel (LU-SGS) solu- tion approach has been applied to a high order spectral volume (SV) method for un- structured tetrahedral grids. The LU-SGS solver is preconditioned by the block ele- ment matrix, and the system of equations is then solved with a LU decomposition. The compact feature of SV reconstruction facilitates the efficient solution algorithm even for high order discretizations. The developed implicit solver has shown more than an order of magnitude of speed-up relative to the Runge-Kutta explicit scheme for typical inviscid and viscous problems. A convergence to a high order solution for high Reynolds number transonic flow over a 3D wing with a one equation turbulence model is also indicated. AMS subject classifications: 65M70, 76M12, 76M22

A unifying discontinuous formulation named the correction procedure via reconstruction (CPR) for conservation laws is extended to solve the Navier–Stokes equations for mixed grids. The CPR framework can unify several popular high order methods including the discontinuous Galerkin and the spectral volume methods into a differential formulation without explicit volume or surface integrations. Several test cases are computed to demonstrate its performance.

The newly developed unifying discontinuous formulation based on flux reconstruction and lifting collocation penalty (LCP) approaches for conservation laws is extended to solve the Navier-Stokes equations for 3D mixed grids. In the current development, tetrahedrons and triangular prisms are considered. The LCP formulation is an extension of the flux reconstruction (FR) method to arbitrary element types. As with the FR method, it can unify several popular high order methods including the discontinuous Galerkin and the spectral volume methods into a more efficient differential form without any explicit integration. By selecting the solution points to coincide the flux points, solution reconstruction can be completely avoided. Numerical accuracy of the scheme is assessed for grid refinement studies. Several typical test cases are computed by solving the Euler equations and the compressible Navier-Stokes equations to demonstrate its performance.

An aerodynamic s simulation code for the Reynolds averaged Navier -Stokes (RANS) equations is developed using the spectral volu me method for unstructured tetrahedral meshes . The turbulent viscousity is modeled by the Spalart -Allmaras (SA) one -equation model. Distributions of the state variables and their gradients in each grid cell are reconstructed using piece -wise linear polynomials with a modified SV limiter near shocks and steep gradients. For time integration, the non -linear LU -SGS scheme is employed to r educe total CPU times. The developed scheme is validated for turbulent flow over a flat plate and assessed for transonic flowfield over a wing compared with availab le CFD results . Then, RANS simulation around the JAXA high -lift configuration model is perfo rmed using developed scheme. By comparing with the numerical solutions obtained by the conventional finite volume solver, we examine the accuracy and the computational cost of the present SV method. Comparisons with the available experimental data are also attempted.