Matematical Modeling of Supersonic Turbulent Flows in Inlets with Rotating Cowl

Abstract

A new, high-order, conservative, and efficient method for conservation laws on unstructured grids is developed. The concept of discontinuous and high-order local representations to achieve conservation and high accuracy is utilized in a manner similar to the Discontinuous Galerkin (DG)[1] and the Spectral Volume (SV)[2] methods, but while these methods are based on the integrated forms of the equations, the new method is based on the differential form to attain a simpler formulation and higher efficiency. Conventional unstructured finite-difference (FD)[3] and finite-volume (FV)[4] methods require data reconstruction based on the least-squares formulation using neighboring point or cell data. Since each unknown employs a different stencil, one must repeat the least-squares inversion for every point or cell at each time step, or store the inversion coefficients. In a high-order, three-dimensional computation, the former would involve impractically large CPU time, while for the latter the memory requirement becomes prohibitive. In addition, the finite-difference method does not satisfy the integral conservation in general. By contrast, the DG and SV methods employ a local, universal reconstruction of a given order of accuracy in each cell in terms of internally defined conservative unknowns. Since the solution is discontinuous across cell boundaries, a Riemann solver is necessary to evaluate boundary flux terms and maintain conservation. In the DG method, a Galerkin finite-element method is employed to update the nodal unknowns within each cell. This requires the inversion of a mass matrix, and the use of quadratures of twice the order of accuracy of the reconstruction to evaluate the surface integrals and additional volume integrals for non-linear flux functions. In the SV method, the integral conservation law is used to update volume averages over subcells defined by a geometrically similar partition of each grid cell. As the order of accuracy increases, the partitioning for 3D requires the introduction of a large number of parameters, whose optimization to achieve convergence becomes increasingly more difficult. Also, the number of interior facets required to subdivide non-planar faces, and the additional increase in the number of quadrature points for each facet, increases the computational cost greatly. In the spectral difference (SD) method, the conservative unknowns in each cell are the number of nodal values required to support a reconstruction of a given order of accuracy. Their locations are chosen so that a quadrature approximation for the volume integral exists at least to the same order of accuracy. The fluxes are calculated at a different set of nodes, whose number will support a reconstruction of one order higher accuracy, since the flux derivatives are used to update the conservative unknowns. They are located so that quadrature approximations for surface integrals over the cell boundaries exist to a required order of accuracy. In addition, the locations of the conservative nodes and the flux nodes must be such that the integral conservation law is satisfied for the cell to the desired order of accuracy. If the nodes are distributed in a geometrically similar manner for all cells, the discretizations become universal, and can be expressed as the same weighted sums of the products of the local metrics and fluxes. These metrics are constants for the line, triangle, and tetrahedron elements, and can be computed analytically for curved elements. We can also show that the number of flux nodes is far less than the number of quadrature points in the SV method. Since all unknowns are decoupled, no mass matrix inversion is required. The SD formulation for the line element is a generalization of the multidomain spectral method[5]. Its tensor products can be used for quadrilateral and hexahedral elements.