LAUSR

We propose a dynamical vortex definition (the ‘ $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D70C}}$ definition’) for flows dominated by density variation, such as compressible and multi-phase flows. Based on the search of the pressure minimum in a plane, $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D70C}}$ defines a vortex to be a connected region with two negative eigenvalues of the tensor $\unicode[STIX]{x1D64E}^{M}+\unicode[STIX]{x1D64E}^{\unicode[STIX]{x1D717}}$ . Here, $\unicode[STIX]{x1D64E}^{M}$ is the symmetric part of the tensor product of the momentum gradient tensor $\unicode[STIX]{x1D735}(\unicode[STIX]{x1D70C}\unicode[STIX]{x1D66A})$ and the velocity gradient tensor $\unicode[STIX]{x1D735}\unicode[STIX]{x1D66A}$ , with $\unicode[STIX]{x1D64E}^{\unicode[STIX]{x1D717}}$ denoting the symmetric part of momentum-dilatation gradient tensor $\unicode[STIX]{x1D735}(\unicode[STIX]{x1D717}\unicode[STIX]{x1D70C}\unicode[STIX]{x1D66A})$ , and $\unicode[STIX]{x1D717}\equiv \unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D66A}$ , the dilatation rate scalar. The $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D70C}}$ definition is examined and compared with the $\unicode[STIX]{x1D706}_{2}$ definition using the analytical isentropic Euler vortex and several other flows obtained by direct numerical simulation (DNS) – e.g. liquid jet breakup in a gas, a compressible wake, a compressible turbulent channel and a hypersonic turbulent boundary layer. For low Mach number ( $M\lesssim 5$ ) compressible flows, the $\unicode[STIX]{x1D706}_{2}$ and $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D70C}}$ structures are nearly identical, so that the $\unicode[STIX]{x1D706}_{2}$ method is still valid for low $M$ compressible flows. But, the $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D70C}}$ definition is needed for studying vortex dynamics in highly compressible and strongly varying density flows.