Algebraic Reductibility Experiments of RANS-Inspired Equations

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.