## Validation cases

So far, only the original Roe scheme has been implemented, so the validations are done with this scheme.

The first validation case is resolving the oblique shock over the wedge of course:

#### Mach 2.36 Flow over a 20° wedge (Inviscid flow)

The wedge angle and Mach number are chosen such that the shock angle becomes 45° according to the 1-D oblique shock analysis (κ=1.4). Since the domain is a square, it is seen that the shock extends from the tip of the wedge (the domain center) to the domain corners.

Mach number after the shock is 1.44

A close-up of the shock region is shown below, with the grids clearly seen.

{{:wiki:compressible:mach-zoom.png?600}}

An adaptive grid is used, where the refinement is perfromed according to the gradients of Mach number, every 10 time-steps. The smallest grid length is 1/4096 of the domain length, and about 1.6 Million cells are used. A uniform mesh simulation of the same resolution would require 4096x4096 or more than 16 Million cells.

The pressure is plotted in the shock-normal direction, normalized with the pre-shock pressure. It is also noticed that increasing the refinement, the plots will collapse on each other if the distance is normalized by the smallest grid size, Δx. Since the Euler equations are solved, this is expected behavior, as the shock thickness is theoretically zero.

The pressure ratio of 3.08, exactly matches the analytical value.