The rheometry of free surface flows

Classically, open channel design for Newtonian turbulent flow is conducted using the pipe flow paradigm, incorporating the classical concept of the hydraulic radius. Although this approach has merit in turbulent flow, where inertial stresses dominate, it becomes problematic in laminar flow, where the tailings rheology dominates the flow behaviour. Arguably, the pipe flow paradigm for open channel flow has been taken as far as it is useful. The objective of this paper is to develop the fundamentals of sheet flow, and translate these into a sheet flow paradigm. Previous work on the pipe flow paradigm is reviewed, and the limits of its usefulness are exposed. A sheet flow approach is presented, developing a bulk shear rate which can be related directly to the tailings' rheology. This paradigm is then extended to a generalised laminar flow open channel flow model, incorporating the classical concept of the hydraulic radius. It is shown analytically that bulk shear rate is a unique function of wall shear stress for a given rheology and for all values of flow depth and slope. The significance of this is that in general the bulk shear rate is a unique function of the rheogram and the wall shear stress, and can be used for scale-up and design at any required slope and depth in laminar flow. Conversely, reversing the process provides the link between the pseudo shear diagram and the rheogram. This is validated using experimental results which show that the laminar data are collinear, the turbulent branches are succinct, and different for each slope, reminiscent of the pipe flow diagram. Experimental validation provides convincing evidence that the new sheet flow paradigm model for open channel flow can provide a competent basis for the analysis, flow behaviour prediction and design of open channel flow.