Shock wave on a ramp

The objective is to use StarCD to calculate the shock wave in the case of a simple compression, this part presents the developpement of a shock on a ramp.
We considered a two-dimension domain, 4m long and 2 m wide, with a ramp at 10 degres.

Mesh structure

The mesh is generated on Gambit and then imported on StarCD (c.f. the work that was done in a previous MICP task named SCCC).

The domain is composed of 40 cells on the horizontal part, 40 cells on the ramp and 40 cells on the vertical.
Finally there is one cell on Z direction. This is because StarCD does not compute 2D-flows, however, the Z component of the variables are not calculated.
The size of the cells is 0.05 m.

Simulation on StarCD

The domain consists in:

• an inlet, with a speed of U=686m/s (Mach=2)
• a wall at the bottom of the domain (no slip)
• a free stream region at the top (non confined domain)
• an oulet
• Other regions (Z-sides of the model) are defined as symetry.
• Density if chosen as Ideal-f(T,p)
• The fluid in inviscid.
• We used the MARS differencing scheme for variables U, V and density, and  UD for Temperature (See StarCD's help and tutorials to justify this choice).

Mach number

Pressure

Density

Temperature

Total temperature

As we can see on the previous profiles, a shock appears at the bottom of the ramp.
The Mach number goes from 2 before the shock, to 1.629 after the shock.
The density varies from 1.189 to 1.757 .
The shock generates a rise in temperature.
The simulation is done with constant enthalpy, which is here called total temperature. We can notice on the last graph that this value does remain constant( the all scale of color correspond to the value 526.9)

We can notice that the shock is pretty thick, effectivly there is only 4 or 5 cells where the shock occurs , what might not be enough.
As a consequence we decided to use a fine mesh.

Pressure Profil

The pressure is represented as a relative value taken from 1.10^5.
On the graph, the minimum value is equal to the reference.
As we can see on the pressure graph, there is a higher pressure zone behind the shock. This can also be observed on a 2D profil. This is not physical but it is a consequence of the differencing schemes.
The pressure is drawn at x=3.625m (middle of a cell). There is a first constant value behind the shock, than comes the higher pressure (between y=1.0 and 1.2). During the shock, the pressure gets smaller to get to a constant value 0, above the shock .

Angle of the shock

We can now calculate the angle of the shock.
Effectivly,  we consider the middle of the difference of pressure during the shock, this value appears for y=1.33.
We now know two points on the shock:
Xa=2 ; Ya=0 (Bottom of the ramp)
Xb=3.625 (middle of a cell) ; Yb=1.33

The angle of the shock is given by:
alpha=arctan[(Yb-Ya)/(Xb-Xa)]=39.299

Comparision with the theory

We can compare these results with the ones given by a fortran program found on the internet (listing) to the following address :

http://www.grc.nasa.gov/www/wind/valid/wedge/wedge.html

 Angle of the shock Outflow Mach Pressure ratio Temperature ratio Density ratio Theory 39.306 1.64052 1.70658 1.17015 1.45843 StarCD simulation 39.299 1.629 1.7162 1.1704 1.4407

You can now have a look at the next page