In [1]:
import pandas as pd
import numpy as np
import matplotlib as plt
from IPython.display import Image


# Mach 3.0 wind tunnel with a step¶

This validation case consideres the Mach 3.0 wind tunnel with a step as presented in the study of Woodward and Colella (1984) The numerical simulation of two-dimensional fluid flow with strong shocks. A rarefraction fan forms at the corner of the step resulting a singularity. With most numerical formulations this singularity introduces an erroneous entropy layer along the wall of the step and subsequently a spurious Mach stem at the bottom wall. For this analysis no special numerical treatment is applied; instead, an additional two levels of mesh refinement are applied at the step corner to reduce these numerical artifacts.

• Cockburn and Shu (1998) The Runge–Kutta Discontinuous Galerkin Method for Conservation Laws
• Greenshields et al. (2009) Implementation of semi-discrete, non-staggered central schemes in a colocated, polyhedral, finite volume framework, for high-speed viscous flows

### Initial conditions¶

The test case is non-dimensionalised so that an acoustic velocity of 1 is recovered for a temperature of 1. The non-dimensional material properties are

In [2]:
mP = pd.read_csv('./input/matProp.csv')
mP

Out[2]:
Gas constant Specific heat ratio
0 0.714 1.4

and the specified, non-dimensional inlet conditions are

In [3]:
iCI = pd.read_csv('./input/intialCond.csv')
iCI

Out[3]:
Pressure Temperature Velocity
0 1.0 1.0 3.0

### Grid¶

For the purpose of the analysis a two-dimensional structured mesh is generated using the blockMesh utility. For the fine mesh the cell size corresponds to a characteristic length of $\frac{1}{120}$. As noted previously, two additional levels of h-refinement are applied in the region of the step corner.

In [4]:
from IPython.display import Image, HTML, display
from glob import glob
imagesList=''.join( ["<img style='width: 600px; margin: 10px; float: left; border: 0px solid black;' src='%s' />" % str(s)
for s in sorted(glob('screenShot/mesh/mesh*.png')) ])
display(HTML(imagesList))


### Running¶

A script is provided to set up the case and run the simulation. The script ./runSim generates the mesh, applies additional refinement at the corner and runs the simulation.

### Results¶

To validate the numerical prediction, the density contours for the AUSM$^{+}$up and Kurganov shock capturing schemes are compared with the results from Woodward and Colella (1984) at 4 seconds. Although the various methods show good correlation when comparing the principal flow features, both the Woodward and Colella scheme (top) and AUSM$^{+}$up (middle) show some numerical artifacts from the entropy layer at the Mach stem on the step wall while Kurganov and Tadmor's scheme does not seem to exibit this.

In [5]:
from IPython.display import Image, HTML, display
from glob import glob
imagesList=''.join( ["<img style='width: 510px; margin: 10px; float: left; border: 0px solid black;' src='%s' />" % str(s)
for s in sorted(glob('screenShot/results/rho*.png')) ])
display(HTML(imagesList))

In [6]:
from IPython.core.display import HTML

def css_styling():