Nian at 08:01, 1 April 2026

2026-04-01T08:01:35Z

Nian at 19:39, 20 March 2026

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← Older revision		Revision as of 19:39, 20 March 2026
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	[[Category:Compressible flow]]		[[Category:Compressible flow]]
	~~[[Category:Topic]]~~
	[[Category:One-dimensional flow]]		[[Category:One-dimensional flow]]
	[[Category:Inviscid flow]]		[[Category:Inviscid flow]]
	[[Category:Wave solution]]		[[Category:Wave solution]]
			<noinclude>
			[[Category:Compressible flow:Topic]]
			</noinclude>

	__TOC__		__TOC__

Nian at 17:12, 20 March 2026

2026-03-20T17:12:05Z

← Older revision		Revision as of 17:12, 20 March 2026
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	[[Category:Compressible flow]]		[[Category:Compressible flow]]
			[[Category:Topic]]
	[[Category:One-dimensional flow]]		[[Category:One-dimensional flow]]
	[[Category:Inviscid flow]]		[[Category:Inviscid flow]]

Nian: /* Normal Shock Relations */

2026-03-17T13:38:12Z

Normal Shock Relations

← Older revision		Revision as of 13:38, 17 March 2026
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	__TOC__		__TOC__

	~~==Normal Shock Relations==~~

	Rewriting the continuity equation (Eqn. \ref{eq:governing:cont})		Rewriting the continuity equation (Eqn. \ref{eq:governing:cont})

Nian: /* The Hugoniot equation */

2026-03-17T13:23:50Z

The Hugoniot equation

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Nian: /* The Hugoniot Equation */

2026-03-16T10:15:14Z

The Hugoniot Equation

← Older revision		Revision as of 10:15, 16 March 2026
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	Now, one question remains. How come that we by analyzing the control volume using the upstream and downstream states get the normal shock relations. There is no way that the governing equations could have known about the fact that we assumed that there would be a shock inside of the control volume, or is it? The answer is that we have assumed that there will be a change in flow properties from upstream to downstream. We have further assumed that the flow is adiabatic (we are using the adiabatic energy equation) so there is no heat exchange. We are, however, allowing for irreversibilities in the flow. The only way to accomplish a change in flow properties under those constraints is a formation of a normal shock (a discontinuity in flow properties - a sudden flow compression) between station 1 and station 2.		Now, one question remains. How come that we by analyzing the control volume using the upstream and downstream states get the normal shock relations. There is no way that the governing equations could have known about the fact that we assumed that there would be a shock inside of the control volume, or is it? The answer is that we have assumed that there will be a change in flow properties from upstream to downstream. We have further assumed that the flow is adiabatic (we are using the adiabatic energy equation) so there is no heat exchange. We are, however, allowing for irreversibilities in the flow. The only way to accomplish a change in flow properties under those constraints is a formation of a normal shock (a discontinuity in flow properties - a sudden flow compression) between station 1 and station 2.

	==The Hugoniot ~~Equation~~==		===The Hugoniot equation===

	\noindent The Hugoniot equation is an alternative normal shock relation based on thermodynamic quantities only. It is derived from the governing equations and relates the change in energy to the change in pressure and specific volume. The starting point of the derivation of the Hugoniot equation is the governing equations (Eqns~\ref{eq:governing:cont} - \ref{eq:governing:energy}).		\noindent The Hugoniot equation is an alternative normal shock relation based on thermodynamic quantities only. It is derived from the governing equations and relates the change in energy to the change in pressure and specific volume. The starting point of the derivation of the Hugoniot equation is the governing equations (Eqns~\ref{eq:governing:cont} - \ref{eq:governing:energy}).

Nian at 10:11, 16 March 2026

2026-03-16T10:11:07Z

← Older revision		Revision as of 10:11, 16 March 2026
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			[[Category:Compressible flow]]
			[[Category:One-dimensional flow]]
			[[Category:Inviscid flow]]
			[[Category:Wave solution]]

			__TOC__

	==Normal Shock Relations==		==Normal Shock Relations==

Nian: Created page with " ==Normal Shock Relations== Rewriting the continuity equation (Eqn. \ref{eq:governing:cont}) $\frac{\rho_2}{\rho_1}=\frac{u_1}{u_2}=\frac{u_1^2}{u_1 u_2}=\left\{{a^}^2=u_1u_2\right\}=\frac{u_1^2}{{a^}^2}={M^*_1}^2$ Eqn. \ref{eq:MachStar} in Eqn. \ref{eq:Normal:density:a} gives $\frac{\rho_2}{\rho_1}=\frac{(\gamma+1)M_1^2}{2+(\gamma-1)M_1^2}$ To get a corresponding relation for the pressure ratio over the s..."

2026-03-16T10:10:25Z

Created page with " ==Normal Shock Relations== Rewriting the continuity equation (Eqn. \ref{eq:governing:cont}) <math display="block"> \frac{\rho_2}{\rho_1}=\frac{u_1}{u_2}=\frac{u_1^2}{u_1 u_2}=\left\{{a^*}^2=u_1u_2\right\}=\frac{u_1^2}{{a^*}^2}={M^*_1}^2 </math> Eqn. \ref{eq:MachStar} in Eqn. \ref{eq:Normal:density:a} gives <math display="block"> \frac{\rho_2}{\rho_1}=\frac{(\gamma+1)M_1^2}{2+(\gamma-1)M_1^2} </math> To get a corresponding relation for the pressure ratio over the s..."

New page

==Normal Shock Relations==

Rewriting the continuity equation (Eqn. \ref{eq:governing:cont})

<math display="block">
\frac{\rho_2}{\rho_1}=\frac{u_1}{u_2}=\frac{u_1^2}{u_1 u_2}=\left\{{a^*}^2=u_1u_2\right\}=\frac{u_1^2}{{a^*}^2}={M^*_1}^2
</math>

Eqn. \ref{eq:MachStar} in Eqn. \ref{eq:Normal:density:a} gives

<math display="block">
\frac{\rho_2}{\rho_1}=\frac{(\gamma+1)M_1^2}{2+(\gamma-1)M_1^2}
</math>

To get a corresponding relation for the pressure ratio over the shock, we go back to the momentum equation (Eqn. \ref{eq:governing:mom})

<math display="block">
p_2-p_1=\rho_1 u^2_1 - \rho_2 u^2_2=\left\{\rho_1 u_1=\rho_2 u_1\right\}=\rho_1 u_1(u_1-u_2)=\rho_1 u^2_1\left(1-\frac{u_2}{u_1}\right)
</math>

<math display="block">
\frac{p_2-p_1}{p_1}=\frac{\rho_1 u^2_1}{p_1}\left(1-\frac{u_2}{u_1}\right)=\left\{a_1=\sqrt{\frac{\gamma p_1}{\rho_1}}\right\}=\gamma\frac{u^2_1}{a^2_1}\left(1-\frac{u_2}{u_1}\right)=\gamma M^2_1\left(1-\frac{u_2}{u_1}\right)
</math>

<math display="block">
\frac{p_2}{p_1}-1=\gamma M^2_1\left(1-\frac{u_2}{u_1}\right)=\left\{\frac{u_2}{u_1}=\frac{\rho_1}{\rho_2}\right\}=\gamma M^2_1\left(1-\frac{2+(\gamma-1)M_1^2}{(\gamma+1)M_1^2}\right)
</math>

<math display="block">
\frac{p_2}{p_1}=1+\frac{2\gamma}{\gamma+1}(M^2_1-1)
</math>

Figure~\ref{fig:shock:pressure:ratio} shows that the pressure must increase over the shock due to the fact that, based on the discussion above, the upstream Mach number must be greater than one and thus the shock is a discontinuous compression process.



The temperature ratio over the shock can be obtained using the already derived relations for pressure ratio and density ratio together with the equation of state <math>p=\rho RT</math>

<math display="block">
\frac{T_2}{T_1}=\left(\frac{p_2}{p_1}\right)\left(\frac{\rho_1}{\rho_2}\right)
</math>

<math display="block">
\frac{T_2}{T_1}=\left[1+\frac{2\gamma}{\gamma+1}(M^2_1-1)\right]\left[\frac{(\gamma+1)M_1^2}{2+(\gamma-1)M_1^2}\right]
</math>

Figure~\ref{fig:normal:shock:relations} below shows how different flow properties change over a normal shock as a function of upstream Mach number.



Now, one question remains. How come that we by analyzing the control volume using the upstream and downstream states get the normal shock relations. There is no way that the governing equations could have known about the fact that we assumed that there would be a shock inside of the control volume, or is it? The answer is that we have assumed that there will be a change in flow properties from upstream to downstream. We have further assumed that the flow is adiabatic (we are using the adiabatic energy equation) so there is no heat exchange. We are, however, allowing for irreversibilities in the flow. The only way to accomplish a change in flow properties under those constraints is a formation of a normal shock (a discontinuity in flow properties - a sudden flow compression) between station 1 and station 2.

==The Hugoniot Equation==

\noindent The Hugoniot equation is an alternative normal shock relation based on thermodynamic quantities only. It is derived from the governing equations and relates the change in energy to the change in pressure and specific volume. The starting point of the derivation of the Hugoniot equation is the governing equations (Eqns~\ref{eq:governing:cont} - \ref{eq:governing:energy}).

\noindent The continuity equation is rewritten and inserted into the momentum equation\\

\begin{equation}
u_1=\left(\frac{\rho_2}{\rho_1}\right) u_2
\label{eq:governing:cont:b}
\end{equation}\\

\noindent Replace $u_1$ in Eqn. \ref{eq:governing:mom} using Eqn. \ref{eq:governing:cont:b}

\[\rho_1 \left(\frac{\rho_2}{\rho_1}\right)^2 u^2_2 +p_1=\rho_2 u^2_2 + p_2\]\\

\[u^2_2\left(\rho_1\left(\frac{\rho_2}{\rho_1}\right)^2-\rho_2\right)=\left(p_2-p_1\right)\]\\

\[u^2_2\left(\left(\frac{\rho_2}{\rho_1}\right)\left(\rho_2-\rho_1\right)\right)=\left(p_2-p_1\right)\]\\

\begin{equation}
u^2_2=\left(\frac{\rho_1}{\rho_2}\right)\frac{p_2-p_1}{\rho_2-\rho_1}
\label{eq:governing:mom:b}
\end{equation}\\

\noindent Eqn. \ref{eq:governing:cont:b} and \ref{eq:governing:mom:b} gives\\

\begin{equation}
u^2_1=\left(\frac{\rho_2}{\rho_1}\right)\frac{p_2-p_1}{\rho_2-\rho_1}
\label{eq:governing:mom:c}
\end{equation}\\

\noindent Eqn. \ref{eq:governing:mom:b} and Eqn. \ref{eq:governing:mom:c} inserted in the energy equation (Eqn. \ref{eq:governing:energy}) gives\\

\begin{equation}
h_1 + \frac{1}{2}\left(\frac{\rho_2}{\rho_1}\right)\left(\frac{p_2-p_1}{\rho_2-\rho_1}\right)=h_2 + \frac{1}{2}\left(\frac{\rho_1}{\rho_2}\right)\left(\frac{p_2-p_1}{\rho_2-\rho_1}\right)
\label{eq:governing:energy:b}
\end{equation}\\

\[h_2-h_1=\frac{p_2-p_1}{2}\left[\left(\frac{\rho_2}{\rho_1}\right)\left(\frac{1}{\rho_2-\rho_1}\right)-\left(\frac{\rho_1}{\rho_2}\right)\left(\frac{1}{\rho_2-\rho_1}\right)\right]\]\\

\[h_2-h_1=\frac{p_2-p_1}{2}\left[\frac{\rho^2_2-\rho^2_1}{\rho_1\rho_2(\rho_2-\rho_1)}\right]=\frac{p_2-p_1}{2}\left[\frac{\rho_2+\rho_1}{\rho_1\rho_2}\right]\]\\

\begin{equation}
h_2-h_1=\frac{p_2-p_1}{2}\left(\frac{1}{\rho_1}+\frac{1}{\rho_2}\right)
\label{eq:governing:energy:c}
\end{equation}\\

\noindent Now, replacing the enthalpies with internal energies using $h=e+p/\rho$ gives\\

\[e_2-e_1=\frac{p_1}{\rho_1}-\frac{p_2}{\rho_2}+\frac{p_2-p_1}{2}\left(\frac{1}{\rho_1}+\frac{1}{\rho_2}\right)\]\\

\noindent which after some rewriting becomes the Hugoniot equation\\

\begin{equation}
e_2-e_1=\frac{p_2+p_1}{2}\left(\frac{1}{\rho_1}-\frac{1}{\rho_2}\right)=\dfrac{p_2+p_1}{2}(\nu_1-\nu_2)
\label{eq:governing:hogoniot}
\end{equation}\\

%\noindent The Hugoniot equation relates thermodynamic properties over the normal shock

\noindent To give an idea about how the normal shock relates to an isentropic compression (a flow compression process without losses) the change in flow density as a function of pressure ratio is compared in Figure~\ref{fig:normal:shock:compression:vs:isentropic}. One can see that the normal-shock compression is more effective but less efficient than the corresponding isentropic process.

\begin{figure}[ht!]
\begin{center}
\includegraphics[]{figures/standalone-figures/Chapter03/pdf/hugoniot-vs-isentropic.pdf}
\caption{Comparison of a normal shock compression process and an isentropic compression}
\label{fig:normal:shock:compression:vs:isentropic}
\end{center}
\end{figure}

\noindent Introducing C as the massflow per unit area (which is a constant)

\[\rho_1 u_1 = \rho_2 u_2 = C\]

\noindent Inserted into the momentum equation this gives

\[p_1+\dfrac{C^2}{\rho_1}=p_2+\dfrac{C^2}{\rho_2}\]

or

\[\dfrac{p_2-p_1}{\nu_2-\nu_1}=-C^2\]

\noindent which implies that all possible solutions to the governing equations must be located on a line in $p\nu$-space (the so-called Rayleigh line). If we add the Hugoniot relation to this we will find that there are two possible solutions, the upstream condition and the condition downstream of the normal shock and the flow cannot be in any of the intermediate stages. The normal-process is a so-called wave solution to the governing equations where the flow state must jump directly from one flow state to another without passing the intermediate conditions. If we add heat or friction to the problem we will instead get continuous solutions as we will see in the following sections. Figures \ref{fig:shock:pv} and \ref{fig:shock:Ts} shows a normal shock process in a $p\nu$- and $Ts$-diagram, respectively. Note that the flow passes the characteristic conditions as it is going through the shock, which means that the flow goes from supersonic to subsonic.

\begin{figure}[ht!]
\begin{center}
\includegraphics[]{figures/standalone-figures/Chapter03/pdf/shock-wave-pv.pdf}
\caption{The normal-shock process illustrated in a $p\nu$-diagram}
\label{fig:shock:pv}
\end{center}
\end{figure}

\begin{figure}[ht!]
\begin{center}
\includegraphics[]{figures/standalone-figures/Chapter03/pdf/shock-wave-Ts.pdf}
\caption{The normal-shock process illustrated in a $Ts$-diagram}
\label{fig:shock:Ts}
\end{center}
\end{figure}