Prediction of depth-averaged velocity in an open channel flow
- 565 Downloads
Abstract
This paper presents a new methodology to predict the depth-averaged velocity along the lateral direction in an open channel flow. The novelty of this work is to determine the point velocity and estimate the discharge capacity by knowing the geometrical parameters at a section of an open channel flow. Experimental investigations have been undertaken in trapezoidal and rectangular channels to observe the variation of local velocities along both the vertical and transverse directions at testing sections. For different geometry, hydraulic and roughness conditions, the measurements are taken for several flow conditions. Multi-variable regression analysis has been adopted to develop five models to predict the point velocities in terms of non-dimensional geometric and flow parameters at any desired location. The present method is favourably compared with the analytical method of Shiono and Knight with reasonable accuracy. The performance of mathematical model is also validated with two natural river data sets. Further, statistical error analysis is carried out to know the degree of accuracy of the present models.
Keywords
Open channel flow Velocity profiles Regression analysis Depth-averaged velocity Error analysisList of symbols
- b
Total width and top width of rectangular and trapezoidal channel, respectively
- b/2
Half width of the channel
- H
Flow depth
- n
Manning’s roughness coefficient
- z
Vertical coordinate above the bed along depth of flow
- y
Lateral coordinate along the width of the channel
- U
Point/local velocity
- \(U_{\text{mean}}\)
Mean velocity
- S_{0}
Bed slope/longitudinal slope
- Q
Discharge of the channel
- A
Wetted area of cross section
- P
Wetted perimeter of the channel
- R = A/P
Hydraulic radius of channel
- τ
Boundary shear stress
- ρ
Water density
- g
Acceleration due to gravity
- \(u, v, w\)
Components of velocity along x, y, z directions, respectively
- \(u^{{\prime }} ,v^{{\prime }} w^{{\prime }}\)
Components of turbulence intensity along x, y, z directions, respectively
- \(\bar{u},\bar{v},\bar{w}\)
Time-averaged mean velocity along x, y, z directions, respectively
- f
Darcy–Weisbach friction factor
- λ
Dimensionless eddy viscosity
- Γ
Secondary flow parameter
- s
Side slope
- R^{2}
Coefficient of deterministic
- MAPE
Mean absolute percentage error
- RMSE
Root-mean-square error
- FCF
Flood channel facility
Introduction
Rivers have been used as a source of water for procuring food, transport, navigation and as a source to generate hydropower to operate machinery. Generally, the water in a river is restricted to a channel, assembled with stream bed and side banks. Understanding the flow velocity of these rivers is most crucial for river engineers for a broad range of application in different exercises such as the meticulous study of water quantity and quality. Further, vertical and lateral velocity distributions are the fundamental understanding of the state of flow in channels, as required for flow modelling, extremity spill management and for different technical aspects related to living organisms and human beings. Generally, the velocity in a cross section differs from point to point, due to the effects of water surface and shear stress at the bed. As the velocity distribution in an open channel is complex, modelling the velocity is not an easy task (Maghrebi and Givehchi 2009). Flow prediction of natural rivers and urban channels are accurately evaluated from the vertical and lateral velocity distributions in association with depth-averaged velocity for several geometric conditions. Hydraulic engineers are always searching for suitable methods of calculating mean discharge in the channels having different shapes and sizes with minimal need of substantial measurement (Jan et al. 2009).
Sarma et al. (1983) studied velocity distributions in a smooth rectangular channel by dividing the channel into four regions for different ranges of hydraulic parameters. Steffler et al. (1985) measured mean velocity as well as turbulence for uniform flow in a smooth rectangular channel for three different aspect ratios such as 5.08, 7.83 and 12.3. They studied the logarithmic law of velocity distribution in the respective channels. Tominaga et al. (1989) studied the secondary currents and also modified the turbulence anisotropy which is affected by the boundary conditions of the bed, the side walls, the free surface as well as the aspect ratio and geometry of the channels. Blumberg et al. (1992) conducted experiments in both smooth and rough open channels and incorporated second-moment turbulence closure model to simulate turbulent flows with various geophysical and engineering boundary layers. Nezu et al. (1997) conducted experiments to measure the turbulence successfully over a smooth bed with non-uniform and unsteady flow. They utilised the two-component laser Doppler anemometer (LDA) to measure the two components of velocity. Shiono and Feng (2003) presented the turbulence measurements of velocity and tracer concentration in rectangular and compound channels using a combination of Laser Doppler anemometer (LDA) and laser induced fluorescence (LIF). Liao and Knight (2007) derived three analytical models which were suitable for hand calculation to find out the stage-discharge relationship in simple channels as well as in symmetric and asymmetric compound channels. Zarrati et al. (2008) modelled semi-analytical equations for distribution of shear stress in straight open channels with rectangular, trapezoidal, and compound cross-sectional areas. Ansari et al.(2011) exhibited the utilization of computational fluid dynamics (CFD) to estimate the bed shear and wall shear stresses in trapezoidal channels. Specifically for low aspect ratio channels, the variety of inclination angle and aspect ratio conveyed significant changes to the distribution of the shear stress at the boundaries as well as in the flow structures as already shown by De Cacqueray et al. (2009). Jesson et al.(2012) simulated the open channel flow over a heterogeneous roughened bed and also analysed it both physically and numerically. The velocity field was mapped at four distinctive cross sections by utilizing an Acoustic Doppler Velocimeter (ADV) and the boundary shear stress is obtained by using the Preston tube technique. Yang et al. (2012) proposed a depth-averaged equation of flow by analysing the forces acting on the natural water body and utilizing the Newton’s second law. Khuntia et al. (2016) investigated experimentally the variation of global and local friction factor based on the measurement of depth-averaged velocity and boundary shear stress over the cross section in channels of different geometries.
Shiono and Knight (1990) simplified the momentum equation to estimate the lateral depth-averaged velocity and boundary shear distribution, and their method is popularly known as Shiono and Knight Method (SKM). The SKM method offers an improved analytical solution to predict the flow parameters in an open channel flow; however, it depends on the three calibrating parameters i.e. f, λ, and Γ before its application.
So, considering the importance of the velocity distribution for the estimation of a number of hydraulic parameters, it is necessary to derive a common, precise and user-friendly method to evaluate the local velocity at any desired point. This local velocity at every point helps to estimate the distribution of depth-averaged velocity and overall flow in a channel. The objective of this paper is to develop expressions to predict local velocities at any desired location of homogeneous roughness channels, which in turn help to estimate the flow distribution and stage-discharge relationships.
Experimental setup and procedure
Details of geometric and hydraulic parameters of the experimental setup
Series name | Shape | Surface condition | Bed width B(m) | Flow depth H(m) | Roughness value (n) | Bed slope S_{0} | Discharge Q (m^{3}/s) |
---|---|---|---|---|---|---|---|
NITR1 | Rectangular | Smooth | 0.34 | 0.076–0.107 | 0.01 | 0.0015 | 0.012–0.020 |
NITR2 | Trapezoidal | Smooth | 0.33 | 0.08–0.11 | 0.011 | 0.001 | 0.016–0.026 |
NITR3 | Trapezoidal | Rough | 0.33 | 0.07–0.09 | 0.02 | 0.001 | 0.006–0.01 |
FCF | Trapezoidal | Smooth | 1.5 | 0.049–0.149 | 0.01 | 0.00103 | 0.029–0.202 |
Tominaga et al. (1989) S1 | Rectangular | Smooth | 0.4 | 0.05–0.199 | 0.01 | 0.000937 | 0.008–0.015 |
Tominaga et al. (1989) S2 | Trapezoidal | Smooth | 0.152 | 0.071 | 0.01 | 0.000594 | 0.0062 |
Tominaga et al. (1989) S3 | Trapezoidal | Smooth | 0.2 | 0.091 | 0.01 | 0.000594 | 0.010 |
Tominaga et al. (1989) S4 | Trapezoidal | Smooth | 0.248 | 0.11 | 0.01 | 0.000594 | 0.011 |
The velocities were measured by a SonTek Micro 16-MHz Acoustic Doppler Velocimeter (ADV). The sampling rate is 50 Hz (the maximum). Sampling volume of ADV is located approximately 5 cm below the down looking probe and was set to be minimum of 0.09 cm^{3}. The 5 cm distance between the probe and sampling volume minimizes the flow interference. A total of 2, 97,000 data points were recorded (at 50 Hz) for a total recording length of 99 min for rectangular channel and 3, 87,000 data points were recorded (at 50 Hz) for a total recording length of 129 min for trapezoidal channel. Correlation has been used to monitor data quality during collection and to edit data in post-processing. Ideally, correlation should be between 70 and 100%. Signal-to-noise ratio (i.e. SNR) is a measure that compares the level of a desired signal to the level of background noise. It can be accessed as signal amplitude in internal logarithmic units called signal-to-noise ratio (SNR) in dB. The range of SNR (signal-to-noise ratio) value should be higher than 20 dB for 16-MHz micro-ADV. So, it was necessary to maintain the value of SNR for each data points reordered using micro-ADV.
Other two data sets from Flood Channel Facility (FCF) and Tominaga et al. (1989) have been considered for present analysis. The UK Flood Channel Facility is a large-scale national facility for undertaking experimental investigations of in-bank and overbank flows in rivers. The FCF (Series A) in-bank dataset was used for this present analysis. The FCF was 56 m long and 10 m wide with a usable length of 45 m. The longitudinal bed slope was 1.027 × 10^{3}. Further, two experimental data sets of Tominaga et al. (1989) were used in this study. The experiments were conducted in a tilting flume with 12.5 m length but having different cross-sectional geometries and longitudinal slopes as given in Table 1.
Model development
Multi-variable regression analysis
In this present study, a number of possible single regression models considering different one to one relationships (e.g. exponential, power, linear or logarithmic) between the dependent parameter and independent parameters were tested. The selection of best regression models was achieved based on the highest coefficient of determination (R^{2}) values. Two preferred input independent variables have been used for this study since these variables are found to control the shear distribution. Multi-variable regression analysis compiles these two independent variables to model up the dependent variable. Finally, through multi-variable regression analysis, five models have been derived with high coefficient of determinations for five vertical positions.
Application of Shiono and Knight model (SKM)
Shiono and Knight integrated the Navier–Stokes equation that is the momentum equation over the flow depth H, mainly to find out the lateral depth-averaged velocity distribution. The method of solving this equation is known as Shiono and Knight method.
Before evaluation of \(U_{\text{d}}\) in the lateral direction, three important calibration coefficients f, λ, Г need to be calibrated. Analytically, the SKM can be executed effectively if the channel is partitioned into reasonable panels where the calibrating coefficients are portrayed enough with fitting boundary conditions.
Results and discussions
The lateral velocity profiles at different planes are found to be power in function, and vertical velocity profiles are found to be logarithmic in nature. The SKM model is found to provide satisfactory velocity profile results; however, it fails to predict velocities at the junction between the constant flow depth domain and variable flow depth domain. Equations 1–5, based on multi-variable regression analysis, are found to provide good results for predicting velocity profiles in both lateral and vertical directions. The values of regression coefficient (R^{2}) for 0.2H, 0.4H, 0.6H, 0.8H and 0.95H profiles were found between 0.85 and 0.97. For applying Eqs. 1–5, there are certain ranges of non-dimensional independent and dependent parameters. The ranges of the parameters are: for U/U_{mean}= 0.4 to 1.87, for z/H = 0.2, 0.4, 0.6, 0.8 and 0.95 and for y/(b/2) = 0 to 1.
RMSE results of depth-averaged velocity distribution by different models
Channel type | RMSE | |
---|---|---|
Present model | SKM | |
NITR1 (H = 0.11 m) | 0.043 | 0.066 |
NITR1 (H = 0.10 m) | 0.056 | 0.074 |
NITR2 (H = 0.10 m) | 0.012 | 0.033 |
NITR2 (H = 0.11 m) | 0.064 | 0.065 |
NITR3 (H = 0.08 m) | 0.014 | 0.015 |
NITR3 (H = 0.09 m) | 0.007 | 0.008 |
FCF (H = 0.10 m) | 0.045 | 0.070 |
FCF (H = 0.15 m) | 0.066 | 0.093 |
Tominaga et al. (1989) S1 (H = 0.10 m) | 0.056 | 0.074 |
Tominaga et al. (1989) S4 (H = 0.11 m) | 0.041 | 0.018 |
The present models have been verified through the data sets of large channel facility of FCF, data of Tominaga et al. (1989) and rough channel data of NITR (NITR3). The error in terms of MAPE for these three channels has been found to be 5%, 5.62%, and 3.78%, respectively, and for SKM model these values are found 8.16%, 5.75% and 4.45%, respectively. The average RMSE values for channels are found to be 0.055, 0.05 and 0.011, and for SKM these values are 0.08, 0.05 and 0.012, respectively, showing the strength of the model.
A reasonable error has been observed when the observed experimental results compared with SKM results and present model results for simple open channels with both smooth and rough case. The Shiono and Knight method (SKM) has shown satisfactory results for the prediction of depth-averaged velocity distribution in the lateral direction for both rectangular and trapezoidal channels. The philosophy of the SKM is based on using three calibrating coefficients for each panel. But the present model is showing better results than SKM as proved from the results of MAPE and RMSE.
Application of model to natural river
Geometric properties and surface conditions used for natural river data
Geometrical properties | River Senggi (B) | River Senggai |
---|---|---|
Bank full depth, H(m) | 1.306 | 1.068 |
Top width, T(m) | 5.285 | 5.5 |
Bed slope (S_{0}) | 0.001 | 0.001 |
Surface condition (main channel) | Erodible Soil | Erodible Soil |
Surface condition (side bank) | Long Vegetation | Erodible Soil |
Manning’s n (main channel) | 0.082 | 0.082 |
Manning’s n (side bank) | 0.25 | 0.082 |
Conclusions
- 1.
An experimental investigation has been carried out to find out the lateral depth-averaged velocity distribution for different flow depths of a smooth open channel flow.
- 2.
The lateral velocity profiles at different horizontal planes are found to be power function, and vertical velocity profiles are found to be logarithmic in nature. Local velocity along the stream-wise direction for a given horizontal and vertical dimensions has been modelled.
- 3.
The new expressions are found to be well matching with the observed values by providing less error. The results of the predicted models have also been compared well with the popular Shiono and Knight Model (SKM). The SKM model is found to provide good velocity profile results; however, it fails to predict velocities at the junction between the constant flow depth domain and variable flow depth domain as occurred in trapezoidal cases.
- 4.
The present models which are based on multi-variable regression analysis are found to provide very good results of velocity profiles for different laboratory channels.
- 5.
Present models have been verified through the data sets of large channel facility of FCF, data of Tominaga et al. (1989) and NITR3. The errors in terms of MAPE for these three channels have been found to be 5%, 5.62%, and 3.78%, respectively. The MAPE values for SKM model are found 8.16%, 5.75%, and 4.45%, respectively, for these channels. The average RMSE values for these three channels are found to be 0.055, 0.05 and 0.011 and for SKM these values are 0.08, 0.05 and 0.012. So, the model is believed to predict the local velocity as well as the depth-averaged velocity for a channel with homogenous roughness.
- 6.
The predicted model has also been well validated against natural river datasets of river Senggi B and river Senggai with reasonable accuracy.
Notes
References
- Ansari K, Morvan HP, Hargreaves DM (2011) Numerical investigation into secondary currents and wall shear in trapezoidal channels. J Hydraul Eng 137(4):432–440CrossRefGoogle Scholar
- Blumberg AF, Galperin B, O’Connor DJ (1992) Modeling vertical structure of open-channel flows. J Hydraul Eng 118(8):1119–1134CrossRefGoogle Scholar
- De Cacqueray N, Hargreaves DM, Morvan HP (2009) A computational study of shear stress in smooth rectangular channels. J Hydraul Res 47(1):50–57CrossRefGoogle Scholar
- Devi K, Khatua KK (2016) Prediction of depth averaged velocity and boundary shear distribution of a compound channel based on the mixing layer theory. Flow Meas Instrum 50:147–157CrossRefGoogle Scholar
- Hin LS, Bessaih N, Ling LP, Ghani AA, Zakaria NA, Seng MY (2008) Discharge estimation for equatorial natural rivers with overbank flow. Int J River Basin Manag 6(1):13–21CrossRefGoogle Scholar
- Jan CD, Chang CJ, Kuo FH (2009) Experiments on discharge equations of compound broad-crested weirs. J Irrig Drain Eng 135(4):511–515CrossRefGoogle Scholar
- Jesson M, Sterling M, Bridgeman J (2012) Modeling flow in an open channel with heterogeneous bed roughness. J Hydrau Eng 139(2):195–204CrossRefGoogle Scholar
- Khuntia JR, Devi K, Khatua KK (2016) Variation of local friction factor in an open channel flow. Indian J Sci Technol 9(46):1–6CrossRefGoogle Scholar
- Liao H, Knight DW (2007) Analytic stage-discharge formulas for flow in straight prismatic channels. J Hydraul Eng 133(10):1111–1122CrossRefGoogle Scholar
- Maghrebi MF, Givehchi M (2009) Estimation of depth-averaged velocity and boundary shear stress in a triangular open channel. J Water Wastewater 2:71–80Google Scholar
- Nezu I, Kadota A, Nakagawa H (1997) Turbulent structure in unsteady depth-varying open-channel flows. J Hydraul Eng 123(9):752–763CrossRefGoogle Scholar
- Sarma KV, Lakshminarayana P, Rao NL (1983) Velocity distribution in smooth rectangular open channels. J Hydraul Eng 109(2):270–289CrossRefGoogle Scholar
- Shiono K, Feng T (2003) Turbulence measurements of dye concentration and effects of secondary flow on distribution in open channel flows. J Hydraul Eng 129(5):373–384CrossRefGoogle Scholar
- Shiono K, Knight DW (1990) Mathematical models of flow in two or multi stage straight channels. In: Proceedings of the international conference on river flood hydraulics. Wiley, New York, pp 229–238Google Scholar
- Steffler PM, Rajaratnam N, Peterson AW (1985) LDA measurements in open channel. J Hydraul Eng 111(1):119–130CrossRefGoogle Scholar
- Tominaga A, Nezu I, Ezaki K, Nakagawa H (1989) Three-dimensional turbulent structure in straight open channel flows. J Hydraul Res 27(1):149–173CrossRefGoogle Scholar
- Yang K, Nie R, Liu X, Cao S (2012) Modeling depth-averaged velocity and boundary shear stress in rectangular compound channels with secondary flows. J Hydraul Eng 139(1):76–83CrossRefGoogle Scholar
- Zarrati AR, Jin YC, Karimpour S (2008) Semianalytical model for shear stress distribution in simple and compound open channels. J Hydraul Eng 134(2):205–215CrossRefGoogle Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.