Moving shock waves: Difference between revisions

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\section{Moving Normal Shock Waves}
=== Moving Normal Shock Waves ===


\noindent The starting point is the governing equations for stationary normal shocks (repeated here for convenience).
The starting point is the governing equations for stationary normal shocks (repeated here for convenience).


\begin{equation}
<math display="block">
\rho_1 u_1 = \rho_2 u_2
\rho_1 u_1 = \rho_2 u_2
\label{eq:stationary:cont}
</math>
\end{equation}


\begin{equation}
<math display="block">
\rho_1 u_1^2+p_1 = \rho_2 u_2^2 + p_2
\rho_1 u_1^2+p_1 = \rho_2 u_2^2 + p_2
\label{eq:stationary:mom}
</math>
\end{equation}


\begin{equation}
<math display="block">
h_1 + \frac{1}{2}u_1^2 = h_2 + \frac{1}{2}u_2^2
h_1 + \frac{1}{2}u_1^2 = h_2 + \frac{1}{2}u_2^2
\label{eq:stationary:energy}
</math>
\end{equation}


\noindent Shock moving to the right with the constant speed $W$ into a gas that is standing still. Moving with the shock, we would see a gas velocity ahead of the shock $u_1=W$, and the gas behind the shock moves to the right with the velocity $u_2=W-u_p$. Now, let's insert $u_1$ and $u_2$ in the stationary shock relations \ref{eq:stationary:cont} - \ref{eq:stationary:energy}.
Shock moving to the right with the constant speed $W$ into a gas that is standing still. Moving with the shock, we would see a gas velocity ahead of the shock <math>u_1=W</math>, and the gas behind the shock moves to the right with the velocity <math>u_2=W-u_p</math>. Now, let's insert <math>u_1</math> and <math>u_2</math> in the stationary shock relations \ref{eq:stationary:cont} - \ref{eq:stationary:energy}.


\begin{equation}
<math display="block">
\rho_1 W = \rho_2 (W-u_p)
\rho_1 W = \rho_2 (W-u_p)
\label{eq:unsteady:cont}
</math>
\end{equation}


\begin{equation}
<math display="block">
\rho_1 W^2+p_1 = \rho_2 (W-u_p)^2 + p_2
\rho_1 W^2+p_1 = \rho_2 (W-u_p)^2 + p_2
\label{eq:unsteady:mom}
</math>
\end{equation}


\begin{equation}
<math display="block">
h_1 + \frac{1}{2}W^2 = h_2 + \frac{1}{2}(W-u_p)^2
h_1 + \frac{1}{2}W^2 = h_2 + \frac{1}{2}(W-u_p)^2
\label{eq:unsteady:energy}
</math>
\end{equation}


%\newpage
Rewriting Eqn. \ref{eq:unsteady:cont}


\noindent Rewriting Eqn. \ref{eq:unsteady:cont}
<math display="block">
 
\begin{equation}
(W-u_p) = W \frac{\rho_1}{\rho_2}
(W-u_p) = W \frac{\rho_1}{\rho_2}
\label{eq:unsteady:cont:mod}
</math>
\end{equation}\\


\noindent Inserting Eqn. \ref{eq:unsteady:cont:mod} in Eqn. \ref{eq:unsteady:mom} gives\\
Inserting Eqn. \ref{eq:unsteady:cont:mod} in Eqn. \ref{eq:unsteady:mom} gives


\begin{equation*}
<math display="block">
p_1+\rho_1 W^2 = p_2+\rho_2 W^2\left(\frac{\rho_1}{\rho_2}\right)^2 \Rightarrow p_2-p_1 = \rho_1W^2\left(1-\frac{\rho_1}{\rho_2}\right)
p_1+\rho_1 W^2 = p_2+\rho_2 W^2\left(\frac{\rho_1}{\rho_2}\right)^2 \Rightarrow p_2-p_1 = \rho_1W^2\left(1-\frac{\rho_1}{\rho_2}\right)
%\label{eq:unsteady:mom:mod}
</math>
\end{equation*}\\


\begin{equation}
<math display="block">
W^2=\frac{p_2-p_1}{\rho_2-\rho_1}\left(\frac{\rho_2}{\rho_1}\right)
W^2=\frac{p_2-p_1}{\rho_2-\rho_1}\left(\frac{\rho_2}{\rho_1}\right)
\label{eq:unsteady:mom:mod}
</math>
\end{equation}\\


\noindent From the continuity equation \ref{eq:unsteady:cont}, we get \\
From the continuity equation \ref{eq:unsteady:cont}, we get


\begin{equation}
<math display="block">
W = (W-u_p) \left(\frac{\rho_2}{\rho_1}\right)
W = (W-u_p) \left(\frac{\rho_2}{\rho_1}\right)
\label{eq:unsteady:cont:modb}
</math>
\end{equation}\\


\noindent Inserting Eqn. \ref{eq:unsteady:cont:modb} in Eqn. \ref{eq:unsteady:mom:mod} gives\\
Inserting Eqn. \ref{eq:unsteady:cont:modb} in Eqn. \ref{eq:unsteady:mom:mod} gives


\begin{equation}
<math display="block">
(W-u_p)^2=\frac{p_2-p_1}{\rho_2-\rho_1}\left(\frac{\rho_1}{\rho_2}\right)
(W-u_p)^2=\frac{p_2-p_1}{\rho_2-\rho_1}\left(\frac{\rho_1}{\rho_2}\right)
\label{eq:unsteady:mom:modb}
</math>
\end{equation}\\


\noindent Now, let's insert Eqns. \ref{eq:unsteady:mom:mod} and \ref{eq:unsteady:mom:modb} in the energy equation (Eqn. \ref{eq:unsteady:energy}).\\
Now, let's insert Eqns. \ref{eq:unsteady:mom:mod} and \ref{eq:unsteady:mom:modb} in the energy equation (Eqn. \ref{eq:unsteady:energy}).


\begin{equation}
<math display="block">
h_1 + \frac{1}{2}\left[\frac{p_2-p_1}{\rho_2-\rho_1}\left(\frac{\rho_2}{\rho_1}\right)\right] = h_2 + \frac{1}{2}\left[\frac{p_2-p_1}{\rho_2-\rho_1}\left(\frac{\rho_1}{\rho_2}\right)\right]
h_1 + \frac{1}{2}\left[\frac{p_2-p_1}{\rho_2-\rho_1}\left(\frac{\rho_2}{\rho_1}\right)\right] = h_2 + \frac{1}{2}\left[\frac{p_2-p_1}{\rho_2-\rho_1}\left(\frac{\rho_1}{\rho_2}\right)\right]
\label{eq:unsteady:energy:mod}
</math>
\end{equation}


\begin{equation}
<math display="block">
h=e+\frac{p}{\rho}
h=e+\frac{p}{\rho}
\label{eq:enthalpy}
</math>
\end{equation}


\begin{equation}
<math display="block">
e_1 + \frac{p_1}{\rho_1} +  \frac{1}{2}\left[\frac{p_2-p_1}{\rho_2-\rho_1}\left(\frac{\rho_2}{\rho_1}\right)\right] = e_2 + \frac{p_2}{\rho_2} + \frac{1}{2}\left[\frac{p_2-p_1}{\rho_2-\rho_1}\left(\frac{\rho_1}{\rho_2}\right)\right]
e_1 + \frac{p_1}{\rho_1} +  \frac{1}{2}\left[\frac{p_2-p_1}{\rho_2-\rho_1}\left(\frac{\rho_2}{\rho_1}\right)\right] = e_2 + \frac{p_2}{\rho_2} + \frac{1}{2}\left[\frac{p_2-p_1}{\rho_2-\rho_1}\left(\frac{\rho_1}{\rho_2}\right)\right]
\label{eq:unsteady:energy:mod:b}
</math>
\end{equation}\\


\noindent which can be rewritten as\\
which can be rewritten as


\begin{equation}
<math display="block">
e_2-e_1=\frac{p_1+p_2}{2}\left(\frac{1}{\rho_1}-\frac{1}{\rho_2}\right)
e_2-e_1=\frac{p_1+p_2}{2}\left(\frac{1}{\rho_1}-\frac{1}{\rho_2}\right)
\label{eq:unsteady:hugonoit}
</math>
\end{equation}\\


\noindent Eqn \ref{eq:unsteady:hugonoit} is the same Hugoniot equation as we get for a stationary normal shock. The Hugoniot equation is a relation of thermodynamic properties over a shock. As the shock in the unsteady case is moving with a constant velocity, the frame of reference moving with the shock is an inertial frame and thus the same physical relations apply in the moving shock case as in the stationary shock case. The fact that the Hugoniot relation does not include any velocities or Mach numbers but only thermodynamic properties, the relation will be unchanged for a moving shock.
Eqn \ref{eq:unsteady:hugonoit} is the same Hugoniot equation as we get for a stationary normal shock. The Hugoniot equation is a relation of thermodynamic properties over a shock. As the shock in the unsteady case is moving with a constant velocity, the frame of reference moving with the shock is an inertial frame and thus the same physical relations apply in the moving shock case as in the stationary shock case. The fact that the Hugoniot relation does not include any velocities or Mach numbers but only thermodynamic properties, the relation will be unchanged for a moving shock.


\subsection{Moving Shock Relations}
=== Moving Shock Relations ===


\noindent For a calorically perfect gas we have $e=C_v T$. Inserted in the Hugoniot relation above this gives\\
For a calorically perfect gas we have <math>e=C_v T</math>. Inserted in the Hugoniot relation above this gives


\begin{equation}
<math display="block">
C_v(T_2-T_1)=\frac{p_1+p_2}{2}\left(\nu_1-\nu_2\right)
C_v(T_2-T_1)=\frac{p_1+p_2}{2}\left(\nu_1-\nu_2\right)
\label{eq:unsteady:hugonoit:b}
</math>
\end{equation}\\


where $\nu=1/\rho$\\
where <math>\nu=1/\rho</math>


\noindent Now, using the ideal gas law $T=p\nu/R$ and $C_v/R=1/(\gamma-1)$ gives\\
Now, using the ideal gas law <math>T=p\nu/R</math> and <math>C_v/R=1/(\gamma-1)</math> gives


\begin{equation*}
<math display="block">
\left(\frac{1}{\gamma-1}\right)(p_2\nu_2-p_1\nu_1)=\frac{p_1+p_2}{2}\left(\nu_1-\nu_2\right)
\left(\frac{1}{\gamma-1}\right)(p_2\nu_2-p_1\nu_1)=\frac{p_1+p_2}{2}\left(\nu_1-\nu_2\right)
\end{equation*}
</math>


\begin{equation*}
<math display="block">
\Leftrightarrow  
\Leftrightarrow  
\end{equation*}
</math>


\begin{equation*}
<math display="block">
p_2\left(\frac{\nu_2}{\gamma-1}-\frac{\nu_1-\nu_2}{2}\right)=p_1\left(\frac{\nu_1}{\gamma-1}+\frac{\nu_1-\nu_2}{2}\right)
p_2\left(\frac{\nu_2}{\gamma-1}-\frac{\nu_1-\nu_2}{2}\right)=p_1\left(\frac{\nu_1}{\gamma-1}+\frac{\nu_1-\nu_2}{2}\right)
\end{equation*}\\
</math>


\noindent From this result, we can derive a relation for the pressure ratio over the shock as a function of density ratio\\
From this result, we can derive a relation for the pressure ratio over the shock as a function of density ratio


\begin{equation}
<math display="block">
\frac{p_2}{p_1}=\frac{\left(\dfrac{\gamma+1}{\gamma-1}\right)\left(\dfrac{\nu_1}{\nu_2}\right)-1}{\left(\dfrac{\gamma+1}{\gamma-1}\right)-\left(\dfrac{\nu_1}{\nu_2}\right)}
\frac{p_2}{p_1}=\frac{\left(\dfrac{\gamma+1}{\gamma-1}\right)\left(\dfrac{\nu_1}{\nu_2}\right)-1}{\left(\dfrac{\gamma+1}{\gamma-1}\right)-\left(\dfrac{\nu_1}{\nu_2}\right)}
\label{eq:unsteady:hugonoit:c}
</math>
\end{equation}\\


\noindent $\nu=RT/p$ and thus
<math>\nu=RT/p</math> and thus


\begin{equation}
<math display="block">
\frac{\nu_1}{\nu_2}=\frac{T_1}{T_2}\frac{p_2}{p_1}
\frac{\nu_1}{\nu_2}=\frac{T_1}{T_2}\frac{p_2}{p_1}
\label{eq:unsteady:density:ratio}
</math>
\end{equation}\\


\noindent Eqn. \ref{eq:unsteady:density:ratio} in Eqn. \ref{eq:unsteady:hugonoit:c} gives\\
Eqn. \ref{eq:unsteady:density:ratio} in Eqn. \ref{eq:unsteady:hugonoit:c} gives


\begin{equation}
<math display="block">
\frac{p_2}{p_1}=\frac{\left(\dfrac{\gamma+1}{\gamma-1}\right)\left(\dfrac{T_1}{T_2}\dfrac{p_2}{p_1}\right)-1}{\left(\dfrac{\gamma+1}{\gamma-1}\right)-\left(\dfrac{T_1}{T_2}\dfrac{p_2}{p_1}\right)}
\frac{p_2}{p_1}=\frac{\left(\dfrac{\gamma+1}{\gamma-1}\right)\left(\dfrac{T_1}{T_2}\dfrac{p_2}{p_1}\right)-1}{\left(\dfrac{\gamma+1}{\gamma-1}\right)-\left(\dfrac{T_1}{T_2}\dfrac{p_2}{p_1}\right)}
\label{eq:unsteady:hugonoit:c}
</math>
\end{equation}\\


\noindent Now, we can get a relation for calculation of the temperature ratio over the moving shock as function of the shock pressure ratio\\
Now, we can get a relation for calculation of the temperature ratio over the moving shock as function of the shock pressure ratio


\begin{equation}
<math display="block">
\frac{T_2}{T_1}=\frac{p_2}{p_1}\left[\frac{\left(\dfrac{\gamma+1}{\gamma-1}\right)+\left(\dfrac{p_2}{p_1}\right)}{1+\left(\dfrac{\gamma+1}{\gamma-1}\right)\left(\dfrac{p_2}{p_1}\right)}\right]
\frac{T_2}{T_1}=\frac{p_2}{p_1}\left[\frac{\left(\dfrac{\gamma+1}{\gamma-1}\right)+\left(\dfrac{p_2}{p_1}\right)}{1+\left(\dfrac{\gamma+1}{\gamma-1}\right)\left(\dfrac{p_2}{p_1}\right)}\right]
\label{eq:unsteady:temperature:ratio}
</math>
\end{equation}\\


\noindent Once again using the ideal gas law\\
Once again using the ideal gas law


\begin{equation}
<math display="block">
\frac{\rho_2}{\rho_1}=\frac{\left(\dfrac{\gamma+1}{\gamma-1}\right)+\left(\dfrac{p_2}{p_1}\right)}{1+\left(\dfrac{\gamma+1}{\gamma-1}\right)\left(\dfrac{p_2}{p_1}\right)}
\frac{\rho_2}{\rho_1}=\frac{\left(\dfrac{\gamma+1}{\gamma-1}\right)+\left(\dfrac{p_2}{p_1}\right)}{1+\left(\dfrac{\gamma+1}{\gamma-1}\right)\left(\dfrac{p_2}{p_1}\right)}
\label{eq:unsteady:density:ratio}
</math>
\end{equation}\\


\noindent Going back to the momentum equation\\
Going back to the momentum equation


\begin{equation*}
<math display="block">
p_2-p_1 = \rho_1W^2\left(1-\frac{\rho_1}{\rho_2}\right)=\left\{W=M_s a_1\right\}=\rho_1M_s^2a_1^2\left(1-\frac{\rho_1}{\rho_2}\right)
p_2-p_1 = \rho_1W^2\left(1-\frac{\rho_1}{\rho_2}\right)=\left\{W=M_s a_1\right\}=\rho_1M_s^2a_1^2\left(1-\frac{\rho_1}{\rho_2}\right)
\end{equation*}\\
</math>


\noindent with $a_1^2=\gamma p_1/\rho_1$, we get\\
with <math>a_1^2=\gamma p_1/\rho_1</math>, we get


\begin{equation}
<math display="block">
\frac{p_2}{p_1} = \gamma M_s^2\left(1-\frac{\rho_1}{\rho_2}\right)+1
\frac{p_2}{p_1} = \gamma M_s^2\left(1-\frac{\rho_1}{\rho_2}\right)+1
\label{eq:unsteady:Mach:a}
</math>
\end{equation}\\


\noindent From the normal shock relations, we have\\
From the normal shock relations, we have


\begin{equation}
<math display="block">
\frac{\rho_1}{\rho_2} = \frac{2+(\gamma-1)M_s^2}{(\gamma+1)M_s^2}
\frac{\rho_1}{\rho_2} = \frac{2+(\gamma-1)M_s^2}{(\gamma+1)M_s^2}
\label{eq:unsteady:Mach:b}
</math>
\end{equation}\\


\noindent Eqn. \ref{eq:unsteady:Mach:b} in \ref{eq:unsteady:Mach:a} gives\\
Eqn. \ref{eq:unsteady:Mach:b} in \ref{eq:unsteady:Mach:a} gives


\begin{equation}
<math display="block">
\frac{p_2}{p_1} = 1 + \left(\frac{2\gamma}{\gamma+1}\right)(M_s^2-1)
\frac{p_2}{p_1} = 1 + \left(\frac{2\gamma}{\gamma+1}\right)(M_s^2-1)
\label{eq:unsteady:Mach:c}
</math>
\end{equation}\\


or\\
or


\begin{equation}
<math display="block">
M_s=\sqrt{\left(\frac{\gamma+1}{2\gamma}\right)\left(\frac{p_2}{p_1}-1\right)+1}
M_s=\sqrt{\left(\frac{\gamma+1}{2\gamma}\right)\left(\frac{p_2}{p_1}-1\right)+1}
\label{eq:unsteady:Mach}
</math>
\end{equation}\\


\noindent Eqn. \ref{eq:unsteady:Mach} with $M_s=W/a_1$\\
Eqn. \ref{eq:unsteady:Mach} with <math>M_s=W/a_1</math>


\begin{equation}
<math display="block">
W=a_1\sqrt{\left(\frac{\gamma+1}{2\gamma}\right)\left(\frac{p_2}{p_1}-1\right)+1}
W=a_1\sqrt{\left(\frac{\gamma+1}{2\gamma}\right)\left(\frac{p_2}{p_1}-1\right)+1}
\label{eq:unsteady:W}
</math>
\end{equation}\\


\subsection{Induced Flow Behind Moving Shock}
==== Induced Flow Behind Moving Shock ====


\noindent Let's try to find a relation for calculation of the induced velocity behind the moving shock. Once again, the starting point is the continuity equation for moving shocks (Eqn. \ref{eq:unsteady:cont}) repeated here for convenience\\
Let's try to find a relation for calculation of the induced velocity behind the moving shock. Once again, the starting point is the continuity equation for moving shocks (Eqn. \ref{eq:unsteady:cont}) repeated here for convenience


\begin{equation*}
<math display="block">
\rho_1 W = \rho_2 (W-u_p)
\rho_1 W = \rho_2 (W-u_p)
\end{equation*}\\
</math>


\noindent The induced velocity appears on the right side of the continuity equation\\
The induced velocity appears on the right side of the continuity equation


\begin{equation*}
<math display="block">
W (\rho_1-\rho_2) = -\rho_2 u_p
W (\rho_1-\rho_2) = -\rho_2 u_p
\end{equation*}\\
</math>


\begin{equation}
<math display="block">
u_p = W \left(1-\frac{\rho_1}{\rho_2}\right)  
u_p = W \left(1-\frac{\rho_1}{\rho_2}\right)  
\label{eq:unsteady:up:a}
</math>
\end{equation}\\


\noindent From before we have a relation for $W$ as a function of pressure ratio and one for $\rho_1/\rho_2$, also as a function of pressure ratio.\\
From before we have a relation for $W$ as a function of pressure ratio and one for <math>\rho_1/\rho_2</math>, also as a function of pressure ratio.


Eqn. \ref{eq:unsteady:up:a} togheter with Eqns. \ref{eq:unsteady:W} and \ref{eq:unsteady:density:ratio} gives\\
Eqn. \ref{eq:unsteady:up:a} togheter with Eqns. \ref{eq:unsteady:W} and \ref{eq:unsteady:density:ratio} gives


\begin{equation}
<math display="block">
u_p=a_1\underbrace{\sqrt{\left(\frac{\gamma+1}{2\gamma}\right)\left(\frac{p_2}{p_1}-1\right)+1}}_{I}\underbrace{\left[1-\dfrac{\left(\dfrac{\gamma+1}{\gamma-1}\right)+\left(\dfrac{p_2}{p_1}\right)}{1+\left(\dfrac{\gamma+1}{\gamma-1}\right)\left(\dfrac{p_2}{p_1}\right)}\right]}_{II}
u_p=a_1\underbrace{\sqrt{\left(\frac{\gamma+1}{2\gamma}\right)\left(\frac{p_2}{p_1}-1\right)+1}}_{I}\underbrace{\left[1-\dfrac{\left(\dfrac{\gamma+1}{\gamma-1}\right)+\left(\dfrac{p_2}{p_1}\right)}{1+\left(\dfrac{\gamma+1}{\gamma-1}\right)\left(\dfrac{p_2}{p_1}\right)}\right]}_{II}
\label{eq:unsteady:up:b}
</math>
\end{equation}\\


\noindent The equation subsets I and II can be rewritten as:\\
The equation subsets I and II can be rewritten as:


Term I:
Term I:


\begin{equation*}
<math display="block">
\sqrt{\left(\frac{\gamma+1}{2\gamma}\right)\left(\frac{p_2}{p_1}-1\right)+1}=\sqrt{\frac{\gamma+1}{2\gamma}\left[\left(\frac{p_2}{p_1}\right)+\left(\frac{\gamma-1}{\gamma+1}\right)\right]}
\sqrt{\left(\frac{\gamma+1}{2\gamma}\right)\left(\frac{p_2}{p_1}-1\right)+1}=\sqrt{\frac{\gamma+1}{2\gamma}\left[\left(\frac{p_2}{p_1}\right)+\left(\frac{\gamma-1}{\gamma+1}\right)\right]}
\end{equation*}\\
</math>




Term II:
Term II:


\begin{equation*}
<math display="block">
\left[1-\dfrac{\left(\dfrac{\gamma+1}{\gamma-1}\right)+\left(\dfrac{p_2}{p_1}\right)}{1+\left(\dfrac{\gamma+1}{\gamma-1}\right)\left(\dfrac{p_2}{p_1}\right)}\right]=\frac{1}{\gamma}\left(\frac{p_2}{p_1}-1\right)\frac{\left(\dfrac{2\gamma}{\gamma+1}\right)}{\left(\dfrac{\gamma-1}{\gamma+1}\right)+\left(\dfrac{p_2}{p_1}\right)}
\left[1-\dfrac{\left(\dfrac{\gamma+1}{\gamma-1}\right)+\left(\dfrac{p_2}{p_1}\right)}{1+\left(\dfrac{\gamma+1}{\gamma-1}\right)\left(\dfrac{p_2}{p_1}\right)}\right]=\frac{1}{\gamma}\left(\frac{p_2}{p_1}-1\right)\frac{\left(\dfrac{2\gamma}{\gamma+1}\right)}{\left(\dfrac{\gamma-1}{\gamma+1}\right)+\left(\dfrac{p_2}{p_1}\right)}
\end{equation*}\\
</math>


\noindent With the rewritten terms I and II implemented, Eqn. \ref{eq:unsteady:up:b} becomes\\
the rewritten terms I and II implemented, Eqn. \ref{eq:unsteady:up:b} becomes


\begin{equation}
<math display="block">
u_p=\frac{a_1}{\gamma}\left(\frac{p_2}{p_1}-1\right)\sqrt{\frac{\left(\dfrac{2\gamma}{\gamma+1}\right)}{\left(\dfrac{\gamma-1}{\gamma+1}\right)+\left(\dfrac{p_2}{p_1}\right)}}
u_p=\frac{a_1}{\gamma}\left(\frac{p_2}{p_1}-1\right)\sqrt{\frac{\left(\dfrac{2\gamma}{\gamma+1}\right)}{\left(\dfrac{\gamma-1}{\gamma+1}\right)+\left(\dfrac{p_2}{p_1}\right)}}
\label{eq:unsteady:up}
</math>
\end{equation}\\


\noindent Since the region behind the moving shock is region 2, the induced flow Mach number is obtained as\\
Since the region behind the moving shock is region 2, the induced flow Mach number is obtained as


\begin{equation*}
<math display="block">
M_p=\frac{u_p}{a_2}=\frac{u_p}{a_1}\frac{a_1}{a_2}=\frac{u_p}{a_1}\sqrt{\frac{\gamma R T_1}{\gamma R T_2}}=\frac{u_p}{a_1}\sqrt{\frac{T_1}{T_2}}
M_p=\frac{u_p}{a_2}=\frac{u_p}{a_1}\frac{a_1}{a_2}=\frac{u_p}{a_1}\sqrt{\frac{\gamma R T_1}{\gamma R T_2}}=\frac{u_p}{a_1}\sqrt{\frac{T_1}{T_2}}
\end{equation*}\\
</math>


\noindent With $up/a_1$ from Eqn. \ref{eq:unsteady:up}  and $T_1/T_2$ from Eqn. \ref{eq:unsteady:temperature:ratio}\\
With <math>up/a_1</math> from Eqn. \ref{eq:unsteady:up}  and <math>T_1/T_2</math> from Eqn. \ref{eq:unsteady:temperature:ratio}


\begin{equation}
<math display="block">
M_p=\frac{1}{\gamma}\left(\frac{p_2}{p_1}-1\right)\left(\frac{\left(\dfrac{2\gamma}{\gamma+1}\right)}{\left(\dfrac{\gamma-1}{\gamma+1}\right)+\left(\dfrac{p_2}{p_1}\right)}\right)^{1/2}
M_p=\frac{1}{\gamma}\left(\frac{p_2}{p_1}-1\right)\left(\frac{\left(\dfrac{2\gamma}{\gamma+1}\right)}{\left(\dfrac{\gamma-1}{\gamma+1}\right)+\left(\dfrac{p_2}{p_1}\right)}\right)^{1/2}
\left(\frac{1+\left(\dfrac{\gamma+1}{\gamma-1}\right)\left(\dfrac{p_2}{p_1}\right)}{\left(\dfrac{\gamma+1}{\gamma-1}\right)\left(\dfrac{p_2}{p_1}\right)+\left(\dfrac{p_2}{p_1}\right)^2}\right)^{1/2}
\left(\frac{1+\left(\dfrac{\gamma+1}{\gamma-1}\right)\left(\dfrac{p_2}{p_1}\right)}{\left(\dfrac{\gamma+1}{\gamma-1}\right)\left(\dfrac{p_2}{p_1}\right)+\left(\dfrac{p_2}{p_1}\right)^2}\right)^{1/2}
\label{eq:unsteady:Mp}
</math>
\end{equation}\\


\noindent There is a theoretical upper limit for the induced Mach number $M_p$\\
There is a theoretical upper limit for the induced Mach number <math>M_p</math>


\begin{equation*}
<math display="block">
\lim_{p_2/p_1\rightarrow\infty} M_p\left(\frac{p_2}{p_1}\right)=\sqrt{\frac{2}{\gamma(\gamma-1)}}
\lim_{p_2/p_1\rightarrow\infty} M_p\left(\frac{p_2}{p_1}\right)=\sqrt{\frac{2}{\gamma(\gamma-1)}}
\end{equation*}\\
</math>


\noindent As can be seen, at the upper limit the induced Mach number is a function of $\gamma$ and for air ($\gamma=1.4$) we get\\
As can be seen, at the upper limit the induced Mach number is a function of <math>\gamma</math> and for air (<math>\gamma=1.4</math>) we get


\begin{equation*}
<math display="block">
\lim_{p_2/p_1\rightarrow\infty} M_p\left(\frac{p_2}{p_1}\right)\simeq 1.89
\lim_{p_2/p_1\rightarrow\infty} M_p\left(\frac{p_2}{p_1}\right)\simeq 1.89
\end{equation*}\\
</math>


\section{Shock Wave Reflection}
\section{Shock Wave Reflection}


\noindent When the incident shock wave reaches the wall, a shock propagating in the opposite direction is generated with a shock strength such that the velocity of the induced flow behind the incident shock is reduced to zero. The flow can not go through the wall and thus the velocity must be zero in the vicinity of the wall. The properties of the incident shock wave are directly related to the pressure ratio over the shock wave. Therefore, it would be convenient to have a relation between the reflected shock wave and incident shock wave.
When the incident shock wave reaches the wall, a shock propagating in the opposite direction is generated with a shock strength such that the velocity of the induced flow behind the incident shock is reduced to zero. The flow can not go through the wall and thus the velocity must be zero in the vicinity of the wall. The properties of the incident shock wave are directly related to the pressure ratio over the shock wave. Therefore, it would be convenient to have a relation between the reflected shock wave and incident shock wave.
 
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\subsection{The Incident Shock Wave}
==== The Incident Shock Wave ====


\noindent The pressure ratio over the incident shock in Fig.~\ref{fig:reflection} can be obtained as \\
The pressure ratio over the incident shock in Fig.~\ref{fig:reflection} can be obtained as


\begin{equation}
<math display="block">
\frac{p_2}{p_1}=1+\frac{2\gamma}{\gamma+1}\left(M_s^2-1\right)
\frac{p_2}{p_1}=1+\frac{2\gamma}{\gamma+1}\left(M_s^2-1\right)
\label{eq:incident:pr}
</math>
\end{equation}\\


\noindent where $M_s$ is the wave Mach number, which is calculated as \\
where <math>M_s</math> is the wave Mach number, which is calculated as


\begin{equation}
<math display="block">
M_s=\frac{W}{a_1}
M_s=\frac{W}{a_1}
\label{eq:incident:Mach:def}
</math>
\end{equation}\\




\noindent In Eqn.~\ref{eq:incident:Mach:def}, $W$ is the speed with which the incident shock wave travels into region 1 and $a_1$ is the speed of sound in region 1 (see Fig.~\ref{fig:reflection}).
In Eqn.~\ref{eq:incident:Mach:def}, <math>W</math> is the speed with which the incident shock wave travels into region 1 and <math>a_1</math> is the speed of sound in region 1 (see Fig.~\ref{fig:reflection}).


\noindent Solving Eqn.~\ref{eq:incident:pr} for $M_s$, we get \\
Solving Eqn.~\ref{eq:incident:pr} for <math>M_s</math>, we get


\begin{equation}
<math display="block">
M_s=\sqrt{\frac{\gamma+1}{2\gamma}\left(\frac{p_2}{p_1}-1\right)+1}
M_s=\sqrt{\frac{\gamma+1}{2\gamma}\left(\frac{p_2}{p_1}-1\right)+1}
\label{eq:incident:Mach}
</math>
\end{equation}\\


\noindent Anderson derives the relations for calculation of the ratio $T_2/T_1$ \\
Anderson derives the relations for calculation of the ratio <math>T_2/T_1</math>


\begin{equation}
<math display="block">
\frac{T_2}{T_1}=\frac{p_2}{p_1}\left(\dfrac{\dfrac{\gamma+1}{\gamma-1}+\dfrac{p_2}{p_1}}{1+\dfrac{\gamma+1}{\gamma-1}\dfrac{p_2}{p_1}}\right)
\frac{T_2}{T_1}=\frac{p_2}{p_1}\left(\dfrac{\dfrac{\gamma+1}{\gamma-1}+\dfrac{p_2}{p_1}}{1+\dfrac{\gamma+1}{\gamma-1}\dfrac{p_2}{p_1}}\right)
\label{eq:incident:tr}
</math>
\end{equation}\\


\noindent From Eqn.~\ref{eq:incident:tr} it is easy to get the corresponding relation for $\rho_2/\rho_1$\\
From Eqn.~\ref{eq:incident:tr} it is easy to get the corresponding relation for <math>\rho_2/\rho_1</math>


\begin{equation}
<math display="block">
\frac{\rho_2}{\rho_1}=\dfrac{1+\dfrac{\gamma+1}{\gamma-1}\dfrac{p_2}{p_1}}{\dfrac{\gamma+1}{\gamma-1}+\dfrac{p_2}{p_1}}
\frac{\rho_2}{\rho_1}=\dfrac{1+\dfrac{\gamma+1}{\gamma-1}\dfrac{p_2}{p_1}}{\dfrac{\gamma+1}{\gamma-1}+\dfrac{p_2}{p_1}}
\label{eq:incident:rr}
</math>
\end{equation}\\


\noindent Anderson also shows how to obtain the induced velocity, $u_p$, behind the incident shock wave, {\emph{i.e.}} the velocity in region 2 (see Fig.~\ref{fig:reflection}).\\
Anderson also shows how to obtain the induced velocity, <math>u_p</math>, behind the incident shock wave, {\emph{i.e.}} the velocity in region 2 (see Fig.~\ref{fig:reflection}).


\begin{equation}
<math display="block">
u_p=W\left(1-\frac{\rho_1}{\rho_2}\right)=M_s a_1 \left(1-\frac{\rho_1}{\rho_2}\right)
u_p=W\left(1-\frac{\rho_1}{\rho_2}\right)=M_s a_1 \left(1-\frac{\rho_1}{\rho_2}\right)
\label{eq:incident:up}
</math>
\end{equation}\\


\subsection{The Reflected Shock Wave}
==== The Reflected Shock Wave ====


\noindent The pressure ratio over the reflected shock can be obtained from Eqn.~\ref{eq:incident:pr} by analogy\\
The pressure ratio over the reflected shock can be obtained from Eqn.~\ref{eq:incident:pr} by analogy


\begin{equation}
<math display="block">
\frac{p_5}{p_2}=1+\frac{2\gamma}{\gamma+1}\left(M_r^2-1\right)
\frac{p_5}{p_2}=1+\frac{2\gamma}{\gamma+1}\left(M_r^2-1\right)
\label{eq:reflected:pr}
</math>
\end{equation}\\


\noindent where $M_r$ is the Mach number of the reflected shock wave defined as\\
where <math>M_r</math> is the Mach number of the reflected shock wave defined as


\begin{equation}
<math display="block">
M_r=\frac{W_r+u_p}{a_2}
M_r=\frac{W_r+u_p}{a_2}
\label{eq:reflected:Mach:def}
</math>
\end{equation}\\


\noindent where $W_r$ is the speed of the reflected shock wave and $a_2$ is the speed of sound in region 2 (see Fig.~\ref{fig:reflection}).\\
where <math>W_r</math> is the speed of the reflected shock wave and <math>a_2</math> is the speed of sound in region 2 (see Fig.~\ref{fig:reflection}).


\noindent Solving Eqn.~\ref{eq:reflected:pr} for $M_r$ gives\\
Solving Eqn.~\ref{eq:reflected:pr} for <math>M_r</math> gives


\begin{equation}
<math display="block">
M_r=\sqrt{\frac{\gamma+1}{2\gamma}\left(\frac{p_5}{p_2}-1\right)+1}
M_r=\sqrt{\frac{\gamma+1}{2\gamma}\left(\frac{p_5}{p_2}-1\right)+1}
\label{eq:reflected:Mach}
</math>
\end{equation}\\


\noindent The ratios $T_5/T_2$ and $\rho_5/\rho_2$ can be obtained from Eqns.~\ref{eq:incident:tr} and \ref{eq:incident:rr} by analogy\\
The ratios <math>T_5/T_2</math> and <math>\rho_5/\rho_2</math> can be obtained from Eqns.~\ref{eq:incident:tr} and \ref{eq:incident:rr} by analogy


\begin{equation}
<math display="block">
\frac{T_5}{T_2}=\frac{p_5}{p_2}\left(\dfrac{\dfrac{\gamma+1}{\gamma-1}+\dfrac{p_5}{p_2}}{1+\dfrac{\gamma+1}{\gamma-1}\dfrac{p_5}{p_2}}\right)
\frac{T_5}{T_2}=\frac{p_5}{p_2}\left(\dfrac{\dfrac{\gamma+1}{\gamma-1}+\dfrac{p_5}{p_2}}{1+\dfrac{\gamma+1}{\gamma-1}\dfrac{p_5}{p_2}}\right)
\label{eq:reflected:tr}
</math>
\end{equation}\\


\begin{equation}
<math display="block">
\frac{\rho_5}{\rho_2}=\dfrac{1+\dfrac{\gamma+1}{\gamma-1}\dfrac{p_5}{p_2}}{\dfrac{\gamma+1}{\gamma-1}+\dfrac{p_5}{p_2}}
\frac{\rho_5}{\rho_2}=\dfrac{1+\dfrac{\gamma+1}{\gamma-1}\dfrac{p_5}{p_2}}{\dfrac{\gamma+1}{\gamma-1}+\dfrac{p_5}{p_2}}
\label{eq:reflected:rr}
</math>
\end{equation}\\


\noindent The velocity in region 2 which is the same as the induced flow velocity behind the incident shock wave can be obtained as\\
The velocity in region 2 which is the same as the induced flow velocity behind the incident shock wave can be obtained as


\begin{equation}
<math display="block">
u_p=W_r\left(\frac{\rho_5}{\rho_2}-1\right)=M_r a_2 \left(1-\frac{\rho_2}{\rho_5}\right)
u_p=W_r\left(\frac{\rho_5}{\rho_2}-1\right)=M_r a_2 \left(1-\frac{\rho_2}{\rho_5}\right)
\label{eq:reflected:up}
</math>
\end{equation}\\


\subsection{Reflected Shock Relation}
\subsection{Reflected Shock Relation}
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