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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2406.00881 |
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Table of Contents:
- Prior to any statistical averaging we derive a rotational form of the Reynolds-Averaged Navier-Stokes (RANS) equations, eliminating the pressure and exposing a velocity--vorticity interplay governed by \[ \partial_t(\boldsymbolω+\boldsymbol{\tildeω}) +(\mathbf{v}\cdot\nabla)\boldsymbolω +(\mathbf{\tilde{v}}\cdot\nabla)\boldsymbol{\tildeω} +(\mathbf{v}\cdot\nabla)\boldsymbol{\tildeω} +(\mathbf{\tilde{v}}\cdot\nabla)\boldsymbolω -νΔ(\boldsymbolω+\boldsymbol{\tildeω})=\mathbf{0}. \] All terms are differential polynomials; hence the system generates a differential--algebraic ideal. Using the Rosenfeld--Groebner algorithm we obtain an equivalent triangular hierarchy whose first equation involves a single variable, the second at most two, and so on. This decoupling clarifies how prescribed mean-flow data drive the turbulent fluctuations and provides a hierarchy-ready foundation for physics-informed or physics-embedded neural networks. Energy estimates in Sobolev spaces complement the algebraic reduction and establish local well-posedness when the initial kinetic energy of the velocity and its curl is finite. The joint algebraic--energetic framework thus offers a pressure-free, computationally economical platform for data-driven turbulence analysis.