Modeling Earth Systems and Environment https://doi.org/10.1007/s40808-018-0548-4
ORIGINAL ARTICLE
Flow field within rectangular lateral intakes in the subcritical flow regimes Hamed Azimi1,2 · Saeid Shabanlou3 · Saeid Kardar4 Received: 21 October 2018 / Accepted: 9 November 2018 © Springer Nature Switzerland AG 2018
Abstract Lateral intakes are used for water transmission and distribution on farms and irrigation networks. In present study, the flow field within a rectangular lateral intake in subcritical flow regime is numerically simulated in a three-dimensional. In this analysis of CFD, turbulence of flow field is simulated using standard k − 𝜀 , and RNG k − 𝜀 turbulence models, and changes of free surface is simulated by volume of fluid scheme. Comparison between CFD and experimental results showed that the numerical model simulated the free surface and velocity field with high accuracy. Root mean square error for longitudinal and transverse profiles of the flow free surface in the main channel is respectively calculated as 0.132% and 0.094%. Based on simulation results, with the flow progress towards the downstream diversion channel, the secondary circulation cell is developed. A relation is presented for calculating strength of the secondary flow with a nonlinear regression method and using Minitab software. Also, a relation is proposed for calculating the flow energy head within the downstream channel and lateral channel (E3, E2) compared to the flow energy head in the upstream intake (E1). Keywords Lateral intake · Numerical simulation · Secondary flow · Energy head · Subcritical flow regimes Abbreviations A Cross-sectional area of flow (m2) b Width of main and lateral channel E1 Specific energy in the main channel on the upstream of the main channel (m) E2 Specific energy in the main channel on the downstream of the main channel (m) E3 Specific energy in the lateral channel (m) F Fluid volume fraction in a cell (–) F1 Froude number in the main channel on the upstream of the main channel (–) F2 Froude number in the main channel on the downstream of the main channel (–) F3 Froude number in the lateral channel (–) * Saeid Shabanlou
[email protected] 1
Department of Civil Engineering, Razi University, Kermanshah, Iran
2
Water and Wastewater Research Center, Razi University, Kermanshah, Iran
3
Department of Water Engineering, Kermanshah Branch, Islamic Azad University, Kermanshah, Iran
4
Department of Architecture, Science and Research Branch, Islamic Azad University, Tehran, Iran
g Acceleration gravity (m/s2) p Pressure (N/m2) Q1 Discharge in the main channel on the upstream of the junction (m3/s) Q2 Discharge in the main channel on the downstream of the junction (m3/s) Q3 Discharge in the lateral channel (m3/s) Rq Discharge ratio (Q3/Q1) t Time (s) Ui,j Velocity components (m/s) u∗ Wall shear velocity (m/s) VX Longitudinal component of velocity over crest of side weir (m/s) VY Lateral component of velocity over crest of side weir (m/s) VZ Vertical component of velocity over crest of side weir (m/s) xi,j or X, Y, Z Direction of Cartesian coordinates (m) y1 Distance of the cell center from the solid wall (m) y+ Non-dimensional parameter (–) Z Flow depth (m) δ Secondary flow strength (m/s) 𝛿i,j Kronecker delta (–) 𝜈 Kinematic viscosity (m2/s)
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Vol.:(0123456789)
𝜈t Turbulent eddy-viscosity (m2/s) 𝜌 Fluid density (kg/m3) Ω Streamwise vorticity (1/s)
Introduction Lateral intakes are used by hydraulic engineers for flow diversion in irrigation networks and water supply systems. A 90-degree branching is the easiest way to divert water from main channels. With approaching of the flow to the intake, the flow accelerates in the transverse direction and is divided into two parts. Part of the flow is directed into the lateral intake and the rest goes into the main channel. The overall flow pattern in the lateral intakes is fully three-dimensional and complicated. That is why many laboratory, theoretical and numerical studies are performed on the flow pattern of such hydraulic structures. For instance, Taylor (1944) was one of the first researchers to study the flow characteristics in rectangular open channels at a junction. Law and Reynolds (1966) studied the diversion flow in a lateral intake experimentally. Kasthuri and Pundarikanthan (1987) conducted an experimental study on the flow characteristics in rectangular lateral intakes in subcritical flow conditions. They examined the changes of flow free surface and velocity field in this type of hydraulic structures. Ramamurthy and Satish (1988) proposed a theoretical model for flow diversion inside short lateral intakes making a 90° angle with the main channel. Using momentum relationships, Ramamurthy et al. (1990) presented an analytic method for calculating the discharge ratio of rectangular lateral intakes. Neary and Odgaard (1993) performed an experimental study on the flow inside rectangular lateral intakes. They investigated the flow field in both channels of smooth bed and rough bed. Ramamurthy et al. (1996) conducted an experimental study on the flow pattern inside lateral intakes. They offered an analytical method to calculate the energy loss coefficient of the flow deviation in 90-degree lateral intakes. Barkdoll (1997) did an experimental study on the secondary flow and velocity field inside the direct channels connected to a lateral intake in both conditions of rigid and moving beds. Hsu et al. (2002) proposed relationships to calculate the depth-discharge and energy-loss coefficient of subcritical flow in rectangular channels attached to a 90-degree lateral intake. In recent years, numerical models are used extensively as a powerful and inexpensive tool for simulating various problems (Azimi et al. 2014, 2015, 2017; Azimi and Shabanlou 2015; Parsaie 2016; Mirzaei et al. 2017, 2018; Kassem 2018). Issa and Oliveira (1994) developed a three-dimensional model for simulating the flow deviation from a straight channel. They simulated the flow field turbulence using a standard k − 𝜀 turbulence model. Shettar and Murthy (1996) modeled a
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two-dimensional flow field of a 90-degree intake using depthaveraged forms of the continuity and momentum equations. Neary and Sotiropoulos (1996) developed a three-dimensional model for simulating the flow field of rectangular channels connected to lateral intakes. Using the Finite Volume Method (FVM), Neary and Sotiropoulos (1996) solved Navier–Stokes equations for steady and laminar flows in a branch of 90 degrees. Neary et al. (1999) offered a three-dimensional numerical model for simulation of the turbulent flow pattern in a 90-degree branch. Also, they used the turbulence model of k − ω to simulate the flow turbulence. Ramamurthy et al. (2007) conducted an experimental and numerical study on the flow characteristics and velocity field in a rectangular channel connected to a lateral intake with branching angle of 90°. They modeled the flow field turbulence with a turbulence model of k − ω and also modeled the changes in flow free surface using the VOF scheme. Using Spalart–Allmaras turbulence, Li and Zeng (2009) developed a three-dimensional model to predict the flow field inside rectangular lateral intakes. Additionally, Azimi et al. (2016) simulated free surface and flow field turbulence in a circular channel along the side weir in supercritical regime. They presented an equation to estimate discharge coefficient of side weir on circular channels. Kadaverugu (2016) modeled the dynamic behavior of sub-surface horizontal flow wetland systems using a CFD model. Due to complex, three-dimensional flow behavior in the lateral intakes, studying the flow pattern in this type of hydraulic structures is quite importance. With review past studies on flow field of lateral intakes, it is observed that no precise numerical simulation was done on strength of the secondary flow and relation between the flow energy head at the upstream, downstream main channel and lateral channel in the subcritical flow regimes. In this numerical study, variations of free surface and flow field turbulence within a rectangular intake are simulated using FLOW-3D software, volume of fluid (VOF) scheme and turbulence models of standard k − 𝜀 and RNG k − 𝜀 in the subcritical flow regime.
Governing equations To solve the flow field the continuity equation and Reynoldsaveraged Navier–Stokes equations are used as follows:
𝜕Ui = 0.0 𝜕xi
(1)
[ ( )] 𝜕Ui 𝜕Ui 𝜕Uj 𝜕Ui 1 𝜕 = −p𝛿ij + 𝜌𝜈t + + Uj 𝜕t 𝜕xj 𝜌 𝜕xj 𝜕xj 𝜕xi
(2)
where Ui , Uj and (i, j = 1, 2, 3) are the velocity and coordinate system components, respectively.
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Fig. 1 The applied boundary conditions to the numerical model
Where t = time, 𝜌 = fluid density, P = pressure, 𝛿ij (i, j = 1, 2 and 3) = Kronecker delta, 𝜈t = turbulent eddy-viscosity. In numerical simulations of the open channels, simulation of the free surface is very important. One of the most important methods which simulate the flow free surface by interface capturing method is the volume of fluid (VOF) scheme. This method for the first time was provided by Hirt and Nichols (1981). The free surface variations have been modeled using VOF model. In VOF model to calculate the volume component the following continuity equation is solved:
𝜕F 𝜕F = 0.0 + Ui 𝜕t 𝜕xi
(3)
where F is the volume component of fluid in a specified computational cell. If F = 0, the cell is empty, and if F = 1 the computational cell is filled with fluid and if 0 < F < 1 the cell is filled with both phases of fluid and air.
Fig. 2 Schematic of the Kasthuri and Pundarikanthan (1987) model
Table 1 The hydraulic characteristics of the Kasthuri and Pundarikanthan (1987) model
/ Rq = Q3 Q1 F1
0.32
F2
F3
0.23 0.54 0.52
Experimental apparatus Boundary condition In this numerical study, to validate the results of the CFD analysis we used of laboratory measurements of Kasthuri and Pundarikanthan (1987). The applied boundary conditions for the numerical model are consistent with the laboratory conditions of Kasthuri and Pundarikanthan (1987). Thus, the specified amounts of flow and depth are used at the inlet section of the main channel. Also, at the outlet section of the main channel, the specified amounts of depth and pressure are used. All solid walls including the side walls and the main channel bed and also the side channel are defined as the wall boundary conditions and the entire upper surface of the air phase is introduced as the symmetry boundary conditions. The applied boundary conditions for the numerical model are shown in Fig. 1.
To validate the numerical model results we used of the laboratory measurements of Kasthuri and Pundarikanthan (1987). The laboratory facilities are including an open rectangular channel which has a length of 6 m and a side channel with a length of 3 m is connected to its middle. The main and lateral channels have width and height of 0.3 and 0.25 m, respectively. Schematic of the experimental model is shown in Fig. 2. The horizontal channel bed is smoothed by cement plaster and the walls are made of Perspex sheets. In Table 1 the hydraulic characteristics of the laboratory model of Kasthuri and Pundarikanthan Q (1987) are provided where, Rq = Q3 is the ratio of dis1
charge of the lateral channel to the main channel, and F1, F2 and F3 are the Froude numbers of the junction upstream, junction downstream and within the lateral channel, respectively.
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Numerical model In this CFD analysis, a rectangular channel with a length of 6 m, height of 0.25 m and width of 0.3 m is defined which is attached with a lateral channel with length of 2 m, width of 0.25 m and height of 0.3 m. For gridding the flow field we used of a non-uniform mesh block composed of rectangular elements and the first node is selected so that to avoid from calculations under the viscose layer. Therefore, the first node is placed where the dimensionless y+ parameter which is introduced by Eq. (4) be larger than 30.
y+ =
y1 u∗ 𝜈
(4)
where y1 is distance between cell center and solid wall, u∗ is shear velocity of wall and 𝜈 is kinematic viscosity of fluid. In Table 2 the characteristics of an independent sample of the used gridding in simulation of the flow free surface profile along the outer bank of the main channel are provided. In order to investigation the accuracy of the numerical model in predict various parameters of flow, the average percent error (APE) and root mean square error (RMSE) using Eqs. (5) and (6), respectively. √ √ ) N ( √1 ∑ R(measured) − R(simulated) 2 √ (5) RMSE = 100 × N i=1 R(measured) N
MAE = Table 2 Results of the gridding for the simulated free surface profile along the outer bank of the main channel Meshing
Number of cells
RMSE (%)
MAE (%)
1 2 3 4 5 6
161,700 215,625 277,200 369,600 408,480 504,000
4.835 2.724 2.106 0.738 0.200 0.184
4.140 2.083 1.027 0.311 0.183 0.150
Fig. 3 Views of the computational field. a Cross-section. b Plan. c 3D view
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100 ∑ | | − R(simulated) | |R | N i=1 | (measured)
(6)
where R(measured) and R(simulated) are the laboratory and simulation results, respectively. According to Table 2 results, it’s concluded that the difference between the gridding (5) and (6) results is negligible and gridding (5) is selected. Thus, the whole computational domain is gridded in X, Y and Z directions with 148, 92 and 30 elements, respectively (Fig. 3).
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Fig. 4 Changes of the flow free surface along the inner bank of the main channel for k − ε turbulence and k − ε RNG models
free surface along the inner bank of the main channel and the outer bank of the lateral channel for both turbulence models of standard k − ε and RNG k − ε. Also, comparison of the depth-averaged velocity results is depicted at X = 0.0 of the main channel for different turbulence models. In Table 3, RMSE and MAE values of the free surface changes and the depth-averaged velocity are arranged for turbulence models of standard k − ε and RNG k − ε. According to simulation results (Table 3), the results of both standard k − ε and RNG k − ε turbulence models were approximately similar. Therefore, the standard k − ε turbulence model is chosen for modeling the flow field turbulence (Figs. 4, 5 and 6).
Results and discussion Validation
Fig. 5 Changes of the flow free surface along the outer bank of the lateral channel for k − ε turbulence and k − ε RNG models
Fig. 6 Comparison of depth-averaged velocity at X = 0.0 of the main channel for k − ε turbulence and k − ε RNG models
In this numerical study, we studied the effect of turbulence models of standard k − ε and RNG k − ε on the results of numerical simulation. In the turbulence model of standard k − ε, two differential equations of kinetic energy and kinetic energy loss rate are used. The RNG k − ε turbulence model is based on normalized Reynolds groups involving a statistical perspective to extract an averaged equation for turbulence quantities. Figures 4 and 5 are shown the variations of flow
Figure 7 depicts the comparison of flow free surface between experimental results and numerical model along the banks of the main channel. Due to input effects of the lateral intake, a loss occurs early in the upstream intake and by advancing towards the downstream intake, the flow depth increases and a hump like rise occurs at the end of intake downstream (Fig. 7a). As can be seen, by moving away from the inner bank and in the vicinity of the outer bank, effects of the lateral intake are reduced. Figure 8 shows the comparison between experimental results and numerical model for transverse profiles across the flow free surface inside the main channel. According to Fig. 8, with advance of the flow from beginning of the junction towards the end of it, the flow depth increases along the inner bank of the main channel. Figure 9 shows the comparison between experimental and numerical results for changes in flow free surface along the banks of the lateral channel. At the beginning of the lateral intake, the flow depth in the vicinity of the inner bank is less than the flow depth in the vicinity of the outer bank. Also, the flow free surface in the vicinity of the outer bank of the lateral channel is reduced rapidly and by advancing downstream of the lateral channel, the water depth continues constantly. Comparison of the depth-averaged velocity
Table 3 RMSE and MAE values for the changes of the free surface and depth-averaged velocity for k − ε turbulence and k − ε RNG models RMSE Water surface profile along the inner bank of the main channel Water surface profile along the outer bank of the lateral channel Depth-averaged velocity profile of the main channel at X = 0.0
k − ε Std RNG k − ε k − ε Std RNG k − ε k − ε Std RNG k − ε
MAE 0.182% 0.200% 0.164% 0.185% 4.023% 4.206%
k − ε Std RNG k − ε k − ε Std RNG k − ε k − ε Std RNG k − ε
0.166% 0.183% 0.158% 0.178% 3.497% 3.745%
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Fig. 7 Comparison of the flow free surface changes between the numerical and laboratory results within the main channel a along the inner bank, b along the outer bank
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Fig. 9 Comparison of the flow free surface changes between the laboratory and numerical results within the lateral channel a along the inner bank, b along the outer bank
secondary flow generated inside the lateral channel. Based on the results of numerical simulation, a flow circulation is generated inside the intake channel. By advancing towards downstream of the diversion channel, the secondary circulation cell is developed and magnitude of the transverse velocity vectors is reduced (Fig. 11). and Odgaard (1995) introduced the circulation flow ( Neary ) ΩY as an indicator of secondary flow strength within a rectangular lateral intake (Issa and Oliveira 1994). Circulation flow in the Y direction (longitudinal axis of the lateral channel) is defined as a function of velocity gradients: ) ( 1 𝜕VY 𝜕VZ ΩY = − (7) 2 𝜕Z 𝜕Y Here, 𝜕YZ and 𝜕ZY are respectively the vertical and transverse velocity gradients along Y and Z directions. In their studies on the secondary flow strength, Neary and Odgaard (1993, 1995) noted that the vertical velocity gradient term in 𝜕V the transverse direction ( 𝜕YZ ) is negligible versus the trans𝜕V verse velocity gradient term in the vertical direction ( 𝜕ZY ). They also stated that the secondary flow strength (𝛿) can be calculated as the difference between the transverse components of velocity in the vicinity of the flow free surface and the lateral channel bed VY(Surface) − VY(Bed) in Eq. (8) (Barkdoll 1997; Mirzaei et al. (2018): 𝜕V
Fig. 8 Profiles across the flow free surface. a X = −0.15 , b X = 0.0 and c X = 0.15
profiles for experimental and numerical models is shown in Fig. 10. Table 4 presents the RMSE and MAE values for the numerical model results. According to the results of numerical simulation, an acceptable agreement is obtained between the laboratory measurements and numerical modeling results.
Secondary flow Generally, when the flow approaches the lateral intake, the acceleration transverse component produces a secondary flow inside the lateral channel. Figure 11 shows the
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𝜕V
(8) In this numerical study, variations of 𝛿 in different discharge ratios (Rq = 0.1 − 0.9) are calculated in the lateral channel and the results of 𝛿 are analyzed using Minitab statistical software. Relation (9) is presented for calculating 𝛿 with a nonlinear regression method. This relation is a function of discharge ratio (Rq ) and the longitudinal component of the flow in the diversion channel (Y):
𝛿 = VY(Surface) − VY(Bed)
𝛿 = 0.063 × (Y)0.174 × (Rq )4.897
(9)
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Fig. 10 Depth-averaged velocity profiles in the main channel a X = −1.65 m , b X = −0.5 m , c X = 0.0 m , d X = 0.5 m , e X = 0.45 m , f X = 2.1 m Table 4 RMSE and MAE of the numerical simulation results
Water surface profiles in the main channel Water surface profiles across the main channel Water surface profiles in the lateral channel Depth-averaged velocity profiles in the main channel
Figure 12 presents the 𝛿variation along the lateral channel for three discharge ratios of Rq = 0.7, 0.8 and 0.9 . In Table 5, the 𝛿 values calculated using Eq. (9) and simulated for different discharge ratios are arranged. In order to show the accuracy of the Eq. (9), the relative error percent (REP) of the calculated and simulated values is presented. According to Table 5, accuracy of the Eq. (9) in predicting the 𝛿 value is acceptable.
|𝛿 − 𝛿calculated || REP% = 100 × || simulated | 𝛿 | | simulated
(10)
Variations of energy head
The flow energy head in each section of the main channel and lateral channel is calculated from the following equation (Hager 1992):
E=Z+
Q2 2gA2
(11)
Here, Z, Q, g, and A respectively represent depth of flow, discharge, gravity and cross-sectional area of the flow in any
RMSE (%)
MAE (%)
0.132 0.094 0.220 3.848
0.120 0.064 0.201 3.072
cross-section of flow. Due to turbulence mixing in the lateral intakes, the flow ( energy ) in the downstream channel and lateral channel E3 , E2 will( drop ) compared to the flow energy in the upstream intake E1 . In this numerical study, the energy variations in the downstream of main channel and the lateral channel is examined and compared to the upstream of the main channel for various discharge ratios of Kasthuri and Pundarikanthan experimental model. In Figs. 13 and 14, variations ( ) in flow energy within downstream of the main channel E2 and ( the ) variations of flow energy within the lateral channel E3 are depicted for various discharge ratios and compared ( ) to flow energy in the upstream of the main channel E1 . By analyzing the simulated data using nonlinear regression, two relationships are obtained between E1 , E2 , and E3 . Coefficients of determination (R-squared) of the mentioned relationships are respectively calculated as 0.9917 and 0.9774.
E2 = 1.66E12 + 0.85E1
(12)
E3 = 3.85E12 + 0.64E1
(13)
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Modeling Earth Systems and Environment
Fig. 11 The produced secondary flow within the lateral channel for Rq = 0.52 a Y = 0.3 m , b Y = 0.45 m , c Y = 0.6 m , d Y = 0.75 m
Conclusion
Fig. 12 Variations of 𝛿 along the lateral channel for different discharge ratios Table 5 Comparison of the calculated 𝛿 using Eq. (9) and simulated results
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In general, a lateral intake is used for diverting the flow on the agriculture fields and distributing water in irrigation and drainage systems. In this numerical study, turbulence and free surface of flow field within a rectangular lateral intake are simulated three-dimensionally in subcritical flow conditions. Changes of the free surface flow are simulated using VOF method. Also, flow turbulence is modeled by using different turbulence models. Comparison between simulation results and experimental measurements showed that the
Rq
Y (m)
𝛿 = VY(Surface) − VY(Bed) (simulated)
𝛿 = 0.063 × (Y)0.174 × (Rq )4.897 (calculated)
REP (%)
0.3 0.4 0.5 0.6 0.7 0.7 0.8 0.8 0.8 0.8 0.9 0.9
0.9 0.15 1.35 1.5 1.05 1.8 0.3 0.6 0.9 1.2 0.3 0.9
0.000157 0.00058 0.00234 0.007 0.0118 0.0121 0.016 0.018 0.0193 0.021 0.035 0.039
0.00017 0.0005 0.0022 0.006 0.0111 0.0122 0.0171 0.0193 0.0207 0.0218 0.0305 0.0369
8.28 13.79 5.98 14.28 5.93 0.83 6.88 7.22 7.25 3.81 12.86 5.38
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CFD model predicted the flow free surface and the velocity field accurately. Root mean square error (RMSE) for longitudinal profiles of the flow free surface in the lateral channel was equal to 0.220% and the mean average error (MAE) for depth-averaged velocity profiles in the main channel is calculated as 3.072%. In this analysis of CFD, the effects of standard k − ε and RNG k − ε turbulence models are examined on the simulation results; and the results of both turbulence models was similar. Due to entrance effects of the lateral intake, a loss occurs early in the upstream of the main channel and advancing towards main channel downstream, the flow depth increases and a hump like rise occurs on the downstream of the main channel. With the advance of flow from the initial junction towards the end of it, the flow depth increases along the inner bank of the main channel. Based on simulation results, as vorticity flow is generated inside the intake channel and by advancing downstream of the channel diversion, the secondary circulation cell is developed. Using non-linear regression method and Minitab statistical software, an equation is presented for calculating 𝛿 (secondary flow strength). Also, some relationships are presented
Fig. 13 Variations of the flow energy within the lateral channel (E2) to the flow energy within the upstream of the main channel (E1) for different discharge ratios
Fig. 14 Variations of the energy within the lateral channel (E3) to the flow energy within the upstream of the main channel (E1) for different discharge ratios
for calculating the flow energy head within the ( ) downstream main channel and the lateral channel E3 , E2 (compared to ) the flow energy upstream of the main channel E1 .
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