A general equation to calculate sequent depth ratio of different types of jump in sloping beds

Hydraulic jumps in sloping channels were first classified by Kindsvater (1944). An A-jump is the jump that starts at the foot of the chute. For the B-jump, the toe of the jump forms on a positive slope and the roller ending on the downstream stilling basin. The C-jump begins on a positive slope and the roller length ending at the foot of the chute and finally the D-jump that its roller length entirely occurs on the chute. The most common type of jump in practice is the B-jump which from the point of hydraulic calculation, is more complicated than others (Hager, 1988). For a jump on sloping beds, application of the one-dimensional (1-D) momentum equation is not easy to compute the sequent depth ratio because some additional information is needed to estimate both the weight component of the jump and the bottom pressure acting on the sloping channel portion (Carollo et al, 2011). Hence, for B-jump sequent depth ratio, previous researchers developed several empirical equations to compute the sequent flow depth of the jump. Although, many researches have been done about jump on smooth sloping beds, so far there has not been a study on roughened sloping beds. The primary aim of the present study was to evaluate a general equation to calculate the sequent depth ratio of different types of jump on the roughened sloping bed. For this purpose, different types of jump on both the smooth and the roughened sloping bed were investigated experimentally in a wide range of relative roughness (0.032 – 0.54), different slopes of chute (14.5, 20.5 and 27.5 degree) and Froude number (1.3 – 7.4). Using -Buckingham theory of dimensional analysis, the effective parameters for determination of sequent depth ratio on smooth and rough bed were obtained and by applying of the incomplete self - similarity theory, a general equation was derived. Then, the coefficients of the general equation were derived and its accuracy was determined, using experimental data from present study and previous published resources.
One of the main reasons of using roughened beds is to dissipate some of the kinetic energy of the flowing water over the chute. In order to examine the effect of the roughness size, the Froude number at the start of the jump and the sequent depth were calculated. In this study, The variation between the sequent depth ratio and upstream Froude number for different types of jumps and for relative roughness sizes (ks/yc) was figured. A parameter E is introduced by Hager (1944) and defines as the location where the jump begins. The results showed for a given range of E, by increase of the relative roughness, the upstream Froude number and sequent depth ratio decreased. In order to develop a functional relationship for computing the B-jump sequent depth ratio on sloping smooth beds, the function m should be determined from experimental data. Therefore, the procedure of Carollo et al (2011) was followed and the values of m(α,E) were calculated for different slopes. In this function, for a given chute angle α, the function m decreases as E increases, and for a fixed E value, m decreases as the angle α increases.
The form of relation between the function m and the variable E is exponential. The empirical relationship ( ) was fitted to each experimental series corresponding to a given α value. In which a and b are coefficients that have to be experimentally determined. For smooth beds, the coefficients a and b depend only on the chute slope (α). In order to illustrate the effect of roughness height on the function m (or the sequent depth ratio), for each slope, the m values were plotted versus E values for various roughness heights. For example, the results of chute with bed angle of 14.5 illustrated that the coefficients a and b depended not only on the slope of the chute, but also on the roughness height. Also, the measurements, for different roughness sizes, indicated that m decreased as E increased and, for a fixed E value, m slightly decreased when the roughness size increased. A comparison between the rough and hydraulically smooth bed conditions shows that boundary roughness reduced m and the sequent depth ratio. This result confirmed the findings of previous investigations. The coefficients a and b was obtained from the investigations of Carollo et al (2011) in the analysis of the data in this study. Using experimental data, two equations obtained for a and b coefficients. Thus, a general equation for sequent depth ratio (Y) obtained. A comparison between experimental Y values and those calculated by main Equation was illustrasted in this study. The lines representing the 15% error range are also illustrated.
The results showed that the general equation has high accuracy and the average percentage error is about 7.5%. Also, the result showed that increasing relative roughness decreased the sequent depth ratio and the Froude number of the upstream flow and when relative roughness is about 0.54 the upstream Froude number and sequent depth ratio decreased about 50%.