Toward vortex identification based on local pressure-minimum criterion in compressible and variable density flows

Abstract

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.