Reproduction of Velocity Profiles in Estuaries by some One-Dimensional Mathematical Models

Abstract

For the prediction of dispersion phenomena and of changes in the morphology of an alluvial bottom, a detailed description of the water flow is necessary. The flow in estuaries is a complicated one, partly because of the time-dependence. To isolate this aspect of tidal flow a simplifying one-dimensional (vertical) flow model 1S used. This one-dimensional model is obtained by the neglect of convective derivatives of the longitudinal velocity and the use of the rigid lid approximation, i.e. the replacement of the free surface by a flat frictionless plate. The error introduced by these approximations is not large for the flow in most tidal channels. Convective derivatives are generally of minor importance. The rigid lid approximation is inaccurate for tidal waves with a large ratio between wave height and water depth. Tidal flow is usually described by simple eddy viscosity models in which various simple distributions of the eddy viscosity are prescribed. Recently models with an eddy viscosity depending on the turbulence energy have gained wide acceptance for all kinds of boundary layer flow. In this investigation the k-model and the k-e:-model are compared to an eddy viscosity model with an appropriate distribution of the eddy viscosity and to the mixing-length model for the case of steady and of time dependent free surface flow. The time dependent free surface flows, considered, represent flows in a tidal channel without a nett discharge over the tidal period. The roughness values and the velocities are typical for tidal channels. The results of the various models differ hardly. The only appreciable difference is around slack water, where all models used are, however, less reliable. The close correspondence is explained by the short adjustment times of the turbulence energy and its dissipation compared to the tidal period and by the small relative roughness height. The flow in a tidal channel can be considered as slowly varying, showing almost logarithmic velocity profiles except around slack water. The hysteresis effect of the shear stress with respect to the surface velocity calculated with all these models is therefore small, in contradiction to the large hysteresis effect as found in some of the prototype measurements.