Inclined Ramp Example

This page covers a simple test case of an inclined ramp.

Problem Definition

The parameters of this problem are \(\underline{\theta} = [\theta, \, L, \, W]\), shown diagramatically in the figure below. The geometry is built from two cells, sharing a common edge.

Inclined Ramp Example Study

Free Stream Conditions

The freestream conditions are defined below.

\[\gamma = 1.4\]

\[ Mach_{freestream} = 6\]

\[ P_{freestream} = 700 Pa\]

\[ T_{freestream} = 70 K\]

Analytical Solution

This problem can be solved analytically using simple isentropic flow relations and shock expansion theory. This solution will later serve as validation for the results obtained using PySAGAS.

Oblique Shock Relations

The nominal ramp conditions are solved using oblique shock wave theory.

\[ \frac{p_2}{p_1} = 1 + \frac{2\gamma}{\gamma+1} (M_1^2 \sin^2\beta - 1) \]

\[ \frac{\rho_2}{\rho_1} = \frac{(\gamma + 1)M_1^2 \sin^2 \beta} {(\gamma - 1)M_1^2 \sin^2 \beta + 2} \]

\[ \frac{T_2}{T_1} = \frac{p_2}{p_1} \frac{\rho_1}{\rho_2} \]

\[ M_2 = \frac{1}{\sin(\beta - \theta)} \sqrt{ \frac{1 + \frac{\gamma-1}{2}M_1^2 \sin^2\beta} {\gamma M_1^2\sin^2\beta - \frac{\gamma-1}{2}} } \]

The oblique shock angle is found by solving the \(\theta-\beta-M\) equation, shown below for reference.

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

Calculating Ramp Conditions

Using the oblique shock relations presented above, the following conditions on the ramp surface can be calculated.

Property	Ramp Value
\(M_{ramp}\)	4.65
\(P_{ramp}\)	2567.41 Pa
\(T_{ramp}\)	107.89 K
\(\rho_{ramp}\)	0.0829 kg/m^3

Sensitivity Results

The force sensitivities for each ramp design parameter can be calculated using the method of finite differencing. Performing such a study produces the results shown in the table below.

\[ F = P \times A \times \underline{n}_{ramp} \cdot \underline{u} \]

Parameter	\(dF_x/dp\)	\(dF_y/dp\)
\(\theta\)	1.07	-3.11
\(L\)	8.92	-50.57
\(W\)	4.46	-25.28

PySAGAS Solution

Given the surface properties on the ramp calculated using the analytical solution, the parameter sensitivities can be approximated using PySAGAS. Note, the error of each sensitivitiy, as calculated using the analytical solution for reference, is shown in brackets.

Parameter	\(dF_x/dp\)	\(dF_y/dp\)
\(\theta\)	1.09 (-1.6%)	-3.20 (-3.1%)
\(L\)	8.92 (0%)	-50.57 (0%)
\(W\)	4.46 (0%)	-25.28 (0%)