Stochastic Model of Turbulent Channel Flow

A stochastic model of incompressible turbulent channel flow is developed. The model is based solely upon knowledge of the mean velocity profile and energy considerations derived from the Navier-Stokes' equations, yet provides a description of the entire turbulent field. In the model, turbulence is presumed to occur as highly structured elements of fluid or eddies. These eddies are initiated along the walls as a Poisson Process and diffuse outward into the flow. The diffusion of eddies is described by a Wiener process in the high shear regions near the walls and an Ornstein-Uhlenbeck process in the center region. A mathematical form for the eddies is postulated which is a random function of the entire past history of the eddy in relation to the mean shear. The eddies grow and then decay as the flow progresses. From such information, a stochastic description of the velocity at any point in the flow is obtained. All spatial correlations of the velocity field may thus be determined through an analysis in which Eulerian statistics are derived from Lagrangian densities. Second moments are computed and compared with available experimental data.