A locally implicit scheme for Navier-Stokes equations

A locally implicit relaxation scheme, is developed for the steady-state solution of the Navier-Stokes Equations. The method uses finite volume spatial discretization, implicit time integration, Jameson-type artificial dissipation terms and a modified Gauss-Seidel iteration. The scheme does not require any matrix solvers. The convergence rate is significantly enhanced by incorporating the multigrid technique. Applications to the transonic flows show that the method is efficient and robust. The basic scheme can be implemented in two modes, namely, single point scheme and multi-point scheme. Both these methods have been analyzed in detail along with their local linear stability analysis for model elliptic Equation in single as well as in three-dimensions, and the scheme is shown to be unconditionally stable. Subsequently, the single point scheme has been applied to thin-layer Navier-Stokes Equations for a variety of test cases, namely, subsonic and transonic flows over NACA0012 and RAE2822 airfoils and the numerical results have been compared with the available experimental results. An algebraic eddy viscosity turbulence model has been used for turbulent flows. Both elliptic and hyperbolic grid generators have been used to generate C-grids for the above mentioned applications.