What is the derivation of the Rankine Hugoniot relations in fluid dynamics?

[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..

 

[BobbyBear]

is u_d the component of u in the direction of n_d or in the tangential direction to the discontinuity? (I suppose n_d 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.