Boundary sources of velocity gradient tensor and its invariants

TL;DR

This paper investigates the boundary behaviors of the velocity gradient tensor and its invariants in compressible flow near a rigid wall, providing new theoretical insights into boundary fluxes and their physical interpretations.

Contribution

It derives explicit boundary flux expressions for the velocity gradient tensor invariants and clarifies the physical mechanisms influencing boundary vorticity and flow structures.

Findings

Boundary flux of Q is due to competition among dilatation, enstrophy, and strain-rate fluxes.

Boundary R flux must vanish on a stationary rigid wall.

The boundary flux of the third strain invariant relates to vortex stretching derivatives.

Abstract

The present work elucidates the boundary behaviors of the velocity gradient tensor ($\bm{A}\equiv\bm{\nabla}\bm{u}$) and its principal invariants ($P,Q,R$) for compressible flow interacting with a stationary rigid wall. Firstly, it is found that the well-known Caswell formula exhibits an inherent physical structure being compatible with the normal-nilpotent decomposition, where both the strain-rate and rotation-rate tensors contain the physical effects from the spin component of the vorticity. Secondly, we derive the kinematic and dynamic forms of the boundary $\bm{A}$-flux from which the known boundary fluxes can be recovered by applying the symmetric-antisymmetric decomposition. Then, we obtain the explicit expression of the boundary $Q$ flux as a result of the competition among the boundary fluxes of squared dilatation, enstrophy and squared strain-rate. Importantly, we emphasize that both the coupling between the spin and surface pressure gradient, and the spin-curvature quadratic interaction, are \textit{not} responsible for the generation of the boundary $Q$ flux, although they contribute to both the boundary fluxes of enstrophy and squared strain-rate. Moreover, we prove that the boundary $R$ flux must vanish on a stationary rigid wall. Finally, the boundary fluxes of the invariants of the strain-rate and rotation-rate tensors are also discussed. It is revealed that the boundary flux of the third invariant of the strain-rate tensor is proportional to the wall-normal derivative of the vortex stretching term, which serves as a source term accounting for the the spatiotemporal evolution rate of the wall-normal enstrophy flux. These theoretical results provide a unified description of boundary vorticity and vortex dynamics, which could be valuable in understanding the formation mechanisms of complex near-wall coherent structures and the boundary sources of flow noise.