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.

  • 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.

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).

Derivation of shear force coefficient based on momentum equation

Newton's second law states that the sum of all the external forces acting on any system is equal to the time rate of change of linear momentum of that system:
(1)
The resultant force includes all surface and body forces acting on the control volume, and rate of change of momentum can be solved through the Reynolds transport theorem. Considering a hydraulic jump over horizontal rough beds (Figure 1), the resultant momentum equation in the flow direction was given by Rajaratnam (1966) and Govinda & Ramaprasad (1966) as,
(2)
where and represent the hydrostatic pressure forces at the inflow and outflow of the control volume (Figure 1) respectively, and represents the integrated bed shear stress on the horizontal plane coinciding with the rough surface. The right-hand side of Equation (2) represents momentum forces, in which and are the supercritical and the subcritical flow velocities, respectively. The constant flow discharge is defined as Q and is the density of the water. Extracting the integrated bed shear stress from Equation (2) and writing all the other terms in the form of upstream and downstream flow depths, and , channel width, B, and the supercritical Froude number, results in Equation (3) for rectangular open channels.
(3)
Figure 1

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.

Figure 1

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.

Close modal
Rajaratnam (1966) and Ead & Rajaratnam (2002) proposed that the integrated bed shear stress, can be assumed to be proportional to the inflow hydrostatic pressure force, with a proportionality constant , . In both studies the main assumption was that integrated bed shear stress depends only on the upstream flow conditions and the proportionality constant is termed as the shear force coefficient, . Replacing the integrated bed shear stress with inflow hydrostatic pressure force and the shear force coefficient modifies Equation (3) as,
(4)
On the other hand, the integrated bed shear stress, is directly proportional to the velocity of the flow. This is an experimentally proven fact. Mathematically, it makes sense that any reasonable function, such as integrated bed shear stress, is expected to have a Taylor series expansion given as,
(5)
So is a polynomial of degree at most N where the constant term, can be replaced with (). Finally, Equation (5) can be re-written in its simplest form for as in Equation (6), since the first three terms should give a good approximation.
(6)
In the case of no-flow conditions, the integrated bed shear stress does not exist ( when ) and the constant term has to be zero. As a conclusion, integrated bed shear stress can be defined in terms of speed of flow, therefore, in general it can be shown as:
(7)
Here and depend on the shape and size of the rough surface, and the type of the fluid. It is important to analyze Equation (7) in two parts and decide on whether the first term or the second term reflect the dominant effect of the integrated bed shear stress. The integrated bed shear stress which is initiated by the interaction of fluid and the rough bed surface depends on and is proportional to the density of the fluid, , mean flow velocity, v, gravitational acceleration, g, and the surface area, A, where, the friction force acts parallel to the bottom of the channel:
(8)
Generally, it is accepted that the second term used in Equation (7) is highly related with a dynamic pressure effect and therefore, Equation (8) can be evaluated in the form of,
(9)
where is the dimensionless skin friction coefficient. This coefficient is used not just for hydraulic jumps but for boundary layers in general. On the other hand, the quadratic relationship between integrated bed shear stress and flow velocity during the hydraulic jumps can be re-considered and tested. To achieve this goal, the experimental data including conjugate depths, supercritical flow velocities and Froude numbers from the work conducted by Carollo et al. (2007) was used. All these variables were inserted into Equation (3) in order to calculate the integrated bed shear stress. Consequently, for a wide range of Froude numbers (1.87 ≤ ≤ 8.72), a linear relationship is observed between the integrated bed shear stress and the supercritical flow velocity (Figure 2). This observation is also statistically confirmed by Pearson's correlation, one of the most common statistical tests used to determine the degree of linear dependence between two random variables. Pearson's correlation coefficient, r always varies between −1 and +1. As the coefficient approaches to zero, it means there is a weak or no relationship between the two random variables. As the coefficient approaches to +1, it is concluded that there is a strong positive linear relationship between the variables, while −1 indicates a strong negative one. Supercritical flow velocity is associated with integrated bed shear stress on rough surfaces. Pearson's correlation analysis showed a significant positive linear correlation between and at different equivalent sand roughness (ks) levels (r = 0.943 for ks = 0.46 cm; r = 0.908 for ks = 0.82 cm; r = 0.971 for ks = 1.46 cm; r = 0.943 for ks = 2.39 cm; r = 0.935 for ks = 3.2 cm). This statistical approach verifies that supercritical flow velocity is linearly related to integrated bed shear stress.
Figure 2

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).

Figure 2

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).

Close modal
The negative observed in Figure 2 is indicative of a reverse flow region close to the rough bed surface, especially at high Froude numbers, where the difference between conjugate depths is high (Imai & Nakagawa 1992; Dey & Sarkar 2008). The analysis done for the experimental observations conducted by Carollo et al. (2007) shows that the integrated bed shear stress along the hydraulic jump can also be accepted to vary linearly with flow velocity as,
(10)
Reynolds coefficient, has a dimension of [L/T] and ascribes the effect of both viscosity of the medium, and the size of the bed roughness on the magnitude of the integrated bed shear stress. Due to its definition, the Reynolds coefficient is expected to have a strong relationship with Reynolds number. Equating Equation (10) with Equation (3) and simplifying the result provides Equation (11) as,
(11)
The right hand side of Equation (11) comprises the definition of the integrated bed shear stress () and according to Equation (4), it can be replaced with the shear force coefficient, as
(12)
The constant K is a dimensionless scale factor and the effect of integrated bed shear stress depends only on the flow conditions at the upstream section of the hydraulic jump (Ead & Rajaratnam 2002). This assumption tolerates the use of upstream flow velocity in terms of velocity used in Equation (12). Rewriting Equation (11) in terms of upstream Froude number, conjugate depths, and shear force coefficient results in a useful relationship, which was previously studied by Rajaratnam (1968),
(13)

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

In natural channels a hydraulic jump is generally retarded by resistive forces associated with the bed roughness. On the other hand, these resistive forces are balanced by the driving forces, satisfying the conservation of momentum. The relationship between the resisting and the driving forces can be defined simply as:
(14)
where m is the mass and is the acceleration that can be rewritten in terms of the fluid volume, density, and the rate of change of mean velocity along the hydraulic jump. The integrated bed shear stress, previously defined by Equation (10), balances the driving forces.
(15)
where V is the infinitesimal volume that can be defined in terms of dimensions in x, y and z directions as and the bed surface area, A can be described in terms of dimensions in x and z directions as . Rewriting Equation (15) and eliminating the common terms in both sides of the equation results in
(16)
Integrating both sides of Equation (16) and assigning the limits of the left-hand side of the equation as supercritical and subcritical flow velocities, and the limits of the right hand side of the equation as the initial and final time steps, Equation (16) emerges as Equation (17)
(17)
where is the subcritical flow velocity after the jump and is the supercritical flow velocity before the jump, and is the time lag required along the hydraulic jump. The subcritical flow velocity can be defined as the infinitesimal channel segment, , traveled by fluid particles per unit time,; . Finally, Equation (17) simplifies into
(18)
Integrating both sides of the equation by inserting and into Equation (18) results in:
(19)
in which represents the length of the roller along the hydraulic jump. The exponential term of Equation (19) approaches 1 since the magnitude of is negligibly smaller than . Therefore, Equation (19) can be re-written as:
(20)
and in a dimensionless form as:
(21)

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.

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.

Table 1

The granulometric sizes of bed surface material used in the flume experiments

Material CodeNominal diameter range (mm)d50 (mm)d84 (mm)
smooth surface – – 
1.40–2.35 1.8 2.3 
2.00–3.35 2.1 2.5 
4.75–8.00 6.3 7.5 
8.00–12.7 10.5 12.0 
Material CodeNominal diameter range (mm)d50 (mm)d84 (mm)
smooth surface – – 
1.40–2.35 1.8 2.3 
2.00–3.35 2.1 2.5 
4.75–8.00 6.3 7.5 
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.

Table 2

Characteristic data of the experimental runs and dimensionless variables

ks (cm)d84 (mm)h1 (cm)h2 (cm)Q (l/s)Lr (cm)F1
5.91 8.81 17.8 52.6 1.32 
6.92 9.17 20.1 59 1.18 
6.25 12.07 22.3 43.5 1.52 
5.54 12.20 23.6 53.3 1.93 
6.84 8.44 18.6 48.7 1.11 
5.86 9.78 21.2 44.8 1.59 
5.23 11.92 22.2 43.5 1.97 
4.69 13.58 23.6 46.2 2.47 
6.52 7.81 17.5 51.1 1.12 
5.76 9.32 19.6 49.3 1.51 
5.32 11.49 20.9 47.2 1.81 
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
5.91 8.81 17.8 52.6 1.32 
6.92 9.17 20.1 59 1.18 
6.25 12.07 22.3 43.5 1.52 
5.54 12.20 23.6 53.3 1.93 
6.84 8.44 18.6 48.7 1.11 
5.86 9.78 21.2 44.8 1.59 
5.23 11.92 22.2 43.5 1.97 
4.69 13.58 23.6 46.2 2.47 
6.52 7.81 17.5 51.1 1.12 
5.76 9.32 19.6 49.3 1.51 
5.32 11.49 20.9 47.2 1.81 
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.

Figure 3

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.

Figure 3

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.

Close modal

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.

Figure 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).

Figure 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).

Close modal

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

Carollo & Ferro (2004) stated that, for hydraulic jump over rough beds, the integrated bed shear stress per unit width of a rectangular channel can be defined in terms of the difference between the momentum flux before and after the jump and the shear force coefficient, . According to various studies (Carollo et al. (2007), Ead & Rajaratnam (2002) and Hughes & Flack (1984)), the shear force coefficient can be defined in terms of relative roughness (ks/h1). Carollo & Ferro (2004) described this ratio with a constant 0.42, characterized by a determination coefficient R2 = 0.44. On the other hand, Pagliara & Palermo (2015) conducted experimental analyses and proposed a new relationship for the shear force coefficient, as follows:
(22)
where k is the critical depth and d50 is the mean diameter of channel bed material. The ratio of the shear force coefficient, , to relative roughness, ks/h1, is strongly implicated in the lack of agreement between different analyses of laboratory studies. The dependency of the shear force coefficient can also be attributed to the magnitude of the modified Reynolds number, R*, whose effect has been neglected in previous studies.
For hydraulic jumps over rough beds the modified Reynolds number can be defined in terms of supercritical velocity, kinematic viscosity of the fluid and the representative grain size of sand and gravel particles, .
(23)

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.

Figure 5

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).

Figure 5

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).

Close modal
Figure 6

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.

Figure 6

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.

Close modal

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.

Figure 7

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).

Figure 7

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).

Close modal

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.

Figure 8

The variation of Reynolds coefficient with modified Reynolds number using experimental data retrieved from the research of Hughes & Flack (1984).

Figure 8

The variation of Reynolds coefficient with modified Reynolds number using experimental data retrieved from the research of Hughes & Flack (1984).

Close modal

Roller length

The roller length is defined in the form of a non-dimensional ratio in which the complementary length scale is the supercritical flow depth, h1. Previously proposed roller length equations are defined as a function of the Froude number, where the Froude number is limited in the range of . Hager et al. (1990) suggested that the roller length depends linearly on the Froude number as and Carollo et al. (2007) proposed Equation (24) as,
(24)
where is the dimensionless relative roller length ( and is the numerical coefficient depending on the bed roughness. Figure 9 shows the variation of relative roller length with respect to the upstream Froude number, compiled from the conducted experiments in this study. Regression analysis confirms that the relative roller length is dependent on Froude number with a determination coefficient , for = 2.5. Therefore, Equation (24) can then be used to determine the roller length on rough beds in the range of .
Figure 9

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

Figure 9

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

Close modal
Hence, the experimental results of this study are in good agreement with Equation (24), where the impact of Reynolds coefficient and the conjugate depth ratio is neglected. On the other hand, it is physically plausible that the bed roughness strongly influences roller length. Therefore, analytically derived Equation (21) is proposed to define roller length, since it includes the effect of both the Reynolds coefficient and the conjugate depth ratio, along with supercritical flow velocity. Figures 7 and 8 depict the linear relationship between Reynolds coefficient and the modified Reynolds number. Therefore, Equation (21) can also be written as
(25)
and the data obtained from the conducted experiments of Hughes & Flack (1984) and Carollo et al. (2007) can be used for assessing the validity of Equation (25).
Assigning a variable in terms of can result in a new form of Equation (25):
(26)
Figure 10 compares the calculated versus the measured (experimentally observed) roller lengths for various rough surfaces during the hydraulic jump. In general, the results indicate that for small roughness height, the calculated roller length is less than the measured length. For such rough surfaces with small and values the calculated roller length is underestimated, whereas as the surface roughness increases, the calculated roller length is overestimated. Quality of fit analysis is applied to the calculated and measured roller length data. The results show that Equation (26) fits the data quite well. The goodness of fit between the calculated and measured results are indicated by,
(27)
where and are the measured and calculated roller lengths, respectively. The value corresponds to a perfect fit between the two sets of data, and increasing value of refers to poorer fit. The results of Equation (27) show that there is a good fit between the calculated and measured roller length values since .
Figure 10

Calculated versus measured roller lengths for hydraulic jumps on various rough surfaces.

Figure 10

Calculated versus measured roller lengths for hydraulic jumps on various rough surfaces.

Close modal

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.

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.

All relevant data are included in the paper or its Supplementary Information.

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