Moving expansion waves: Difference between revisions
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==== Moving Expansion Waves ==== | ==== Moving Expansion Waves ==== | ||
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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. | 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. | ||
<math | {{NumEqn|<math> | ||
J^+_a=J^+_b | J^+_a=J^+_b | ||
</math> | </math>}} | ||
<math>J^+</math> invariants constant along <math>C^+</math> characteristics | <math>J^+</math> invariants constant along <math>C^+</math> characteristics | ||
<math | {{NumEqn|<math> | ||
J^+_a=J^+_c=J^+_e | J^+_a=J^+_c=J^+_e | ||
</math> | </math>}} | ||
<math | {{NumEqn|<math> | ||
J^+_b=J^+_d=J^+_f | J^+_b=J^+_d=J^+_f | ||
</math> | </math>}} | ||
Since <math>J^+_a=J^+_b</math> this also implies <math>J^+_e=J^+_f</math>. In fact, since the flow properties ahead of the expansion are constant, all <math>C^+</math> lines will have the same <math>J^+</math> value. | Since <math>J^+_a=J^+_b</math> this also implies <math>J^+_e=J^+_f</math>. In fact, since the flow properties ahead of the expansion are constant, all <math>C^+</math> lines will have the same <math>J^+</math> value. | ||
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<math>J^-</math> invariants constant along <math>C^-</math> characteristics | <math>J^-</math> invariants constant along <math>C^-</math> characteristics | ||
<math | {{NumEqn|<math> | ||
J^-_c=J^-_d | J^-_c=J^-_d | ||
</math> | </math>}} | ||
<math | {{NumEqn|<math> | ||
J^-_e=J^-_f | J^-_e=J^-_f | ||
</math> | </math>}} | ||
<math | {{NumEqn|<math> | ||
\left. | \left. | ||
\begin{aligned} | \begin{aligned} | ||
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\end{aligned} | \end{aligned} | ||
\right\}\Rightarrow u_e=u_f \Rightarrow a_e=a_f | \right\}\Rightarrow u_e=u_f \Rightarrow a_e=a_f | ||
</math> | </math>}} | ||
Due to the fact the <math>J^+</math> is constant in the entire expansion region, <math>u</math> and <math>a</math> will be constant along each <math>C^-</math> line. | Due to the fact the <math>J^+</math> is constant in the entire expansion region, <math>u</math> and <math>a</math> will be constant along each <math>C^-</math> line. | ||
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The constant <math>J^+</math> value can be used to obtain relations for the variation of flow properties through the expansion region. Evaluation of the <math>J^+</math> invariant at any position within the expansion region should give the same value as in region 4. | The constant <math>J^+</math> value can be used to obtain relations for the variation of flow properties through the expansion region. Evaluation of the <math>J^+</math> invariant at any position within the expansion region should give the same value as in region 4. | ||
<math | {{NumEqn|<math> | ||
u+\frac{2a}{\gamma-1}=u_4+\frac{2a_4}{\gamma-1}=0+\frac{2a_4}{\gamma-1} | u+\frac{2a}{\gamma-1}=u_4+\frac{2a_4}{\gamma-1}=0+\frac{2a_4}{\gamma-1} | ||
</math> | </math>}} | ||
and thus | and thus | ||
<math | {{NumEqn|<math> | ||
\frac{a}{a_4}=1-\frac{\gamma-1}{2}\left(\frac{u}{a_4}\right) | \frac{a}{a_4}=1-\frac{\gamma-1}{2}\left(\frac{u}{a_4}\right) | ||
</math> | </math>}} | ||
Eqn. \ref{eq:expansion:a} and <math>a=\sqrt{\gamma RT}</math> gives | Eqn. \ref{eq:expansion:a} and <math>a=\sqrt{\gamma RT}</math> gives | ||
<math | {{NumEqn|<math> | ||
\frac{T}{T_4}=\left[1-\frac{\gamma-1}{2}\left(\frac{u}{a_4}\right)\right]^2 | \frac{T}{T_4}=\left[1-\frac{\gamma-1}{2}\left(\frac{u}{a_4}\right)\right]^2 | ||
</math> | </math>}} | ||
Using isentropic relations, we can get pressure ratio and density ratio | Using isentropic relations, we can get pressure ratio and density ratio | ||
<math | {{NumEqn|<math> | ||
\frac{p}{p_4}=\left[1-\frac{\gamma-1}{2}\left(\frac{u}{a_4}\right)\right]^{2\gamma/(\gamma-1)} | \frac{p}{p_4}=\left[1-\frac{\gamma-1}{2}\left(\frac{u}{a_4}\right)\right]^{2\gamma/(\gamma-1)} | ||
</math> | </math>}} | ||
<math | {{NumEqn|<math> | ||
\frac{\rho}{\rho_4}=\left[1-\frac{\gamma-1}{2}\left(\frac{u}{a_4}\right)\right]^{2/(\gamma-1)} | \frac{\rho}{\rho_4}=\left[1-\frac{\gamma-1}{2}\left(\frac{u}{a_4}\right)\right]^{2/(\gamma-1)} | ||
</math> | </math>}} | ||
