On the Non-Markovian Navier-Stokes Framework for Turbulence Modeling -- A Preliminary Analysis

This study explores a formulation of the Navier Stokes equations (NSE) using fractional calculus in modeling turbulence. By generalizing the stress strain constitutive relation to incorporate nonlocal spatial interactions and memory effects, we redefine a fractional Navier Stokes equation (fNSE). Regarding the inertial range scaling, the fractional Laplacian of order 1/3 and time fractional derivative capture non Markovian energy transfer. The one dimensional advection diffusion equation, for the purpose of initial validation and Burgers non-linear equation for the energy spectrum behavior are employed to investigate numerically the fNSE formulation. Moreover, the transient one-dimensional heat equation and the Caputo derivative embedded Burgers equations are solved, demonstrating the solution behavior regarding temporal memory effects. To simulate turbulent kinetic energy decay, we numerically solve the incompressible NSE using a pseudo spectral method in a 3D periodic domain, demonstrating the fNSE solution behavior. Key unresolved challenges include: Enforcing boundary conditions in fractional models. Hybridizing fNSE with Large Eddy Simulation (LES) or Reynolds Averaged Navier Stokes (RANS) approaches. Bridging fNSE with Lagrangian averaged models like Navier Stokes alpha. Calibrating fractional parameters and developing robust numerical strategies (e.g., preconditioning). These directions remain critical for future research.