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Pre-Analysis & Start-Up

Pre-Analysis

In order to calculate the expected results behind the shock, we recommend using a oblique shock wave calculator (link grc.nasa.gov). At Mach 3 and an angle of 15 degrees, we find the following:

Image Removed

To calculate this by hand: 

Hand Calculations

Flow flow with M = 3 comes straight on in the x-direction towards the wedge. We know the wedge angle theta from our geometry of the wedge . From this we can calculate the normal component of our free stream Mach number. 

as 15 degrees.

Step 1: We then look at the Theta-Beta-M chart here we can find what the shock angle is corresponding to our conditions. The line M = 3 with wedge angle theta at 15 degrees corresponds to a shock angle beta of about 32 degrees. 

Step 2: We calculate the value of the free stream Mach Number normal to the shock so we can use normal shock relations to relate quantities upstream and downstream of the shock. 

\Large
\begin{equation}\nonumber
M_{1N} = M_1sin(\beta)
\end{equation}
\\

Where the shock angle Beta comes from the theta-beta-M chart. 

Now Step 3: Now we can relate the normal Mach numbers to each other through the normal shock relations

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\Large
\begin{equation}\nonumber
M_{2N}^2 = M_{1N}^2(\frac{(\gamma -1)M_{1N}+2}{2\gamma M_{1N}-(\gamma -1)})
\end{equation}
\\
\\
\begin{equation}\nonumber
M_2 =\frac{M_{2N}}{sin(\beta-\theta)}
\end{equation}
\\
\\
\begin{equation}\nonumber
\frac{p2}{p1} = \frac{2\gamma M_{1N}^2 - (\gamma - 1)}{\gamma + 1}
\end{equation}
\\
\\
\begin{equation}\nonumber
\frac{T_2}{T_1} = \frac{(2\gamma M_{1N}^2 - (\gamma - 1))((\gamma -1)M_{1N}^2 +2)}{(\gamma +1)^2 M_{1N}^2}
\end{equation}
\\

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From the equations above it is quite easy to relate upstream quantities. 

We also expect that the flow downstream of the shock will still be supersonic as the flow experiences only a weak oblique shock, evident from looking at the theta-beta-M chart. This also becomes clear in the hand calculations. 

Alternate Procedure:

In order to calculate the expected results behind the shock, you can also use an oblique shock wave calculator (link grc.nasa.gov). At Mach 3 and an angle of 15 degrees, we find the following:

Image Added

Open ANSYS Workbench

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