Normal-shock relations: Difference between revisions

\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\\

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

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

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

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

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

\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\\

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

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

\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\\

 

\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.

\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

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

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

\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.