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In the hand calculations we will applying the conservation of energy, mass and momentum equations for a 1D inviscid compressible flow. This differs from the way that FLUENT solves the problem as FLUENT instead uses the 2D inviscid compressible flow equations.

The equations can be written as: 

\Large
\begin{equation}\nonumber
\frac{\partial e}{\partial t} + \textbf u\cdot\nabla e + \frac{p}{\rho}\nabla\cdot\textbf u = 0
\end{equation}
\\
\\
\begin{equation}\nonumber
\frac{\partial\rho}{\partial t}+\textbf u\cdot\nabla\rho+\rho\nabla\cdot\textbf u = 0
\end{equation}
\\
\\
\begin{equation}\nonumber
\frac{\partial\textbf u}{\partial t} + {\textbf u}\cdot\nabla\textbf u = - \frac{\nabla p}{\rho}
\end{equation}

Hand Calculations

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 as to be 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. 

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