Numerical solution of 2D Euler equations for the transonic flow past over NACA0012 and RAE2822 airfoils using high order accurate Runge-Kutta Discontinuous Galerkin method

Abstract

The Runge-Kutta Discontinuous Galerkin (RKDG) finite element method has the several advantages. First, the method is better suited than finite difference methods to handle complicated geometric domains. Second, the method can easily realized for adaptive strategies since the refining or coarsening grid can be done without taking account continuous restrictions that the conforming finite element methods need. Third, the method is highly parallel. Hence, this method is applied to solve Euler equations. This paper deals with a high-order accurate RKDG method for the numerical solution of the 2D Euler equations for the transonic in-viscid flow past over NACA0012 and RAE2822 airfoils. Nodal discontinuous Galerkin method is used for the space discretization of the 2D Euler governing PDEs and resulting ODE is further solved using Low Storage Explicit Runge Kutta method for temporal discretization. ROE upwind flux schemes have been opted for computing flux between the element interfaces. Since, the problem is nonlinear therefore exponential filter is used with RKDG method instead of slope limiters, to achieve the stability of the RKDG scheme and to curb the oscillation in the vicinity of shock waves for the nonlinear problems. 2D Euler equations have been solved for 3rd order accuracy i.e. DG (p=2). The results are also compared with the finite volume method and experimental data. Finally it is revealed that the RKDG method is stable, high-order accurate and well suited to achieve high order accurate solution for the problems governed by hyperbolic PDEs.