Mixing Induced by an Internal Hydraulic Jump1

Abstract

The efficiency of an inverted internal hydraulic jump as a mixing and dispersion mechanism in an aquatic environment was examined. The flow considered was a two‐dimensional buoyant flow from a shallow channel over a sloping bottom into a deep reservoir. It could be seen that a rapidly varied flow associated with violent turbulent mixing occurred near the point of discharge if specific discharge conditions and downstream controls were met. Downstream from the mixing zone the flow was stably stratified. The main object of the study was to find the conditions under which a turbulent mixing zone occurred and the rate of turbulent entrainment. Energy loss and length of the mixing zone were also investigated. The independent variables were the outlet densimetric Froude number, the density differential between outfall water and receiving water, the relative depths of the upper and lower layers in the stratified flow portion, the total depth, and the slope of the transition. Theoretical calculations had to be confined to a step increase in depth, but experiments in a laboratory flume showed that results obtained with slopes of 23° and 90° were quite similar. Both theory and experiments showed, for example, that dilution (entrainment rates) up to 2:1 (2 parts heated water to one part cold water) can be achieved at very little energy expense and with downstream depths approximately 8 times the outlet depth. Theoretically, any amount of dilution can be obtained, but large depths may be required. Experimental results also indicate that the length of the mixing zone was frequently ten times the value of the outlet densimetric Froude number.

Plane turbulent buoyant surface jets and jumps

“…The depth bj at the end of the jump, which was equal to ba>, was also measured. Since it was found that there is no entrainment in the jump region, the two subsequent depths b\ and bj are related by the Belanger type equation (17 (14) U\ being the mean (or depth averaged) velocity and Ae the mean density deficiency at the beginning of the surface jump. Knowing b\ and b\, F\ can be obtained from equation (13).…”

“…But one finds in the literature a number of rather inconclusive (or even contradictory) statements regarding this surface jump. Some investigators indicate that under certain circumstances, the surface jumps entrain the ambient fluid whereas under certain other circumstances, the jumps are of the non-entraining type (14,16). If the jumps did indeed entrain the ambient fluid, then downstream sequent depth ratio is not a function (anymore) of only the upstream, Richardson (or densimetric Froude) number.…”

“…It could also rise as a buoyant surface stratified flow with the invasion of the ambient water in the form of a wedge into the (upstream) channel which carries the warm water discharge (4). Experimental observations on plane surface jets in stagnant ambients have been made by Ellison and Turner [3], Wilkinson and Wood [16], Stefan [13], Stefan and Hayakawa [14], Chu and Vanvari [1] and Rajaratnam and Humphries [10]. Based on the work of Rajaratnam and Humphries [11] it appears that if the surface jet is non-buoyant, that is Rio = 0, it grows linearly and if b\s its characteristic thickness, equal to the vertical distance ƒ from the surface where the time-averaged longitudinal velocity «is equal to one halfofthe maximum value u m of «at that x station, dA/dx= 0.07.…”

“…greater than a few thousand) the flow in the surface jet becomes turbulent very quickly and this study deals with only turbulent jets. Depending upon Richardson number at the source Ri 0 (equal to gjflAgo/ga Revision wherein g is the acceleration due to gravity and AQ 0 = (Q 3 -Q 0 )) and the downstream control, the jet might grow in thickness like a non-buoyant surface jet, or grow in thickness for some longitudinal distance and then become a stratified surface layer or form a surface jump or form a surface jump at the outlet itself or even form a drowned surface jump ( 1,2,11,(13)(14)(15)(16). It could also rise as a buoyant surface stratified flow with the invasion of the ambient water in the form of a wedge into the (upstream) channel which carries the warm water discharge (4).…”

Calculation of the near field of velocities and temperatures beyond a surface discharge of heated water

“…A solution similar to that above of the problem of contact of a buoyant surface jet with a pool of a water body based on introducing characteristic scales of the jet and direct solution of a simplified equation of momentum without changing to an equation of form (6) was proposed in [7] independently of the authors, and in this work the relation between variables h and q is not symmetric and the schemes of contact are not included, and therefore a comparison of the theoretical solution of Eq~ (6) with the experimental data, which is given in Fig. 4, is of great interest.…”