9  Supersonic Wedge — Oblique-Shock Verification

Comparison with the exact theta–beta–Mach and Rankine–Hugoniot relations

9.1 Run Provenance and Artifacts

Item Value
Wake CFD baseline version 0.22.0
Solver DBNS
Compute device CPU
Parallel execution 1 MPI rank
Arithmetic precision FP64
Run date 2026-08-20
Note

Results depend on the solver build that produced them. Read every figure in this report together with the version and date above — re-running the same input on a later build may give different numbers.

9.1.1 Downloadable Files

Not available: standalone input decks and solver result files have not yet been published for download.

9.2 Verification Objective

This is a code-verification case, not an experimental validation case. The two-dimensional Euler solution over a 15° wedge is compared directly with the exact oblique-shock relations. It verifies shock capturing, inviscid wall treatment, and supersonic inflow and outflow boundary conditions.

The geometry and freestream condition reproduce the NASA GRC WIND verification case, Oblique Shock on a 15 Degree Wedge at Mach 2.5.

9.3 Reference Solution and Data

9.3.1 Analytical Solution

For an upstream Mach number M_1, wedge deflection theta, and specific-heat ratio gamma, the weak-shock angle beta is obtained from the theta–beta–Mach relation:

\[ \tan\theta = 2\cot\beta\; \frac{M_1^2\sin^2\beta-1}{M_1^2(\gamma+\cos2\beta)+2}. \]

The downstream state follows from the normal-shock Rankine–Hugoniot relations using M_1n=M_1\sin\beta. For M_1=2.5, theta=15°, and gamma=1.4, the reference values are:

Quantity Reference value
Weak-shock angle, beta 36.94490°
Downstream Mach number, M_2 1.873526
Pressure ratio, p_2/p_1 2.467500
Density ratio, rho_2/rho_1 1.866549
Temperature ratio, T_2/T_1 1.321958

9.4 Simulation Conditions

Item Setting
Geometry 2D wedge; 15° half-angle
Physics Steady Euler equations, ideal-gas air (gamma=1.4)
Freestream M=2.5, p=101325 Pa, T=288.15 K, zero angle of attack

9.5 Mesh

Item Setting
Mesh Unstructured triangular mesh; 8,789 nodes

Triangular mesh for the supersonic wedge

9.6 Boundary and Initial Conditions

Item Setting
Wall Free-slip inviscid wall
Inlet Supersonic farfield inflow
Outlet and top boundary Supersonic pressure outlet; interior state extrapolated

Initial field: Not reported.

9.7 Numerical Methods

Item Setting
Convective schemes AUSM, AUSM+up2, AUSM+up, HLLC, JST, Roe, SLAU2, and SLAU (compared below)
Spatial method Green–Gauss gradients with Venkatakrishnan limiter
Time integration Implicit, adaptive CFL ramp

9.8 Results

The Roe result captures the attached oblique shock and the uniform post-shock region expected for this inviscid flow.

Pressure and Mach contours for the Roe result

Eight convective schemes were evaluated on the same mesh and boundary conditions. All reached the specified convergence criterion. Post-shock values were sampled downstream of the shock and compared with the exact solution.

Scheme p_2/p_1 Error in p_2/p_1 M_2 Error in M_2
AUSM 2.46752 +0.0008% 1.86655 −0.372%
AUSM+up2 2.46750 −0.0001% 1.86539 −0.434%
AUSM+up 2.46744 −0.0025% 1.86402 −0.507%
HLLC 2.46746 −0.0017% 1.86528 −0.440%
JST 2.46729 −0.0086% 1.87445 +0.049%
Roe 2.46749 −0.0003% 1.86451 −0.481%
SLAU2 2.46747 −0.0012% 1.86295 −0.564%
SLAU 2.46744 −0.0023% 1.86152 −0.641%

The pressure-ratio error is below 0.01% for every scheme, and the downstream-Mach error is below 0.7%. This case is deliberately simple: its purpose is to verify the exact shock jump and boundary-condition treatment, not to differentiate scheme robustness for complex shock interactions.

9.9 Conclusions and Applicable Scope

  • Wake CFD reproduces the exact pressure jump for the attached oblique shock to within 0.01% across the tested schemes.
  • The downstream Mach number agrees with the exact solution to within 0.7%.
  • These results apply to a single attached shock in steady, two-dimensional inviscid flow on one mesh. They do not establish accuracy for viscous effects, separated flow, shock interaction, or mesh convergence.

9.10 References