  Main    Overview    Validation    Benchmark    FAQ's    Links    
Steady Verification Data for a 2D Supersonic Wedge This page presents a set of steady verification data obtained for the simulation of a supersonic flow over a wedge with a half angle of 15°. The supersonic flow develops an oblique shock at the leading edge of the wedge. The results will be compared to the exact solution computed using the perfect gas equations. The first test will involve an upstream Mach number of 2.5, for which our theory predicts a downstream Mach number of M_{2} = 1.87353, an oblique shock angle of θ = 36.94°, a pressure ratio across the shock of p_{2}/p_{1} = 2.46750, and a density ratio across the shock of r_{2}/r_{1} = 1.86655. 
Geometry We take advantage of symmetry when defining this problem and represent only the upper half of the wedge. The layout of the computational domain, which covers the area 0 < x < 1.5 and 0 < y < 1, is presented in Figure 1. Boundary conditions for the five boundary curves of this domain are specified as follows: the leftmost, rightmost and top curves are farfield boundaries, the lower curve from x = 0 to x = 0.5 is a symmetry boundary, and the lower curve from x = 0.5 to x = 1.5 is a wall boundary. Furthermore, the intersection of the two lower boundaries at x = 0.5 is specified as a singular point to avoid ambiguity in the enforcement of boundary conditions at that point.
To solve this problem, we employ three grids that are refined successively using a constant refinement ratio of r = 2. Table 1 presents the grid spacing h, number of nodes nnd, number of elements nel, and the average computational time required per iteration Δt_{cpu} for each grid. All three grids are made up of linear triangular elements and are generated using a uniform grid spacing. Figure 2 shows the coarse grid generated for this study. Table 1: Summary of grid parameters for oblique shock problem.
The computational times presented in Table 1 are intended to show the relative increase in computer resources required as the grid resolution increases. With this information, the total computational time required for an analysis of any grid is computed by multiplying the total number of iterations required for convergence, nstp, by Δt_{cpu} for that grid. Unfortunately, the total computational time increases as the grid resolution increases not only because of an increase in Δt_{cpu}, but also because more iterations are required for convergence on the finer grids due to greater restrictions placed on local time steps for smaller elements. 
Theoretical Solution The theoretical solution for this problem is well known and is typically available in any textbook on compressible flow. An exact solution can be found for a wedge angle δ and freestream Mach number M by numerically solving Equation (1) for the shock angle θ.
The shock angle is then used to compute the downstream Mach number and the pressure and density ratios across the shock. The program
Note that the nondimensional upstream pressure given above was computed using the following relation:

Computational Results

Downloads Geometry Data: Solution Data: 
References 1. John, J.E.A., Gas Dynamics, Second Edition, Prentice Hall, New Jersey, 1984. 
 
URL: http://www.caselab.okstate.edu/research/verification/compr2d.html Revised: March 20, 2001 [TJC] Webmaster: caselab@gmail.com 