Author(s): Robert F. Dressler

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Abstract: New equations are derived which are a generalization of the nonlinear unsteady shallow-flow partial differential equations, usually called the Saint-Venant equations. The new equations possess terms containing explicitly the curvature of the channel bottom, and the derivative of the curvature. The resulting streamlines are therefore curved, and the pressure expression contains terms, in addition to the hydrostatic term, which describe the effect of the streamline curvatures. The new equations should give more accurate flow solutions than Saint-Venant's whenever the channel bottom is curved or non-horizontal; for a flat horizontal channel they reduce to the Saint-Venant equations. The equations show that the velocity is no longer constant over any cross-section orthogonal to a curved bottom. The method of derivation starts with a specific set of curvilinear coordinates, and follows boundarylayer asymptotic methods requiring stretching of the flow domain in transformed independent variables. For steady flow, the new equations define a generalization of the Bresse profile equation, when a Chezy resistance term, modified for curvature, is added. A curve is presented showing when flow separation, defined in terms of the local Froude number, from a convex bottom will occur as a function of channel curvature. The new equations are hyperbolic, and the directions of the real characteristic curves are derived. The locus for critical flow is graphed as a function of curvature. Although our equations contain more terms and more complicated coefficients than the Saint-Venant equations, they are identical in structure, and therefore should be just as easy to solve by computer for unsteady flows as the Saint-Venant equations have proven to be. Our equations are immediately applicable to hydraulic flows. With some slight modifications, they should also become applicable to new non-linear analyses of certain meteorogical problems such as stratified flow and interfacial wave propagation over mountainous terrain.

DOI: https://doi.org/10.1080/00221687809499617

Year: 1978