A transport equation for the one-point velocity probability density function (pdf) of turbulence is derived, modelled and solved. The new pdf equation is obtained by two modeling steps. In the first step, a dynamic equation for the fluid elements is proposed in terms of the fluctuating part of Navier-Stokes equation. A transition probability density function (tpdf) is extracted from the modelled dynamic equation. Then the pdf equation of Fokker-Planck type is obtained from the tpdf. In the second step, the Fokker-Planck type pdf equation is modified by Lundgren's formal pdf equation to ensure it can properly describe the turbulence intrinsic mechanism. With the new pdf equation, the turbulent plane Couette flow is solved by the direct finite difference method coupled with dimensionality reduction and QUICKER scheme. A simple boundary treatment is proposed such that the near-wall solution is tractable and then no refined grid is required. The calculated mean velocity, friction coefficient, and turbulence structure are in good agreement with available experimental data. In the region departed from the center of flow field, the contours of isojoint pdf of V-1 and V-2 is very similar to that of experimental result of channel flow. These agreements show the validity of the new pdf model and the availability of the boundary treatment and QUICKER scheme for solving the turbulent plane Couette flow.
INTERNATIONAL JOURNAL OF NUMERICAL METHODS FOR HEAT & FLUID FLOW