A new vortex definition for compressible and stratified flows

We propose an objective vortex identification method (call it `$\lambda _{\rho } $ criterion') for flows dominated by compressibility or density variation effects, where the standard $\lambda_{2} $ method is not expected to be valid. The new $\lambda_{\rho } $ criterion - which is a direct extension of $\lambda_{2} $ criterion for incompressible flow - defines a vortex to be the region where the second eigenvalue of the tensor $\mathbf{S}^{m}\mathbf{+S}^{\vartheta }$ is negative. Here, $\mathbf{S}^{m}$ is the symmetric part of the tensor product of the momentum gradient tensor $\nabla (\rho \mathbf{u})$ and the velocity gradient tensor $\nabla (\mathbf{u})$ ; $\mathbf{S}^{\vartheta }$ is the symmetric part of dilatation-momentum gradient tensor $\nabla (\vartheta \rho \mathbf{u})$; and $\vartheta \equiv \nabla \bullet \mathbf{u}$ is the dilatation rate. We demonstrate the difference between $\lambda_{\rho } $ and $\lambda_{2} $ boundaries for the compressible isentropic vortex column. We also compare the $\lambda_{\rho } $ and $\lambda_{2} $ structures for several numerically simulated flows, e.g., liquid jet breakup in air, compressible jet, compressible wake, and shock-turbulent boundary layer interactions. For low Mach number ($Ma<2)$ compressible flows, we find that the structures identified by $\lambda_{2} $ and $\lambda_{\rho } $ definitions are nearly identical - indicating that $\lambda_{2} $ method can still be used for low Mach number compressible flows.