Weighted averaged equations for modeling velocity profiles of 1D steady open channel flows

Abstract

SUMMARYA new mathematical algorithm is proposed to address the essential details of vertical distributions of horizontal velocity for one‐dimensional steady open‐channel flow. This new algorithm comprises a system of weighted averaged equations developed from corresponding Reynolds equations by performing weighted average operations instead of conventional depth average operations. It is the system of weighted averaged equations, instead of the vertical grids, that allows for more hydraulic coefficients identifiable. It can be thought of as an extension of the St. Venant equations to address the vertical distributions of horizontal velocities, as well as the water surface profiles.To avoid the difficult expansion of governing partial differential equations in high order, an indirect scheme is proposed to solve hydraulic variables through their weighted average values. The governing partial differential equations are generated by using a variety of weight functions, and the weighted averages of relevant hydraulic variables are taken as the unknown independent variables to be solved first. Then, on the basis of the values and polynomial expansions of these weighted averaged velocities, a system of linear algebraic equations is generated and the unknown hydraulic variables or their coefficients are easily solved.Note that the new model is not proposed to compete with any three‐dimensional models in modeling accuracy or accommodation ability to all conditions. It just provides a valuable option to study the vertical structure of flow in open channels where only essential detail and reasonable accuracy of vertical distributions are required, and the data availability and other conditions limit the application of fully three‐dimensional models. The performance of the model is evaluated with experimental data of flows in two different flumes. It is shown that the model well predicted the velocity profiles of sections along the centerlines of these flumes with reasonable accuracy and essential details of vertical distributions of horizontal velocity. Copyright © 2013 John Wiley & Sons, Ltd.

Depth‐integrated nonhydrostatic free‐surface flow modeling using weighted‐averaged equations

“…These can be generated using the method of weighted residuals. () The weighted average of a function f is defined by where F is the weight function. If the weighting function is F = 1, Equation degenerates into the standard depth‐averaged operation.…”

“…These can be generated using the method of weighted residuals. 1,54 The weighted average of a function f is defined by…”

“…The assumed parabolic profiles for pressure and vertical velocity() are reasonable, but the linear distribution used to model the horizontal velocity component is quite crude, although being the next degree of improvements over the widely used uniform velocity. The weighted residual method has been applied recently by Xia et al to model the steady turbulent velocity profile in open‐channel flow with hydrostatic pressure.…”