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**is an increasing function of Mach number M1. Hence, at the flow around a concave fixed angle_{max}**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

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

*Peter Deak*

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