The log-wake law for turbulent current has been developed and tested with laboratory data on turbulent flow in smooth pipes. However, flow with turbulence and vortices in a stepped spillway have not been described. Therefore, in this study, a log-wake law has been developed for use in stepped spillway systems. It can be divided into three parts. The first part, a logarithmic equation, describes the effect of shear stress between the flow layers with a von Kármán constant of 0.41. The second part, a third-degree polynomial, describes the effect of the shear stress on the wall. The last part, a fourth-degree polynomial, describes the effect of changing the flow pressure distribution, similar to the wall-free shear stress. Calibration tests (68 datasets) are used with a flow rate between 0.0233 and 3.285 m3/s, a spillway slope of 14–30°, and a step height of 0.0380–0.610 m. The developed log-wake law characterized the flow in a stepped spillway well. The limitation of the equation is a maximum flow velocity of 4 m/s; the accuracy of this equation decreases as the step height increases.

• The developed log-wake law for the spillway is provided.

• Related experimental data are included.

River mechanics are complex processes consisting of flow and sediment transport. Flow in rivers is generally variable in time and is typically unsteady. Sections of rivers can be classified into two types: steep slope upriver and mild slope downstream or at the estuary, and the flow characteristics in each type are different. Moreover, the flow velocity at steep slopes upriver is always higher than downstream. The equation used to explain the flow velocity profile is assumed to be in accordance with the classic logarithmic law of the wall (Cheng & Gulliver 2011). Hence, the flow velocity profile has been explained using a power law for a long time.

Recently, a study comparing the basic logarithmic equation and the power law equation to clarify the flow velocity profile of a stepped spillway has shown that the basic logarithmic equation is more accurate than the power law equation (George 2007; Cheng & Gulliver 2011). The basic logarithmic equation, developed from the flow in a smooth pipe, is widely used to calculate the flow velocity profile and the variables related to the flow. However, the actual flow is non-ideal as there is friction between the flow surface and the eddy viscosity of the flow layer. For this reason, the basic logarithmic equation has been developed to simulate the flow (Cheng & Gulliver 2011). This equation is referred to as the ‘log-wake law equation’ and is first found and applied to the flow in a smooth pipe.

Turbulent flow is characterized by irregularities or disturbances, which lead to the formation of eddies and diffuses across the flow, as well as moving in the mean direction of the flow. Pulsing cross-current velocity fluctuations cause individual fluid parcels to move in irregular patterns, while the overall flow moves downstream. Turbulent flow can be considered a complex motion where random velocity fluctuations are superimposed on the average motion of translation. Turbulent eddies can transfer momentum normal to the flow boundary at a rate much higher than the molecular transfer that occurs in laminar flows. In this case, additional resistance to the flow arises, expressed by the eddy viscosity, which is not constant for a given fluid and temperature. In the turbulent flow, the water particles move in very irregular paths, causing an exchange of momentum from one portion of the fluid to another, and hence, the turbulent shear (Reynolds) stress.

The spillway is the important facility designed to prevent the overtopping of dams and release flood flow. An appropriate design of spillways for both drought and rainy seasons is another idea for preventing overtopping flow. To be safe, the spillway must allow a high flow without jeopardizing the dam. Although the general spillway shape and its flow are well-studied, it is also understood that even small changes to the design can change the flow properties. When a large flow is required, a specific spillway must be designed. Due to the high flow over the spillway, the design can be very complicated, introducing cavitation and high-flow kinetic energy problems. Some structural forms along the spillway, such as a stepped spillway, have been used for energy dissipation. A good energy-dissipating design should reduce the flow velocity without destroying other hydraulic structures. One recent popular approach to energy reduction is a spillway with high bed roughness rather than a smooth spillway with baffles or a stilling basin (Cheng & Gulliver 2011). This kind of spillway can also reduce velocities at the toe of the spillway.

A log-wake law equation aligned with the theory of a stepped spillway, or the high flow velocity, has not yet been proposed. Therefore, this research develops a log-wake law equation and the turbulent flow equation for a stepped spillway. Then, the collected data are analysed to determine the Coles wake strength (Π) suitable for such a flow pattern.

Velocity profile for a stepped spillway

The distribution of the flow velocity in a steep stepped spillway is different from the distribution of the flow velocity in an open channel, as shown in Figure 1. The flow in an open stepped spillway can be explained as the flow in a rough open channel with mild or steep slopes. The channel bed is considered rough, and the turbulent boundary layer increases in a gradient from the channel bed to the water surface. The flow in the open channel stepped spillway can be a nappe flow with low discharge, a transition flow with medium discharge, or a skimming flow with high discharge. The velocity distribution of the turbulent flow region can be estimated using a power law equation and the basic logarithmic equation (George 2007; Cheng & Gulliver 2011). The data obtained from roughened-bed flows show that the velocity distribution over the rough layer can also be used to describe the flow velocity distribution, as shown in Figures 1 and 2. The wake function influences the flow with turbulent flow conditions. The changes in the eddy current are independent of the distance of the flow , where W is the wake function and x is the direction of the flow. The wake function distribution is shown in Figure 2.
Figure 1

Velocity profile in the open channel flow. (a) Velocity profile on the stepped spillway and (b) velocity profile on the open channel flow.

Figure 1

Velocity profile in the open channel flow. (a) Velocity profile on the stepped spillway and (b) velocity profile on the open channel flow.

Close modal
Figure 2

Velocity-dip phenomenon on the stepped spillway. (a) Velocity distribution on the stepped spillway, (b) turbulent shear stress distribution, and (c) dip effect from the wake law.

Figure 2

Velocity-dip phenomenon on the stepped spillway. (a) Velocity distribution on the stepped spillway, (b) turbulent shear stress distribution, and (c) dip effect from the wake law.

Close modal
Figures 1 and 2 show the flow velocity profile of the flow in a stepped spillway where no air enters the flow, the turbulent shear stress distribution, and the dip effect on the wake law, where H is the depth of the flow, y is the depth perpendicular to the flow, l is the horizontal length of the steps, h is the height of the steps, θis the slope angle of the stepped spillway, is the height of the steps in the same direction as the flow or the roughness height of the spillway, is the displacement height, is the turbulent shear stress, is the bed shear stress, and is the surface roughness length. The roughness height is the reference bottom level; thus, the displacement height is used as the mean roughness because the pseudo-bottom layer is not affected by the roughness height of the spillway (Jackson 1981). A study by Perry et al. (1969) using the equation of Millikan (1938) with the logarithmic flow equation has described the turbulent flow that occurs above the steps. This test revealed that the equation is related to the displacement height and can be expressed as Equation (1):
(1)
where is the flow velocity according to the flow depth, is the depth perpendicular to the flow, k is the von Kármán constant, is the shear velocity, is the displacement height, and is the surface roughness length. From the study of a flow on a rough surface, the most important variables in applying these logarithmic equations were and (Jackson 1981).

Log-wake law equation

The logarithmic equation that is currently used, which is based on the flow in a smooth pipe, can be written as follows:
(2)
where u is the flow velocity at the depth of flow, is the shear velocity, is the von Kármán constant, is the depth perpendicular to the flow, ν is the kinematic viscosity, and B is the flow integration constant related to the bed roughness. Later, a basic logarithmic equation was developed to suit the actual flow under the condition that the flow is influenced by the variables related to the wake flow at the wall (Coles 1956). This equation is as follows:
(3)
where Π is the Coles wake strength, is the boundary layer thickness, and is the wake function. From the study of the wake function (Hinze 1975), it is written as follows:
(4)
where ξ is the normalized distance from the wall . Thus, the logarithmic equation, including the wake function based on the flow in a smooth pipe, is as follows:
(5)
The above equation is called the log-wake law. However, as it is not consistent with the flow in a smooth or rough pipe, this theory cannot completely explain the flow in a pipe. As a result, an equation explaining the profile of the flow velocity in a smooth pipe has been proposed as follows (Guo 1998; Guo & Julien 2001):
(6)
Then, an equation to explain the flow in the pipe has been developed by measuring the flow data in the pipe and analysing the ratio between the viscosity of the turbulence and the flow friction (Guo & Julien 2003; Guo et al. 2005). This equation is as follows:
(7)

Guo & Julien (2008) have studied Equation (7), the log-wake law equation, to model open channel flow with sediments using data from Coleman (1986), Kironoto & Graf (1994), Lyn (1986), and Sarma et al. (2000). Moreover, Guo (2022) has re-derived the log-law of the wall in the overlap using pure dimensional analysis and a functional equation. This study reveals three important variables in the functional equation of any turbulence model: (i) the log-law of the wall, which is a pure mathematical law and is independent; (ii) the von Kármán constant, a pure mathematical constant that is independent of the Reynolds number; and (iii) a wake law correction, which is necessary in the outer flow region.

As mentioned earlier, the Coles log-wake law is often suitable for use with the velocity distribution of turbulent pipe flow. However, it lacks consideration for the pipe symmetry principle and the effect of wall-induced turbulence at the obvert. It only includes the effects of wall-induced turbulence at the invert (the log-law) and free turbulence at the pipe centreline (the wake law). To address this limitation, Guo (2020) proposed an application of the log-wake law for mean velocity distributions. This innovation involves adding an additional log term due to the pipe obvert, resulting in a second log-wake law that can be effectively used with data from pipes, symmetric channel flow, and antisymmetric channel flow. This enhancement allows for a more comprehensive and accurate modelling of turbulent flows in these different scenarios. Patel et al. (2021) applied the second log-wake law to an open channel flow study. The research focused on understanding the impact of water surface shear stress and wind-induced turbulent mixing on velocity distributions in open channels. The study observed both laboratory flumes and natural rivers, as they have similarities near the channel beds but exhibit differences near the water surfaces. The findings indicated distinct velocity distributions between laboratory flow and natural river flow. In laboratory flumes, water surface shear stress was present, but wind-induced turbulent mixing was negligible. In contrast, natural river flow experienced both water surface shear stress and wind-induced turbulent mixing. The second log-wake law demonstrated its applicability to both types of open channel flow, although some parameters varied depending on the specific scenario. This research sheds light on the differences between laboratory and natural river flow dynamics, providing valuable insights for future studies in open channel hydraulics.

Shan et al. (2022) studied the effects of ice cover and wind-induced water surface shear stress through laboratory experiments. The second log-wake law is a tool used for wall-bounded turbulent open channel flows. The velocity distributions obtained from the experiments exhibit three distinct patterns: a bowl-shaped velocity distribution with a dip phenomenon, a typical boundary layer velocity distribution, or an S-shaped velocity distribution with an inflection. Based on these findings, it can be concluded that the second log-wake law, an innovative three-point method, can be utilized for estimating river discharge using three-point velocity measurements.

Physical model of flow in a stepped spillway

This study collected and used data from the experiments of Frizell (1992), Ruff & Ward (2002), Boes & Hager (2003), Hunt & Kadavy (2008), and Lurker et al. (2008) to understand the flow behaviour and propose a new equation. Each dataset has a different step angle and height for various analysis criteria.

Frizell (1992) used a physical model with a length of 4.72 m, a total height of 2.36 m, anda width of 0.457 m. The angle of the gutter was 26.6°, the height of the steps was 0.051 m, and the width of the steps was 0.102 m. The flow rate used in the test was 0.245 m3/s. The results showed that steps with a slope of 0° were better able to dissipate energy than steps with a slope of 15°.

Ruff & Ward (2002) used a physical model with a length of 34.14 m, a total height of 15.24 m, and a track width of 1.22 m. The angle was 26.6°, the height of the steps was 0.61 m, the length of the steps was 0.305 m, and the flow rate used ranged from 0.57 to 3.28 m3/s. The results showed that the flow energy of the open-stepped waterway could dissipate up to 32% of the input energy.

Boes & Hager (2003) used a physical model with a length of 5.7 m and a width of 0.5 m. The heights of the steps were 23.10, 46.20, and 92.40 mm for a spillway slope of 30°. The heights of the steps were 31.10 and 93.30 mm for a spillway slope of 50°, and the height of the step was 26.10 mm for a spillway slope of 40°. The flow rate in the experiment was between 0.023 and 0.19 m3/s. This study proposed a power equation to describe the flow velocity distribution, which has an accuracy of 80%.

Hunt & Kadavy (2008) used a physical model with a length of 10.82 m and a width of 1.8 m. The total height of the 40-step spillway was 1.52 m; each step was 3.8 cm high and 15.2 cm wide. The slope of the gutter was 14°. The flow rate in the experiment was between 0.18 and 1.48 m3/s. The test results showed that the energy loss from the flow was about 30%.

Lurker et al. (2008) studied a physical model with a length of 40.2 m and a width of 2.74 m, consisting of a flume with a slope of 22°, a length of 23.46 m, and steps with a total length of 2.35 m. The flow rate in the experiment was between 0.19 and 2.56 m3/s. The results showed that the overflow channel dissipated up to 45% of the energy. All of the data are shown in Table 1.

Table 1

Experimental investigation of flow over the stepped spillway

Physical modelDischargeSpecific dischargeSlopeSpillwayStepMaximum roughnessNumber of steps
Q (m3/s)q (m3/s/m)(degree)Height H (m)Height h (m)Kshcosθ (m)N
Frizell (1992)  0.245 0.577 26.6  0.051 0.046 47
Ruff & Ward (2002)  0.5663 0.4645 26.6 15.24 0.61 0.545 25
1.1327 0.9291 26.6 15.24 0.61 0.545
1.699 1.3935 26.6 15.24 0.61 0.545
2.2653 1.858 26.6 15.24 0.61 0.545
2.8317 2.3226 26.6 15.24 0.61 0.545
3.2848 2.6942 26.6 15.24 0.61 0.545
Boes & Hager (2003)  0.0233 0.0466 30 2.85 0.0231 0.02
0.066 0.1319 30 2.85 0.0462 0.04
0.1866 0.3732 30 2.85 0.0924 0.08
0.0328 0.0656 50 4.36 0.0311 0.02
0.1705 0.3409 50 4.36 0.0933 0.06
Hunt & Kadavy (2008)  0.18 0.1 14 1.52 0.038 0.037 40
0.378 0.21 14 1.52 0.038 0.037
0.504 0.28 14 1.52 0.038 0.037
0.756 0.42 14 1.52 0.038 0.037
1.116 0.62 14 1.52 0.038 0.037
1.476 0.82 14 1.52 0.038 0.037
Lurker et al. (2008)  0.19 0.07 22 3.66 0.241–0.031 0.223–0.029 68
0.74 0.27 22 3.66 0.241–0.031 0.223–0.029
0.94 0.34 22 3.66 0.241–0.031 0.223–0.029
1.31 0.48 22 3.66 0.241–0.031 0.223–0.029
2.56 0.94 22 3.66 0.241–0.031 0.223–0.029
Physical modelDischargeSpecific dischargeSlopeSpillwayStepMaximum roughnessNumber of steps
Q (m3/s)q (m3/s/m)(degree)Height H (m)Height h (m)Kshcosθ (m)N
Frizell (1992)  0.245 0.577 26.6  0.051 0.046 47
Ruff & Ward (2002)  0.5663 0.4645 26.6 15.24 0.61 0.545 25
1.1327 0.9291 26.6 15.24 0.61 0.545
1.699 1.3935 26.6 15.24 0.61 0.545
2.2653 1.858 26.6 15.24 0.61 0.545
2.8317 2.3226 26.6 15.24 0.61 0.545
3.2848 2.6942 26.6 15.24 0.61 0.545
Boes & Hager (2003)  0.0233 0.0466 30 2.85 0.0231 0.02
0.066 0.1319 30 2.85 0.0462 0.04
0.1866 0.3732 30 2.85 0.0924 0.08
0.0328 0.0656 50 4.36 0.0311 0.02
0.1705 0.3409 50 4.36 0.0933 0.06
Hunt & Kadavy (2008)  0.18 0.1 14 1.52 0.038 0.037 40
0.378 0.21 14 1.52 0.038 0.037
0.504 0.28 14 1.52 0.038 0.037
0.756 0.42 14 1.52 0.038 0.037
1.116 0.62 14 1.52 0.038 0.037
1.476 0.82 14 1.52 0.038 0.037
Lurker et al. (2008)  0.19 0.07 22 3.66 0.241–0.031 0.223–0.029 68
0.74 0.27 22 3.66 0.241–0.031 0.223–0.029
0.94 0.34 22 3.66 0.241–0.031 0.223–0.029
1.31 0.48 22 3.66 0.241–0.031 0.223–0.029
2.56 0.94 22 3.66 0.241–0.031 0.223–0.029

Ikinciogullari (2022, 2023) conducted a study on different types of stepped spillways, utilizing both experimental data from previous research and numerical models, including trapezoidal and circular designs. The primary objective was to evaluate the energy dissipation rate and propose new designs for enhanced energy dissipation. The study found that the trapezoidal stepped spillway proved to be more effective, providing up to 30% more energy dissipation compared to classical stepped spillways. Additionally, the circular stepped spillway performed better when the step radius was smaller. For models with step depths of 0.50, 1.50, and 2.50 cm, energy dissipation increased by 21.4, 38, and 50.7%, respectively, compared to the flat stepped spillway.

Data analysis of flow on a stepped spillway

Generally, the velocity distribution of the flow in the open channel flow of a stepped spillway is affected by the water depth and is similar to a parabola (Guo & Julien 2003). In addition, this equation can be determined by taking the depth from the steps to the water surface as the x-axis and the flow velocity along the depth of flow as the y-axis. It can be written as follows:
(8)
where B1, B2, and C are constants. Thus, the vertex, maximum, and minimum points of the parabola can be determined to identify the boundary layer thickness and the maximum velocity as shown in the following equations:
(9)
(10)
Then, when the boundary layer thickness and the maximum velocity have been identified, the variables can be analysed using the log-wake law using Equation (7) as the velocity-defect law deviation to determine the variables of flow with sediments. Variables similar to the collected data were obtained (Guo & Julien 2008). The equation is as follows:
(11)
From Equation (11), those parameters are shown in a logarithmic equation below:
(12)
where
(13)
(14)
where . Then, Equations (12) and (14) can be applied to find the values of and Π.

The results of studying the layer of turbulent flow on smooth and rough surfaces indicated that (Krogstad et al. 1992) where is the height of steps in the same direction with the flow . The flow data analysis from Equations (8) to (14) using some data (Frizell 1992) shows an example and exhibits the calculation steps by arranging some data collected from the experiment into Table 2.

Table 2

Flow velocity and measurement location

 0.019 0.021 0.027 0.034 0.039 0.046 0.052 0.064 0.077 0.09 0.102 0.115 0.127 0.14 2.055 2.247 2.7 3.083 3.274 3.449 3.605 4.058 4.11 4.145 4.163 4.215 4.337 4.389
 0.019 0.021 0.027 0.034 0.039 0.046 0.052 0.064 0.077 0.09 0.102 0.115 0.127 0.14 2.055 2.247 2.7 3.083 3.274 3.449 3.605 4.058 4.11 4.145 4.163 4.215 4.337 4.389

The data from Table 2 were plotted; the x-axis was the elevation above the channel bed and the y-axis was the flow velocity by the depth of the flow , as shown in Figure 3.
Figure 3

Relationship between the flow velocity and elevation above the channel bed.

Figure 3

Relationship between the flow velocity and elevation above the channel bed.

Close modal
From Figure 3, Equation (8) takes the form:
(15)
From Equation (15), the boundary layer thickness and the maximum flow velocity were calculated using the values −220.34, 51.46, and1.43, and Equations (9) and (10), respectively.
(16)
(17)

Equations (8)–(10) have been used to calculate the boundary layer thickness and the maximum flow velocity (um) from the data of Frizell (1992), Ruff & Ward (2002), Boes & Hager (2003), Hunt & Kadavy (2008), and Lurker et al. (2008), as shown in Table 3.

Table 3

Boundary layer thickness and maximum flow velocity (

Physical modelDatasetBoundary layer thicknessMaximum flow velocityPhysical modelDatasetBoundary layer thicknessMaximum flow velocity
δ (m) (m/s)δ (m) (m/s)
Frizell (1992)  0.117 4.436 Boes & Hager (2003)  0.688 15.777
0.101 5.134 0.685 15.665
0.103 5.467 0.604 16.547
Ruff & Ward (2002)  0.455 6.869 0.488 15.032
0.379 6.639 0.428 15.010
0.377 7.346 0.517 17.416
0.362 7.069 0.545 18.312
0.326 7.582 Hunt & Kadavy (2008)  0.080 3.204
0.450 5.783 0.086 4.020
0.494 8.087 Lurker et al. (2008)  0.165 7.354
0.518 10.028 0.232 12.170
0.461 9.401 0.255 13.060
10 0.419 11.418 0.316 14.244
11 0.336 6.569 0.507 15.735
12 0.463 8.566 0.168 7.308
13 0.519 12.184 0.293 13.345
14 0.494 9.571 0.331 13.900
15 0.474 11.360 0.389 15.305
16 0.380 7.502 10 0.667 16.701
17 0.558 10.833 11 0.134 7.365
18 0.572 14.525 12 0.291 13.235
19 0.535 16.415 13 0.318 13.907
20 0.545 16.484 14 0.419 15.160
21 0.504 8.263 15 0.648 16.827
22 0.710 10.298 16 0.162 7.302
23 0.682 15.674 17 0.289 13.119
24 0.662 17.723 18 0.319 13.851
25 0.590 16.871 19 0.399 15.092
26 0.689 9.596 20 0.651 16.832
27 0.676 11.824 21 0.154 7.019
28 0.639 16.267 22 0.258 11.920
29 0.689 15.908 23 0.279 12.673
30 0.696 13.494 24 0.369 13.938
31 0.663 15.167 25 0.560 15.256
Physical modelDatasetBoundary layer thicknessMaximum flow velocityPhysical modelDatasetBoundary layer thicknessMaximum flow velocity
δ (m) (m/s)δ (m) (m/s)
Frizell (1992)  0.117 4.436 Boes & Hager (2003)  0.688 15.777
0.101 5.134 0.685 15.665
0.103 5.467 0.604 16.547
Ruff & Ward (2002)  0.455 6.869 0.488 15.032
0.379 6.639 0.428 15.010
0.377 7.346 0.517 17.416
0.362 7.069 0.545 18.312
0.326 7.582 Hunt & Kadavy (2008)  0.080 3.204
0.450 5.783 0.086 4.020
0.494 8.087 Lurker et al. (2008)  0.165 7.354
0.518 10.028 0.232 12.170
0.461 9.401 0.255 13.060
10 0.419 11.418 0.316 14.244
11 0.336 6.569 0.507 15.735
12 0.463 8.566 0.168 7.308
13 0.519 12.184 0.293 13.345
14 0.494 9.571 0.331 13.900
15 0.474 11.360 0.389 15.305
16 0.380 7.502 10 0.667 16.701
17 0.558 10.833 11 0.134 7.365
18 0.572 14.525 12 0.291 13.235
19 0.535 16.415 13 0.318 13.907
20 0.545 16.484 14 0.419 15.160
21 0.504 8.263 15 0.648 16.827
22 0.710 10.298 16 0.162 7.302
23 0.682 15.674 17 0.289 13.119
24 0.662 17.723 18 0.319 13.851
25 0.590 16.871 19 0.399 15.092
26 0.689 9.596 20 0.651 16.832
27 0.676 11.824 21 0.154 7.019
28 0.639 16.267 22 0.258 11.920
29 0.689 15.908 23 0.279 12.673
30 0.696 13.494 24 0.369 13.938
31 0.663 15.167 25 0.560 15.256

Then, the shear velocity and the Coles wake strength (Π) were calculated using Equations (12) and (14). Moreover, the data from Table 1 were also applied by using the boundary layer thickness to determine the normalized distance from the wall, which is the ratio between the point where the data were collected and the boundary layer thickness . From the boundary layer theory, the scope of the analysed data is limited to the range of , where is the ratio between the point where the data were collected and the boundary layer thickness . Then, the graph can be plotted, where the x-axis is the normalized distance from the wall and the y-axis is the flow velocity as shown in Figure 4.
Figure 4

Relationship between the flow velocity and the normalized distance from the wall.

Figure 4

Relationship between the flow velocity and the normalized distance from the wall.

Close modal
From Figure 4, an equation can be written in the form of Equation (12):
(18)
From Equation (18), the slope was 1.235. The shear velocity was determined from the slope of Equation (18), where the von Kármán constant 0.41. Hence, the shear velocity was 0.506 m/s, calculated as follows:
(19)
From Equation (18), the constant . The Coles wake strength (Π) was calculated using , and in Equation (14) and found to be 0.139, as follows:
(20)

Equations (12)–(14) have been used to calculate the shear velocity ( ) and the Coles wake strength (Π) using data from the experiments of Frizell (1992), Ruff & Ward (2002), Boes & Hager (2003), Hunt & Kadavy (2008), and Lurker et al. (2008). The results are shown in Table 4.

Table 4

Shear velocity ( ) and Coles' wake strength (Π)

Physical modelDatasetShear velocity (m/s)Coles' wake strength Physical modelDatasetShear velocity (m/s)Coles’ wake strength
Frizell (1992)  0.507 0.139 Boes & Hager (2003)  1.536 0.127
0.547 0.125 1.020 0.112
0.561 0.128 1.290 0.121
Ruff & Ward (2002)  1.719 0.088 1.071 0.138
1.406 0.106 1.140 0.109
1.739 0.118 1.313 0.143
1.885 0.110 1.332 0.139
1.860 0.106 Hunt & Kadavy (2008)  0.243 0.117
0.711 0.139 0.370 0.125
1.409 0.167 Lurker et al. (2008)  1.055 0.124
1.795 0.113 1.334 0.088
1.506 0.124 1.195 0.108
10 2.179 0.108 1.218 0.124
11 0.071 0.527 1.020 0.0627
12 1.710 0.127 0.948 0.102
13 2.700 0.190 1.189 0.125
14 1.889 0.126 1.273 0.118
15 1.939 0.159 1.234 0.158
16 1.115 0.122 10 0.829 0.132
17 1.809 0.143 11 1.114 0.162
18 2.951 0.220 12 1.139 0.127
19 4.027 0.179 13 1.140 0.115
20 3.318 0.150 14 1.119 0.144
21 0.874 0.117 15 0.900 0.160
22 1.227 0.088 16 1.005 0.117
23 2.977 0.188 17 1.166 0.132
24 3.797 0.137 18 1.210 0.117
25 2.668 0.173 19 1.163 0.142
26 1.154 0.158 20 0.787 0.176
27 1.717 0.157 21 0.875 0.111
28 2.742 0.181 22 1.113 0.113
29 2.187 0.106 23 0.882 0.116
30 2.729 0.193 24 0.803 0.166
31 3.171 0.192 25 1.067 0.095
Physical modelDatasetShear velocity (m/s)Coles' wake strength Physical modelDatasetShear velocity (m/s)Coles’ wake strength
Frizell (1992)  0.507 0.139 Boes & Hager (2003)  1.536 0.127
0.547 0.125 1.020 0.112
0.561 0.128 1.290 0.121
Ruff & Ward (2002)  1.719 0.088 1.071 0.138
1.406 0.106 1.140 0.109
1.739 0.118 1.313 0.143
1.885 0.110 1.332 0.139
1.860 0.106 Hunt & Kadavy (2008)  0.243 0.117
0.711 0.139 0.370 0.125
1.409 0.167 Lurker et al. (2008)  1.055 0.124
1.795 0.113 1.334 0.088
1.506 0.124 1.195 0.108
10 2.179 0.108 1.218 0.124
11 0.071 0.527 1.020 0.0627
12 1.710 0.127 0.948 0.102
13 2.700 0.190 1.189 0.125
14 1.889 0.126 1.273 0.118
15 1.939 0.159 1.234 0.158
16 1.115 0.122 10 0.829 0.132
17 1.809 0.143 11 1.114 0.162
18 2.951 0.220 12 1.139 0.127
19 4.027 0.179 13 1.140 0.115
20 3.318 0.150 14 1.119 0.144
21 0.874 0.117 15 0.900 0.160
22 1.227 0.088 16 1.005 0.117
23 2.977 0.188 17 1.166 0.132
24 3.797 0.137 18 1.210 0.117
25 2.668 0.173 19 1.163 0.142
26 1.154 0.158 20 0.787 0.176
27 1.717 0.157 21 0.875 0.111
28 2.742 0.181 22 1.113 0.113
29 2.187 0.106 23 0.882 0.116
30 2.729 0.193 24 0.803 0.166
31 3.171 0.192 25 1.067 0.095

Development of the log-wake law equations for flow in a stepped spillway

A study to develop the log-wake law equations for flow in stepped spillway can be summarized as:

• – An incompressible flow has been used with the continuity and momentum equations.

• – Wall-free shear stress analysis was performed by considering the extent of the pressure distribution of the flow on the step. The boundary of the wake function of the stepped spillway () can be analysed using the direct approach. This is because the pressure distribution of the flow on the step is not very complicated.

• – Shear stress analysis near the wall was conducted by considering the influence of the shear stress on the wall or the surface of the flow on the step. The boundary of the wake function of the stepped spillway () can be analysed using the Newton-like approach, an accurate interpolation theory. It is more accurate than the direct approach because it uses the second derivative to analyse the equation.

Comparison of the observed and calculated values

After the development of the log-wake law equation in Section 3.2, the results from the modified log-wake law equations for a stepped spillway proposed in this study and the log-wake law equations for flow in smooth pipes were compared with the experimentally observed values to prove that the log-wake law equations for a stepped spillway are accurate. This study uses the correlation coefficient (Rcc) and R2 as statistical values for comparison.

Data analysis of the Coles wake strength

When the variables related to the flow in a stepped spillway and the variables related to the log-wake law equation using 68 datasets from stepped spillways were analysed, the Coles wake strength was between 0.085 and 0.175. Figure 5 shows the Coles wake strength and the Reynolds number where is the density, u is the flow velocity, is the boundary layer thickness, is the dynamic viscosity, and is the kinematic viscosity. Therefore, from the calculation using all datasets to determine the Coles wake strength, when the von Kármán constant was set at 0.41, the Coles wake strength was found to be 0.126.
Figure 5

Relationship between Coles’ wake strength and Reynolds’ number.

Figure 5

Relationship between Coles’ wake strength and Reynolds’ number.

Close modal

Development of the log-wake law equations for a stepped spillway

Theoretical analysis of a stepped spillway

This study considers a turbulent flow to be incompressible. The flow through the cross-section was steady along the x-axis, and the variables did not vary on the z-axis. The continuity and momentum equations were applied as follows:

• 1.
Continuity equation
(21)
• 2.
Momentum equation
(22)
(23)

From Figure 6, when the flow was analysed in the x-direction, the equation is as follows:
(24)
Figure 6

Analysis of the momentum equation in the flow x-direction.

Figure 6

Analysis of the momentum equation in the flow x-direction.

Close modal
The flow surface of the control volume along the , and directions was analysed. The surface was affected by the pressure and shear stress of the flow as described by the following:
(25)
where is the shear stress. Because the flow in a stepped spillway was classified as sub-layer flow, the analysis of the flow cross-section required the analysis of the viscosity between the boundary layer thickness and the analysis of the eddy viscosity of the boundary layer thickness in the steep slope stream23. This can be written as:
(26)
where , is the viscosity of dimensionless flow, and is the eddy viscosity. From Equation (26), the shear velocity is related to the bed shear stress. Therefore, the shear stress of the channel should be determined by integrating Equation (25) to define the constant of integration, which was . Then, the constant was substituted into Equation (25), which was arranged into a new equation, as shown in Equation (27).
(27)
Next, were substituted into Equation (27), and the equation was arranged into a form similar to that of the log-wake law equation, as shown in Equation (28).
(28)
Substituting into Equation (28) and arranging the equation in an integral form yielded the following equation:
(29)
Because one variable could not be determined in Equation (29), mathematical methods and the characteristics of the stepped spillway flow were applied to impose a boundary condition and resolve the problem. Equation (29) shows that two parts impact the flow velocity profile. The first part is , which is affected by the shear stress at the wall. The second part is which is affected by changes in pressure. Consequently, from the influence of shear stress and the wake function that affects the stepped flow in Figures 2 and 6, Guo & Julien (2008) have assumed that the first term could be used to determine the value of the logarithmic equation as follows:
(30)
where and were the wake functions of the stepped spillway.

Boundary condition for wall-free shear stress analysis

In order to consider the pressure distribution of the stepped spillway flow , which is the wall-free shear stress that was similar to the flow through a curved surface, the boundary of was analysed using the boundary condition as follows:
(31)
(32)
because was similar to the wall-free shear. Thus, from Figure 2, at the water area, or , is
(33)
Thus, the log-wake law Equation (6) suggests the boundary conditions of the wake function of the open channel flow when are:
(34)
Because the cross-sectional flow was inconsistent with a second-order polynomial equation, the cubic equation was used in the calculation. Therefore, with the chosen boundary condition, the following equation was obtained:
(35)

The analysis results from these boundary conditions using the direct approach showed that the wall-free shear stress equation was similar to the equation proposed by White (1991).

4.2.3. Boundary conditions for the near-wall shear stress analysis

In order to consider the influence of the near-wall shear stress of the stepped spillway flow using the mathematical method, the boundary condition of could be determined from Equation (35). Then, the mathematical method was applied, and the results were as follows:
(36)
(37)
(38)
(39)
because was similar to the near-wall shear stress. Thus, from Figure 2, when , at the water surface area is:
(40)
Then, substituting and in Equation (35), the equation is as follows:
(41)
Then, Equations (35) and (41) can be re-written as the velocity-defect law deviation as follows:
(42)
Figure 2 shows the flow profile friction and wake function distributions of a stepped spillway; the wake function has a maximum value between and has a maximum value when as follows:
(43)
Then, the flow variable is substituted in Equation (43) to find the value of , which is between − 0.4 and − 1.0. Figure 7 is a comparison graph of the term and the normalized distance from the wall.
Figure 7

Relationship between wake function w1 (1) and normalized distance from the wall.

Figure 7

Relationship between wake function w1 (1) and normalized distance from the wall.

Close modal
Therefore, when the wake function where the von Kármán constant was set at 0.41 was calculated using all the datasets, the wake function was − 0.73. Thus, interpolation theory was applied to this analysis using the boundary conditions resulting from the wall shear stress, and the equation can be presented as follows:
(44)
Using the above equation, the value differed from the log-wake law equation in previous studies since Equation (44) was adjusted to fit the stepped spillway flow well. Moreover, Equation (44) can be rearranged by substituting and and merged with Equation (35). The resulting log-wake law equation is as follows:
(45)

Comparison of data using the developed log-wake law equation

When testing Equation (45) with the datasets of Frizell (1992), Ruff & Ward (2002), Boes & Hager (2003), Hunt & Kadavy (2008), and Lurker et al. (2008) compare the results of the log-wake law equation of Guo & Julien (2008), Equation (11) and the developed log-wake law equation proposed in this study, Equation (45). Example results are shown in Figures 812.
Figure 8

Relationship between the measured data (x-axis) (1992) compared with the calculated data from equations (y-axis). (a) Calculated data from Equation (11) and (b) calculated data from Equation (45).

Figure 8

Relationship between the measured data (x-axis) (1992) compared with the calculated data from equations (y-axis). (a) Calculated data from Equation (11) and (b) calculated data from Equation (45).

Close modal
Figure 9

Relationship between the measured data (x-axis) (Ruff & Ward 2002) compared with the calculated data from equations (y-axis). (a) Calculated data from Equation (11) and (b) calculated data from Equation (45).

Figure 9

Relationship between the measured data (x-axis) (Ruff & Ward 2002) compared with the calculated data from equations (y-axis). (a) Calculated data from Equation (11) and (b) calculated data from Equation (45).

Close modal
Figure 10

Relationship between the measured data (x-axis) (2003) compared with the calculated data from equations (y-axis). (a) Calculated data from Equation (11) and (b) calculated data from Equation (45).

Figure 10

Relationship between the measured data (x-axis) (2003) compared with the calculated data from equations (y-axis). (a) Calculated data from Equation (11) and (b) calculated data from Equation (45).

Close modal
Figure 11

Relationship between the measured data (x-axis) (2008) compared with the calculated data from equations (y-axis). (a) Calculated data from Equations (11) and (b) calculated data from Equation (45).

Figure 11

Relationship between the measured data (x-axis) (2008) compared with the calculated data from equations (y-axis). (a) Calculated data from Equations (11) and (b) calculated data from Equation (45).

Close modal
Figure 12

Relationship between the measured data (x-axis) (2008) compared with the calculated data from equations (y-axis). (a) Calculated data from Equations (11) and (b) calculated data from Equation (45).

Figure 12

Relationship between the measured data (x-axis) (2008) compared with the calculated data from equations (y-axis). (a) Calculated data from Equations (11) and (b) calculated data from Equation (45).

Close modal

Thus, the log-wake law (Equation (11)) and the developed log-wake law equation from this study (Equation (45)) could be compared using statistics, as shown in Table 5.

Table 5

Results of comparison between log-wake law Equation (11) and developed log-wake law Equation (45)

AuthorsModified log-wake law Equation (11)
Modified log-wake law Equation (45)
Correlation coefficient (Rcc)R2Correlation coefficient (Rcc)R2
Frizell (1992)  0.947 0.992 0.993 0.992
Ruff & Ward (2002)  0.873 0.927 0.948 0.938
Boes & Hager (2003)  0.966 0.977 0.975 0.978
Hunt & Kadavy (2008)  0.950 0.972 0.994 0.975
Lurker et al. (2008)  0.910 0.964 0.951 0.961
AuthorsModified log-wake law Equation (11)
Modified log-wake law Equation (45)
Correlation coefficient (Rcc)R2Correlation coefficient (Rcc)R2
Frizell (1992)  0.947 0.992 0.993 0.992
Ruff & Ward (2002)  0.873 0.927 0.948 0.938
Boes & Hager (2003)  0.966 0.977 0.975 0.978
Hunt & Kadavy (2008)  0.950 0.972 0.994 0.975
Lurker et al. (2008)  0.910 0.964 0.951 0.961

From the study on the log-wake law equation with the turbulent flow equation for the simulation of stepped spillway applying the theoretical, physical and mathematic analysis, it was discovered that the flow in the stepped spillway was affected by two factors, the wall shear stress and the pressure gradient (wall-free shear). Thus, in order to adjust the basic logarithmic equation and log-wake law equation to fit well with the flow in stepped spillway, Equation (45) is proposed as the modified log-wake law equation. From the analysis results, it could be summarized as follows:

• – The proposed equation comprised of three parts,

• (i)

the logarithmic part where the von Kármán constant was set at 0.41,

• (ii)

the polynomial cube equation where the Coles wake strength was 0.126, and

• (iii)

the adjustment to explain the flow in the stepped spillway which was the polynomial to the fourth power equation.

It was obvious that the equation contained the Coles wake strength to be consistent with the flow in stepped spillway, which was influenced by the wake flow on the step area.

• – From the statistical comparison result of log-wake law equation, which was the cosine equation and the developed log-wake law equation, Equation (45) which was the polynomial equation, with a total of 68 sets of data, it was found out that the correlation coefficient (Rcc) of the developed log-wake law equation was higher than that of Equation (11) about 2–10% as shown in Table 5. It could be concluded that the use of developed log-wake law equation for calculation would obtain the data that was consistent with the observed data. Moreover, the calculated data are also getting closer to the observed data than that from the log-wake law equation. Consequently, the application of modified log-wake law equation is more appropriate for the stepped spillway.

Therefore, from the above conclusions, it was found that for flow in open channel, Equation (45) shows more accurate computational results than the logarithmic Equation (11). The introduction of the swirling flow equations presented in this research to be used to calculate the vertical flow velocity distribution or variables related to flow in open-stepped waterways. The flow data should have a water flow rate in the range of 0.0233–3.285 m3/s or a flow velocity of not more than 4 m/s for the accuracy of the calculations. The Coles wake strength (Π) presented in this paper is suitable for flow in open-stepped waterways only.

This research was funded by Petchra Pra Jom Klao master's Degree Research Scholarship from King Mongkut's University of Technology Thonburi

The authors declare there is no conflict.

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