Turbulent Flow in Geophysical Channels

The problem of turbulent flow in a rough pipe of arbitrary shape is considered. The classical Izakson-Millikan argument for a logarithmic velocity profile is presented, and matched asymptotic expansions are introduced. Scaled, dimensionless equations are produced and simplified. A simple mixing length turbulence model is presented, which closes the problem. To calibrate the model, the mechanical problem is solved in the case of a circular pipe. Excellent agreement with engineering relations is obtained. The mechanical problem for a non-circular pipe is posed, and the boundary layer problem is solved. This leaves unknown the wall stress, which is sought through approximate methods of solution in the outer region. These are presented and the approximate solutions thus obtained are compared to full numerical solutions and data for a square, elliptical, and semi-elliptical pipe. The approximations are vindicated, but agreement between the numerical solutions and data is only moderate. Discrepancies are explained in terms of the neglected secondary flow. The thermal problem is posed, with scalings taken for intended application in glaciology. The problem is solved for a circular pipe. Heat transfer results are presented and compared with empirical relations. The general problem for a non-circular pipe is posed, and approximate methods of solution are motivated, in analogy to those used for the mechanical problem. These are used to obtain approximate solutions, which are compared with numerical solutions, to good agreement. Possible applications of these solutions are discussed.