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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: 

the hand calculations we will be 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 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.to be 15 degrees. See the figure below:

Image Added

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. 

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

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

\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{p_2}{p_1} = \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}
\\

We 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

We are ready to do a simulation in ANSYS Workbench! Open ANSYS Workbench by going to Start > ANSYS > Workbench. This will open the start up screen seen as seen below:

Screen Management

This tutorial is designed such that the user can have both ANSYS Workbench and the tutorial open. As shown below, this online tutorial should fill approximately 1/3 of the screen, while ANSYS Workbench fills the remaining 2/3 of the screen.

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To begin, we need to tell ANSYS what kind of simulation we are doing. If you look to the left of the start up window, you will see the Toolbox Window. Take a look through the different selections. We will be using FLUENT to complete the simulation. Load the Fluid Flow (FLUENT) box by dragging and dropping it into the Project Schematic.

Right-click the top box of the project schematic and go to Rename, and name the project Supersonic Flow Over a Wedge. You are ready to create the geometry for the simulation.

Go to Step 2: Geometry

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