Abstract
The aim of this study was to assess and quantify the effect of channel bed roughness on hydraulic jumps based on sound physical theories. Assuming that integrated bed shear stress due to surface roughness changes linearly with supercritical velocity, a novel definition for the shear force coefficient and for roller length were obtained. Experimental findings and Pearson's correlation verify that the developed equations perform reasonably well and they prove that a linear correlation assumption between integrated bed shear stress and supercritical velocity is valid for a Froude number between 1.1 and 9.8. The shear force coefficient is defined in terms of the Reynolds coefficient and the supercritical flow velocity is directly related to the modified Reynolds number. A new analytical equation for roller length as a function of the modified Reynolds number was also developed and validated by using data from the experimental study.
HIGHLIGHTS
The tendency of the Reynolds coefficient to approach a fixed value increases as the modified Reynolds number increases.
The results of this study showed that the modified Reynolds number is important in roller length analysis and cannot be ignored.
Pearson's correlation analysis showed a significant positive linear correlation between integrated bed shear stress and upstream supercritical velocity.
INTRODUCTION
The well-known Belanger equation, which defined the relationship between conjugate depths along hydraulic jumps, does not take into account the effect of the bed roughness along the jumps. On the other hand, due to their morphological evolution, natural channels like rivers and creeks are lined with rough surfaces. Therefore, the pertinent literature on hydraulic jump along rough beds has aimed to model the influence of bed surface roughness on the ratio of conjugate depths and the length of the hydraulic jump. The systematic analysis of hydraulic jumps along hydraulically rough beds was first initiated by Rajaratnam (1968), defining the conjugate depth ratio as a function of the supercritical Froude number and the equivalent relative roughness. Later, Hager & Bremen (1989) designed a laboratory model to observe the effect of bed friction on the conjugate depth ratio and provided a fair agreement with the experimental results. The contribution of Hager & Bremen (1989) showed that the sequent depth ratio is influenced not only by the Froude number, but also by the inflow Reynolds number and the aspect ratio. Ead & Rajaratnam (2002) investigated hydraulic jumps on corrugated beds at a predefined range of Froude numbers (from 4 to 10). They found that the tailwater depths of hydraulic jumps along the corrugated beds were considerably smaller than those along smooth beds. They also added that the length of jumps on corrugated beds is half length when compared to smooth beds.
Another important contribution to the thematically relevant literature is the experimental work carried out by Carollo et al. (2007) performing 200 experiments in a horizontal rectangular flume measuring flow rate, upstream depth, tailwater depth and jump length on an artificially roughened bed varying from 0.46 cm to 3.2 cm. Carollo et al. (2007) solved the momentum equation for deriving a new conjugate depth ratio equation as a function of the upstream Froude number and the ratio between the roughness height and the upstream supercritical flow depth. Later, Carollo et al. (2009) proposed a new generalized solution for the sequent depth ratio over both smooth and rough horizontal beds. Karbasi & Azamathulla (2016) developed new equations based on gene expression programming in order to determine the characteristics of hydraulic jump over rough beds and to compare their results with other soft computing techniques.
Saghebian (2019) used support vector machine as an intelligence approach to relate conjugate depth ratio and the energy dissipation for rough and smooth channels. The performance of a support vector machine model was good for predicting energy dissipation in smooth and rough bed channels. Kumar et al. (2019) proposed an artificial neural network model for rounded and crushed aggregates reporting that reductions in sequent depth ratios were greater in the case of crushed aggregates than rounded aggregates. Recently, Mahtabi et al. (2020) classified a hydraulic jump over rough beds (natural and artificial) based on Froude number by using a decision tree and neural network classifiers. They aimed to produce simple, understandable, and practical results with high accuracy and reliability.
The roller length is one of the major geometric features defining a hydraulic jump and is generally accepted as the horizontal distance from the toe of a jump to the exit of that jump, where the stagnation point separates the region of backward surface flow in the roller and the forward subcritical flow. Previous studies on roller length consistently defined it as a function of the conjugate depth ratio and the Froude number. In some of these studies (Roushangar & Ghasempour 2018; Saadatnejadgharahassanlou et al. 2020), the roller length is analyzed for hydraulic jumps occurring on smooth surfaces, while others (Carollo et al. 2007; Pagliara & Palermo 2015) investigated roller length on rough beds. Palermo & Pagliara (2017) performed experiments on small channel slopes (both negative and positive), and concluded that the effect of channel bed roughness on the sequent depth ratio depends mainly on the dimensionless ratio between the sediment mean diameter and the critical flow depth.
The motivation of this study is to explore and illustrate the effect of bed roughness on the characteristics of a hydraulic jump, such as the conjugate depth ratio, the roller length and the shear force coefficient. The proposed model is developed to calculate the integrated bed shear stress by assuming that it linearly changes according to the supercritical flow velocity. A momentum equation is then solved: (a) describing the shear force coefficient in terms of the flow variables like the supercritical flow velocity and the depth; (b) relating the shear force coefficient with the modified Reynolds numbers; and (c) solving the roller length analytically in terms of the Reynolds coefficient. The proposed model is validated by using the outcomes of the conducted experiments and the experimental data provided by pertinent literature like Carollo et al. (2007) and Hughes & Flack (1984).
MATERIALS AND METHODS
Derivation of shear force coefficient based on momentum equation











Conceptual sketch for hydraulic jump over rough beds. is the integrated bed shear stress,
and
are the hydrostatic forces at the inflow and outflow of the control volume, respectively,
is the flow depth at the toe of the jump,
, is the flow depth at the end of the jump and
is the roller length.
Conceptual sketch for hydraulic jump over rough beds. is the integrated bed shear stress,
and
are the hydrostatic forces at the inflow and outflow of the control volume, respectively,
is the flow depth at the toe of the jump,
, is the flow depth at the end of the jump and
is the roller length.














Plot of the supercritical flow velocity and the integrated bed shear stress on rough surfaces, based on the experimental data of Carollo et al. (2007). The x-axis represents the velocity of the flow before the jump and the y-axis represents the integrated bed shear stress calculated by using the experimental data of Carollo et al. (2007) in Equation (3).
Plot of the supercritical flow velocity and the integrated bed shear stress on rough surfaces, based on the experimental data of Carollo et al. (2007). The x-axis represents the velocity of the flow before the jump and the y-axis represents the integrated bed shear stress calculated by using the experimental data of Carollo et al. (2007) in Equation (3).


Equation (13) reduces to the well-known Belanger equation since the integrated bed shear stress is negligibly small . However, in the case of natural channels, the shear force coefficient is not equivalent to zero. The bed roughness, at constant upstream velocity, reduces the conjugate depth ratio while increasing the integrated bed shear stress. The increase in integrated bed shear stress is directly related with an increase in the Reynolds coefficient. Any reduction on conjugate depth ratio proportionally reduces the length of the roller. Therefore, it becomes essential to solve analytically the relationship between the roller length and Reynolds coefficient.
Analytical solution for the roller length














Analytically derived Equation (21) reflects the effect of bed roughness on the magnitude of the roller length. The roller length definitions derived in previous studies assumed that was only proportional to the conjugate depths, upstream Froude number, and sometimes to the supercritical flow velocity.
LABORATORY EXPERIMENTS
Experimental installation and methods
The experimental work was conducted in a 0.3 m wide by 0.4 m deep glass walled recirculating tilting flume, at the Water Engineering laboratory of the University of Glasgow. The flume was 15 m long and the channel slope was 1 in 160. The channel bottom was roughened with varying sizes of sand and gravel particles. The sand and gravel particles were glued with epoxy to a 2.45 m long by 0.3 m wide, 8 mm thick timber panel located on the channel floor. Four different timber panels were covered with different granulometric sand and gravel sizes, and the nominal diameter of sand and gravel particles used ranged from 1.4 mm to 12.7 mm. The granulometric characteristics of the bed materials are given in Table 1.
The granulometric sizes of bed surface material used in the flume experiments
Material Code . | Nominal diameter range (mm) . | d50 (mm) . | d84 (mm) . |
---|---|---|---|
A | smooth surface | – | – |
B | 1.40–2.35 | 1.8 | 2.3 |
C | 2.00–3.35 | 2.1 | 2.5 |
D | 4.75–8.00 | 6.3 | 7.5 |
E | 8.00–12.7 | 10.5 | 12.0 |
Material Code . | Nominal diameter range (mm) . | d50 (mm) . | d84 (mm) . |
---|---|---|---|
A | smooth surface | – | – |
B | 1.40–2.35 | 1.8 | 2.3 |
C | 2.00–3.35 | 2.1 | 2.5 |
D | 4.75–8.00 | 6.3 | 7.5 |
E | 8.00–12.7 | 10.5 | 12.0 |
The experiments were conducted with different sluice gate openings ranging from 2.1 cm to 2.5 cm. The test area was located 4.45 m away from the inflow of the flume. The water surface elevation across the channel was controlled by accurately fixing the angle of the flume's tailgate to 153° from the horizontal. The mean flow velocity was estimated via two independent methods: the first was calculated from the cross-sectional average of the measured flow velocities across the channel; the second was calculated from the mean flow discharge Q, which was calculated from the average measurements of electromagnetic flow meters attached to the inlet pipes, via which water was circulated in and around the flume. The flow conditions were chosen appropriately, so that the glued sand and gravel particles were not eroded away during the experiments, and the hydraulic jump occurred at the vena contracta below the sluice gate. The depth of flow before and after the hydraulic jump was measured with the aid of two electronic distance measuring gauges with sub-millimeter precision.
Experimental conditions and outcomes
The bed roughness can be interpreted by using the granulometric characteristics of the sand and gravel particles, such as the median diameter (d50), grain diameter for which 84% (d84) or 99% (d99) of grains are finer, and the equivalent sand roughness (ks). Cheng (2016), after analyzing previous studies, states that a grain diameter greater than at least d50 should be used to reflect the dominant effect of sand and gravel particles on the flow and to limit this size to vary from d65 to d90. Since larger particles have a relatively higher effect on flow retardation than smaller particles, d84 is suggested in many studies as a useful scale for bed roughness (Ferguson 2007; Rickenmann & Recking 2011). Thus in this study, d84 was selected as the representative granulometric characteristic of the sand and gravel particles to be used during the experiments on hydraulic jump.
During this study, 60 experiments were conducted in total. The flow discharges varied between 17.2 and 24.9 l/sec and the Froude number at supercritical flow condition ranged from 1.1 to 9.8. Table 2 provides fundamental outcomes of the conducted experiments such as the discharge, Q, the upstream and downstream flow depths h1 and h2, and the roller length, Lr.
Characteristic data of the experimental runs and dimensionless variables
ks (cm) . | d84 (mm) . | h1 (cm) . | h2 (cm) . | Q (l/s) . | Lr (cm) . | F1 . |
---|---|---|---|---|---|---|
0 | 0 | 5.91 | 8.81 | 17.8 | 52.6 | 1.32 |
0 | 0 | 6.92 | 9.17 | 20.1 | 59 | 1.18 |
0 | 0 | 6.25 | 12.07 | 22.3 | 43.5 | 1.52 |
0 | 0 | 5.54 | 12.20 | 23.6 | 53.3 | 1.93 |
0 | 0 | 6.84 | 8.44 | 18.6 | 48.7 | 1.11 |
0 | 0 | 5.86 | 9.78 | 21.2 | 44.8 | 1.59 |
0 | 0 | 5.23 | 11.92 | 22.2 | 43.5 | 1.97 |
0 | 0 | 4.69 | 13.58 | 23.6 | 46.2 | 2.47 |
0 | 0 | 6.52 | 7.81 | 17.5 | 51.1 | 1.12 |
0 | 0 | 5.76 | 9.32 | 19.6 | 49.3 | 1.51 |
0 | 0 | 5.32 | 11.49 | 20.9 | 47.2 | 1.81 |
0 | 0 | 4.13 | 14.54 | 22.3 | 51.9 | 2.83 |
1.4–2.35 | 2.3 | 5.91 | 8.21 | 17.7 | 42.9 | 1.31 |
1.4–2.35 | 2.3 | 5.04 | 10.72 | 20.7 | 41.9 | 1.94 |
1.4–2.35 | 2.3 | 4.41 | 10.96 | 22.2 | 38.9 | 2.55 |
1.4–2.35 | 2.3 | 3.64 | 16.32 | 23.9 | 41.2 | 3.66 |
1.4–2.35 | 2.3 | 5.52 | 10.36 | 18.9 | 36.9 | 1.55 |
1.4–2.35 | 2.3 | 4.63 | 10.84 | 21.1 | 37.1 | 2.25 |
1.4–2.35 | 2.3 | 3.76 | 14.07 | 22.1 | 39.7 | 3.22 |
1.4–2.35 | 2.3 | 2.98 | 17.80 | 24.0 | 44.6 | 4.96 |
1.4–2.35 | 2.3 | 5.12 | 10.36 | 18.3 | 39.8 | 1.68 |
1.4–2.35 | 2.3 | 4.46 | 11.05 | 20.4 | 44.2 | 2.30 |
1.4–2.35 | 2.3 | 3.24 | 15.23 | 21.6 | 37 | 3.95 |
1.4–2.35 | 2.3 | 2.21 | 17.15 | 23.4 | 53.5 | 7.59 |
2.0–3.35 | 2.5 | 5.72 | 9.88 | 19.0 | 32.6 | 1.48 |
2.0–3.35 | 2.5 | 4.35 | 10.12 | 20.6 | 34.9 | 2.41 |
2.0–3.35 | 2.5 | 3.52 | 13.23 | 22.9 | 36.7 | 3.69 |
2.0–3.35 | 2.5 | 2.14 | 16.65 | 24.3 | 51.9 | 8.26 |
2.0–3.35 | 2.5 | 3.95 | 9.83 | 18.9 | 34.9 | 2.56 |
2.0–3.35 | 2.5 | 2.48 | 15.11 | 20.8 | 35.4 | 5.68 |
2.0–3.35 | 2.5 | 2.11 | 15.24 | 22.0 | 40 | 7.64 |
2.0–3.35 | 2.5 | 1.87 | 16.41 | 23.2 | 44.6 | 9.63 |
2.0–3.35 | 2.5 | 3.34 | 9.67 | 18.3 | 39.3 | 3.18 |
2.0–3.35 | 2.5 | 2.32 | 14.13 | 20.1 | 42.4 | 6.06 |
2.0–3.35 | 2.5 | 2.41 | 15.29 | 21.6 | 38.1 | 6.15 |
2.0–3.35 | 2.5 | 1.85 | 17.21 | 22.7 | 46.5 | 9.60 |
4.75–8.0 | 7.5 | 4.95 | 9.83 | 18.3 | 32.8 | 1.76 |
4.75–8.0 | 7.5 | 2.95 | 12.85 | 21.1 | 37.5 | 4.43 |
4.75–8.0 | 7.5 | 2.43 | 14.28 | 22.9 | 47.2 | 6.43 |
4.75–8.0 | 7.5 | 2.25 | 13.93 | 24.4 | 50.9 | 7.69 |
4.75–8.0 | 7.5 | 3.61 | 9.42 | 19.7 | 33.2 | 3.05 |
4.75–8.0 | 7.5 | 2.15 | 13.89 | 21.5 | 40.2 | 7.27 |
4.75–8.0 | 7.5 | 1.95 | 13.67 | 22.6 | 52.2 | 8.84 |
4.75–8.0 | 7.5 | 1.90 | 14.75 | 24.2 | 58.8 | 9.84 |
4.75–8.0 | 7.5 | 3.53 | 9.62 | 17.4 | 34.2 | 2.79 |
4.75–8.0 | 7.5 | 2.42 | 11.87 | 19.2 | 39.6 | 5.44 |
4.75–8.0 | 7.5 | 2.11 | 12.54 | 20.7 | 51.3 | 7.20 |
4.75–8.0 | 7.5 | 1.92 | 14.05 | 22.7 | 48.1 | 9.08 |
8.0–12.7 | 12 | 3.71 | 9.43 | 18.7 | 37.8 | 2.78 |
8.0–12.7 | 12 | 2.40 | 9.97 | 21.5 | 36.6 | 6.16 |
8.0–12.7 | 12 | 2.60 | 10.52 | 23.0 | 47.1 | 5.83 |
8.0–12.7 | 12 | 2.30 | 12.18 | 24.9 | 55.8 | 7.61 |
8.0–12.7 | 12 | 3.81 | 9.34 | 19.2 | 37.1 | 2.75 |
8.0–12.7 | 12 | 2.30 | 10.06 | 24.0 | 50.2 | 7.34 |
8.0–12.7 | 12 | 2.40 | 11.60 | 21.5 | 44.7 | 6.17 |
8.0–12.7 | 12 | 2.90 | 9.98 | 22.8 | 46 | 4.91 |
8.0–12.7 | 12 | 4.80 | 9.15 | 17.2 | 32.6 | 1.74 |
8.0–12.7 | 12 | 2.50 | 9.60 | 19.5 | 40 | 5.25 |
8.0–12.7 | 12 | 2.30 | 10.19 | 20.9 | 42.8 | 6.38 |
8.0–12.7 | 12 | 2.25 | 10.09 | 23.1 | 47.3 | 7.27 |
ks (cm) . | d84 (mm) . | h1 (cm) . | h2 (cm) . | Q (l/s) . | Lr (cm) . | F1 . |
---|---|---|---|---|---|---|
0 | 0 | 5.91 | 8.81 | 17.8 | 52.6 | 1.32 |
0 | 0 | 6.92 | 9.17 | 20.1 | 59 | 1.18 |
0 | 0 | 6.25 | 12.07 | 22.3 | 43.5 | 1.52 |
0 | 0 | 5.54 | 12.20 | 23.6 | 53.3 | 1.93 |
0 | 0 | 6.84 | 8.44 | 18.6 | 48.7 | 1.11 |
0 | 0 | 5.86 | 9.78 | 21.2 | 44.8 | 1.59 |
0 | 0 | 5.23 | 11.92 | 22.2 | 43.5 | 1.97 |
0 | 0 | 4.69 | 13.58 | 23.6 | 46.2 | 2.47 |
0 | 0 | 6.52 | 7.81 | 17.5 | 51.1 | 1.12 |
0 | 0 | 5.76 | 9.32 | 19.6 | 49.3 | 1.51 |
0 | 0 | 5.32 | 11.49 | 20.9 | 47.2 | 1.81 |
0 | 0 | 4.13 | 14.54 | 22.3 | 51.9 | 2.83 |
1.4–2.35 | 2.3 | 5.91 | 8.21 | 17.7 | 42.9 | 1.31 |
1.4–2.35 | 2.3 | 5.04 | 10.72 | 20.7 | 41.9 | 1.94 |
1.4–2.35 | 2.3 | 4.41 | 10.96 | 22.2 | 38.9 | 2.55 |
1.4–2.35 | 2.3 | 3.64 | 16.32 | 23.9 | 41.2 | 3.66 |
1.4–2.35 | 2.3 | 5.52 | 10.36 | 18.9 | 36.9 | 1.55 |
1.4–2.35 | 2.3 | 4.63 | 10.84 | 21.1 | 37.1 | 2.25 |
1.4–2.35 | 2.3 | 3.76 | 14.07 | 22.1 | 39.7 | 3.22 |
1.4–2.35 | 2.3 | 2.98 | 17.80 | 24.0 | 44.6 | 4.96 |
1.4–2.35 | 2.3 | 5.12 | 10.36 | 18.3 | 39.8 | 1.68 |
1.4–2.35 | 2.3 | 4.46 | 11.05 | 20.4 | 44.2 | 2.30 |
1.4–2.35 | 2.3 | 3.24 | 15.23 | 21.6 | 37 | 3.95 |
1.4–2.35 | 2.3 | 2.21 | 17.15 | 23.4 | 53.5 | 7.59 |
2.0–3.35 | 2.5 | 5.72 | 9.88 | 19.0 | 32.6 | 1.48 |
2.0–3.35 | 2.5 | 4.35 | 10.12 | 20.6 | 34.9 | 2.41 |
2.0–3.35 | 2.5 | 3.52 | 13.23 | 22.9 | 36.7 | 3.69 |
2.0–3.35 | 2.5 | 2.14 | 16.65 | 24.3 | 51.9 | 8.26 |
2.0–3.35 | 2.5 | 3.95 | 9.83 | 18.9 | 34.9 | 2.56 |
2.0–3.35 | 2.5 | 2.48 | 15.11 | 20.8 | 35.4 | 5.68 |
2.0–3.35 | 2.5 | 2.11 | 15.24 | 22.0 | 40 | 7.64 |
2.0–3.35 | 2.5 | 1.87 | 16.41 | 23.2 | 44.6 | 9.63 |
2.0–3.35 | 2.5 | 3.34 | 9.67 | 18.3 | 39.3 | 3.18 |
2.0–3.35 | 2.5 | 2.32 | 14.13 | 20.1 | 42.4 | 6.06 |
2.0–3.35 | 2.5 | 2.41 | 15.29 | 21.6 | 38.1 | 6.15 |
2.0–3.35 | 2.5 | 1.85 | 17.21 | 22.7 | 46.5 | 9.60 |
4.75–8.0 | 7.5 | 4.95 | 9.83 | 18.3 | 32.8 | 1.76 |
4.75–8.0 | 7.5 | 2.95 | 12.85 | 21.1 | 37.5 | 4.43 |
4.75–8.0 | 7.5 | 2.43 | 14.28 | 22.9 | 47.2 | 6.43 |
4.75–8.0 | 7.5 | 2.25 | 13.93 | 24.4 | 50.9 | 7.69 |
4.75–8.0 | 7.5 | 3.61 | 9.42 | 19.7 | 33.2 | 3.05 |
4.75–8.0 | 7.5 | 2.15 | 13.89 | 21.5 | 40.2 | 7.27 |
4.75–8.0 | 7.5 | 1.95 | 13.67 | 22.6 | 52.2 | 8.84 |
4.75–8.0 | 7.5 | 1.90 | 14.75 | 24.2 | 58.8 | 9.84 |
4.75–8.0 | 7.5 | 3.53 | 9.62 | 17.4 | 34.2 | 2.79 |
4.75–8.0 | 7.5 | 2.42 | 11.87 | 19.2 | 39.6 | 5.44 |
4.75–8.0 | 7.5 | 2.11 | 12.54 | 20.7 | 51.3 | 7.20 |
4.75–8.0 | 7.5 | 1.92 | 14.05 | 22.7 | 48.1 | 9.08 |
8.0–12.7 | 12 | 3.71 | 9.43 | 18.7 | 37.8 | 2.78 |
8.0–12.7 | 12 | 2.40 | 9.97 | 21.5 | 36.6 | 6.16 |
8.0–12.7 | 12 | 2.60 | 10.52 | 23.0 | 47.1 | 5.83 |
8.0–12.7 | 12 | 2.30 | 12.18 | 24.9 | 55.8 | 7.61 |
8.0–12.7 | 12 | 3.81 | 9.34 | 19.2 | 37.1 | 2.75 |
8.0–12.7 | 12 | 2.30 | 10.06 | 24.0 | 50.2 | 7.34 |
8.0–12.7 | 12 | 2.40 | 11.60 | 21.5 | 44.7 | 6.17 |
8.0–12.7 | 12 | 2.90 | 9.98 | 22.8 | 46 | 4.91 |
8.0–12.7 | 12 | 4.80 | 9.15 | 17.2 | 32.6 | 1.74 |
8.0–12.7 | 12 | 2.50 | 9.60 | 19.5 | 40 | 5.25 |
8.0–12.7 | 12 | 2.30 | 10.19 | 20.9 | 42.8 | 6.38 |
8.0–12.7 | 12 | 2.25 | 10.09 | 23.1 | 47.3 | 7.27 |
For the hydraulic jump on rough beds, the ratio of conjugate depth to Froude number always diverges from a linear trend, as proposed by Belanger's equation. This is due to the high rates/amounts of energy dissipation along rough beds, which is completely neglected in Belanger's equation. At small Froude numbers (undular or weak jumps) this divergence is low, whereas it increases as the Froude number increases (Figure 3). The experimental observations show that at constant Froude number, sequent depth ratio decreases as bed roughness (d84) increases.
Conjugate depth ratio versus upstream Froude number for different granulometric sizes. As d84 increases at constant Froude number, the conjugate depth ratio is underestimated from the expected outcome calculated by the Belanger equation.
Conjugate depth ratio versus upstream Froude number for different granulometric sizes. As d84 increases at constant Froude number, the conjugate depth ratio is underestimated from the expected outcome calculated by the Belanger equation.
RESULTS AND DISCUSSION
Integrated bed shear stress versus upstream flow velocity
As shown in Figure 2 and discussed above, the experimental results obtained by Carollo et al. (2007) validate the assumption proposed in Equation (10). In light of this result, the integrated bed shear stress can be defined in terms of the surface area of the rough bed, the density of the fluid, flow velocity and the Reynolds coefficient, .
Aiming to observe the relationship between the integrated bed shear stress and the flow velocity further, the characteristic data of the experimental work conducted in this study was tested through Equation (3) and plotted with the experimentally captured supercritical flow velocity (Figure 4). The plots lead to a spurious linear correlation for the velocity, ranging from 0.89 to 4.25 m/sec. Notably, Pearson's correlation coefficients were robust with respect to different bed roughness compositions when the supercritical velocity was correlated with integrated bed shear stress (r = 0.880 for d84 = 0.23 cm; r = 0.979 for d84 = 0.25 cm; r = 0.987 for d84 = 0.75 cm; r = 0.989 for d84 = 1.2 cm). Therefore, in a predefined flow velocity range, the integrated bed shear stress increases linearly with the flow velocity (Figures 2 and 4). The increase in the magnitude of integrated bed shear stress with respect to an increase in the representative granulometric characteristic (d84) of the sand and gravel particles can also be observed clearly in Figures 2 and 4.
Plot of supercritical flow velocity with respect to the integrated bed shear stress according to various granulometric sizes. For each graph, the x-axis represents the velocity of the flow before the jump and the y-axis represents the friction force calculated by Equation (3).
Plot of supercritical flow velocity with respect to the integrated bed shear stress according to various granulometric sizes. For each graph, the x-axis represents the velocity of the flow before the jump and the y-axis represents the friction force calculated by Equation (3).
Consequently, it can be outlined that the relationship as proposed in Equation (10) is robust for hydraulic jump over rough beds. Although this has been validated through the experimental data of Carollo et al. (2007) and the analysis conducted in this research, additional experimental tests may be necessary to delineate the upper and lower limits of the flow velocity to extend the validity of this relationship.
Shear force coefficient


In an attempt to define the relationship between the shear force coefficient, , and the modified Reynolds number,
, the experimental data given in Table 2 and the results of the previous studies are plotted in Figures 5 and 6. While the exponential relationship with a low modified Reynolds number shows sharp changes in
, slight changes are obvious with the increase of the modified Reynolds number. This indicates that the effect of the modified Reynolds number decreases as the roughness increases. Consequently, the findings depict that
is not only dependent on the mean grain diameter and critical water depth as is given in Equation (23), but also on the supercritical flow velocity. Although there is a similarity between the range of modified Reynolds numbers for the experimental results conducted in this study and in Carollo et al.’s (2007), the scale of the experimental results of Hughes & Flack (1984) present slight deviations. Instead, all the experimental data shown in Figures 5 and 6 follow an exponential relationship between the modified Reynolds number and the shear force coefficient.
The relationship between the shear force coefficient and the modified Reynolds number. The variables are derived from the data gathered from Table 2 and Carollo et al. (2007). For the calculation of shear force coefficient, the scale factor K is taken as 1 × 10−3. The legend shows the granulometric size of sand and gravel particles in which d84 represents the sediment particles used in conducted experiments, and ks stands for those received from Carollo et al. (2007).
The relationship between the shear force coefficient and the modified Reynolds number. The variables are derived from the data gathered from Table 2 and Carollo et al. (2007). For the calculation of shear force coefficient, the scale factor K is taken as 1 × 10−3. The legend shows the granulometric size of sand and gravel particles in which d84 represents the sediment particles used in conducted experiments, and ks stands for those received from Carollo et al. (2007).
Plot of the relationship between the shear force coefficient and the modified Reynolds number. The variables are retrieved from the experimental data of Hughes & Flack (1984). For the calculation of shear force coefficient, the scale factor K is 1 × 10−1.
Plot of the relationship between the shear force coefficient and the modified Reynolds number. The variables are retrieved from the experimental data of Hughes & Flack (1984). For the calculation of shear force coefficient, the scale factor K is 1 × 10−1.
The results show that at large values (high resisting forces), the modified Reynolds number approaches a constant value which indicates a uniform flow condition rather than a hydraulic jump. On the other hand, as the modified Reynolds number increases, the turbulence in the flow increases, reducing the roller length of the hydraulic jump.
Reynolds coefficient
The magnitude of the Reynolds coefficient for each experiment was derived through Equation (21), in which roller length, supercritical flow depth and velocity were measured from the conducted experiments.
Based on the experimental results it was found that the modified Reynolds number increases with the increase of value. The higher the magnitude of
, the more agitated the flow, indicating higher turbulence levels for a given flow rate. Further increase in turbulence leads to an increase in energetic events that may be associated with bedload transport and scouring processes downstream of the sluice gate.
Figure 7 illustrates the Reynolds coefficient versus the modified Reynolds number for two different experimental conditions: from Carollo et al. (2007) and the data obtained from the conducted experiments of this study. It is shown that the tendency of the modified Reynolds number is to increase as the Reynolds coefficient increases. Further increase in the Reynolds coefficient can then be accepted as strip roughness, which in turn can be treated and calculated as a drag coefficient (Habibzadeh et al. 2011; Mudgal & Pani 2011). For large values of the modified Reynolds number (R* → ∞), the bed roughness approaches a fixed value depending on the diameter of sand and gravel particles which generate the bed roughness.
The variation of Reynolds coefficient with modified Reynolds number. As the Reynolds number increases, the gradient of the Reynolds coefficient changes from sharp to mild. The legend shows the granulometric sizes of sand and gravel particles in which d84 represent the sediment particles used in conducted experiments and ks stands for data from experiments of Carollo et al. (2007).
The variation of Reynolds coefficient with modified Reynolds number. As the Reynolds number increases, the gradient of the Reynolds coefficient changes from sharp to mild. The legend shows the granulometric sizes of sand and gravel particles in which d84 represent the sediment particles used in conducted experiments and ks stands for data from experiments of Carollo et al. (2007).
In the data presented by Hughes & Flack (1984), a linear relationship was observed between the modified Reynolds number and the Reynolds coefficient with a high degree of accuracy (R2 = 1.0; Figure 8). Hughes & Flack (1984) conducted experiments for two different types of rough surfaces: strip roughness and gravel roughness. In Figure 8, only the results of the gravel roughness tests were used.
The variation of Reynolds coefficient with modified Reynolds number using experimental data retrieved from the research of Hughes & Flack (1984).
The variation of Reynolds coefficient with modified Reynolds number using experimental data retrieved from the research of Hughes & Flack (1984).
Roller length








Linear correlation between the relative roller length and the upstream Froude number.
Linear correlation between the relative roller length and the upstream Froude number.







Calculated versus measured roller lengths for hydraulic jumps on various rough surfaces.
Calculated versus measured roller lengths for hydraulic jumps on various rough surfaces.
CONCLUSIONS
Hydraulic jumps are generally defined in terms of Froude number and subcritical/supercritical flow depths. The difference between the conjugate depths and the corresponding velocities and roller length increases with the upstream Froude number. While such trends can be observed and analytically solved for hydraulically smooth surfaces, this may not always be the case for hydraulically rough surfaces, resulting in limitations for the implementation of the well-known Belanger equation.
Based on the novel assumption that the integrated bed shear stress which is associated with bed roughness along the hydraulic jump is linearly proportional to the flow velocity, one dimensional momentum and continuity equations were solved using the Reynolds transport theorem. Based on this, a novel definition for the shear force coefficient was derived and expressed as a function of integrated bed shear stress, Reynolds coefficient and modified Reynolds number. The new analytical roller length formula for rough surfaces was derived in terms of the modified Reynolds number, reflecting the turbulence effect on the roller length.
The obtained definition of the shear force coefficient in terms of Reynolds coefficient indirectly describes the effect of a modified Reynolds number on the integrated bed shear stress. For a certain roughness, as the turbulence increased, the shear force coefficient tended to increase and motivate the integrated bed shear stress to increase.
The reliability of the proposed analytical solution for Reynolds coefficient, shear force coefficient and roller length was tested by using the results of conducted experiments and the results of experiments by Hughes & Flack (1984) and Carollo et al. (2007). The important observations of the flow characteristics along hydraulic jumps over rough beds are summarized below:
Any increase in representative sand and gravel sizes,
, induces a reduced conjugate depth ratio along the hydraulic jump.
A linear correlation is observed between the integrated bed shear stress and supercritical flow velocity, when the flow velocity ranges from 0.89 to 4.25 m/sec. The integrated bed shear stress increases with the supercritical flow velocity and bed roughness.
The shear force coefficient which is used in the modified Belanger equation depends not only on the representative sand and gravel sizes, but also on the supercritical flow velocity, which dominates the flow momentum.
When bed roughness is increased, the shear force coefficient can be conserved only if the modified Reynolds number increases. On the other hand, as the modified Reynolds number increases, the number of eddies in the flow increases accompanied by an increase in resisting forces. As a result, the length of the hydraulic jump is reduced.
The tendency of the Reynolds coefficient to reach a constant value increases as the modified Reynolds number increases. This indicates that, as the bed roughness increases the turbulence increases, and thus the resistance to flow tends to approach to a fixed value.
The bed roughness strongly influences the length of the roller in the range 1.1 < F < 9.8.
As given in the pertinent literature, the roller length is a function of the upstream Froude number. The results of this study show that the effect of the modified Reynolds number is also important in roller length analysis and cannot be ignored.
ACKNOWLEDGEMENTS
The authors are grateful for the equipment funding from the Research Division of Infrastructure and Environment, at the University of Glasgow. Umut Türker acknowledges funding by the Eastern Mediterranean University and EuropeAid/133886/L/ACT/CY Scholarships program to join the University of Glasgow as a visiting researcher.
DATA AVAILABILITY STATEMENT
All relevant data are included in the paper or its Supplementary Information.