Moving shock waves: Difference between revisions
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Created page with "Category:Compressible flow Category:Unsteady waves Category:Inviscid flow __TOC__ \section{Moving Normal Shock Waves} \noindent The starting point is the governing equations for stationary normal shocks (repeated here for convenience). \begin{equation} \rho_1 u_1 = \rho_2 u_2 \label{eq:stationary:cont} \end{equation} \begin{equation} \rho_1 u_1^2+p_1 = \rho_2 u_2^2 + p_2 \label{eq:stationary:mom} \end{equation} \begin{equation} h_1 + \frac{1}{2}u_1^2 =..." |
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\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*}\\ | \end{equation*}\\ | ||
\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. | |||
\vspace{1cm} | |||
\begin{figure}[ht!] | |||
\begin{center} | |||
\includegraphics[]{figures/standalone-figures/Chapter07/pdf/shock-reflection.pdf} | |||
\caption{shock reflection at solid wall (located at $x$=4.0)} | |||
\label{fig:reflection} | |||
\end{center} | |||
\end{figure} | |||
\subsection{The Incident Shock Wave} | |||
\noindent The pressure ratio over the incident shock in Fig.~\ref{fig:reflection} can be obtained as \\ | |||
\begin{equation} | |||
\frac{p_2}{p_1}=1+\frac{2\gamma}{\gamma+1}\left(M_s^2-1\right) | |||
\label{eq:incident:pr} | |||
\end{equation}\\ | |||
\noindent where $M_s$ is the wave Mach number, which is calculated as \\ | |||
\begin{equation} | |||
M_s=\frac{W}{a_1} | |||
\label{eq:incident:Mach:def} | |||
\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}). | |||
\noindent Solving Eqn.~\ref{eq:incident:pr} for $M_s$, we get \\ | |||
\begin{equation} | |||
M_s=\sqrt{\frac{\gamma+1}{2\gamma}\left(\frac{p_2}{p_1}-1\right)+1} | |||
\label{eq:incident:Mach} | |||
\end{equation}\\ | |||
\noindent Anderson derives the relations for calculation of the ratio $T_2/T_1$ \\ | |||
\begin{equation} | |||
\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} | |||
\end{equation}\\ | |||
\noindent From Eqn.~\ref{eq:incident:tr} it is easy to get the corresponding relation for $\rho_2/\rho_1$\\ | |||
\begin{equation} | |||
\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} | |||
\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}).\\ | |||
\begin{equation} | |||
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} | |||
\end{equation}\\ | |||
\subsection{The Reflected Shock Wave} | |||
\noindent The pressure ratio over the reflected shock can be obtained from Eqn.~\ref{eq:incident:pr} by analogy\\ | |||
\begin{equation} | |||
\frac{p_5}{p_2}=1+\frac{2\gamma}{\gamma+1}\left(M_r^2-1\right) | |||
\label{eq:reflected:pr} | |||
\end{equation}\\ | |||
\noindent where $M_r$ is the Mach number of the reflected shock wave defined as\\ | |||
\begin{equation} | |||
M_r=\frac{W_r+u_p}{a_2} | |||
\label{eq:reflected:Mach:def} | |||
\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}).\\ | |||
\noindent Solving Eqn.~\ref{eq:reflected:pr} for $M_r$ gives\\ | |||
\begin{equation} | |||
M_r=\sqrt{\frac{\gamma+1}{2\gamma}\left(\frac{p_5}{p_2}-1\right)+1} | |||
\label{eq:reflected:Mach} | |||
\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\\ | |||
\begin{equation} | |||
\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} | |||
\end{equation}\\ | |||
\begin{equation} | |||
\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} | |||
\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\\ | |||
\begin{equation} | |||
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} | |||
\end{equation}\\ | |||
\subsection{Reflected Shock Relation} | |||
\noindent With the relations for the incident shock wave and reflected shock wave defined, we now have the tools to derive a relation between the incident and reflected shock waves. The induced flow velocity $u_p$ calculated using the relation obtained for the incident shock wave must of course be the same as when calculated using reflected wave properties, {\emph{i.e.}} the result of Eqn.~\ref{eq:incident:up} is identical to that of Eqn.~\ref{eq:reflected:up}\\ | |||
\begin{equation} | |||
M_r a_2 \left(1-\frac{\rho_2}{\rho_5}\right)=M_s a_1 \left(1-\frac{\rho_1}{\rho_2}\right) | |||
\label{eq:relation:a} | |||
\end{equation}\\ | |||
\noindent rewriting gives \\ | |||
\begin{equation} | |||
M_r \left(1-\frac{\rho_2}{\rho_5}\right)=M_s \left(1-\frac{\rho_1}{\rho_2}\right) \frac{a_1}{a_2} | |||
\label{eq:relation:b} | |||
\end{equation}\\ | |||
\noindent Assuming calorically perfect gas gives $a=\sqrt{\gamma RT}$ and thus\\ | |||
\begin{equation} | |||
M_r \left(1-\frac{\rho_2}{\rho_5}\right)=M_s \left(1-\frac{\rho_1}{\rho_2}\right) \sqrt{\frac{T_1}{T_2}} | |||
\label{eq:relation:c} | |||
\end{equation}\\ | |||
\noindent Let's first look at the term on the left hand side of Eqn.~\ref{eq:relation:c}\\ | |||
\begin{equation*} | |||
M_r \left(1-\frac{\rho_2}{\rho_5}\right) | |||
\end{equation*}\\ | |||
\noindent Using the $\rho_5/\rho_2$ and $p_2/p_5$ from Eqns.~\ref{eq:reflected:rr} and~\ref{eq:reflected:pr} and simplifying gives\\ | |||
\begin{equation} | |||
M_r \left(1-\frac{\rho_2}{\rho_5}\right)=\left(\frac{2}{\gamma+1}\right)\left(\frac{M_r^2-1}{M_r}\right) | |||
\label{eq:relation:d} | |||
\end{equation}\\ | |||
\noindent Using the same approach on the corresponding term for the incident shock wave on the right hand side of Eqn.~\ref{eq:relation:c} gives\\ | |||
\begin{equation} | |||
M_s \left(1-\frac{\rho_1}{\rho_2}\right)=\left(\frac{2}{\gamma+1}\right)\left(\frac{M_s^2-1}{M_s}\right) | |||
\label{eq:relation:e} | |||
\end{equation}\\ | |||
\noindent Now, inserting~\ref{eq:relation:d} and~\ref{eq:relation:e} in Eqn.~\ref{eq:relation:c} gives\\ | |||
\begin{equation} | |||
\left(\frac{2}{\gamma+1}\right)\left(\frac{M_r^2-1}{M_r}\right)=\left(\frac{2}{\gamma+1}\right)\left(\frac{M_s^2-1}{M_s}\right)\sqrt{\frac{T_1}{T_2}} | |||
\label{eq:relation:f} | |||
\end{equation}\\ | |||
\noindent Simplifying and inverting gives\\ | |||
\begin{equation} | |||
\left(\frac{M_r}{M_r^2-1}\right)=\left(\frac{M_s}{M_s^2-1}\right)\sqrt{\frac{T_2}{T_1}} | |||
\label{eq:relation:g} | |||
\end{equation}\\ | |||
\noindent The rightmost term in Eqn.~\ref{eq:relation:g} ($\sqrt{T_2/T_1}$) needs to be rewritten. Inserting~\ref{eq:incident:pr} in~\ref{eq:incident:tr} and expanding all terms gives\\ | |||
\begin{align*} | |||
\frac{T_2}{T_1} & =\frac{2(\gamma+1) + (\gamma+1)(\gamma-1)M_s^2+4\gamma(M_s^2-1)+2\gamma(\gamma-1)M_s^2(M_s^2-1)}{(\gamma+1)^2M_s^2} = \\ | |||
& \\ | |||
& =\frac{2(\gamma+1) + (\gamma+1)(\gamma-1)M_s^2+4\gamma(M_s^2-1)}{(\gamma+1)^2M_s^2}+\frac{2(\gamma-1)}{(\gamma+1)^2}(M_s^2-1)\gamma = \\ | |||
& \\ | |||
& =\dfrac{2(\gamma+1) + (\gamma+1)(\gamma-1)M_s^2+4\gamma(M_s^2-1)-(2(\gamma-1)(M_s^2-1))}{(\gamma+1)^2M_s^2}+\nonumber\\ | |||
& \dfrac{2(\gamma-1)}{(\gamma+1)^2}(M_s^2-1)\left(\gamma+\dfrac{1}{M_s^2}\right) | |||
\end{align*}\\ | |||
\noindent Finally we end up with the following relation\\ | |||
\begin{equation} | |||
\frac{T_2}{T_1}=1+\frac{2(\gamma-1)}{(\gamma+1)^2}(M_s^2-1)\left(\gamma+\frac{1}{M_s^2}\right) | |||
\label{eq:relation:tr} | |||
\end{equation}\\ | |||
\noindent The temperature ratio over the incident shock wave is now totally defined by the incident Mach number $M_s$ and the ratio of specific heats $\gamma$. With~\ref{eq:relation:tr} in~\ref{eq:relation:g} we get the sought relation between the reflected and incident Mach numbers.\\ | |||
\begin{equation} | |||
\left(\frac{M_r}{M_r^2-1}\right)=\left(\frac{M_s}{M_s^2-1}\right)\sqrt{1+\frac{2(\gamma-1)}{(\gamma+1)^2}(M_s^2-1)\left(\gamma+\frac{1}{M_s^2}\right)} | |||
\label{eq:relation:final} | |||
\end{equation}\\ | |||
\noindent It should be noted that Eqn.~\ref{eq:relation:final} is valid for calorically perfect gases only. | |||
