28.1. Irrotational Euler Equation

Irrotationality of the flow can be evident in the compressible flowfield for weak or zero entropy gradients as per the Croco's theorem. Supersonic flows shocked in the presence of weakly curved oblique shocks can be treated as irrotational flows. Lets consider such irrotational flow. We know that the curl of velocity vector being zero is the irrotationality condition. Hence,

                                                                                       

 

Componentwise equality gives,
 

This gradient equality will be used for simplifing the momentum equation. The momentum equation for the inviscid compressible can be written as,

  

 
The u-mometum or x-directional momentum equation is,

 
Multiplication of dx on either side gives,

 However, from the irrotationality condition, we know that,

Using this, u-mometum equation gets modified as,

 Further simplification of this equation leads to,

Similarly,  v and w mometum equations can be obtained,

 

Adding all the momentum equations, we get,
 

Where   

This leads to the irrotational form of the Euler equation as,  

----------------(28.1)