Normal-shock relations: Difference between revisions

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 shock, we go back to the momentum equation (Eqn. \ref{eq:governing:mom})

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)

\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)

\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)

\frac{p_2}{p_1}=1+\frac{2\gamma}{\gamma+1}(M^2_1-1)

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>

\frac{T_2}{T_1}=\left(\frac{p_2}{p_1}\right)\left(\frac{\rho_1}{\rho_2}\right)

\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]

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

The continuity equation is rewritten and inserted into the momentum equation

u_1=\left(\frac{\rho_2}{\rho_1}\right) u_2

Replace <math>u_1</math> 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)

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

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

u^2_1=\left(\frac{\rho_2}{\rho_1}\right)\frac{p_2-p_1}{\rho_2-\rho_1}

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_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)

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]

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

Now, replacing the enthalpies with internal energies using <math>h=e+p/\rho</math> 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)

which after some rewriting becomes the Hugoniot 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)

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.

Introducing <math>C</math> as the massflow per unit area (which is a constant)

\rho_1 u_1 = \rho_2 u_2 = C

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

which implies that all possible solutions to the governing equations must be located on a line in <math>p\nu</math>-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 <math>p\nu</math>- and <math>Ts</math>-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.