7.2 Moving Normal Shock WavesNiklas Andersson
Chalmers University of Technology
Department of Mechanical Engineering
Division of Fluid Dynamics
The starting point is the governing equations for stationary normal shocks (repeated here for convenience).
ρ 1 u 1 = ρ 2 u 2 \rho_1 u_1 = \rho_2 u_2 ρ 1 u 1 = ρ 2 u 2 ρ 1 u 1 2 + p 1 = ρ 2 u 2 2 + p 2 \rho_1 u_1^2+p_1 = \rho_2 u_2^2 + p_2 ρ 1 u 1 2 + p 1 = ρ 2 u 2 2 + p 2 h 1 + 1 2 u 1 2 = h 2 + 1 2 u 2 2 h_1 + \frac{1}{2}u_1^2 = h_2 + \frac{1}{2}u_2^2 h 1 + 2 1 u 1 2 = h 2 + 2 1 u 2 2 Shock moving to the right with the constant speed W W W u 1 = W u_1=W u 1 = W u 2 = W − u p u_2=W-u_p u 2 = W − u p u 1 u_1 u 1 u 2 u_2 u 2 (7.1) (7.3)
ρ 1 W = ρ 2 ( W − u p ) \rho_1 W = \rho_2 (W-u_p) ρ 1 W = ρ 2 ( W − u p ) ρ 1 W 2 + p 1 = ρ 2 ( W − u p ) 2 + p 2 \rho_1 W^2+p_1 = \rho_2 (W-u_p)^2 + p_2 ρ 1 W 2 + p 1 = ρ 2 ( W − u p ) 2 + p 2 h 1 + 1 2 W 2 = h 2 + 1 2 ( W − u p ) 2 h_1 + \frac{1}{2}W^2 = h_2 + \frac{1}{2}(W-u_p)^2 h 1 + 2 1 W 2 = h 2 + 2 1 ( W − u p ) 2 Rewriting Eqn. (7.4)
( W − u p ) = W ρ 1 ρ 2 (W-u_p) = W \frac{\rho_1}{\rho_2} ( W − u p ) = W ρ 2 ρ 1 Inserting Eqn. (7.7) (7.5)
p 1 + ρ 1 W 2 = p 2 + ρ 2 W 2 ( ρ 1 ρ 2 ) 2 ⇒ p 2 − p 1 = ρ 1 W 2 ( 1 − ρ 1 ρ 2 ) p_1+\rho_1 W^2 = p_2+\rho_2 W^2\left(\frac{\rho_1}{\rho_2}\right)^2 \Rightarrow p_2-p_1 = \rho_1W^2\left(1-\frac{\rho_1}{\rho_2}\right) p 1 + ρ 1 W 2 = p 2 + ρ 2 W 2 ( ρ 2 ρ 1 ) 2 ⇒ p 2 − p 1 = ρ 1 W 2 ( 1 − ρ 2 ρ 1 ) W 2 = p 2 − p 1 ρ 2 − ρ 1 ( ρ 2 ρ 1 ) W^2=\frac{p_2-p_1}{\rho_2-\rho_1}\left(\frac{\rho_2}{\rho_1}\right) W 2 = ρ 2 − ρ 1 p 2 − p 1 ( ρ 1 ρ 2 ) From the continuity equation (7.4)
W = ( W − u p ) ( ρ 2 ρ 1 ) W = (W-u_p) \left(\frac{\rho_2}{\rho_1}\right) W = ( W − u p ) ( ρ 1 ρ 2 ) Inserting Eqn. (7.10) (7.8)
( W − u p ) 2 = p 2 − p 1 ρ 2 − ρ 1 ( ρ 1 ρ 2 ) (W-u_p)^2=\frac{p_2-p_1}{\rho_2-\rho_1}\left(\frac{\rho_1}{\rho_2}\right) ( W − u p ) 2 = ρ 2 − ρ 1 p 2 − p 1 ( ρ 2 ρ 1 ) Now, let’s insert Eqns. (7.8) (7.11) (7.6)
h 1 + 1 2 [ p 2 − p 1 ρ 2 − ρ 1 ( ρ 2 ρ 1 ) ] = h 2 + 1 2 [ p 2 − p 1 ρ 2 − ρ 1 ( ρ 1 ρ 2 ) ] h_1 + \frac{1}{2}\left[\frac{p_2-p_1}{\rho_2-\rho_1}\left(\frac{\rho_2}{\rho_1}\right)\right] = h_2 + \frac{1}{2}\left[\frac{p_2-p_1}{\rho_2-\rho_1}\left(\frac{\rho_1}{\rho_2}\right)\right] h 1 + 2 1 [ ρ 2 − ρ 1 p 2 − p 1 ( ρ 1 ρ 2 ) ] = h 2 + 2 1 [ ρ 2 − ρ 1 p 2 − p 1 ( ρ 2 ρ 1 ) ] h = e + p ρ h=e+\frac{p}{\rho} h = e + ρ p e 1 + p 1 ρ 1 + 1 2 [ p 2 − p 1 ρ 2 − ρ 1 ( ρ 2 ρ 1 ) ] = e 2 + p 2 ρ 2 + 1 2 [ p 2 − p 1 ρ 2 − ρ 1 ( ρ 1 ρ 2 ) ] e_1 + \frac{p_1}{\rho_1} + \frac{1}{2}\left[\frac{p_2-p_1}{\rho_2-\rho_1}\left(\frac{\rho_2}{\rho_1}\right)\right] = e_2 + \frac{p_2}{\rho_2} + \frac{1}{2}\left[\frac{p_2-p_1}{\rho_2-\rho_1}\left(\frac{\rho_1}{\rho_2}\right)\right] e 1 + ρ 1 p 1 + 2 1 [ ρ 2 − ρ 1 p 2 − p 1 ( ρ 1 ρ 2 ) ] = e 2 + ρ 2 p 2 + 2 1 [ ρ 2 − ρ 1 p 2 − p 1 ( ρ 2 ρ 1 ) ] which can be rewritten as
e 2 − e 1 = p 1 + p 2 2 ( 1 ρ 1 − 1 ρ 2 ) e_2-e_1=\frac{p_1+p_2}{2}\left(\frac{1}{\rho_1}-\frac{1}{\rho_2}\right) e 2 − e 1 = 2 p 1 + p 2 ( ρ 1 1 − ρ 2 1 ) Eqn (7.15)
Moving Shock Relations ¶ For a calorically perfect gas we have e = C v T e=C_v T e = C v T
C v ( T 2 − T 1 ) = p 1 + p 2 2 ( ν 1 − ν 2 ) C_v(T_2-T_1)=\frac{p_1+p_2}{2}\left(\nu_1-\nu_2\right) C v ( T 2 − T 1 ) = 2 p 1 + p 2 ( ν 1 − ν 2 ) where ν = 1 / ρ \nu=1/\rho ν = 1/ ρ
Now, using the ideal gas law T = p ν / R T=p\nu/R T = p ν / R C v / R = 1 / ( γ − 1 ) C_v/R=1/(\gamma-1) C v / R = 1/ ( γ − 1 )
( 1 γ − 1 ) ( p 2 ν 2 − p 1 ν 1 ) = p 1 + p 2 2 ( ν 1 − ν 2 ) \left(\frac{1}{\gamma-1}\right)(p_2\nu_2-p_1\nu_1)=\frac{p_1+p_2}{2}\left(\nu_1-\nu_2\right) ( γ − 1 1 ) ( p 2 ν 2 − p 1 ν 1 ) = 2 p 1 + p 2 ( ν 1 − ν 2 ) p 2 ( ν 2 γ − 1 − ν 1 − ν 2 2 ) = p 1 ( ν 1 γ − 1 + ν 1 − ν 2 2 ) p_2\left(\frac{\nu_2}{\gamma-1}-\frac{\nu_1-\nu_2}{2}\right)=p_1\left(\frac{\nu_1}{\gamma-1}+\frac{\nu_1-\nu_2}{2}\right) p 2 ( γ − 1 ν 2 − 2 ν 1 − ν 2 ) = p 1 ( γ − 1 ν 1 + 2 ν 1 − ν 2 ) From this result, we can derive a relation for the pressure ratio over the shock as a function of density ratio
p 2 p 1 = ( γ + 1 γ − 1 ) ( ν 1 ν 2 ) − 1 ( γ + 1 γ − 1 ) − ( ν 1 ν 2 ) \frac{p_2}{p_1}=\frac{\left(\dfrac{\gamma+1}{\gamma-1}\right)\left(\dfrac{\nu_1}{\nu_2}\right)-1}{\left(\dfrac{\gamma+1}{\gamma-1}\right)-\left(\dfrac{\nu_1}{\nu_2}\right)} p 1 p 2 = ( γ − 1 γ + 1 ) − ( ν 2 ν 1 ) ( γ − 1 γ + 1 ) ( ν 2 ν 1 ) − 1 ν = R T / p \nu=RT/p ν = RT / p
ν 1 ν 2 = T 1 T 2 p 2 p 1 \frac{\nu_1}{\nu_2}=\frac{T_1}{T_2}\frac{p_2}{p_1} ν 2 ν 1 = T 2 T 1 p 1 p 2 Eqn. (7.21) (7.20)
p 2 p 1 = ( γ + 1 γ − 1 ) ( T 1 T 2 p 2 p 1 ) − 1 ( γ + 1 γ − 1 ) − ( T 1 T 2 p 2 p 1 ) \frac{p_2}{p_1}=\frac{\left(\dfrac{\gamma+1}{\gamma-1}\right)\left(\dfrac{T_1}{T_2}\dfrac{p_2}{p_1}\right)-1}{\left(\dfrac{\gamma+1}{\gamma-1}\right)-\left(\dfrac{T_1}{T_2}\dfrac{p_2}{p_1}\right)} p 1 p 2 = ( γ − 1 γ + 1 ) − ( T 2 T 1 p 1 p 2 ) ( γ − 1 γ + 1 ) ( T 2 T 1 p 1 p 2 ) − 1 Now, we can get a relation for calculation of the temperature ratio over the moving shock as function of the shock pressure ratio
T 2 T 1 = p 2 p 1 [ ( γ + 1 γ − 1 ) + ( p 2 p 1 ) 1 + ( γ + 1 γ − 1 ) ( p 2 p 1 ) ] \frac{T_2}{T_1}=\frac{p_2}{p_1}\left[\frac{\left(\dfrac{\gamma+1}{\gamma-1}\right)+\left(\dfrac{p_2}{p_1}\right)}{1+\left(\dfrac{\gamma+1}{\gamma-1}\right)\left(\dfrac{p_2}{p_1}\right)}\right] T 1 T 2 = p 1 p 2 ⎣ ⎡ 1 + ( γ − 1 γ + 1 ) ( p 1 p 2 ) ( γ − 1 γ + 1 ) + ( p 1 p 2 ) ⎦ ⎤ Once again using the ideal gas law
ρ 2 ρ 1 = ( γ + 1 γ − 1 ) + ( p 2 p 1 ) 1 + ( γ + 1 γ − 1 ) ( p 2 p 1 ) \frac{\rho_2}{\rho_1}=\frac{\left(\dfrac{\gamma+1}{\gamma-1}\right)+\left(\dfrac{p_2}{p_1}\right)}{1+\left(\dfrac{\gamma+1}{\gamma-1}\right)\left(\dfrac{p_2}{p_1}\right)} ρ 1 ρ 2 = 1 + ( γ − 1 γ + 1 ) ( p 1 p 2 ) ( γ − 1 γ + 1 ) + ( p 1 p 2 ) Going back to the momentum equation
p 2 − p 1 = ρ 1 W 2 ( 1 − ρ 1 ρ 2 ) = { W = M s a 1 } = ρ 1 M s 2 a 1 2 ( 1 − ρ 1 ρ 2 ) p_2-p_1 = \rho_1W^2\left(1-\frac{\rho_1}{\rho_2}\right)=\left\{W=M_s a_1\right\}=\rho_1M_s^2a_1^2\left(1-\frac{\rho_1}{\rho_2}\right) p 2 − p 1 = ρ 1 W 2 ( 1 − ρ 2 ρ 1 ) = { W = M s a 1 } = ρ 1 M s 2 a 1 2 ( 1 − ρ 2 ρ 1 ) with a 1 2 = γ p 1 / ρ 1 a_1^2=\gamma p_1/\rho_1 a 1 2 = γ p 1 / ρ 1
p 2 p 1 = γ M s 2 ( 1 − ρ 1 ρ 2 ) + 1 \frac{p_2}{p_1} = \gamma M_s^2\left(1-\frac{\rho_1}{\rho_2}\right)+1 p 1 p 2 = γ M s 2 ( 1 − ρ 2 ρ 1 ) + 1 From the normal shock relations, we have
ρ 1 ρ 2 = 2 + ( γ − 1 ) M s 2 ( γ + 1 ) M s 2 \frac{\rho_1}{\rho_2} = \frac{2+(\gamma-1)M_s^2}{(\gamma+1)M_s^2} ρ 2 ρ 1 = ( γ + 1 ) M s 2 2 + ( γ − 1 ) M s 2 Eqn. (7.27) (7.26)
p 2 p 1 = 1 + ( 2 γ γ + 1 ) ( M s 2 − 1 ) \frac{p_2}{p_1} = 1 + \left(\frac{2\gamma}{\gamma+1}\right)(M_s^2-1) p 1 p 2 = 1 + ( γ + 1 2 γ ) ( M s 2 − 1 ) or
M s = ( γ + 1 2 γ ) ( p 2 p 1 − 1 ) + 1 M_s=\sqrt{\left(\frac{\gamma+1}{2\gamma}\right)\left(\frac{p_2}{p_1}-1\right)+1} M s = ( 2 γ γ + 1 ) ( p 1 p 2 − 1 ) + 1 Eqn. (7.29) M s = W / a 1 M_s=W/a_1 M s = W / a 1
W = a 1 ( γ + 1 2 γ ) ( p 2 p 1 − 1 ) + 1 W=a_1\sqrt{\left(\frac{\gamma+1}{2\gamma}\right)\left(\frac{p_2}{p_1}-1\right)+1} W = a 1 ( 2 γ γ + 1 ) ( p 1 p 2 − 1 ) + 1 Induced Flow Behind Moving Shock ¶ Let’s try to find a relation for calculation of the induced velocity behind the moving shock. Once again, the starting point is the continuity equation for moving shocks (Eqn. (7.4)
ρ 1 W = ρ 2 ( W − u p ) \rho_1 W = \rho_2 (W-u_p) ρ 1 W = ρ 2 ( W − u p ) The induced velocity appears on the right side of the continuity equation
W ( ρ 1 − ρ 2 ) = − ρ 2 u p W (\rho_1-\rho_2) = -\rho_2 u_p W ( ρ 1 − ρ 2 ) = − ρ 2 u p u p = W ( 1 − ρ 1 ρ 2 ) u_p = W \left(1-\frac{\rho_1}{\rho_2}\right) u p = W ( 1 − ρ 2 ρ 1 ) From before we have a relation for W W W ρ 1 / ρ 2 \rho_1/\rho_2 ρ 1 / ρ 2
Eqn. (7.33) (7.30) (7.24)
u p = a 1 ( γ + 1 2 γ ) ( p 2 p 1 − 1 ) + 1 ⏟ I [ 1 − ( γ + 1 γ − 1 ) + ( p 2 p 1 ) 1 + ( γ + 1 γ − 1 ) ( p 2 p 1 ) ] ⏟ I I u_p=a_1\underbrace{\sqrt{\left(\frac{\gamma+1}{2\gamma}\right)\left(\frac{p_2}{p_1}-1\right)+1}}_{I}\underbrace{\left[1-\dfrac{\left(\dfrac{\gamma+1}{\gamma-1}\right)+\left(\dfrac{p_2}{p_1}\right)}{1+\left(\dfrac{\gamma+1}{\gamma-1}\right)\left(\dfrac{p_2}{p_1}\right)}\right]}_{II} u p = a 1 I ( 2 γ γ + 1 ) ( p 1 p 2 − 1 ) + 1 II ⎣ ⎡ 1 − 1 + ( γ − 1 γ + 1 ) ( p 1 p 2 ) ( γ − 1 γ + 1 ) + ( p 1 p 2 ) ⎦ ⎤ The equation subsets I and II can be rewritten as:
Term I:
( γ + 1 2 γ ) ( p 2 p 1 − 1 ) + 1 = γ + 1 2 γ [ ( p 2 p 1 ) + ( γ − 1 γ + 1 ) ] \sqrt{\left(\frac{\gamma+1}{2\gamma}\right)\left(\frac{p_2}{p_1}-1\right)+1}=\sqrt{\frac{\gamma+1}{2\gamma}\left[\left(\frac{p_2}{p_1}\right)+\left(\frac{\gamma-1}{\gamma+1}\right)\right]} ( 2 γ γ + 1 ) ( p 1 p 2 − 1 ) + 1 = 2 γ γ + 1 [ ( p 1 p 2 ) + ( γ + 1 γ − 1 ) ] Term II:
[ 1 − ( γ + 1 γ − 1 ) + ( p 2 p 1 ) 1 + ( γ + 1 γ − 1 ) ( p 2 p 1 ) ] = 1 γ ( p 2 p 1 − 1 ) ( 2 γ γ + 1 ) ( γ − 1 γ + 1 ) + ( p 2 p 1 ) \left[1-\dfrac{\left(\dfrac{\gamma+1}{\gamma-1}\right)+\left(\dfrac{p_2}{p_1}\right)}{1+\left(\dfrac{\gamma+1}{\gamma-1}\right)\left(\dfrac{p_2}{p_1}\right)}\right]=\frac{1}{\gamma}\left(\frac{p_2}{p_1}-1\right)\frac{\left(\dfrac{2\gamma}{\gamma+1}\right)}{\left(\dfrac{\gamma-1}{\gamma+1}\right)+\left(\dfrac{p_2}{p_1}\right)} ⎣ ⎡ 1 − 1 + ( γ − 1 γ + 1 ) ( p 1 p 2 ) ( γ − 1 γ + 1 ) + ( p 1 p 2 ) ⎦ ⎤ = γ 1 ( p 1 p 2 − 1 ) ( γ + 1 γ − 1 ) + ( p 1 p 2 ) ( γ + 1 2 γ ) With the rewritten terms I and II implemented, Eqn. (7.34)
u p = a 1 γ ( p 2 p 1 − 1 ) ( 2 γ γ + 1 ) ( γ − 1 γ + 1 ) + ( p 2 p 1 ) u_p=\frac{a_1}{\gamma}\left(\frac{p_2}{p_1}-1\right)\sqrt{\frac{\left(\dfrac{2\gamma}{\gamma+1}\right)}{\left(\dfrac{\gamma-1}{\gamma+1}\right)+\left(\dfrac{p_2}{p_1}\right)}} u p = γ a 1 ( p 1 p 2 − 1 ) ( γ + 1 γ − 1 ) + ( p 1 p 2 ) ( γ + 1 2 γ ) Since the region behind the moving shock is region 2, the induced flow Mach number is obtained as
M p = u p a 2 = u p a 1 a 1 a 2 = u p a 1 γ R T 1 γ R T 2 = u p a 1 T 1 T 2 M_p=\frac{u_p}{a_2}=\frac{u_p}{a_1}\frac{a_1}{a_2}=\frac{u_p}{a_1}\sqrt{\frac{\gamma R T_1}{\gamma R T_2}}=\frac{u_p}{a_1}\sqrt{\frac{T_1}{T_2}} M p = a 2 u p = a 1 u p a 2 a 1 = a 1 u p γ R T 2 γ R T 1 = a 1 u p T 2 T 1 With u p / a 1 up/a_1 u p / a 1 (7.37) T 1 / T 2 T_1/T_2 T 1 / T 2 (7.23)
M p = 1 γ ( p 2 p 1 − 1 ) ( ( 2 γ γ + 1 ) ( γ − 1 γ + 1 ) + ( p 2 p 1 ) ) 1 / 2 ( 1 + ( γ + 1 γ − 1 ) ( p 2 p 1 ) ( γ + 1 γ − 1 ) ( p 2 p 1 ) + ( p 2 p 1 ) 2 ) 1 / 2 M_p=\frac{1}{\gamma}\left(\frac{p_2}{p_1}-1\right)\left(\frac{\left(\dfrac{2\gamma}{\gamma+1}\right)}{\left(\dfrac{\gamma-1}{\gamma+1}\right)+\left(\dfrac{p_2}{p_1}\right)}\right)^{1/2}
\left(\frac{1+\left(\dfrac{\gamma+1}{\gamma-1}\right)\left(\dfrac{p_2}{p_1}\right)}{\left(\dfrac{\gamma+1}{\gamma-1}\right)\left(\dfrac{p_2}{p_1}\right)+\left(\dfrac{p_2}{p_1}\right)^2}\right)^{1/2} M p = γ 1 ( p 1 p 2 − 1 ) ⎝ ⎛ ( γ + 1 γ − 1 ) + ( p 1 p 2 ) ( γ + 1 2 γ ) ⎠ ⎞ 1/2 ⎝ ⎛ ( γ − 1 γ + 1 ) ( p 1 p 2 ) + ( p 1 p 2 ) 2 1 + ( γ − 1 γ + 1 ) ( p 1 p 2 ) ⎠ ⎞ 1/2 There is a theoretical upper limit for the induced Mach number M p M_p M p
lim p 2 / p 1 → ∞ M p ( p 2 p 1 ) = 2 γ ( γ − 1 ) \lim_{p_2/p_1\rightarrow\infty} M_p\left(\frac{p_2}{p_1}\right)=\sqrt{\frac{2}{\gamma(\gamma-1)}} p 2 / p 1 → ∞ lim M p ( p 1 p 2 ) = γ ( γ − 1 ) 2 As can be seen, at the upper limit the induced Mach number is a function of γ \gamma γ γ = 1.4 \gamma=1.4 γ = 1.4
lim p 2 / p 1 → ∞ M p ( p 2 p 1 ) ≃ 1.89 \lim_{p_2/p_1\rightarrow\infty} M_p\left(\frac{p_2}{p_1}\right)\simeq 1.89 p 2 / p 1 → ∞ lim M p ( p 1 p 2 ) ≃ 1.89