Normal Shock Relations

Rewriting the continuity equation (Eqn. (4.1))

\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

(4.79)

Eqn. (4.70) in Eqn. (4.79) gives

\frac{\rho_2}{\rho_1}=\frac{(\gamma+1)M_1^2}{2+(\gamma-1)M_1^2}

(4.80)

To get a corresponding relation for the pressure ratio over the shock, we go back to the momentum equation (Eqn. (4.2))

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)

(4.81)

\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)

(4.82)

\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)

(4.83)

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

(4.84)

Figure Figure 4.8 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.

pressure ratio over a normal shock — Figure 4.8:Pressure ratio over a normal shock ( $p_2/p_1$ ) as function of upstream Mach number ( $M_1$ )

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 $p=\rho RT$

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

(4.85)

\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]

(4.86)

Figure Figure 4.9 below shows how different flow properties change over a normal shock as a function of upstream Mach number.

changes of flow properties over a normal shock — Figure 4.9:Changes of flow properties over a normal shock as a function of the upstream Mach number ( $M_1$ )

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.

4.5 Normal Shock Relations