Open Channel Flow Equations Research Articles

Discussion of “One-Dimensional Equations of Open-Channel Flow”

Discussion of “One-Dimensional Equations of Open-Channel Flow”

One-Dimensional Equations of Open-Channel Flow

One-dimensional equations of continuity, momentum, and energy for unsteady, spatially varied flow in a fixed-bed open channel of arbitrary configuration are derived from the point forms of the corresponding hydrodynamic equations by integrating the latter over a deforming region of space comprising a slice of differential thickness across the flow with a top always coincident with the fluctuating water surface. To bring the equations to the form of Saint Venant in fixed and accelerating reference frames, departures from hydrostatic pressure conditions in a cross section due to lateral acceleration, viscous deformation, and turbulent Reynolds stresses are reproduced exactly; the rate of energy dissipation in a cross section is uniquely related to wall shear over the wetted perimeter; the term accounting for lateral discharge depends upon its nature; bulk outflow is considered.

Flow of Clay-Water Suspensions

Previous publications have shown that suspensions of clay particles in water exhibit non-Newtonian rheological behavior. This behavior can extend the laminar flow regime to extremeley high Reynolds numbers in pipeline flow and demonstrates that laminar flow may be encountered in practice in open channels on rather steep slopes. Laminar and turbulent pipeline flow and plug flow of these suspensions are reviewed, and equations for open-channel flow are introduced. Sample calculations are included to demonstrate the degree of divergence from Newtonian behavior. The existence of a fluid yield stress is proven by open-channel observations that are reported in literature.

THE ANALYSIS OF HARBOR AND ESTUARY SYSTEMS

It is one of the paradoxes of our age that hydraulic engineering is concerned with problems that in many ways exceed in difficulty those encountered in the more glamorous fields of science. One measure of this situation is that classical mathematics, which is a powerful tool when applied to simple systems, has proved to be a rather impotent aid m hydraulic calculations, except when the geometry is simple. Harbor and estuary systems are usually associated with complex geometry, and thus we ordinarily cannot depend on mathematics to give general solutions. Instead of solving the hydraulic equations of flow in the complex geometry, we have sought to reconstruct the geometry of the prototype in a reduced scale, by means of models, and by assuming that the equations governing the full-size and reduced-scale systems are the same, to find specific solutions through direct measurements in the latter. Because of scale effects, the hydraulic model is not perfect, but it does reproduce the complex geometry and some of the complexities of three dimensional flows in the prototype. These cannot at present be described mathematically, and where they are important a properly constructed and adjusted hydraulic model is our most powerful tool in the investigation of estuarial problems, and is likely to remain so.
 However, hydraulic models are very expensive, especially if they are built to a reasonably large scale, so it behooves us to be sure that we have not overlooked other, perhaps less powerful, methods when we are confronted with a particular problem. Within the past decade two methods have been developed that will eventually replace hydraulic models for tidal flow and river flood routing investigations. One is based on the use of digital computers to numerically integrate the differential equations of open channel flow; the realization of this method by a digital computer program for a particular system has been called a "mathematical model". The second is based on the use of analog elements that behave with respect to electrical current in the same way as the prototype behaves with respect to flows of water. When an assembly of such elements is adjusted to duplicate the behavior of a hydraulic system in a way similar to the way a hydraulic model is adjusted during its verification period, the result may be called an analog model.

Discussion of “Tabular Solution of Open Channel Flow Equations”

Discussion of “Tabular Solution of Open Channel Flow Equations”

Tabular Solution of Open Channel Flow Equations

Presentation and explanation of table by which cubic equations which arise in open channel flow problems may be solved without tiresome trial and error method usually necessary; use is limited to rectangular channels.