Computation of gravity currents in estuaries

A great deal of literature has been devoted to gravity currents in estuaries. However, more or less detailed theoretical models of these phenomena are scarce. This is partly due to the fact that the equations have been difficult to solve if they describe the situation with some generality. This difficulty is surmounted by the use of digital computers. A more fundamental drawback is the lack of knowledge concerning the physical processes of turbulent flow in a stratified fluid. This precludes a detailed two- or three-dimensional description of the flow-pattern. Some schematical models exist which give an overall picture of the flow, still taking variations in space (along the estuary) and time (with the tide) into account. One of these is the two-layer model that is the subject of the present study. It is found that a great part of the information required for engineering applications can be obtained from it. A salt water and a fresh water layer are assumed to be present, either with or without mixing between them. Although flow in most estuaries is not strictly stratified, the two-layer schematization can be useful. This follows from an investigation of the approximations involved in the derivation of the equations. Empirically, the same fact is demonstrated by applying the two-layer model to the partly mixed Rotterdam Waterway. Knowledge of the turbulent flow processes, though in a less detailed form, is still required for a two-layer model, mainly to describe turbulent friction and mixing at the interface, as well as convection through it. If the interface is assumed to be impermeable, only the turbulent friction remains as an empirical parameter. Although the dynamical processes are reproduced less well in this case, the applicability is found to be superior, due to the small number of empirical parameters. Too little is yet known concerning the exchange of salt and water between the layers to permit a more detailed reproduction by means of the model with mixing. The latter therefore will be applied only if information on the salinity is required. The two-layer models result in mean velocities in each layer, and for the case with mixing also in mean densities. These parameters can be applied to define a family of velocity and density profiles. Combined with a crude model of the turbulent structure, this turns out to give reasonably realistic profiles. Therefore as an extension of the two-layer model an estimate of the velocity (and density) profiles can be given. The theory is verified by means of the 1956 measurements in the Rotterdam Waterway. A satisfactory correspondence is found, especially for the case without mixing. An estimate of the interfacial frictional coefficient as a function of the global conditions is obtained by hindcasting a number of flume tests. Although the present study is concerned mainly with estuaries, the two-layer model can be applied to several other cases of stratified flow, notably those concerned with thermal stratification. Such applications, however, require specific descriptions of empirical quantities, like mixing, friction, radiation.Hydraulic EngineeringCivil Engineering and Geoscience

The effect of the cell-Reynolds number in the numerical solution of the convection-diffusion equation

Internal memo. In the numerical solution of convection-diffusion equations a difficulty is commonly met in the occurrence of "standing waves" or spatial oscillations of a numerical origin. This has given rise to a considerable number of methods designed to avoid oscillations, often by means of upstream differencing or a similar approach in finite elements. In the present report, a short analysis of the phenomenon and some methods of remedy are given. The discussion is by no means complete, but it gives some general trends. For simplicity, the analysis is restricted to steady-state solutions of a linear convection-diffusion equation with constant coefficients and in one space dimensionHydraulic EngineeringCivil Engineering and Geoscience

Waterloopkundige berekeningen I

Hydraulic EngineeringCivil Engineering and Geoscience

Cursus Turbulentie

Interne cursus van het Waterloopkundig Laboratorium over turbulentie, bevat de onderwerpen: -basisvergelijkingen; -statistische beschrijving; -transportprocessen; -wandturbulentie; -tijdsafhankelijke grenslagen; -vrije turbulentie; -nieuwe turbulentietheorien

Experiences with mathematical models used for water quality and quantity problems

The use of models for solving water quality and quantity problems is not restricted to mathematical modeIs, as other types also have their possibilities and drawbacks. Mathematical methods, however, have certain properties which make them particularly suitable for water management studies. The examples of mathematical models dealt with in this paper involve one or two spatial dimensions (in the horizontal plane) and time. Research on two-dimensional modeIs in the vertical plane and three-dimensional modeIs is not far advanced enough for them to be available for routine use. Quality parameters that do not affect the flow of water are the only ones considered in this paper. Methods of evaluating the effects of buoyancy and stratification are peing developed but are not yet operationaI. Situations which can be investigated by means of present-day mathematical models are: a. Flow of water and dispersion of heat or dissolved substances in canals or systems of canals (networks), either for steady or for unsteady flows, b. The same phenomena in shallow lakes, estuaries or seas characterized by twodimensional flow in the horizontal plane. One-dimensional flow models have been used to study water management in several networks of canals. The studies concern both design problems, such as the location and capacity of new pumping stations, and decision problems connected with the operation of a system of pumping stations and sluices. The one-dimensional technique has also been successfully applied to the Oosterschelde estuary, much of which consists of natural channels. Similar methods are being used to determine the distribution of flow in the major beds of river systems when designing dikes. In the latter type of computation the flow is assumed to be quasi-steady. Once the flow in channels or rivers is known, either from one-dimensional models as described above, or from other sources, the water quality can be studied by means of convection-dispersion equations for the transport of dissolved substances or heat. Examples are the cooling-water circuit of a power plant (steady flow), the quality of water in artificial or natural storage basins (unsteady flow, including tidal effects) and the distribution of waste water in estuaries (considering net-flow only; dispersion is taken to include tidal mixing). The coefficient of dispersion involved in these models depends very greatly on the schematization and is empirical