Computational Fluid Dynamics - Basic (e-4)

Normal shock waves

In the 1-D flow model in nozzles so far, the flow in converging nozzle is discussed in which the shock waves cannot be appeared. But before starting the discussion about the converging-diverging nozzle and talking about supersonic flows, it is better to introduce the shock wave phenomenon.

Normal shock waves

As it is clear from the name, it is a shock wave normal to the flow direction. Assume a volume as shown in Fig. e.4.1 around a normal shock wave. The properties denoted by subscript 1 are the initial conditions and the ones with subscript 2 are the conditions behind the wave.

Fig. e.4.1. Normal shock wave.

We consider an ideal gas in adiabatic, steady-state; so we have:
\begin{equation} p = \rho RT \end{equation} and from the continuity of mass, energy and momentum equation:
\begin{equation} \begin{cases} \rho _1 v_1 A_1 = \rho _2 v_2 A_2\\ \\ A_1 = A_2 \end{cases} \Longrightarrow \rho _1 v_1 = \rho _2 v_2 \end{equation}
\begin{equation} p_1 + \rho _1 v_1^2 = p_2 + \rho _2 v_2^2 \end{equation}
\begin{equation} h_1 + \frac{v_1^2}{2} = h_2 + \frac{v_2^2}{2} \end{equation} where $ h = c_p T $. Furthermore, since shock is not isentropic, $ p_{0,1} \not= p_{0,2} $. However, because the shock is adiabatic, we have $ T_{0,1} \not= T_{0,2} $, where $ p_0 $ and $ T_0 $ are total pressure and total temperature respectively.

Using the compressible relations which are derived in one of the previous sub-sections, we can show that:
\begin{equation} \frac{a^2}{\gamma - 1} + \frac{v^2}{2} = \frac{a_0^2}{\gamma - 1} = \text{constant} \end{equation} in which $ a $ is the speed of sound and $ \gamma $ is isentropic expansion factor. For the sonic conditions in which $ M = v/a = 1 $, we can determine the properties at sonic conditions:
\begin{equation} a^* = \sqrt{\gamma RT^*} \end{equation} where $ a^* $ and $ T^* $ are characteristic speed of sound and characteristic temperature respectively. Furthremore, we can define the characteristic Mach number as:
\begin{equation} M^* = \frac{u}{a^*} \end{equation} For the normal shock wave relation, we can use Prandtl relation, stating that:
\begin{equation} a^{*2} = v_1 v_2 \Rightarrow 1 = M_1^* M_2^* \end{equation} From these relations, it can be seen that $ M_2 = f(M_1) $:
\begin{equation} M_2^2 = \frac{2 + (\gamma -1) M_1^2}{2 \gamma M_1^2 - (\gamma - 1)} \Longrightarrow \begin{cases} M_1 > 1 \rightarrow M_2 < 1\\ M_1 = 1 \rightarrow M_2 = 1\\ M_1 < 1 \rightarrow M_2 > 1 \end{cases} \end{equation} According to the second law of thermodynamics, entropy has to increase for an adiabatic transformation. Therefore, the static pressure has to increase. So the statement $ M_1 < 1 \rightarrow M_2 > 1 $ will not be possible since in that case, the entropy would decrease. As a results, we can conclude that shock waves appears only in supersonic flows. After finding $ M_2 $, all other properties can be determined, using the following jump relations:
\begin{equation} \frac{\rho _2}{\rho _1} = \frac{v_1}{v_2} = \frac{(\gamma + 1) M_1^2}{2 + (\gamma - 1) M_1^2} \end{equation}
\begin{equation} \frac{p_2}{p_1} = 1 + \frac{2 \gamma}{\gamma + 1}(M_1^2 - 1) \end{equation}
\begin{equation} \frac{T_2}{T_1} = \left(1 + \frac{2 \gamma}{\gamma + 1}(M_1^2 - 1)\right) \left(\frac{2 + (\gamma - 1) M_1^2}{(\gamma + 1) M_1^2}\right) \end{equation} Using the provided Matlab code, it is possible to calculate all these properties.

The relevant Matlab codes can be downloaded using the following link:
Matlab Codes Bank

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