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 $${\mathrm{\partial }}_t(\mathit{\omega }+)+(\mathbf{v}\cdot \mathrm{\nabla })\mathit{\omega }+(\tilde{\mathbf{v}}\cdot \mathrm{\nabla })+(\mathbf{v}\cdot \mathrm{\nabla })+(\tilde{\mathbf{v}}\cdot \mathrm{\nabla })\mathit{\omega }-\nu \mathrm{\Delta }(\mathit{\omega }+)=\mathbf{0.}$$ 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.