# Derivation Rankine Hugoniot

1. Mar 10, 2009

### steem84

Hello,

See picture, I do not understand one step w.r.t. to the derivation of the Rankine Hugoniot relations (fluid dynamics): how does one get the second function from the first function? I think I am missing some vector algebra here..

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2. Mar 12, 2009

### BobbyBear

is ud the component of u in the direction of nd or in the tangential direction to the discontinuity? (I suppose nd is normal to the discontinuity?)

Usually the conservation equations across normal discontinuities are what are known as the Rankine Hugoniot equations. For an oblique discontinuity the conservation of energy (assuming Re>>1) would be:

$$\rho_1 (e+v_1^2/2)\vec{v_1} \cdot \vec{n} + p_1\vec{v_1} \cdot \vec{n} = \rho_2 (e+v_2^2/2)\vec{v_2} \cdot \vec{n} + p_2\vec{v_2} \cdot \vec{n}$$

where $$\vec{n}$$ is the unit vector normal to the discontinuity.