Algebraic Reductibility Experiments of RANS-Inspired Equations

TL;DR

This paper derives a pressure-free form of RANS equations, uses algebraic methods to simplify the system, and establishes a foundation for physics-informed neural networks in turbulence modeling.

Contribution

It introduces an algebraic reduction of RANS equations using Rosenfeld-Groebner algorithm, providing a hierarchical, decoupled system for turbulence analysis.

Findings

Derivation of a rotational, pressure-free RANS form.

Application of algebraic algorithms to decouple the equations.

Establishment of local well-posedness with energy estimates.

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 \[ \partial_t(\boldsymbol{\omega}+\boldsymbol{\tilde{\omega}}) +(\mathbf{v}\cdot\nabla)\boldsymbol{\omega} +(\mathbf{\tilde{v}}\cdot\nabla)\boldsymbol{\tilde{\omega}} +(\mathbf{v}\cdot\nabla)\boldsymbol{\tilde{\omega}} +(\mathbf{\tilde{v}}\cdot\nabla)\boldsymbol{\omega} -\nu\Delta(\boldsymbol{\omega}+\boldsymbol{\tilde{\omega}})=\mathbf{0}. \] 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.