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Ideal-gas rarefaction and shock pressure response

Definition

Fix $γ > 1$ and admissible left/right primitive states $(ρ_{K}, u_{K}, p_{K})$ , $K \in {L, R}$ , with $ρ_{K}, p_{K} > 0$ and $c_{K} = \sqrt{γ p_{K} / ρ_{K}}$ . Set $A_{K} = \frac{2}{(γ + 1) ρ_{K}}, B_{K} = \frac{γ - 1}{γ + 1} p_{K} .$ For a trial star pressure $p > 0$ , the wave pressure response from side $K$ is $f_{K} (p) = {\begin{cases} \frac{2 c_{K}}{γ - 1} [{(\frac{p}{p_{K}})}^{(γ - 1) / (2 γ)} - 1], & 0 < p \leq p_{K}, \\ (p - p_{K}) \sqrt{\frac{A_{K}}{p + B_{K}}}, & p \geq p_{K} . \end{cases}$ The two formulas agree at $p = p_{K}$ . The star-pressure residual is $Φ (p) = f_{L} (p) + f_{R} (p) + u_{R} - u_{L}, p > 0.$ This is a scalar root equation; it does not itself define a Riemann solution.

Depends on

Admissible states and pressure for one-dimensional ideal-gas Euler

Used by

Dependency tree · two levels

2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Randall J. LeVeque, Numerical Methods for Conservation Laws (standard reference, not scraped)