Geodesic
Generalization of the Crocco invariant for 3D gas flows behind detached bow shock wave
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 12 (2019), pp. 52-56

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In steady 3D ideal gas flow between bow shock wave and nose part of the body in uniform supersonic flow, isoentropic stream surfaces are extracted which originate at closed lines on the shock and envelope its leading point. It was shown that each vortex line is closed and surrounds the isoentropic stream surface once. The integral invariant of isoentropic stream surfaces was obtained — the circulation vector of vorticity over (closed) vortex lines. This result is 3D generalization of the Crocco invariant for axisymmetric flow.
Keywords: Helmholtz theorems on vortices, Helmholtz–Zorawski criterion, isoenergetic flows, vorticity, detached bow shock, Crocco invariant. 
@article{IVM_2019_12_a4,
     author = {V. N. Golubkin and G. B. Sizykh},
     title = {Generalization of the {Crocco} invariant for {3D} gas flows behind detached bow shock wave},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {52--56},
     publisher = {mathdoc},
     number = {12},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2019_12_a4/}
}
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V. N. Golubkin; G. B. Sizykh. Generalization of the Crocco invariant for 3D gas flows behind detached bow shock wave. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 12 (2019), pp. 52-56. http://geodesic.mathdoc.fr/item/IVM_2019_12_a4/