Second integral generalization of the Crocco invariant for 3D flows behind detached bow shock wave

Stationary flows of an ideal gas behind the detached bow shock are investigated in the general 3D case. The well-known integral invariant (V.N. Golubkin, G.B. Sizykh, 2019), generalizing the axisymmetric invariant of (L. Crocco, 1937) to asymmetric flows, is a curvilinear integral over a closed vortex line (such lines lie on isentropic surfaces), in which the integrand is the pressure divided by the vorticity. This integral takes on the same value on all (closed) vortex lines lying on one isentropic surface. It was obtained after the discovery of the fact that the vortex lines are closed in the flow behind the shock in the general 3D case. Recently, another family of closed lines behind the shock was found, lying on isentropic surfaces (G.B. Sizykh, 2020). It is given by vector lines a — the vector product of the gas velocity and the gradient of the entropy function. In the general 3D case, these lines and vortex lines do not coincide.
In the presented study, an attempt is made to find the integral invariant associated with closed vector lines a. Without using asymptotic, numerical and other approximate methods, the Euler equations are analyzed for the classical model of the flow of an ideal perfect gas with constant heat capacities. The concept of imaginary particles “carrying” the streamlines of a real gas flow, based on the Helmholtz–Zoravsky criterion, is used. A new integral invariant of isentropic surfaces is obtained. It is shown that the curvilinear integral over a closed vector line a, in which the integrand is the pressure divided by the projection of the vorticity on the direction a, has the same values for all lines a lying on one isentropic surface. This invariant, like another previously known integral invariant (V.N. Golubkin, G.B. Sizykh, 2019), in the particular case of non-swirling axisymmetric flows, coincides with the non-integral invariant of L. Crocco and generalizes it to the general spatial case.