Equation of Rankine-Hugoniot

The experiment of Hugoniot makes it possible to connect the variation of section with the variation of Speed according to the field of the flow. It consists in observing the flow of a Fluide in convergent with a speed of lower entry then higher than the propagation velocity of the Onde S in this fluid. The formula of Rankine-Hugoniot can be found starting from the expression of the conservation of movement:

$v.dv+\backslash frac\{dP\}\{\backslash rho\}=0$ where v are speed, FD the variation speed, dP the variation of pressure and ρ the density of gas considered.

where $dP$ can be replaced by $c^\; \{2\}\; .d\; \backslash \; rho$

since celerity $c=\; \backslash \; sqrt\; \{\backslash \; frac\; \{dP\}\; \{D\; \backslash \; rho\}\}$

We obtain then:

$\backslash \; frac\; \{D\; \backslash \; rho\}\; \{\backslash \; rho\}\; =\; \backslash \; frac\; \{v.dv\}\; \{c^\; \{2\}\}\; =\; \backslash \; frac\; \{v^\; \{2\}\; .dv\}\; \{c^\; \{2\}\; .v\}\; =-M^\; \{2\}.\; \backslash \; frac\; \{FD\}\; \{v\}$

By using the law of Pascal in his derived form, $\backslash \; frac\; \{D\; \backslash \; rho\}\; \{\backslash \; rho\}$ can be replaced by $-\; (\backslash \; frac\; \{FD\}\; \{v\}\; +\; \backslash \; frac\; \{ds\}\; \{S\})$

where S is the cross-section of the stream discharge and ds the variation of section.

what gives us the formula of Rankine-Hugoniot:

$\backslash \; frac\; \{ds\}\; \{S\}\; =\; (M^\; \{2\}\; -\; 1).\; \backslash \; frac\; \{FD\}\; \{v\}$

See too

Related articles

• Pierre-Henri Hugoniot
• William Rankine

Random links:

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