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7.7 The Shock Tube Relations

Chalmers University of Technology
Department of Mechanical Engineering
Division of Fluid Dynamics
traveling waves in a shock tube

Figure 7.4:Traveling waves in a shock tube

From the analysis of the incident shock, we have a relation for the induced flow behind the shock

u2=up=a1γ1(p2p1−1)((2γ1γ1+1)(γ1−1γ1+1)+(p2p1))1/2u_2=u_p=\frac{a_1}{\gamma_1}\left(\frac{p_2}{p_1}-1\right)\left(\frac{\left(\dfrac{2\gamma_1}{\gamma_1+1}\right)}{\left(\dfrac{\gamma_1-1}{\gamma_1+1}\right)+\left(\dfrac{p_2}{p_1}\right)}\right)^{1/2}
(7.149)

The velocity in region 3 can be obtained from the expansion relations

p3p4=[1−γ4−12(u3a4)]2γ4/(γ4−1)\frac{p_3}{p_4}=\left[1-\frac{\gamma_4-1}{2}\left(\frac{u_3}{a_4}\right)\right]^{2\gamma_4/(\gamma_4-1)}
(7.150)

Solving for u3u_3 gives

u3=2a4γ4−1[1−(p3p4)(γ4−1)/(2γ4)]u_3=\frac{2a_4}{\gamma_4-1}\left[1-\left(\frac{p_3}{p_4}\right)^{(\gamma_4-1)/(2\gamma_4)}\right]
(7.151)

There is no change in pressure or velocity over the contact surface, which means u2=u3u_2=u_3 and p2=p3p_2=p_3.

u2=2a4γ4−1[1−(p2p4)(γ4−1)/(2γ4)]u_2=\frac{2a_4}{\gamma_4-1}\left[1-\left(\frac{p_2}{p_4}\right)^{(\gamma_4-1)/(2\gamma_4)}\right]
(7.152)

Now, we have two ways of calculating u2u_2. Setting Eqn. (7.149) equal to Eqn. (7.152) leads to the shock tube relation

p4p1=p2p1{1−(γ4−1)(a1/a4)(p2/p1−1)2γ1[2γ1+(γ1+1)(p2/p1−1)]}−2γ4/(γ4−1)\frac{p_4}{p_1}=\frac{p_2}{p_1}\left\{ 1 -\frac{(\gamma_4-1)(a_1/a_4)(p_2/p_1-1)}{\sqrt{2\gamma_1\left[2\gamma_1+(\gamma_1+1)(p_2/p_1-1)\right]}}\right\}^{-2\gamma_4/(\gamma_4-1)}
(7.153)
Shock Waves
Moving Expansion Waves
Shock Waves
High-Temperature Effects