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

Published online by Cambridge University Press:  02 July 2018

Jie Yao Open the ORCID record for Jie Yao [Opens in a new window]  and
Fazle Hussain Open the ORCID record for Fazle Hussain [Opens in a new window]
Jie Yao
Affiliation:
Department of Mechanical Engineering, Texas Tech University, Lubbock, TX 79409, USA
Fazle Hussain*
Affiliation:
Department of Mechanical Engineering, Texas Tech University, Lubbock, TX 79409, USA
*
†Email address for correspondence: fazle.hussain@ttu.edu
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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.

JFM classification

Compressible Flows: Compressible Flows Multiphase and Particle-laden Flows: Multiphase and Particle-laden Flows Vortex Flows: Vortex Flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 850 , 10 September 2018 , pp. 5 - 17
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Copyright
© 2018 Cambridge University Press 

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