A fractional PDE model for turbulent velocity fields near solid walls

Abstract: This paper presents a class of turbulence models written in terms of fractional partial differential equations (FPDEs) with stochastic loads. Every solution of these FPDE models is an incompressible velocity field and the distribution of solutions is Gaussian. Interaction of the turbulence with solid walls is incorporated through the enforcement of various boundary conditions. The various boundary conditions deliver extensive flexibility in the near-wall statistics that can be modelled. Reproduction of both fully-developed shear-free and uniform shear boundary layer turbulence are highlighted as two simple physical applications; the first of which is also directly validated with experimental data. The rendering of inhomogeneous synthetic turbulence inlet boundary conditions is an additional application, motivated by contemporary numerical wind tunnel simulations. Calibration of model parameters and efficient numerical methods are also conferred upon.