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The first fractional model for Reynolds stresses in wall-bounded turbulent flows was proposed by Wen Chen [ 2 ]. Here, we extend this formulation by allowing the fractional order  α ( y ) of the model to vary with the distance from the wall ( y ) for turbulent Couette flow. Using available direct numerical simulation (DNS) data, we formulate an inverse problem for  α ( y ) and design a physics-informed neural network (PINN) to obtain the fractional order. Surprisingly, we found a  universal scaling law  for  α ( y  + ), where  y  +  is the non-dimensional distance from the wall in wall units. Therefore, we obtain a variable-order fractional model that can be used at any Reynolds number to predict the mean velocity profile and Reynolds stresses with accuracy better than 1%.

Discovering a universal variable-order fractional model for turbulent Couette flow using a physics-informed neural network