Fractional Reynolds-averaged Navier Stokes equations (f-RANS) for modeling of separated boundary layers.

The duality of local and non-local regions observed in turbulent flows makes defining mathematically rigorously operators a cumbersome task with literature focused on either local or non-local modeling. Recently, we showed that a variable-order fractional operator can not only model both the regimes but also seamlessly transitions from local to non-local regimes. The effect of pressure gradients further complicates the problem, especially in separated flows, which has been a long standing problem in the turbulence modelling community. Thus, in this work we investigate the role of pressure coupled with Reynolds number effects for adverse pressure gradient boundary layers, including modelling of the recirculation bubble. Upon the formulation of one- and two-sided models using Caputo fractional derivatives, we found that only the two-sided model leads to physical solutions. This is not a surprise as non-locality at a given point is an aggregate effect of all directions and the two-sided model addresses this very fact. The predictive nature of this formulation presents a model free of ad-hoc tuning coefficients thereby providing the basis for a robust engineering tool.