Abstract
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.
HIGHLIGHTS
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.
INTRODUCTION
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.
MATERIALS AND METHODS
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
Channels major elements, dimensions, and the points of measurements, and 3D scanning of the concrete channel using 3D EinScan HX 3D-Scanner.
Channels major elements, dimensions, and the points of measurements, and 3D scanning of the concrete channel using 3D EinScan HX 3D-Scanner.
Designed smooth steel channel to conduct flow rates and velocity rates.
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.
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
is the modelled value of flow or velocity
are explanatory variable (slope and surface roughness)
are the fitting values based on linear relationship
RESULT AND DISCUSSIONS
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.
Pearson correlation for the concrete channel flow in response to slope
Variable . | Slope . | Q1 . | Q2 . | Q3 . | Q4 . |
---|---|---|---|---|---|
Slope | 1 | 0.78 | 0.84 | 0.91 | 0.86 |
Q1 | 0.78 | 1 | 0.70 | 0.70 | 0.68 |
Q2 | 0.84 | 0.70 | 1 | 0.90 | 0.91 |
Q3 | 0.92 | 0.70 | 0.90 | 1 | 0.98 |
Q4 | 0.86 | 0.68 | 0.91 | 0.98 | 1 |
Variable . | Slope . | Q1 . | Q2 . | Q3 . | Q4 . |
---|---|---|---|---|---|
Slope | 1 | 0.78 | 0.84 | 0.91 | 0.86 |
Q1 | 0.78 | 1 | 0.70 | 0.70 | 0.68 |
Q2 | 0.84 | 0.70 | 1 | 0.90 | 0.91 |
Q3 | 0.92 | 0.70 | 0.90 | 1 | 0.98 |
Q4 | 0.86 | 0.68 | 0.91 | 0.98 | 1 |
Pearson correlation for the steel channel in response to slope
Variables . | Slope . | Q1 . | Q2 . | Q3 . | Q4 . |
---|---|---|---|---|---|
Slope | 1 | 0.98 | 0.98 | 0.99 | 0.99 |
Q1 | 0.98 | 1 | 1 | 1 | 1 |
Q2 | 0.98 | 1 | 1 | 1 | 1 |
Q3 | 0.99 | 1 | 1 | 1 | 1 |
Q4 | 0.99 | 1 | 1 | 1 | 1 |
Variables . | Slope . | Q1 . | Q2 . | Q3 . | Q4 . |
---|---|---|---|---|---|
Slope | 1 | 0.98 | 0.98 | 0.99 | 0.99 |
Q1 | 0.98 | 1 | 1 | 1 | 1 |
Q2 | 0.98 | 1 | 1 | 1 | 1 |
Q3 | 0.99 | 1 | 1 | 1 | 1 |
Q4 | 0.99 | 1 | 1 | 1 | 1 |
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:
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.
3D wireframe of steel channel flow and velocity distributions (smooth surface) using Kriging Interpolation method at average slope 0.23%.
3D wireframe of steel channel flow and velocity distributions (smooth surface) using Kriging Interpolation method at average slope 0.23%.
3D wireframe of concrete channel flow rate and velocity distributions (rough surface) using Kriging Interpolation method at slope at average slope 0.23%.
3D wireframe of concrete channel flow rate and velocity distributions (rough surface) using Kriging Interpolation method at slope at average slope 0.23%.
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).
CONCLUSION
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.
ACKNOWLEDGEMENTS
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.
DATA AVAILABILITY STATEMENT
All relevant data are included in the paper or its Supplementary Information.
CONFLICT OF INTEREST
The authors declare there is no conflict.