# Vorticity generation behind curved shock waves

The generation of a curved shock wave can take place in many applications. For instance, with high speed blunt bodies, the bow shock in front of the body will display a general curvature. Below is an example, which is a shadowgraph image of the Project Mercury reentry capsule during a reentry wind tunnel test.

Other examples include the curved shock produced during planar shock diffraction around convex corners. Below is an example of a schlieren image where an initially planar shock wave traveling at approximately Mach 2 diffracts over a 45$^{\circ}$ convex wall, resulting in the curved shock shape.

Now, because the shock waves are curved, each streamline that passes through the shock at a given incidence angle will experience a different change in entropy. Hence, the flowfield behind the curved shock will have an entropy gradient such that $\nabla s \neq 0$. As a result, from the well known Crocco theorem,

$$T \nabla s = \nabla h_0 - \vec{V} \times \left(\nabla \times \vec{V} \right) + \frac{\partial \vec{V}}{\partial t}$$

where $s$ is the entropy, $h_0$ is the total enthalpy, and $\vec{V}$ is the velocity, the non-zero entropy gradient would imply vorticity in the flowfield behind the curved shock. Hence, behind any curved shock wave, we will have the condition,

$$\nabla \times \vec{V} \neq 0$$

This is all well established and is relatively easy to follow and understand. However, I have a difficult time trying to justify the satisfaction of the Helmholtz vortex theorems to this particular class of problems. The one in question is his third theorem. The third theorem gas as follows:

"In the absence of rotational external forces, a fluid that is initially irrotational remains irrotational."

Now the Helmholtz vortex theorems apply to inviscid flows where viscous forces are neglected. However, the curved shock problems described above can easily be established from an invsicid framework, such that the Helmholtz theorems should still apply. So my question is, if the flow starts irrotational ahead of a curved shock wave ($\nabla \times \vec{V} = 0$), yet becomes rotational behind the curved shock wave ($\nabla \times \vec{V} \neq 0$) because of the entropy gradient, then are we in some way violating Helmholtz's third vortex theorem? Is a curved shock wave somehow introducing a rotational external forces on the fluid elements? The common phrase regarding the third vortex theorem states that fluid elements initially free of vorticity remain free of vorticity, but this doesn't seem to be the case here. Anyways, I was hoping to get some clarification or perhaps I am missing something very fundamental.

• Does the piston driving the shock count as the source for an "external force"? Commented Jan 9, 2017 at 14:39
• I don't think so in these cases.
– TRF
Commented Jan 11, 2017 at 6:40
• Are you sure? I would naively assume that the frictional drag alone would introduce a viscosity and thus vorticity, would it not? Regardless, a shock can be thought of as an expanding sound wave in the 2nd image. The change in the cross-section means that the lower part of the shock needs to expand further vertically to stay in contact with the boundary but the horizontal speed should remain constant, right? Thus, after more distance down the slope the shock should weaken (i.e., the normal speed reduces but the horizontal remains constant). I think that's right... Commented Jan 26, 2017 at 22:17
• This problem can be formulated from the context of the Euler equations (i.e. inviscid). You do not need viscosity to go from a rotational to an irrotational flow when dealing with curved shocks.
– TRF
Commented Jan 27, 2017 at 3:51
• @TRF There is the Kutta–Joukowski theorem about force and circulation. Maybe it helps to understand what global circulation is in inviscid flow. Commented Sep 19, 2022 at 4:17

## 2 Answers

The generation of vorticity via a curved shock front indeed comes directly from Crocco's theorem, since curved shocks have variable shock strength and thus variable entropy. However, the entropy generation resulting from "inviscid" shocks are itself induced by viscous diffusion and thermal conductivity (see Liepmann & Roshko or related text). These are neglected in theoretical frameworks, but is by far the most important thing in numerics (numerical viscosity, see for example Toro's book on Riemann solvers or von Neumann & Richtmyer's paper) and even understanding what happens physically.

The Helmholtz vortex theorems are derived from an inviscid framework by consideration of vortex lines and tubes, if vortex filaments follow material lines within a closed curve (irreducible or reducible), then the vortex strength does not change with time (i.e. a flow that is irrotational will remain irrotational). The entropy gradient generated by a nonlinear shock front can only be understood by considering it as a local activation of viscosity and heat transfer, this then makes the Helmholtz theorems invalid in the proximity of shocks.

The Helmholtz vortex theorems cannot be used to draw conclusions that relate the shock wave's front and back. Nonetheless, the Helmholtz theorems can be applied independently to the shock wave's front and back. The difference between normal inviscid flows for which Helmholtz theorem is valid and the one in which there shock waves are present is precisely the entropy jumps that occur in the later cases. As the equation directly suggests, the entropy gradients can lead to vorticity production (not only in shock waves but in any other case where you can generate entropy in inviscid flows such as in local hot spots or flames). From Landau & Lifshitz,