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arXiv · 2406.00881

Algebraic Reductibility Experiments of RANS-Inspired Equations

Carla Valencia·Sebastián Velasco·Manuel Romero de Terreros
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Abstract

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.

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Carla Valencia, Sebastián Velasco, Manuel Romero de Terreros. 2025-07-19. Algebraic Reductibility Experiments of RANS-Inspired Equations. https://arxiv.org/abs/2406.00881

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