Boundary sources of velocity gradient tensor and its invariants

The present work elucidates the boundary behaviors of the velocity gradient tensor (A≡∇u) and its principal invariants (P, Q, R) for compressible flow interacting with a stationary rigid wall. First, it is found that the boundary value of A exhibits an inherent physical structure being compatible with the normal-nilpotent decomposition, where both the strain-rate and rotation-rate tensors contain the physical contributions from the spin component of the vorticity. Second, we derive the kinematic and dynamical forms of the boundary 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 find that both the coupling between the spin and surface pressure gradient, and the spin-curvature quadratic interaction (sw·K·sw), are 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 principal 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 (ω·D·ω), which serves as a source term accounting for the spatiotemporal evolution rate of the wall-normal enstrophy flux. As an example, several relevant surface quantities to the surface curvature are calculated based on global skin friction and surface pressure measurements in a flow over a National Advisory Committee for Aeronautics Fundamental Aeronautics Investigates The Hill model. 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.