Compressible Navier-Stokes computations on unstructured quadrilateral grids by a staggered-grid Chebyshev method

Abstract

We describe a new spectral multidomain method for the solution of the compressible Navier-Stokes equations. Within each subdomain, the method collocates the solution unknowns and the gradients at the nodes of the Gauss-Chebyshev quadrature. The total fluxes are evaluated at the nodes of the Gauss- Lobatto quadrature. Both conforming and non-conforming subdomain grids are allowed. Two examples are included to show the behavior of the method. First, exponential convergence is shown for the Couette flow on an unstructured grid quadrilateral grid. Next, subsonic flow over a backward facing step is solved on a non-conforming grid and a comparison to experiments is made. In this paper, we present a new spectral multidomain method for the solution of the compressible Navier- Stokes equations in two space dimensions. With this method, unlike traditional spectral methods,2 the flow region is subdivided into an unstructured grid of quadri- lateral elements, which can be generated by finite ele- ment meshing programs. Within each element, a vari- able order staggered-grid Chebyshev spectral collocation method is used to approximate the solution and fluxes. The result is a method with the flexibility of a finite volume scheme and the accuracy of a spectral method. The key feature of the approach is the use of a stag- gered Chebyshev grid within a quadrilateral element. It is an extension of the staggered-grid method for the Euler gas-dynamics equations introduced in two papers, Ref. 5 and Ref. 4. In those papers, the solution unknowns were evaluated at the Chebyshev-Gauss quadrature nodes, which are interior to the elements. The fluxes were evaluated at the more common Gauss-Lobatto points, which include the edges. Thus, the solution and fluxes were staggered within an element in a manner analogous to the staggering of solution and fluxes in a cell-centered finite volume method. To further simplify the applica- tion of physical boundary conditions and inter-element connectivity in two space dimensions, the solutions and fluxes were fully staggered so that no quantity was ap- proximated at domain corners. An advantage of the method over fixed-order finite volume methods is that accuracy of the staggered-grid Chebyshev method can be controlled by grid refinement of /i-type, p-type, or both. In /i-type refinement, typical of the usual fixed order finite volume and finite element methods, the grid is refined by increasing the number of elements. In p-type refinement, the most natural refine- ment for spectral methods, the order of the polynomials is increased within the elements. For smooth solutions, the error decays exponentially with the polynomial or- der under p-type refinement. An advantage of p-type refinement is that accuracy can be assessed without re- generating the grid. We will describe the new method and demonstrate its use on two test problems. In the first test problem, we show exponential convergence of the solution of the Couette flow on an unstructured quadrilateral grid typical of grids that are generated by finite element meshing programs. In the second, we solve a low speed flow over a backward facing step, and compare the results with experiments.