Convergence

  We first consider the convergence rate of this new formulation by solving for an inviscid and isentropic flow problem in the geometry shown in figure 7.1. Low-order methods erroneously produce entropy from inlet to outlet for this problem. Here we show in figure 7.1 (bottom) that the entropy error converges exponentially fast to zero with p-refinement. A comparison is shown on the right plot between a fully unstructured and a hybrid discretization; more elements are used in the unstructured grid.

  

Figure 7.1: Density contours obtained on a hybrid grid for an inviscid M=0.3 flow (Top), on a triangle grid (Middle). The bottom plot shows exponential convergence of the error for the unstructured (triangles) and the hybrid (squares) grid.

We repeated this test with the three-dimensional compressible code. In figure 7.2 we show the three-dimensional domain that we extruded from a two-dimensional bump mesh with K=120. Large hexahedra are used at the inlet and outlet, whereas smaller prisms and hexahedra are used around the bump. We also show that the entropy again decreased exponentially fast with increasing expansion order.

  

Figure 7.2: Inviscid M=0.3 flow past a bump in a three-dimensional (periodic in the spanwise direction) domain. From the top: (1) domain, (2) spectral element mesh used in the convergence test K=120, (3) Iso-Contours of density and (4) convergence of entropy to zero with increasing order.