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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 M2 = 1.87353, an oblique shock angle of θ = 36.94°, a pressure ratio across the shock of p2/p1 = 2.46750, and a density ratio across the shock of r2/r1 = 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 left-most, right-most and top curves are far-field 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.

Figure 1
Figure 1: Overview of layout for oblique shock at Mach 2.5.

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 Δtcpu 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.

h nnd nel Δtcpu
coarse 0.04 1016 1912 0.0197 s
medium 0.02 3984 7727 0.1048 s
fine 0.01 15628 30777 0.4296 s

Figure 2
Figure 2: Representative grid for oblique shock geometry.

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 Δtcpu for that grid. Unfortunately, the total computational time increases as the grid resolution increases not only because of an increase in Δtcpu, 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 free-stream Mach number M by numerically solving Equation (1) for the shock angle θ.

(1) Equation 1

The shock angle is then used to compute the downstream Mach number and the pressure and density ratios across the shock. The program oblique.exe, available for download on our codes page, solves this problem using Newton's root finding method. The solution given in the description of this problem was found using this program. For comparison with our CFD solution, we now interpret the theoretical solution in terms of non-dimensional CFD quantities as follows:

  • up-stream conditions: M1 = 2.5, ρ1 = 1.0, u1 = 1.0, p1 = 0.114286
  • down-stream conditions: M2 = 1.87353, ρ2 = 1.866549, u2 = 0.861646, p2 = 0.282000

Note that the non-dimensional up-stream pressure given above was computed using the following relation:

(2) Equation 2

Computational Results

Figure 3
Figure 3: Computational results for oblique shock at Mach 2.5.


Downloads

Geometry Data:

Solution Data:


References

1. John, J.E.A., Gas Dynamics, Second Edition, Prentice Hall, New Jersey, 1984.


CASE Lab URL: http://www.caselab.okstate.edu/research/verification/compr2d.html
Revised: March 20, 2001 [TJC]
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