Open channel flow and velocity behaviour presents a dilemma for drainage systems designers where hydrologic parameters are fluctuating in space and time. The experience of having extreme flash floods almost every year is flustering the need for understanding the flow behaviour at different altitudes. In this study, open channel experimental efforts were made to model flow and velocity profiles. The three-dimensional (3D) open channel flow and velocity profiles are generated at two types of roughness namely steel (smooth) and concrete (rough beds). The experiments included different slope gradients and flow measurements at different distances. The channels slope ranges between 0 and 4.7% with an interval of 0.2%. Multiple linear regression (MLR) was applied to quantify the flow for longer distance while Kriging extrapolation proxy was used to generate 3D surfaces of flow and velocity. The results showed that the flow in concrete channel is decreasing by moving to the end of channel due to higher frictional resistance while it is rising up for the steel channel. In average, the velocity has been increased by 7.4% for steel and 6.1% for concrete channels at a changing slope.

  • Experimental validations of two open channel types are important to predict flow rates and velocities for different lengths with complex slope and roughness conditions.

  • The Pearson correlation shows a strong relation between the flow rates and the slope.

  • The results showed the applicability of multiple linear regression model and 3D kriging model for analyzing real flow structure in a longer distance.

The status of urban drainage facilities as a part of the integrated infrastructure system is still under development in many regions around the world. This, however, depends on the level of development and the society awareness of the drainage system importance. As the accumulated flow reaches the catchment outlet, it may destroy the infrastructure of located city or traffic roads, unless it is accommodated in a proper designed drainage system. The risk of damage caused by this flow is basically based on water magnitudes, rainfall intensity, and storm duration. Therefore, a proper design of drainage network at the catchment outlet section is important to avoid life losses and damages of infrastructure. Flow and velocity distributions in drainage system are needed for a wide range of applications in hydrologic studies such as flood early warning, sediment transport, urban drainage system design and maintenance. The research in this field started even before Darcy, Weisbach, and Manning when ancient nations conveyed rainwater from the mountains into resident areas. The flow rate and velocity in open channel flow have been studied physically and mathematically by numerous researchers, e.g. Alawdi & Prasad (2018); Basu (2019); Benoumessad et al. (2014); Li Zeng & Bai (2020); Wang et al. (2019); Welderufael et al. (2019); and Abushandi & Al Sarihi (2022). In addition to those studies, several experiments have been conducted to model the turbulent flow and velocity in both two- and three-dimensional (2D and 3D) cases using Reynolds-averaged Navier-Stokes equations (RANS) (Kang & Choi 2006; Wu et al. 2018; Mohotti Wijesooriya & Dias-da-Costa 2019; Welderufael et al. 2019). They calculated the kinetic energy and turbulent dissipation at a fixed Reynolds number. Apart from some pilot studies at a catchment scale, e.g. Abdel-Fattah et al. (2017); Abushandi & Merkel (2013); Aceves & Fuamba (2016); and Kordilla et al. (2013); the flow magnitudes have been estimated based on empirical laboratory investigation. The flow attributes in the channels are directly connected to channel slope and surface roughness. Therefore, several investigations in recent years have been conducted to determine the flow and vertical distributions of velocities for turbulent fluids on both ‘smooth’ and ‘rough’ surfaces (Kordilla et al. 2013; Singh Raushan & Debnath 2018; Bormashenko 2019). Computation findings for velocity distribution in turbulent smooth-wall open channel flows were assessed and compared to experimental data (Welderufael et al. 2019). The results concluded that the model based on the Reynolds-averaged Navier-Stokes (RANS) equation provides accurate assessments. An ordinary differential equation (ODE) for velocity distribution in open channel flows is presented by Absi (2011) based on an analysis of the Reynolds-averaged Navier–Stokes equations and a log-wake modified eddy viscosity distribution. The method allows predicting the velocity-dip-phenomenon, i.e. the maximum velocity below the free surface. Pal et al. (2016) attempt to model the impact of particles on a stream wise velocity profile at several flow conditions using mathematical modelling. Their method provided an accurate assessment of velocity vertical and horizontal profiles. In addition, the assumption of steady-state, turbulent flow, or velocity distributions were the core for computational methods (Shi & Yu 2015; Govorukhin & Zhdanov 2018; Kimiaghalam et al. 2018; Luo et al. 2018; Zhang et al. 2018; Zhao et al. 2019). The estimation of sediment transport rates in open channels is of great interest to engineers in addition to slope and vegetation drag forces because of the practical importance of sediment transport in changing flow and velocity rates (Zhong et al. 2015; Palucis et al. 2018; Elgueta-Astaburuaga & Hassan 2019; Tan & Yuan 2019).

The sampling interval or frequency has taken a place for accurate simulation (Ruonan et al. 2016). However, the state of art technology in measurements and simulation were developed to improve the results such as unmanned aerial vehicle (UAV)-based tracer tests using RGB (red, green, blue) images (Baek et al. 2019). Researchers have studied the velocity and flow profiles intensively in open and close conduits. Kirkgoz (1989) measured the velocity using a laser-doppler anemometer for a fully developed, rectangular, subcritical open channel flow on smooth and rough beds. The calculated velocities showed an increasing tendency as the wall roughness increases. The representation of the overall data in terms of law-of-the-wall distribution seems reasonable; however, the velocity-defect distribution is not satisfactory because of small flow depth. A couple of years later Kirkgoz & Ardlboglu (1997) approved better results, they found a strong linear relationship between the channel length and flow depths. Castellarin (2014) used Kriging interpolation method to spatially estimate flow for ungagged basin which dominated by same factors. It is a fact that Kriging method gives better result in a homogenous channel.

The primary objectives of this research are to provide a predictive tool for flow rates and velocities for complex real-world conditions with different lengths and surface roughness. Furthermore, spatially describe flow and velocity distribution and test slope and surface roughness affecting flow and velocity. For this purpose, two open channels were designed, and experimental laboratory efforts have been conducted.

The concrete grade (M15) was used to structure the rough open channel. The mix included Ordinary Portland Cement, sand as fine aggregate and coarse aggregate with a maximum size of 20 mm. The concrete mix ratio of 1:2:4 with 0.5 water–cement ratio. In addition, the compression strength for M15 is 15 MPa (N/mm2) which is suitable for water flow and water direct contact during the experiment's time.

The steel channel was designed from a plate has resistance to general corrosion with a thickness of 2.0 mm.

Experimental set-up

The steel and concrete channels were designed and manufactured at Sohar University/Hydraulics Laboratory with the dimensions of 4 m long, 0.2 m wide and 0.3 m deep. Figure 1 shows the vertical and horizontal dimensions of the two open channels and velocity measurement points. In addition, a 3D scanning was conducted to show the surface of a concrete open channel. However, the dimensions of both channels are the same, in particular, the width of 20 cm, and will have the same surface tension effects. Therefore, the effect on flow generated by scale is neglected. A small, electrically powered centrifugal pump was used to provide water to a closed water circle with a flow rate of 0.5 m3/s. The roughness values used for concrete is 0.2 mm while steel has a value of 0.02 mm (Native Dynamics 2020).
Figure 1

Channels major elements, dimensions, and the points of measurements, and 3D scanning of the concrete channel using 3D EinScan HX 3D-Scanner.

Figure 1

Channels major elements, dimensions, and the points of measurements, and 3D scanning of the concrete channel using 3D EinScan HX 3D-Scanner.

Close modal
The flow rate tests of the two conveyance systems were measured at several lengths and slopes to ensure materials suitability for the channel functionality. The flow rates were measured using float method and compared to ultrasonic flowmeter to ensure the accuracy. The steel channel bed was thought to be suitable for a smooth surface wall experiment (Figure 2), while the concrete is considered for the rough surface (Figure 3). The steel channel was manufactured by bending the sheet without using any inside welding to avoid minor losses of energy. This, however, helps to consider the major losses water flow area and avoid any affect in the velocity values accuracy; one long bolt was fixed as a jack to control the channel slope. Furthermore, the most common used material to construct a drainage system nowadays is the concrete. Therefore, it was an important aspect to test the flow rate through a designed concrete channel. The channels were designed to meet the real-world adjustment. The models were performed for a variety of slope grades while always maintaining a constant flowrate from the source. The system can be applied for agricultural and domestic purposes to collect harvest rainwater at different slope gradients. It is a hand on the development of appropriate techniques for designing drainage system to accommodate the unexpected flood magnitudes.
Figure 2

Designed smooth steel channel to conduct flow rates and velocity rates.

Figure 2

Designed smooth steel channel to conduct flow rates and velocity rates.

Close modal
Figure 3

Designed concrete channel to conduct flow and velocity rates.

Figure 3

Designed concrete channel to conduct flow and velocity rates.

Close modal

The design is similar to the internal height and width of the steel channel in order to get a constant cross-sectional area while conducting the flow tests and the regression model. Flow system was designed including a tank with 100 L capacity and the pipes system was connected to get a water cycle. The slope interval for the two channels was between 0.0% and 4.7%.

Mathematical representation

There are several equations used for this research including Kriging interpolation method, Pearson correlation, multi-linear regression model and Nash-Sutcliffe efficiency equation.

Kriging interpolation method was used to 3D interpolate the observed flow and velocity. Kriging is a method of interpolation named after the South African mining engineer D. G. Krige, who created the technique while investigating correlation parameters in 1951 (Ryu et al. 2002). The method takes into considerations spatial autocorrelation of observed records and the dimension of the channel.

Kriging is the same as inverse distance weighing method (IDW) that considers the value and location of an observed point to find unmeasured values. The general formula of Kriging as follows:
where:
  • is the measured value of flow or velocity at the location of 1–4 m

  • is the unknown weight for flow or velocity at the location at i location

  • is the predicted location

  • N is the number of measured values

To assess the efficiency of Kriging method results Pearson correlation was used in order to evaluate the relationship between the two different flows based on the following formula:
The values of Pearson correlation are always between −1 and 1, and if x and y are not related, the correlation is equal to zero. Furthermore, multiple linear regressions model was used to model the relationships between slope, friction, and flow or velocity by fitting a linear equation to observed data. General formula of multiple linear regressions model is given by:
where
  • is the modelled value of flow or velocity

  • are explanatory variable (slope and surface roughness)

  • are the fitting values based on linear relationship

The Nash-Sutcliffe efficiency (Ef) has frequently been applied to assess the goodness of modelled records in comparison with observed records:
where is observed flow, is simulated flow and is the mean value of observed flow.
The laboratory experiments were conducted to determine the accurate velocity and flow records. The concrete channel is presenting the case of ‘rough’ walls with records obtained at different slope ranges for the same initial flow rates. The velocity and flow behaviours showed continued decline rate by moving from the source of water to the tail water location. However, this was not the case in steel channel which represents the smooth walls. The velocity and flow behaviours show continued rising rates by moving from the source of water to form a hydraulic jump at the end of the steel channel (Figure 4). The hydraulic jump explains dissipation of surplus water energy at the end of the channel.
Figure 4

Hydraulic jump in steel channel.

Figure 4

Hydraulic jump in steel channel.

Close modal
Figures 5 and 6 show the changing rates in both concrete and steel channels for measuring points. (You et al. 2019) studied the turbulent behaviour of flow through a steel channel showed that the turbulent kinetic energy is happening at beginning and the end of the channel. However, there is no major difference of velocity between the top and the bottom of the channel.
Figure 5

Steel vs. concrete channels average velocities in m/s.

Figure 5

Steel vs. concrete channels average velocities in m/s.

Close modal
Figure 6

Steel vs. concrete channels average flow in m3/s.

Figure 6

Steel vs. concrete channels average flow in m3/s.

Close modal
Generally, the centerline average maximum flow and velocity appeared to be at the first meter for the concrete with values of 0.27 m3/s and 0.82 m/s, respectively. While the average maximum flow and velocity appeared to be at the tail of steel channel with values of 0.4 m3/s and 1.1 m/s, respectively. Indeed, there is a linear relationship between the dimensionless length and changing rates of open channel flow and velocity in both channels. The velocity profiles were measured while the slope is changing at four different points along the channels to extract precise quantified relationship between velocity and flow with changing slopes (Figure 7). It is obvious for experts that the velocity profiles will be increased by increasing the slopes. However, the major concern of this experiment is to answer the question of how much the word ‘increase’ means? In fact, this is an important aspect for engineers to construct an accurate linear regression model. However, the prediction of water flow and velocity values agreed well with different experimental test cases by Ahadi et al. (2019). Depending on the wall roughness starting from smooth to very rough surfaces with perfectly diffuse collisions, even when the experimental design includes vegetation or sediments the flow takes a linear behaviour (Kang & Choi 2006; Huai et al. 2012).
Figure 7

Concrete channel velocity.

Figure 7

Concrete channel velocity.

Close modal

It is necessary to apply Pearson's correlation to show the relationship between discharge and slope. Although this observation has been known for nearly two centuries, there is a need to quantify such relationship and understand the flow behaviour for different surfaces and slopes. Based on observations comparisons, Pearson correlation between the flow at different distance and slope in concrete and steel showed a significant relationship between each two variables with an average value of r equal to 0.86 and 0.99 for concrete and steel, respectively. However, when the flow in the concrete channel is declining, it has been rising in the steel channel. While r values between the slope and the concrete channel was 0.88, and for the steel channel flow was 0.98. These values reflect how strong the influence of slope is on the flow. Tables 1 and 2 showed detailed values of correlation between slope gradients and flow rates at different points of measurements. Generally, the correlation is significant, thus, MLR can be successfully applied.

Table 1

Pearson correlation for the concrete channel flow in response to slope

VariableSlopeQ1Q2Q3Q4
Slope 0.78 0.84 0.91 0.86 
Q1 0.78 0.70 0.70 0.68 
Q2 0.84 0.70 0.90 0.91 
Q3 0.92 0.70 0.90 0.98 
Q4 0.86 0.68 0.91 0.98 
VariableSlopeQ1Q2Q3Q4
Slope 0.78 0.84 0.91 0.86 
Q1 0.78 0.70 0.70 0.68 
Q2 0.84 0.70 0.90 0.91 
Q3 0.92 0.70 0.90 0.98 
Q4 0.86 0.68 0.91 0.98 
Table 2

Pearson correlation for the steel channel in response to slope

VariablesSlopeQ1Q2Q3Q4
Slope 0.98 0.98 0.99 0.99 
Q1 0.98 
Q2 0.98 
Q3 0.99 
Q4 0.99 
VariablesSlopeQ1Q2Q3Q4
Slope 0.98 0.98 0.99 0.99 
Q1 0.98 
Q2 0.98 
Q3 0.99 
Q4 0.99 

To simulate flow rates values at 1, 2, 3, and 4 m within different length of channels, an adjustment was performed based on multiple linear regressions to model additional length of channels, Q5 represent the value of modelled flow for one additional meter:

Concrete channel MLR:
Steel channel MLR:

However, the purpose of developing MLR equations is to simulate the flow in a further distance with slope and surface roughness variability.

Based on the statistical analysis, the accuracy of the modelled values is around 97%. In addition, the Nash-Sutcliffe efficiency (Ef) was around 88.4 and 83.1 for concrete and steel channels, respectively.

In a different manner, 3D numerical models for calculating flow and velocity profiles in open channels were based on Kriging formula. The models showed the changes with respect to the distance (Figures 8 and 9). The profile shows the values of unmeasured points. Results indicated that Kriging method has the potential to spatially visualize the flow and velocity. In addition, using MLR equation can further produce extrapolation for longer distances.
Figure 8

3D wireframe of steel channel flow and velocity distributions (smooth surface) using Kriging Interpolation method at average slope 0.23%.

Figure 8

3D wireframe of steel channel flow and velocity distributions (smooth surface) using Kriging Interpolation method at average slope 0.23%.

Close modal
Figure 9

3D wireframe of concrete channel flow rate and velocity distributions (rough surface) using Kriging Interpolation method at slope at average slope 0.23%.

Figure 9

3D wireframe of concrete channel flow rate and velocity distributions (rough surface) using Kriging Interpolation method at slope at average slope 0.23%.

Close modal

The performance of computed velocities and flows are rather good in comparison to observed records. Kriging method was tested against several slopes for similar flow rate. All tests provided very good correspondence between modeled and observed measured profiles. Generally, Kriging extrapolation method seems to be an appropriate tool for analysing open channel spatial data. The accuracy of quantified estimation variance average reached 84.6 and 98.5% for concrete and steel channels, respectively. Despite the interpolation method used, the advantage of velocity and flow is that the effects of roughness and slope over the entire channel distance have been considered at same time, instead of temporal (points) effect. However, this research was conducted at laboratory conditions. Real-world open channels have much more complicated mechanisms due to sedimentation and slope fluctuation along the channel. In addition, natural open channels in arid areas (wadis) are dominated by rocks, and limited vegetation cover. This, however, makes most modelling flow and velocity profiles in vegetated channels based on a single physical concept (Nikora et al. 2013). Furthermore, modelling natural channels includes incident conditions including soil moisture content (Abushandi & Al Sarihi 2022).

Experimental validations of smooth and rough open channels are important to predict flow rates and velocities for different lengths with complex slope and roughness conditions. As the slope is increasing flow rate and velocity of both type channels are increasing. On average, the velocity has been increased by 7.4% for steel and 6.1% for concrete channels at each increasing slope interval. While the velocity tendency over concrete surface was higher due to frictional resistance. However, further studies are required to investigate real-world open channel hydrologic behavior including sediments transport and vegetation cover influence. A hydraulic jump has been created in a steel channel while the flow rates and velocity are declining by moving to the end of channel due to friction. MLR model can be applied as far as a linear relationship is found between different variables. The measurements were conducted for 20 different slope gradients to understand numerically how this parameter is influencing water flow and velocity in open channels. The three-dimensional representations provided a clear image flow and velocity behavior. Further measurements will be conducted to understand vegetation cover and settlements impacts on flow and velocity profiles.

The research leading to these results has received funding from the Research Council (TRC) of the Sultanate of Oman under the Open Research Grant Program #BFP/RGP/EBR/19/164.

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

The authors declare there is no conflict.

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