Moving expansion waves

Moving Expansion Waves

The expansion wave propagation into the driver section in a shock tube can be described using characteristic lines.

The expansion is propagating into stagnant fluid in region four (the driver section), which means that the flow properties ahead of the expansion wave are constant.

$$J_a^{+}=J_b^{+}$$

$J^{+}$ invariants constant along $C^{+}$ characteristics

$$J_a^{+}=J_c^{+}=J_e^{+}$$

$$J_b^{+}=J_d^{+}=J_f^{+}$$

Since $J_a^{+}=J_b^{+}$ this also implies $J_e^{+}=J_f^{+}$. In fact, since the flow properties ahead of the expansion are constant, all $C^{+}$ lines will have the same $J^{+}$ value.

$J^{-}$ invariants constant along $C^{-}$ characteristics

$$J_c^{-}=J_d^{-}$$

$$J_e^{-}=J_f^{-}$$

$$\begin{array}{cc}& u_e=\frac{1}{2}(J_e^{+}+J_e^{-})\\ & u_f=\frac{1}{2}(J_f^{+}+J_f^{-})\\ & J_e^{-}=J_f^{-}\\ & J_e^{+}=J_f^{+}\end{array}\}\Rightarrow u_e=u_f\Rightarrow a_e=a_f$$

Due to the fact the $J^{+}$ is constant in the entire expansion region, $u$ and $a$ will be constant along each $C^{-}$ line.

The constant $J^{+}$ value can be used to obtain relations for the variation of flow properties through the expansion region. Evaluation of the $J^{+}$ invariant at any position within the expansion region should give the same value as in region 4.

$$u+\frac{2a}{\gamma -1}=u_4+\frac{2a_4}{\gamma -1}=0+\frac{2a_4}{\gamma -1}$$

and thus

$$\frac{a}{a_4}=1-\frac{\gamma -1}{2}\left(\frac{u}{a_4}\right)$$

Eqn. \ref{eq:expansion:a} and $a=\sqrt{\gamma RT}$ gives

$$\frac{T}{T_4}={[1-\frac{\gamma -1}{2}\left(\frac{u}{a_4}\right)]}^2$$

Using isentropic relations, we can get pressure ratio and density ratio

$$\frac{p}{p_4}={[1-\frac{\gamma -1}{2}\left(\frac{u}{a_4}\right)]}^{2\gamma /(\gamma -1)}$$

$$\frac{\rho }{{\rho }_4}={[1-\frac{\gamma -1}{2}\left(\frac{u}{a_4}\right)]}^{2/(\gamma -1)}$$