19.3. Normal shocks

Jump conditions. A normal shock is a discontinuity in the physical quantities of the flow, with non-zero mass flux \(\dot{m} \ne 0\) across it and zero tangential component of the velocity. Jump conditions across a normal shock are

\[\begin{split}\begin{aligned} 0 & = [ \dot{m} ] = [ \rho u_{n}^{rel} ] \\ \mathbf{u}_\mathbf{t} & = \mathbf{0} \\ 0 & = \dot{m} [ u_n ] + [ \, p \, ] = \dot{m} [ u_n^{rel} ] + [ \, p \, ] \\ 0 & = \dot{m} [ h^{t,rel} ] = \dot{m} [ h^{t,rel}_n ] \end{aligned}\end{split}\]

19.3.1. Perfect ideal gas

For perfect ideal gas, the ratio between quantities before and after normal shocks can be evaluated analitically.

Notation. Here \(u_n^{rel} = u\) for brevity.

Jump conditions.

\[\begin{split}\begin{cases} \rho_1 u_1 = \rho_2 u_2 \\ \rho_1 u_1^2 + p_1 = \rho_2 u_2^2 + p_2 \\ \frac{u_1^2}{2} + h_1 = \frac{u_2^2}{2} + h_2 \\ \end{cases}\end{split}\]

19.3.1.1. Quantities as a function of Mach numbers \(M_1\), \(M_2\)

Property	…	…
Pressure	\(\frac{p_2}{p_1} = \frac{1 + \gamma M_1^2}{1 + \gamma M_2^2}\)	…
Density	\(\frac{\rho_2}{\rho_1} = \frac{M_1^2}{M_2^2} \frac{1 + \gamma M_2^2}{1+ \gamma M_1^2}\) …	…
Temperature	\(\frac{T_2}{T_1} = \left( \frac{1 + \gamma M_1^2}{1 + \gamma M_2^2} \right)^2 \frac{M_2^2}{M_1^2}\)	…
Velocity	\(\frac{u_2}{u_1} = \frac{M_2^2}{M_1^2} \frac{1 + \gamma M_1^2}{1 + \gamma M_2^2}\)	…
Speed of sound	\(\frac{a_2}{a_1} = \sqrt{\frac{T_2}{T_1}}\)	…
Entropy	\(s_2 - s_1 = \frac{R}{\gamma-1} \ln \left\{ \left( \frac{1+ \gamma M_1^2}{1 + \gamma M_2^2} \right)^{1+\gamma} \left( \frac{M_2^2}{M_1^2} \right)^{\gamma} \right\}\)	…
Stagnation pressure	\(\frac{p_{02}}{p_{01}} =\) …	…
Other stag quantities	…	…

Details

Pressure. From momentum equation

\[\begin{split}\begin{aligned} p_2 & = p_1 + \rho_1 u_1^2 - \rho_2 u_2^2 \\ \end{aligned}\end{split}\]

\[\begin{split}\begin{aligned} \frac{p_2}{p_1} & = 1 + \frac{\rho_1}{p_1} u_1^2 - \frac{p_2}{p_1} \frac{\rho_2}{p_2} u_2^2 = && \left( a^2 = \gamma \frac{p}{\rho} \right) \\ & = 1 + \gamma M_1^2 - \frac{p_2}{p_1} \gamma M_2^2 \ , \end{aligned}\end{split}\]

and thus

\[\frac{p_2}{p_1} = \frac{1 + \gamma M_1^2}{1 + \gamma M_2^2} \ .\]

Velocity. Starting from mass jump condition

\[\begin{split}\begin{aligned} \frac{u_2}{u_1} & = \frac{\rho_1}{\rho_2} = \\ & = \frac{\rho_1}{p_1} \frac{p_2}{\rho_2} \frac{p_1}{p_2} = \\ & = \frac{a_2^2}{a_1^2}\frac{p_1}{p_2} = \\ & = \frac{u_2^2}{u_1^2} \frac{M_1^2}{M_2^2} \frac{p_1}{p_2} \end{aligned}\end{split}\]

and thus

\[\frac{u_2}{u_1} = \frac{M_2^2}{M_1^2} \frac{p_2}{p_1} = \frac{M_2^2}{M_1^2} \frac{1 + \gamma M_1^2}{1 + \gamma M_2^2} \ .\]

Density. Using mass jump condition, and the relation for \(\frac{u_2}{u_1}\),

\[\frac{\rho_2}{\rho_1} = \frac{u_1}{u_2} = \frac{M_1^2}{M_2^2} \frac{1 + \gamma M_2^2}{1+ \gamma M_1^2} \ .\]

Speed of sound.

\[\frac{a_2}{a_1} = \frac{u_2}{M_2} \frac{M_1}{u_1} = \frac{M_2}{M_1} \frac{1 + \gamma M_1^2}{1 + \gamma M_2^2} \ .\]

Temperature.

\[\begin{split}\begin{aligned} \frac{T_2}{T_1} & = \frac{p_2}{R \rho_2} \frac{R \rho_1}{p_1} = \\ & = \frac{1 + \gamma M_1^2}{1 + \gamma M_2^2} \cdot \frac{M_2^2}{M_1^2} \frac{1 + \gamma M_1^2}{1+ \gamma M_2^2} = \\ & = \left( \frac{1 + \gamma M_1^2}{1 + \gamma M_2^2} \right)^2 \frac{M_2^2}{M_1^2} \ . \end{aligned}\end{split}\]

Entropy.

\[\begin{split}\begin{aligned} s_2 - s_1 & = c_v \ln \left( \frac{p_2}{p_1} \right) - c_p \ln \left( \frac{\rho_2}{\rho_1} \right) = \\ & = \frac{R}{\gamma-1} \left\{ \ln \left( \frac{p_2}{p_1} \right) - \gamma \ln \left( \frac{\rho_2}{\rho_1} \right) \right\} = \\ & = \frac{R}{\gamma-1} \ln \left\{ \left( \frac{p_2}{p_1} \right) \left( \frac{\rho_2}{\rho_1} \right)^{-\gamma} \right\} = \\ & = \frac{R}{\gamma-1} \ln \left\{ \left( \frac{1+ \gamma M_1^2}{1 + \gamma M_2^2} \left[ \frac{M_1^2}{M_2^2} \frac{1 + \gamma M_2^2}{1+ \gamma M_1^2} \right]^{-\gamma} \right) \right\} = \\ & = \frac{R}{\gamma-1} \ln \left\{ \left( \frac{1+ \gamma M_1^2}{1 + \gamma M_2^2} \right)^{1+\gamma} \left( \frac{M_2^2}{M_1^2} \right)^{\gamma} \right\} \end{aligned}\end{split}\]

Entropy determines a condition for feasible shocks, i.e. increasing entropy, \(s_2 - s_1 \ge 0\).

19.3.1.2. Relation between Mach numbers \(M_1\), \(M_2\)

\[M_2^2 = \frac{M_1^2 + \frac{2}{\gamma-1}}{\frac{2\gamma}{\gamma-1}M_1^2 - 1}\]

Details

todo isn’t there any cleaner way to derive the desired relation?

Comparing the expression of the temperature ratio obtained above using equation of state and momentum equation, with the following expression obtained using energy equation

\[\begin{split}\begin{aligned} \frac{u_1^2}{2} + h_1 & = \frac{u_2^2}{2} + h_2 \\ \frac{u_1^2}{2} + c_p T_1 & = \frac{u_2^2}{2} + c_p T_2 \\ \frac{a_1^2 M_1^2}{2} + \frac{\gamma}{\gamma-1} R T_1 & = \frac{a_1^2 M_1}{2} + \frac{\gamma}{\gamma-1} R T_2 && \left( a^2 = \gamma R T \right) \\ T_1 \left( \frac{M_1^2}{2} + \frac{1}{\gamma-1} \right) & = T_2 \left( \frac{M_2^2}{2} + \frac{1}{\gamma-1} \right) \\ \quad \rightarrow \quad \frac{T_2}{T_1} & = \frac{\frac{M_1^2}{2} + \frac{1}{\gamma-1}}{\frac{M_2^2}{2} + \frac{1}{\gamma-1}} \end{aligned}\end{split}\]

it follows

\[\left( \frac{1 + \gamma M_1^2}{1 + \gamma M_2^2} \right)^2 \frac{M_2^2}{M_1^2} = \frac{\frac{M_1^2}{2} + \frac{1}{\gamma-1}}{\frac{M_2^2}{2} + \frac{1}{\gamma-1}}\]

…todo do all the required algebra, to solve the equation. Be smart and gather binomial \(M_1^2 - M_2^2\) and simplify

Remarks.

• \(M_2(M_1 = 1) = 1\)

• the derivative reads

  \[\begin{split}\begin{aligned} \frac{d M_2^2}{d M_1^2} & = \frac{\frac{2 \gamma}{\gamma-1}M_1^2 - 1 - \frac{2 \gamma}{\gamma-1} \left( M_1^2 + \frac{2}{\gamma-1} \right)}{\left( \dots \right)^2} = \\ & = \frac{- 1 - \frac{2}{\gamma-1} }{\left( \dots \right)^2} = \\ & = - \frac{\gamma+1}{\gamma-1} \left( 2 \gamma M_1^2 - (\gamma - 1) \right)^{-2} < 0 \ , \end{aligned}\end{split}\]

  and thus if \(M_1^2 > 1\), then \(M_2^2 < 1\), and viceversa. Entropy condition \(s_2 - s_1 \ge 0\) defines the feasible shocks: for a PIG, feasible shocks have \(M_1 > 1\) and \(M_2 < 1\), i.e. supersonic flow becomes subsonic after a normal shock.

• the function is singular for \(M_1^2 = \frac{\gamma-1}{2\gamma}\). This condition only occurs for \(M_1 < 1\), i.e. it never happens for physical shocks.

Entropy condition - PIG

\[\begin{split}\begin{aligned} \frac{\gamma-1}{R} (s_2 - s_1) & = \ln \left\{ \left( \frac{1+ \gamma M_1^2}{1 + \gamma M_2^2} \right)^{1+\gamma} \left( \frac{M_2^2}{M_1^2} \right)^{\gamma} \right\} = \\ & = \ln \left\{ \left(\frac{ 2 \gamma M_1^2 - \gamma + 1}{\gamma + 1} \right)^{\gamma+1} \left( \frac{2 + ( \gamma - 1 ) M_1^2}{2 \gamma M_1^2 - \gamma + 1} \right)^{\gamma} M_1^{-2 \gamma} \right\} = \\ & = \ln \left\{ \frac{ 2 \gamma M_1^2 - \gamma + 1}{(\gamma + 1)^{\gamma+1}} \frac{(2 + ( \gamma - 1 ) M_1^2)^{\gamma}}{M_1^{2 \gamma}} \right\} \ , \end{aligned}\end{split}\]

Remarks.

• For \(M_1 = 1\), \(x(M_1) = \{ \dots \}\)

  \[\{ \dots \}(M_1=1) = \frac{(\gamma+1) (\gamma+1)^{\gamma+1}}{(\gamma+1)^{\gamma+1}} = 1 \ ,\]

  and thus

  \[s_2 - s_1 \propto \ln \{ \dots \} = 0 \ .\]

• the derivative w.r.t. \(M_1^2\) reads

  \[\begin{split}\begin{aligned} & \frac{(\gamma+1)^{\gamma+1}}{2 \gamma M_1^2 - \gamma + 1} \frac{2 \gamma}{(\gamma+1)^{\gamma+1}} + \frac{M_1^{2 \gamma}}{(2 + (\gamma-1)M_1^2)^ \gamma } \left( \gamma (2 +(\gamma-1)M_1^2)^{\gamma-1} (\gamma-1) M_1^{-2\gamma} - (\dots) \gamma M_1^{2(\gamma-1)} \right) = \\ & = \frac{2 \gamma}{2 \gamma M_1^2 - \gamma + 1} + \frac{\gamma (\gamma-1)}{2 + (\gamma-1) M_1^2} - \frac{\gamma}{M_1^2} = \\ & = \dots \text{(see below)} = \\ & = \frac{2 \gamma (\gamma-1) (M_1^2 - 1)^2}{(2 \gamma M_1^2 - \gamma + 1)(2 + (\gamma-1) M_1^2) M_1^2} \ . \end{aligned}\end{split}\]

  The derivative is always non-negative and thus the entropy condition \(s_2 - s_1 \ge 0\) is satisfied only for \(M_1 \ge 1\), i.e. for supersonic inflow becoming subsonic (see above) after a normal shock.

\[\begin{split}\begin{aligned} & 4 \gamma M_1^2 + 2 \gamma (\gamma - 1) M_1^4 + \\ & \quad + \underbrace{2 \gamma^2 (\gamma-1) M_1^4}_{1} - \underbrace{\gamma (\gamma-1)^2 M_1^2}_{0} + \\ & \quad - 4 \gamma^2 M_1^2 - \underbrace{2 \gamma^2 ( \gamma - 1 ) M_1^4}_{1} + 2 \gamma ( \gamma - 1 ) + \underbrace{\gamma ( \gamma - 1 )^2 M_1^2}_{0} = \\ & = 2 \gamma ( \gamma - 1 ) - M_1^2 4 \gamma ( \gamma - 1 ) + 2 \gamma ( \gamma - 1 ) M_1^4 = \\ & = 2 \gamma ( \gamma - 1 ) ( M_1^2 - 1 )^2 \ . \end{aligned}\end{split}\]

19.3.1.3. Quantities as a function of Mach numbers \(M_1\)

Property
Entropy	\(s_2 - s_1 = \frac{R}{\gamma - 1} \ln \left\{ \frac{ 2 \gamma M_1^2 - \gamma + 1}{(\gamma + 1)^{\gamma+1}} \frac{(2 + ( \gamma - 1 ) M_1^2)^{\gamma}}{M_1^{2 \gamma}} \right\}\)	\( \ge 0 \)
Inflow Mach number	\(M_1\)	\( \ge 1 \)
Outflow Mach number	\(M_2 = \frac{M_1^2 + \frac{2}{\gamma-1}}{\frac{2\gamma}{\gamma-1}M_1^2 - 1} \)	\( \le 1 \)
Pressure	\(\frac{p_2}{p_1} = 1 + \frac{2 \gamma}{\gamma + 1} ( M_1^2 - 1 )\)	\( \ge 1 \)
Density	\(\frac{\rho_2}{\rho_1} = \left[ 1 - \frac{2}{\gamma+1}\frac{ M_1^2 - 1 }{M_1^2} \right]^{-1} \)	\( \ge 1 \)
Velocity	\(\frac{u_2}{u_1} = 1 - \frac{2}{\gamma+1}\frac{ M_1^2 - 1 }{M_1^2}\)	\( \le 1 \)
Speed of sound	\(\left(\frac{a_2}{a_1}\right)^2 = \left( 1 + \frac{2 \gamma}{\gamma + 1} ( M_1^2 - 1 ) \right) \left( 1 - \frac{2}{\gamma+1}\frac{ M_1^2 - 1 }{M_1^2} \right)\)	\( \ge 1 \)
Temperature	\(\frac{T_2}{T_1} = \left(\frac{a_2}{a_1}\right)^2 \)	\( \ge 1 \)
Stagnation pressure	\(\frac{p_{02}}{p_{01}} = \)	…
Other stag quantities		…

Details

Mach number \(M_2\) can be recast as

\[M_2^2 = \frac{2 + (\gamma-1)M_1^2}{2 \gamma M_1^2 - (\gamma-1)}\]

Let’s evaluate the ratio of functions of Mach numbers apeparing in ratios of physical quantities

\[\begin{split}\begin{aligned} \frac{1 + \gamma M_1^2}{1 + \gamma M_2^2} & = \frac{1 + \gamma M_1^2}{1 + \gamma \frac{2 + (\gamma-1)M_1^2}{2 \gamma M_1^2 - (\gamma-1)}} = \\ & = \frac{(1+\gamma M_1^2)(2 \gamma M_1^2 - \gamma + 1)}{2 \gamma M_1^2 - \gamma + 1 + 2 \gamma + \gamma ( \gamma - 1 )M_1^2 } = \\ & = \frac{(1+\gamma M_1^2)(2 \gamma M_1^2 - \gamma + 1)}{\gamma M_1^2 + 1 + \gamma + \gamma^2 M_1^2} = \\ & = \frac{(1+\gamma M_1^2)(2 \gamma M_1^2 - \gamma + 1)}{( \gamma + 1 ) ( 1 + \gamma M_1^2 )} = \\ & = \frac{2 \gamma M_1^2 - \gamma + 1}{\gamma + 1} = \\ & = 1 + \frac{2 \gamma ( M_1^2 - 1 )}{\gamma + 1} \ . \end{aligned}\end{split}\]

Pressure.

\[\frac{p_2}{p_1} = 1 + \frac{2 \gamma}{\gamma + 1} ( M_1^2 - 1 )\]

Velocity.

\[\begin{split}\begin{aligned} \frac{u_2}{u_1} & = \frac{M_2^2}{M_1^2} \frac{1+ \gamma M_1^2}{1+\gamma M_2^2} = \\ & = \frac{2 + (\gamma-1) M_1^2}{(\gamma + 1)M_1^2} = \\ & = \frac{2 + (\gamma+1) M_1^2 - 2 M_1^2}{(\gamma + 1)M_1^2} = \\ & = 1 - \frac{2}{\gamma+1}\frac{ M_1^2 - 1 }{M_1^2} \ . \end{aligned}\end{split}\]

Density.

\[\begin{split}\begin{aligned} \frac{\rho_2}{\rho_1} & = \frac{u_1}{u_2} = \left[ 1 - \frac{2}{\gamma+1}\frac{ M_1^2 - 1 }{M_1^2} \right]^{-1} \\ \end{aligned}\end{split}\]

Speed of sound.

\[\begin{split}\begin{aligned} \frac{a_2^2}{a_1^2} & = \frac{\frac{p_2}{p_1}}{ \frac{\rho_2}{\rho_1} } = \\ & = \left( 1 + \frac{2 \gamma}{\gamma + 1} ( M_1^2 - 1 ) \right) \left( 1 - \frac{2}{\gamma+1}\frac{ M_1^2 - 1 }{M_1^2} \right) = \\ & = 1 + \frac{2}{(\gamma+1) M_1^2} \left( \gamma M_1^2 - 1 \right) \left( M_1^2 - 1 \right) + \gamma \left( \frac{2}{\gamma+1} \right)^2 \left( \frac{M_1^2 - 1}{M_1} \right)^2 \end{aligned}\end{split}\]

Temperature.

\[\frac{T_2}{T_1} = \frac{a_2^2}{a_1^2}\]

Entropy.

\[s_2 - s_1 = \ln \left\{ \frac{ 2 \gamma M_1^2 - \gamma + 1}{(\gamma + 1)^{\gamma+1}} \frac{(2 + ( \gamma - 1 ) M_1^2)^{\gamma}}{M_1^{2 \gamma}} \right\}\]