Analysis of Turbulent Hydraulic Jump over a Transitional Rough Bed of a Rectangular Channel: Universal Relations

Abstract

The streamwise flow structure of a turbulent hydraulic jump over a rough bed rectangular channel has been investigated. The flow is divided into inner and outer layers, where upstream supercritical flow changes to downstream subcritical flow. The analysis is based on depth averaged Reynolds momentum equations. The molecular viscosity on the rough bed imposes the no slip boundary condition, but close to the wall the turbulent process in inner layer provides certain matching conditions with the outer layer, where molecular viscosity has no dominant role. It is shown that the bed roughness in the inner layer has a passive role in imposing wall shear stress during formation of hydraulic jump in the outer layer. The Belanger’s jump condition of rectangular channel has been extended to account for the implications of the drag attributable to channel bed roughness, kinetic energy correction factor, and coefficient of the Reynolds normal stresses. For depth averaged Reynolds normal stress, an eddy viscosity model containing gradient of depth averaged axial velocity is considered. Analytical solutions for sequent depth ratio, jump length, roller length, and profiles of jump depth and velocity were found to depend upon the upstream Froude number, drag owing to bed roughness, and kinetic energy correction factor. On the basis of dynamical similarity, the roller length and aeration length were proposed to be of the same order as the jump length. An effective upstream Froude number, introduced in the present work, yields universal predictions for sequent depth ratio, jump length, roller length, jump profile, and other hydraulic jump characteristics that are explicitly independent of bed roughness drag. Thus, results for hydraulic jump over a rough bed channel can be directly deduced from classical smooth bed hydraulic jump theory, provided the upstream Froude number is replaced by the effective upstream Froude number. These findings of universality have been supported by experimental data over a rough bed rectangular channel.