Generalization of the Crocco invariant for 3D gas flows behind detached bow shock wave

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.