Research on a shallow water mixing layer

Abstract

A shallow water mixing layer is a flow pattern which develops between two adjacent streams with a different velocity. The horizontal dimensions of a shallow water mixing layer are much larger than the depth. This results in a turbulent process in the mixing layer which is characterized by the presumable presence of two distinct turbulent length scales. Small scale turbulence is present due to the bottom friction and large scale structures develop horizontally over the whole vertical interface between the two flows. It is assumed that no intermediate length scales can develop due to the limited water depth. After a certain distance downstream, the large eddies appear not to develop any more. This can be attributed to an equilibrium between the energy supply to the large scale eddies, due to the velocity difference across the mixing layer, and a direct energy loss to bottom turbulence. First, a one-dimensional model is developed to describe the development of flow parameters of the shallow water mixing layer, such as the velocities outside the mixing layer, the water depth and the displacement of the mixing layer. A model matrix is derived on the basis of the equation of continuity and the Navier Stokes equations describing two adjacent flows in a wall-limited shallow water flume. The complex flow pattern in the mixing layer is modelled using an approximation for the mixing layer development and assuming an error function velocity profile over the mixing layer width. The longitudinal pressure gradient, caused by the bottom friction, results in a displacement of the mixing layer to the low velocity side, a smaller velocity difference across the mixing layer and a larger slope downstream. Results based on the error function profile are compared to those of a linear velocity profile. Comparison of the error function velocity profile results with propeller measurements and LDA-measurements shows a reasonable correspondence between model results and measurements. The small differences that occur can possibly be attributed to the neglect of a slope in lateral direction in the model. Second, the turbulent process in a shallow water mixing layer is investigated to reveal the possible presence of two distinct turbulent length scales in the mixing layer. The two distinct length scales are determined by means of measuring the spatial correlations in lateral direction of the mixing layer. Since the spatial correlations measured are caused by small and large scale turbulence, it is assumed that the correlation consists of two parts which are independent from each other. One part is assumed to be caused by the small scale turbulence and the other by the large scale eddies. After separating the two parts, the length scales are determined by integrating the two contributions to the correlation function. Measurements were taken at two elevations at 16 meter downstream from the point where the two flows come together. This results in an order of magnitude difference between the two calculated length scales. At a smaller elevation, the small turbulence length scale is significantly smaller than at a larger elevation. The large turbulence length scale is smaller as well, but this decline is relatively less than the small scale turbulence decline. The decline of the large scale turbulence can possibly be attributed to the length scale determination method used.