Cell-vertex adaptive Euler method for cascade flows

Abstract

This paper is concerned with the numerical solution of the two-dimensional steady Euler equations, using a multidimensional upwind cell-vertex residual distribution scheme. A solution-adaptive grid-refinement procedure is proposed, which combines with an efficient multigrid strategy based on an optimally smoothing explicit multistage scheme. A very accurate treatment of solid wall boundaries and a general approach for guaranteeing conservation on patched grids are developed within the present context of cell-vertex residual distribution schemes. The proposed approach is shown to be very efficient and sufficiently accurate to compute the transonic flow past a high-turning turbine cascade. N the last few years, a large effort has been devoted to the development of numerical methods for solving the multidimensional Euler equations with improved discontinuity-capturing capability. Classical upwind methods, based on the application of one-dimensional Riemann solvers along grid dependent directions, experience a loss of resolution in the presence of discontinuiti es not aligned with the mesh. In order to overcome such a difficulty, two different approaches have been proposed that introduce a truly multidimensional modeling of the propagation phenomena dominating the behavior of compressible flows. The Euler system is decomposed either into an equivalent set of five-to-six simple wave equations1 or into a system of four optimally decoupled compatibility equations.2 The scalar equations are solved using one of the newly developed multidimensional fluctuation splitting (FS) space discretization schemes,3 that send the residual calculated over each cell-vertex triangular element to its downstream nodes. The more robust methodology introduced by Roe1 and further developed in Refs. 4—6 has been widely used by the authors in combination with the linear first-order accurate space discretization AT-scheme.3 A standard full approximation scheme (FAS) multigrid strategy with an optimally smoothing explicit three-stage scheme57 has been employed to obtain a very efficient genuinely multidimensional Euler solver. The compact space discretization and the explicit time integration make the resulting methodology ideally suited for adaptive mesh refinement as well as for vector and parallel computers. In particular, the use of a solution-adaptive grid-refinement strategy is expected to further improve the efficiency of the method, as well as to make it sufficiently accurate for computing two-dimensional complex Euler flows. In this paper, after reviewing the basic multidimensional upwind FS Euler solver, a multigrid adaptive strategy is proposed that effectively combines a solution-adapt ive grid-refinement procedure with the FAS multigrid approach based on the optimally smoothing explicit scheme.6'7 Furthermore, the solid wall boundary conditions based on isentropic simple radial equilibrium (ISRE)8 are generalized to the present context of cell-vertex distribution schemes, and a general approach for guaranteeing conservation on patched grids is developed. Finally, results are presented for two transonic cascade flows. A simple cascade of NACA 0012 airfoils is used to demonstrate the accuracy improvement obtained using the ISRE wall boundary conditions with respect to the standard characteristic ones. A severe high-turning turbine blade cascade is employed to