Detached curvilinear shock wave

I would like to first introduce the concept of homoenergetic flow (stationary). In such flows, we have.

Similarly, the flow is homoentropic if. Thus when the flow is homoentropic then the entropy is uniformly distributed in the flow domain. (see in yellow on the figure)

If the flow is homoenergetic and homoentropic, it is automatically barotropic and the Bernoulli constant CB in global. (In 2D flows, it implies that the velocity field is potential, since , they are perpendicular to each other, therefore.

Now, having all these, we can write the Crocco equation:

According to the Crocco equation, any inhomogeneity in the spatial distribution of entropy in the homoenergetic flow immediately leads to vorticity generation.

In the figure you may see the detached shock wave generated at the wedge. The flow in the neighborhood of a nose tip is subsonic (M<1) and then after the intersection of the dashed lines it become supersonic (M>1). 

The value of θ_max is an increasing function of Mach number M1. Hence, at the flow around a concave fixed angle θ or the wedge with a fixed apex angle 2θ, a detached shock wave with increasing free-stream Mach number at critical M1* will arise. (Detached shock wave arising in a supersonic gas flow.) Further increasing M1, the distance between the detached shock wave and the body as well as the subsonic flow region will increase.

In the region where the shock wave is strong, the constant entropy curves have a larger value than further distance from the apex of the wedge, where the shock is weak. (this assumption is given on the figure)


Reference: Foundations of Fluid Mechanics with Applications - Sergey P. Kiselev, Evgenii Vorozhtsov, Vasily M. Fomin

Peter Deak