Evolution of the velocity gradient invariants in homogeneous isotropic turbulence with an inverse energy cascade

An investigation of topological features of homogeneous isotropic turbulence with an inverse energy cascade is performed by using a direct numerical simulation. The inverse energy cascade is induced by reversing the velocity field (i.e., under the transformation from u(i) to -u(i)) of a freely decaying isotropic turbulence. In the backward energy transfer process, the joint probability density function (PDF) of Q and R exhibits a novel shape, which seems to be symmetric with the well-known teardrop shape about the Q-axis (Q and R are the second and third invariants of the velocity gradients tensor, respectively). The predominance of the top-right (R > 0 and Q > 0) and the bottom-left (R < 0 and Q < 0) quadrants is observed in the backward energy transfer process. This observation is different from that of the forward energy cascade, in which the top-left (R < 0 and Q > 0) and the bottom-right (R > 0 and Q < 0) quadrants are dominant. The unexpected shape of the joint PDF of Q and R indicates that the turbulence with an inverse energy cascade is dominated by vortex compression and tube-like structures. The PDF of the intermediate eigenvalue of the strain-rate tensor in the inverse energy cascade is negatively skewed, which is opposite to the universal feature of the forward energy cascade. Nevertheless, the preferential alignment of vorticity with the intermediate eigenvector is rather robust, no matter whether the direction of the energy transfer is forward or backward. In addition, it is universal that the vorticity is mostly perpendicular to the eigenvector, which corresponds to a strain-rate eigenvalue with the maximum absolute value among the three eigenvalues. Since the velocity gradient invariants are closely related to the local flow topology, the numerical results reported in this work are expected to shed light on the intrinsic dynamics and mechanisms of inverse energy cascade.

Automated summary

This study investigated the topological features of homogeneous isotropic turbulence with an inverse energy cascade through direct numerical simulation. The inverse energy cascade was induced by reversing the velocity field of freely decaying isotropic turbulence. The joint probability density function of Q and R, the second and third invariants of the velocity gradients tensor, exhibited a novel shape during the backward energy transfer process. The dominance of specific quadrants in the joint PDF indicated that the turbulence with an inverse energy cascade is characterized by vortex compression and tube-like structures. The PDF of the intermediate eigenvalue of the strain-rate tensor in the inverse energy cascade displayed a negative skewness, opposite to the universal feature of the forward energy cascade. However, the preferential alignment of vorticity with the intermediate eigenvector remained robust regardless of the direction of energy transfer. The numerical results in this study are expected to provide insights into the dynamics and mechanisms of inverse energy cascade by analyzing the relationship between velocity gradient invariants and local flow topology.