View source for Shock-tube relations

[[Category:Compressible flow]]
[[Category:Unsteady waves]]
[[Category:Inviscid flow]]
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[[Category:Compressible flow:Topic]]
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\begin{figure}[ht!]
\begin{center}
\includegraphics[]{figures/standalone-figures/Chapter07/pdf/shock-tube.pdf}
\caption{traveling waves in a shock tube}
\label{fig:shocktube}
\end{center}
\end{figure}
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From the analysis of the incident shock, we have a relation for the induced flow behind the shock

<math display="block">
u_2=u_p=\frac{a_1}{\gamma_1}\left(\frac{p_2}{p_1}-1\right)\left(\frac{\left(\dfrac{2\gamma_1}{\gamma_1+1}\right)}{\left(\dfrac{\gamma_1-1}{\gamma_1+1}\right)+\left(\dfrac{p_2}{p_1}\right)}\right)^{1/2}
</math>

The velocity in region 3 can be obtained from the expansion relations

<math display="block">
\frac{p_3}{p_4}=\left[1-\frac{\gamma_4-1}{2}\left(\frac{u_3}{a_4}\right)\right]^{2\gamma_4/(\gamma_4-1)}
</math>

Solving for <math>u_3</math> gives

<math display="block">
u_3=\frac{2a_4}{\gamma_4-1}\left[1-\left(\frac{p_3}{p_4}\right)^{(\gamma_4-1)/(2\gamma_4)}\right]
</math>

There is no change in pressure or velocity over the contact surface, which means <math>u_2=u_3</math> and <math>p_2=p_3</math>.

<math display="block">
u_2=\frac{2a_4}{\gamma_4-1}\left[1-\left(\frac{p_2}{p_4}\right)^{(\gamma_4-1)/(2\gamma_4)}\right]
</math>

Now, we have two ways of calculating <math>u_2</math>. Setting Eqn. \ref{eq:shocktube:up:a} equal to Eqn. \ref{eq:shocktube:up:d} leads to the shock tube relation

<math display="block">
\frac{p_4}{p_1}=\frac{p_2}{p_1}\left\{ 1 -\frac{(\gamma_4-1)(a_1/a_4)(p_2/p_1-1)}{\sqrt{2\gamma_1\left[2\gamma_1+(\gamma_1+1)(p_2/p_1-1)\right]}}\right\}^{-2\gamma_4/(\gamma_4-1)}
</math>