An Improved LU-SGS Implicit Scheme for High Reynolds Number Flow Computations on Hybrid Unstructured Mesh

The lower-upper symmetric Gauss-Seidel (LU-SGS) implicit relaxation has been widely used because it has the merits of less dependency on grid topology, low numerical complexity and modest memory requirements. In original LU-SGS scheme, the implicit system matrix is constructed based on the splitting of convective flux Jacobian according to its spectral radius. Although this treatment has the merit of reducing computational complexity and helps to ensure the diagonally dominant property of the implicit system matrix, it can also cause serious distortions on the implicit system matrix because too many approximations are introduced by this splitting method if the contravariant velocity is small or close to sonic speed. To overcome this shortcoming, an improved LU-SGS scheme with a hybrid construction method for the implicit system matrix is developed in this paper. The hybrid way is that: on the cell faces having small contravariant velocity or transonic contravariant velocity, the accurate derivative of the convective flux term is used to construct more accurate implicit system matrix, while the original Jacobian splitting method is adopted on the other cell faces to reduce computational complexity and ensure the diagonally dominant property of the implicit system matrix. To investigate the convergence performance of the improved LU-SGS scheme, 2D and 3D turbulent flows around the NACA0012 airfoil, RAE2822 airfoil and LANN wing are simulated on hybrid unstructured meshes. The numerical results show that the improved LU-SGS scheme is significantly more efficient than the original LU-SGS scheme.

[1]  Zhi J. Wang,et al.  Fast, Block Lower-Upper Symmetric Gauss-Seidel Scheme for Arbitrary Grids , 2000 .

[2]  J. S. Shang,et al.  Computational fluid dynamics application to aerospace science , 2009, The Aeronautical Journal (1968).

[3]  Iterative modified approximate factorization , 2001 .

[4]  Seokkwan Yoon,et al.  An LU-SSOR scheme for the Euler and Navier-Stokes equations , 1987 .

[5]  Mario A. Storti,et al.  Physics based GMRES preconditioner for compressible and incompressible Navier-Stokes equations , 1998 .

[6]  Oh Joon Kwon,et al.  Improvement on Block LU-SGS Scheme for Unstructured Mesh Navier-Stokes Computations , 2002 .

[7]  Carl Ollivier-Gooch,et al.  Effect of discretization order on preconditioning and convergence of a high-order unstructured Newton-GMRES solver for the Euler equations , 2008, J. Comput. Phys..

[8]  Meng-Sing Liou,et al.  A sequel to AUSM, Part II: AUSM+-up for all speeds , 2006, J. Comput. Phys..

[9]  Ye Zheng.  Generation of Three Dimensional Mixed and Unstructured Grids and its Application in Solving Navier-Stokes Equations , 2003 .

[10]  Gang Wang,et al.  Simulation of Flow Separation at the Wing-Body Junction with Different Fairings , 2007 .

[11]  Antony Jameson,et al.  How Many Steps are Required to Solve the Euler Equations of Steady, Compressible Flow: In Search of a Fast Solution Algorithm , 2001 .

[12]  A. Jameson,et al.  Lower-upper Symmetric-Gauss-Seidel method for the Euler and Navier-Stokes equations , 1988 .

[13]  N. J. Yu,et al.  Progress toward CFD for full flight envelope , 2005 .

[14]  Xinghua Chang,et al.  A block LU-SGS implicit unsteady incompressible flow solver on hybrid dynamic grids for 2D external bio-fluid simulations , 2009 .

[15]  P. Spalart.  A One-Equation Turbulence Model for Aerodynamic Flows , 1992 .

[16]  P. Roe.  CHARACTERISTIC-BASED SCHEMES FOR THE EULER EQUATIONS , 1986 .

[17]  A. Jameson,et al.  Numerical solution of the Euler equations by finite volume methods using Runge Kutta time stepping schemes , 1981 .