{
  "id": "2406.08760",
  "version": "v1",
  "published": "2024-06-13T02:44:01.000Z",
  "updated": "2024-06-13T02:44:01.000Z",
  "title": "Boundary sources of velocity gradient tensor and its invariants: a unified theory of boundary vorticity and vortex dynamics",
  "authors": [
    "Tao Chen",
    "Jie-Zhi Wu",
    "Tianshu Liu",
    "Jie Yao"
  ],
  "categories": [
    "physics.flu-dyn"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-13T02:44:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "velocity gradient tensor",
      "vortex dynamics",
      "boundary sources",
      "boundary vorticity",
      "boundary flux"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}