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Determination of capacity of labyrinth side weir by CFD
Flow Measurement and Instrumentation 29 (2013) 1–8
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Flow Measurement and Instrumentation journal homepage: www.elsevier.com/locate/flowmeasinst
Determination of capacity of labyrinth side weir by CFD M. Cihan Aydin a,n, M. Emin Emiroglu b a b
Civil Engineering Department, Bitlis Eren University, Bitlis, Turkey Civil Engineering Department, Firat University, Elazig, Turkey
a r t i c l e i n f o
abstract
Article history: Received 11 April 2012 Received in revised form 17 August 2012 Accepted 19 September 2012 Available online 23 October 2012
Side weirs are widely used in irrigation, land drainage, urban sewage systems, flood protection, and forebay pool of hydropower systems by flow diversion or intake devices. The hydraulic behavior of side weirs received considerable interest by many researchers. A large number of these studies are physical model tests of rectangular side weirs. However, in the study, Computational Fluid Dynamics (CFD) models together with laboratory models of labyrinth side weirs were used for determining the discharge capacity of the labyrinth side weir located on the straight channel. The discharges performances obtained from CFD analyses were compared with the observed results for various Froude number, dimensionless nappe height, dimensionless weir width, and weir included angle. The results obtained from both methods are in a good agreement. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Side weir Water discharge Labyrinth weir CFD analysis Free surface flow
1. Introduction A side weir is an overflow weir framed in the side of a channel, which allows lateral overflow when the surface of the liquid in the channel rises above the weir crest [1]. Side weirs are hydraulic structures often in irrigation techniques, sewer networks, land drainage, and flood protection. The crest length of a weir has the greatest influence on the discharge capacity. A labyrinth weir is defined as a weir crest that is not straight in planform. The increased sill length provided by labyrinth weirs effectively reduces upstream head to the particular discharge. A labyrinth weir can therefore be used to particular advantage where the width of a channel is restricted and a weir is required to pass a range of discharges with a limited variation in upstream water level. Hay and Taylor [2] and Tullis et al. [3] carried out experiments on the advantage of labyrinth weirs in case of limited channel widths. There are a lot of prototype labyrinth spillways (for example the Avon Dam (Australia), the Ritschard Dam (USA), the Ute Dam (USA), the Hyrum Dam (USA), the Sarioglan Dam (Turkey), the Kizilcapinar Dam (Turkey)). However the idea of labyrinth side weir was first presented by Emiroglu et al. [4], and there is not any prototype labyrinth side weir in our knowledge. They reported that discharge coefficient of the labyrinth side weir is 1.5–4.5 times higher than normal rectangular side weir in a rectangular channel. However it is noted that one of the disadvantages of
n
Corresponding author. Tel.: þ90 4342285170; fax: þ 90 4342285171. E-mail addresses: [email protected] (M.C. Aydin), [email protected] (M.E. Emiroglu). 0955-5986/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.flowmeasinst.2012.09.008
the labyrinth side weirs is the decrease of discharge capacity with increasing the nappe height when the weir included angle is chosen too small. A review of previous studies indicated that rectangular sharpcrested side weirs have been investigated extensively, including work by Ackers [5], Collinge [6], Frazer [7], Subramanya and Awasthy [8], El-Khashab and Smith [9], Uyumaz and Muslu [10], Helweg [11], Swamee et al. [12], Agaccioglu and Yuksel [13]. The hydraulic behavior and discharge coefficient of side weirs for the different type of weirs, main channel and flow conditions have been studied by many researchers such as: Nadesamoorthy and Thomson [14], Singh et al. [15], Yu-Tek [16], Cheong [17], and others. Ranga Raju et al. [1] investigated on the discharge coefficient of a broad-crested rectangular side weir depending on the main channel, Froude number and head/weir width ratio. Recently, Aydin [18] modeled the free surface flow over the triangular labyrinth side weir by using Volume of Fluids (VOF) method to describe the surface characteristics in subcritical flow conditions. Borghei and Parvaneh [19] studied a new type of oblique side weir with asymmetric geometry on an experimental set-up. The researchers stated that this kind of weir is more efficient than the ordinary oblique weir. Emiroglu et al. [20] carried out a comprehensive study to determine the discharge coefficient of a sharp crested rectangular side weir in a straight channel, and developed an equation for discharge coefficient including all dimensional parameters. Agaccioglu et al. [21] presented a reliable equation based on 1504 experimental runs for discharge coefficient of the rectangular side weir in a curved channel depending on the all dimensionless parameters. Haddadi and Rahimpour [22] investigated an experimental setup to obtain
2
M.C. Aydin, M.E. Emiroglu / Flow Measurement and Instrumentation 29 (2013) 1–8
the relationships of discharge coefficient with the other dimensionless parameters for trapezoidal broad-crested side weir. Bagheri S., Heidarpour [23] investigated the water flow over sharp-crested side weirs obtaining the distribution of the threedimensional velocity. Assuming that the specific energy across the weir is constant in the dynamic equation of spatially varied flow for outflow over a weir [24], the general equation of weirs can be described as q¼
pffiffiffiffiffiffi dQ ¼ C d 2g ½hp3=2 dx
ð1Þ
where q is the discharge per unit length over the weir, Q is the out-flow discharge, x is the longitudinal direction, g is the gravitational acceleration, p is the height of the weir, h is the depth of flow measured from the channel bottom along the channel centerline, and Cd is the discharge coefficient of the weir. Emiroglu et al. [25] used Eq. (1) to estimate the discharge coefficient of sharp-crested triangular labyrinth side weirs on a straight channel, and they proposed the following equation for subcritical flow: 0:4 0:254 0:122 þ 3:214 ‘L 0:684 hp1 C d ¼ 0:4 þ 2:62 þ0:634 bL #3:857 1:982 y þ 0:22F 2:458 þ 0:122 sin 1 4
ð2Þ
In the present study, the discharge capacity of triangular labyrinth side weir with one cycle was determined by using VOF method with Fluent code, and the discharge coefficients obtained from the CFD were compared with the experimental data of Emiroglu et al. [4]. Recently, some of the researchers such as Bridgeman et al. [26], Aydin and Ozturk [27], Schaffrath et al. [28] used Fluent in their studies at various field as nuclear, mechanical and civil engineering. In the other study, Hargreaves et al. [29] studied the validation of CFD for modeling free surface flows over common hydraulic structures using Fluent with VOF (Volume of Fluid) method. They gained the confidence that more complex hydraulic structures should be modeled by using similar techniques.
Fig. 1. Sketch view of labyrinth side weirs. (a) Hydraulic profile of the triangular side weir, (b) Plan view if triangular side weir and (c) Front view of the triangular side weir.
Table 1 Model geomeries and meshing specifications. L (m)
y (1)
p (m)
Grid size (mm)
Number of cells
Meshing type
0.25
45
0.12 0.16 0.20 0.12 0.16 0.20
10 10 10 10 10 10
291800 292325 292325 347610 347610 463480
3D 3D 3D 3D 3D 3D
Hexahedral Hexahedral Hexahedral Hexahedral Hexahedral Hexahedral
0.12 0.16 0.20 0.12 0.16 0.20
10 10 10 10 10 10
561320 561320 561320 340675 408810 646440
3D 3D 3D 3D 3D 3D
Hexahedral Hexahedral Hexahedral Hexahedral Hexahedral Hexahedral
2. Physical model 60
The data used in this study were taken from the experimental studies conducted by Emiroglu et al. [4] on a large model. The authors conducted 2830 laboratory tests for determining discharge coefficient of triangular labyrinth side weirs. The typical hydraulic characteristics and the views of the triangular labyrinth side weir are also illustrated in Fig. 1. Their set-up consisted of a rectangular main channel 0.5 m wide, 0.50 deep and 12 m long, with a discharge collection channel 0.5 m wide and 0.70 m deep parallel to the main channel. A sluice gate was fitted at the end of the main channel to control flow depth. A rectangular weir was placed at the end of collection channel to measure the discharge of the side weir. In order to measure the water level in the collection channel, a digital point gauge was fixed further 0.40 m from the weir. Labyrinth side weirs were fabricated from sharp edged steel plates. The discharges were measured to an accuracy of 70.1 L/s, by means of an electromagnetic flow-meter installed in the supply line. The experiments were conducted for subcritical flow, stable flow and free overflow conditions. Novak and Cabelka [30] stated that minimum nape height should not be less than 30 mm because of the surface tension over the weir crest. Therefore, minimum nape height is taken into account as 30 mm. The data of the experiments were considered for the weir lengths (L) of 0.25 and 0.50 m, the weir heights (p) of 0.12 m, 0.16 m and, 0.20 m, and weir included angles with y ¼451and 601. Labyrinth side weirs were tested for different Froude numbers, different ratios of
0.50
45
60
(h1–p)/p and different weir included angles in order to obtain the variation of the discharge coefficient.
3. CFD model A High Performance Computer (HPC) was used to simulate the CFD models with VOF method in ANSYS-FLUENT1. The VOF model can model two or more immiscible fluids by solving a single set of momentum equations, Eq. (3), and tracking the volume fraction of each of the fluids throughout the domain, Eq. (4) [31]. @ @ @P s @ @ui @uj ðruj Þ þ ðrui uj Þ ¼ þ m þ ð3Þ þ rg j þF j @t @xi @xi @xj @xj @xi
1
License owner: Bitlis Eren University, Customer Number: 618883.
M.C. Aydin, M.E. Emiroglu / Flow Measurement and Instrumentation 29 (2013) 1–8
1
rq
"
n X @ ! _ pq m _ qp Þ ðaq rq Þ þ r:ðaq rq v q Þ ¼ Saq þ ðm @t p¼1
# ð4Þ
where r is the density of fluid, m is the dynamic viscosity of fluid, u is _ qp is the the velocity vectors, Ps is the pressure and F is a body force; m _ pq is the mass transfer from mass transfer from phase q to phase p, m phase p to phase q, aq is the qth fluid’s volume fraction in a cell. The source term Saq of Eq. (4) is zero by default. If aq ¼0, the cell is empty, if aq ¼1, the cell is full (of the qth fluid), and if 0o aq o1, the cell contains the interface between two fluid phases. The VOF model with time-dependent solution in Fluent was used to calculate free surface flow in the labyrinth side weir models. 3.1. Geometry and boundary conditions The geometries and meshes were generated by using Gambit. The parameters of the created models and its meshing specifications were given in Table 1. The typical view of model geometry and boundary conditions are given in Fig. 2. The open channel boundary conditions were applied to the CFD model as the pressure inlet at the main channel inlet, the pressure outlets at the boundaries which open to atmosphere and labyrinth side weir outlet (Fig. 2). The pressure inlet boundary condition presents an option which ensures to easily describe the bottom and the surface levels of the main channel and also the velocity of the main channel flow. A water level of downstream weir is imposed initially by the pressure inlet boundary at the main channel outlet. It is assumed that the free surface levels of the inlet and outlet boundaries in the main channel are approximately equal initially. After the solution is converged, the water levels reach own natural level in the subcritical conditions. For subcritical outlet flows (F1 o1), if there are only two phases, then the pressure is taken from the pressure profile specified over the boundary, otherwise the pressure is taken from the neighboring cell. Based on the Froude number when F1 o1 the flow is known to be subcritical where disturbances can travel upstream as well as downstream. In this case, downstream conditions might affect the flow upstream [31].
3
aspects known as iterative convergence and grid convergence, in terms of time-dependent problems, iterative convergence at every time step should be checked. It must be ensured that iterative convergence is achieved with at least three orders-ofmagnitude decrease in the normalized residuals for each equation solved. Each time steps are maximum 40 iterations and all residuals drop under three orders every time steps. As shown in Fig. 3, iterative convergences were achieved for each Froude number of the each CFD model of labyrinth side weirs (for L¼0.25 m, y ¼601, and p ¼0.20 m). It is pointed out that this is quite time-consuming process despite of using the HPC system. A method for discretization error estimation is the Richardson extrapolation (RE) method which was first used by Richardson [33]. The Grid Convergence Index (GCI) method which is based on Richardson Extrapolation was used to estimate the discretization errors. The procedure of this method was outlined by Celik et al. [34] as following. For three dimensional calculations the representative cell, mesh or grid size l was estimated by "
l¼
N 1X ðDV i Þ Ni¼1
#1=3 ð5Þ
3.2. Solution convergence The numerical verification in this paper is partially based on the American Society of Mechanical Engineers (ASME) editorial policy statement, which provides a framework for CFD uncertainty analysis [32]. Convergence investigation involves two
Fig. 3. Iterative convergence for various Froude numbers. (a) Hydraulic profile of the triangular side weir, (b) Plane view of triangular side weir and (c) Front view of the triangular side weir.
Fig. 2. Model geometry and boundary conditions [18].
4
M.C. Aydin, M.E. Emiroglu / Flow Measurement and Instrumentation 29 (2013) 1–8
4. Results and discussion
where DVi is the volume of ith cell and N is the number of cell. Three different sets of grid are selected and the apparent order is calculated using the equation below. P¼
1
ln j32 =j21 þ qðPÞ
lnðr 21 Þ
The discharge coefficients Cd of CFD models were calculated by Eq. (1) using the data obtained from CFD analyses as calculated by Emiroglu et al. [25] for their physical model. Emiroglu et al. [4] found that the discharge coefficient Cd for labyrinth side weir included angles of 451 and 601 is greater than the Cd coefficient for the other included angles (y ¼901, 1201 and 1501). They stated that the effect of L/b ratio, which plays an important role in the discharges capacity of labyrinth side weir, on Cd for labyrinth side weir included angles of 451 and 601 is very important due to the increase in the intensity of secondary flow created by lateral flow. Therefore, in the CFD analyses, it is only considered that the triangular labyrinth side weir included angles of 451 and 601. To show the effects of parameter (h1 p)/p on discharge coefficients, Cd values obtained from both the CFD and physical models are plotted against the dimensionless nappe height (h1 p)/p for L/b ¼0.5 and 1.0 in Fig. 5(a–d). These figures show that Cd decreases with the increase in (h1 p)/p. The effect of (h1 p)/p on Cd can be explained with the discontinuity region. This discontinuity region has a strong secondary motion next to the boundary of the weir side. Tullis et al. [3] also stated that the values of discharge coefficient decrease with the increasing nappe height. The decreasing trend of Cd with the increase of (h1 p)/p increases with reducing labyrinth side weir included angle. For this reason, higher nappe heights on labyrinth side weir are not selected by hydraulic designers. The discharge capacity of the labyrinth side weir increases with the reducing included angle (y). Thus, the effect of nappe height on discharge of labyrinth side weir is more significant in the small angles. Therefore, the designer must carefully be choosing the nappe height.
ð6Þ
where j21 ¼ f2 f1, j32 ¼f3 f2, r is the grid refinement factor, defined as r21 ¼ l2/l1 for l1 o l 2, f1, f2, f3 are the key variables which play an important role for the considered study. If the grid refinement (r) is constant, q(P) ¼0. Then, it is calculated following approximate relative errors:
f f
e21 ¼
1 2
f1
ð7Þ
Finally, the fine grid convergence index was calculated as [35] GCI21 f ine ¼
1:25e21 r P21 1
ð8Þ
Three various grid sizes which are fine-grid, base grid (10 mm) and coarse-grid were used to control grid convergence. The refinement factors between the coarse and fine grids (r ¼ l coarse/ l fine 41) is 1.11 according to Roache [36] who recommended a minimum 10% change in the grid refinement factor. The values of key variable (f1, f2, f3) were considered as the overflow discharges of side weir for three grid sizes. The numerical errors for L¼0.25 m, y ¼601, p¼0.12 m were estimated in Table 2. According to this table, the fine-grid evaluation of GCI can be given as between 0.77% and 7.88% which corresponds to the weir discharges of 0.12 and 0.52 L/s approximately. The maximum discretization error with the averaged apparent order was also obtained as 2.13% in terms of velocity magnitude in the middle of side-weir by Aydin [18].
3.3. Turbulence model In order to determine the sensitivity of results to turbulence, the six various turbulence models (RSM, standard k–e, RNG k–e, Realizible k–e, k–o, Spalart–Allmaras, where k is the turbulence kinetic energy, e is the turbulence dissipation rate, and o is the specific dissipation rate of turbulence) were compared in Fig. 4 for L¼0.25 m, y ¼601, p ¼0.12 m and F1 ¼0.43. Fig. 4 shows that the variations of dimensionless velocity ratios (V/Vm) at the middle of the weir entrance with the turbulence models are quite slight especially for RSM, Standard k–e, Realizable k–e and Spalart– Allmaras turbulence models besides other two models. Additionally, the RSM is probably the best competent model among all the models to simulate the surface shapes especially the surface fluctuations and vortex occurrence in the side-weir for subcritical condition [18].
Fig. 4. Comparison of different turbulence models.
Table 2 Discretization errors. (h1-p)/h1
0.25 0.30 0.35 0.40 0.45
Discharges (L/s) f1
f2
f3
7.69 9.86 12.52 15.66 19.29
6.68 9.37 12.30 15.46 18.86
6.23 9.34 12.39 15.38 18.30
e21
e32
r21
r32
pave
e21
e32
GCI21 fine (%)
GCI32 fine (%)
1.0066 0.4858 0.2175 0.2018 0.4385
0.4487 0.0283 0.0935 0.0833 0.5587
1.11 1.11 1.11 1.11 1.11
1.11 1.11 1.11 1.11 1.11
11.26 11.26 11.26 11.26 11.26
0.131 0.049 0.017 0.013 0.023
0.067 0.003 0.008 0.005 0.030
7.88 2.96 1.05 0.77 1.37
4.04 0.18 0.46 0.32 1.78
M.C. Aydin, M.E. Emiroglu / Flow Measurement and Instrumentation 29 (2013) 1–8
5
Fig. 5. (a–d) Comparison of Cd coefficients against the dimensionless nappe height.
Fig. 6 (a–d) shows the comparison of Cd values obtained from CFD and experimental studies against the Froude number. The results generally indicated that discharge coefficient increases ¨ with increasing Froude number. Agaccioglu and Yuksel [13] also found the similar tendency. But, the variation of Cd against F1 is not significant for the small width of the side weir as L/b¼ 0.5. Moreover, there has been a little reduction of Cd with increasing Froude number (Fig. 6a) like sharp-crested side weirs. The main reason of this is the low intensity of secondary motion created by lateral flow for small value of L/b. However for higher L/b¼1.0 the increasing of Cd values against F1 is more significant, as shown in Fig. 6(c–d). The strength of the secondary flow was affected by the length of the side weir and Froude number. An increase on the secondary flow causes the growth of the deviation angle and kinetic energy towards the side weir when the relative side-weir length increases. Therefore, an increase of F1 values also increases Cd values. El-Khashab and Smith [9] also mentioned that the secondary flow condition due to lateral flow is dominant when a side weir is relatively long (i.e., L/b41). The development of secondary flow towards the downstream direction inside the labyrinth side weir is presented in Fig. 7. The intensity of secondary flow created by lateral flow is defined as the ratio of the mean kinetic energy of the lateral motion to the total kinetic energy of main flow at a given cross section. The intensity of secondary flow is probably created by lateral flow shown in Fig. 8. Turbulence and velocity streamlines oriented towards the side weir. By orientation of the velocity streamlines towards side weir, the occurrence of the stagnation zone and the vortex plays an important role in flow interactions (see Fig. 8). The strength of the secondary
flow created by the lateral flow was affected by the length of the side weir, crest height of side weir and Froude number. An increase in the secondary flow causes the growth of the deviation angle and kinetic energy towards the side weir when the relative length of the side weir increases. The occurrence of the vortex near to middle of labyrinth side weir in XY-plane gets complicated the secondary flow phenomenon in YZ-plane. Due to the vortex occurrence while the secondary flow moves counter-clockwise at the upstream of middle of the weir, the secondary flow moves clockwise at the downstream of middle of the weir (see Fig. 7). As it can be seen from Figs. 5 and 6, the discharge coefficients from CFD analyses are quietly compatible with the experimental data observed. In Fig. 9, the weir discharges obtained from CFD analyses and computed from Eq. (1) with Cd values of Emiroglu et al. [4] were compared. As seen in this figure, the CFD and the calculated values are in reasonable agreement with 711% errors. Additionally, to evaluate the accuracy between experimental and CFD results, the root mean square error (RMSE) and the average percent error (APE) criteria were used. The RMSE and APE are given as 2.016% and 10.64% respectively for the side weir discharges. These values present a satisfactory agreement between experimental and CFD results. vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N u1 X 2 RMSE ¼ t ð9Þ ðQ ðobservedÞQ s ðCFDÞÞ Ni¼1 s
APE ¼
N
100 X
Q s ðobservedÞQ s ðCFDÞ
N i¼1
Q s ðobservedÞ
ð10Þ
6
M.C. Aydin, M.E. Emiroglu / Flow Measurement and Instrumentation 29 (2013) 1–8
Fig. 6. (a–d) Comparison of Cd coefficients against the Froude number.
Fig. 7. Secondary flows in the different cross sections. (a) Plan, (b) Section II-II, (c) Section III-III, (d) Section IV-IV and (e) Section V-V.
M.C. Aydin, M.E. Emiroglu / Flow Measurement and Instrumentation 29 (2013) 1–8
7
Fig. 8. Velocity vectors on free surface at junction location. (a) XY-plan, (b) X-component and (c) Y-component.
in which N is the number of data set and Qs is the discharge of weir.
5. Conclusions In the present study, the CFD (Fluent) analyses of labyrinth side weir located on a straight channel were performed to investigate the effects of some dimensionless parameters as F1 and (h1 p)/p at the certain values of y and L/b. The open channel boundary conditions used in the CFD models provide an efficient approach for simulation of the flow over the labyrinth side weir. The GCI analysis performed with the fine, base (10 mm) and coarse grid to determine the numerical sensitivity to the grid of
model. It is reported that the fine-grid evaluation of GCI as between 0.77% and 7.88% in terms of overflow discharges. Both the results of CFD and physical model showed that while Cd coefficient decreases with increasing values of (h1 p)/p, it increases with increasing F1 due to the effects of secondary flow inside the labyrinth side weir. Only for small values of L/b as 0.5, Cd values decreases with increase of Froude number, like sharpcrested side weir because of low intensity of secondary flow. For all results, it is concluded that a reasonable agreement was achieved between the CFD results and the experimental observations. Additionally, an interval error of 711%, RMSE¼2.016 and APE¼10.64% respect to the side weir discharges between CFD and experimental results is reported. The presented results in this study can encourage further the researchers in making new different designs of labyrinth side weir by using CFD.
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Fig. 9. Comparison of the discharges values.
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Contents lists available at SciVerse ScienceDirect
Flow Measurement and Instrumentation journal homepage: www.elsevier.com/locate/flowmeasinst
Determination of capacity of labyrinth side weir by CFD M. Cihan Aydin a,n, M. Emin Emiroglu b a b
Civil Engineering Department, Bitlis Eren University, Bitlis, Turkey Civil Engineering Department, Firat University, Elazig, Turkey
a r t i c l e i n f o
abstract
Article history: Received 11 April 2012 Received in revised form 17 August 2012 Accepted 19 September 2012 Available online 23 October 2012
Side weirs are widely used in irrigation, land drainage, urban sewage systems, flood protection, and forebay pool of hydropower systems by flow diversion or intake devices. The hydraulic behavior of side weirs received considerable interest by many researchers. A large number of these studies are physical model tests of rectangular side weirs. However, in the study, Computational Fluid Dynamics (CFD) models together with laboratory models of labyrinth side weirs were used for determining the discharge capacity of the labyrinth side weir located on the straight channel. The discharges performances obtained from CFD analyses were compared with the observed results for various Froude number, dimensionless nappe height, dimensionless weir width, and weir included angle. The results obtained from both methods are in a good agreement. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Side weir Water discharge Labyrinth weir CFD analysis Free surface flow
1. Introduction A side weir is an overflow weir framed in the side of a channel, which allows lateral overflow when the surface of the liquid in the channel rises above the weir crest [1]. Side weirs are hydraulic structures often in irrigation techniques, sewer networks, land drainage, and flood protection. The crest length of a weir has the greatest influence on the discharge capacity. A labyrinth weir is defined as a weir crest that is not straight in planform. The increased sill length provided by labyrinth weirs effectively reduces upstream head to the particular discharge. A labyrinth weir can therefore be used to particular advantage where the width of a channel is restricted and a weir is required to pass a range of discharges with a limited variation in upstream water level. Hay and Taylor [2] and Tullis et al. [3] carried out experiments on the advantage of labyrinth weirs in case of limited channel widths. There are a lot of prototype labyrinth spillways (for example the Avon Dam (Australia), the Ritschard Dam (USA), the Ute Dam (USA), the Hyrum Dam (USA), the Sarioglan Dam (Turkey), the Kizilcapinar Dam (Turkey)). However the idea of labyrinth side weir was first presented by Emiroglu et al. [4], and there is not any prototype labyrinth side weir in our knowledge. They reported that discharge coefficient of the labyrinth side weir is 1.5–4.5 times higher than normal rectangular side weir in a rectangular channel. However it is noted that one of the disadvantages of
n
Corresponding author. Tel.: þ90 4342285170; fax: þ 90 4342285171. E-mail addresses: [email protected] (M.C. Aydin), [email protected] (M.E. Emiroglu). 0955-5986/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.flowmeasinst.2012.09.008
the labyrinth side weirs is the decrease of discharge capacity with increasing the nappe height when the weir included angle is chosen too small. A review of previous studies indicated that rectangular sharpcrested side weirs have been investigated extensively, including work by Ackers [5], Collinge [6], Frazer [7], Subramanya and Awasthy [8], El-Khashab and Smith [9], Uyumaz and Muslu [10], Helweg [11], Swamee et al. [12], Agaccioglu and Yuksel [13]. The hydraulic behavior and discharge coefficient of side weirs for the different type of weirs, main channel and flow conditions have been studied by many researchers such as: Nadesamoorthy and Thomson [14], Singh et al. [15], Yu-Tek [16], Cheong [17], and others. Ranga Raju et al. [1] investigated on the discharge coefficient of a broad-crested rectangular side weir depending on the main channel, Froude number and head/weir width ratio. Recently, Aydin [18] modeled the free surface flow over the triangular labyrinth side weir by using Volume of Fluids (VOF) method to describe the surface characteristics in subcritical flow conditions. Borghei and Parvaneh [19] studied a new type of oblique side weir with asymmetric geometry on an experimental set-up. The researchers stated that this kind of weir is more efficient than the ordinary oblique weir. Emiroglu et al. [20] carried out a comprehensive study to determine the discharge coefficient of a sharp crested rectangular side weir in a straight channel, and developed an equation for discharge coefficient including all dimensional parameters. Agaccioglu et al. [21] presented a reliable equation based on 1504 experimental runs for discharge coefficient of the rectangular side weir in a curved channel depending on the all dimensionless parameters. Haddadi and Rahimpour [22] investigated an experimental setup to obtain
2
M.C. Aydin, M.E. Emiroglu / Flow Measurement and Instrumentation 29 (2013) 1–8
the relationships of discharge coefficient with the other dimensionless parameters for trapezoidal broad-crested side weir. Bagheri S., Heidarpour [23] investigated the water flow over sharp-crested side weirs obtaining the distribution of the threedimensional velocity. Assuming that the specific energy across the weir is constant in the dynamic equation of spatially varied flow for outflow over a weir [24], the general equation of weirs can be described as q¼
pffiffiffiffiffiffi dQ ¼ C d 2g ½hp3=2 dx
ð1Þ
where q is the discharge per unit length over the weir, Q is the out-flow discharge, x is the longitudinal direction, g is the gravitational acceleration, p is the height of the weir, h is the depth of flow measured from the channel bottom along the channel centerline, and Cd is the discharge coefficient of the weir. Emiroglu et al. [25] used Eq. (1) to estimate the discharge coefficient of sharp-crested triangular labyrinth side weirs on a straight channel, and they proposed the following equation for subcritical flow: 0:4 0:254 0:122 þ 3:214 ‘L 0:684 hp1 C d ¼ 0:4 þ 2:62 þ0:634 bL #3:857 1:982 y þ 0:22F 2:458 þ 0:122 sin 1 4
ð2Þ
In the present study, the discharge capacity of triangular labyrinth side weir with one cycle was determined by using VOF method with Fluent code, and the discharge coefficients obtained from the CFD were compared with the experimental data of Emiroglu et al. [4]. Recently, some of the researchers such as Bridgeman et al. [26], Aydin and Ozturk [27], Schaffrath et al. [28] used Fluent in their studies at various field as nuclear, mechanical and civil engineering. In the other study, Hargreaves et al. [29] studied the validation of CFD for modeling free surface flows over common hydraulic structures using Fluent with VOF (Volume of Fluid) method. They gained the confidence that more complex hydraulic structures should be modeled by using similar techniques.
Fig. 1. Sketch view of labyrinth side weirs. (a) Hydraulic profile of the triangular side weir, (b) Plan view if triangular side weir and (c) Front view of the triangular side weir.
Table 1 Model geomeries and meshing specifications. L (m)
y (1)
p (m)
Grid size (mm)
Number of cells
Meshing type
0.25
45
0.12 0.16 0.20 0.12 0.16 0.20
10 10 10 10 10 10
291800 292325 292325 347610 347610 463480
3D 3D 3D 3D 3D 3D
Hexahedral Hexahedral Hexahedral Hexahedral Hexahedral Hexahedral
0.12 0.16 0.20 0.12 0.16 0.20
10 10 10 10 10 10
561320 561320 561320 340675 408810 646440
3D 3D 3D 3D 3D 3D
Hexahedral Hexahedral Hexahedral Hexahedral Hexahedral Hexahedral
2. Physical model 60
The data used in this study were taken from the experimental studies conducted by Emiroglu et al. [4] on a large model. The authors conducted 2830 laboratory tests for determining discharge coefficient of triangular labyrinth side weirs. The typical hydraulic characteristics and the views of the triangular labyrinth side weir are also illustrated in Fig. 1. Their set-up consisted of a rectangular main channel 0.5 m wide, 0.50 deep and 12 m long, with a discharge collection channel 0.5 m wide and 0.70 m deep parallel to the main channel. A sluice gate was fitted at the end of the main channel to control flow depth. A rectangular weir was placed at the end of collection channel to measure the discharge of the side weir. In order to measure the water level in the collection channel, a digital point gauge was fixed further 0.40 m from the weir. Labyrinth side weirs were fabricated from sharp edged steel plates. The discharges were measured to an accuracy of 70.1 L/s, by means of an electromagnetic flow-meter installed in the supply line. The experiments were conducted for subcritical flow, stable flow and free overflow conditions. Novak and Cabelka [30] stated that minimum nape height should not be less than 30 mm because of the surface tension over the weir crest. Therefore, minimum nape height is taken into account as 30 mm. The data of the experiments were considered for the weir lengths (L) of 0.25 and 0.50 m, the weir heights (p) of 0.12 m, 0.16 m and, 0.20 m, and weir included angles with y ¼451and 601. Labyrinth side weirs were tested for different Froude numbers, different ratios of
0.50
45
60
(h1–p)/p and different weir included angles in order to obtain the variation of the discharge coefficient.
3. CFD model A High Performance Computer (HPC) was used to simulate the CFD models with VOF method in ANSYS-FLUENT1. The VOF model can model two or more immiscible fluids by solving a single set of momentum equations, Eq. (3), and tracking the volume fraction of each of the fluids throughout the domain, Eq. (4) [31]. @ @ @P s @ @ui @uj ðruj Þ þ ðrui uj Þ ¼ þ m þ ð3Þ þ rg j þF j @t @xi @xi @xj @xj @xi
1
License owner: Bitlis Eren University, Customer Number: 618883.
M.C. Aydin, M.E. Emiroglu / Flow Measurement and Instrumentation 29 (2013) 1–8
1
rq
"
n X @ ! _ pq m _ qp Þ ðaq rq Þ þ r:ðaq rq v q Þ ¼ Saq þ ðm @t p¼1
# ð4Þ
where r is the density of fluid, m is the dynamic viscosity of fluid, u is _ qp is the the velocity vectors, Ps is the pressure and F is a body force; m _ pq is the mass transfer from mass transfer from phase q to phase p, m phase p to phase q, aq is the qth fluid’s volume fraction in a cell. The source term Saq of Eq. (4) is zero by default. If aq ¼0, the cell is empty, if aq ¼1, the cell is full (of the qth fluid), and if 0o aq o1, the cell contains the interface between two fluid phases. The VOF model with time-dependent solution in Fluent was used to calculate free surface flow in the labyrinth side weir models. 3.1. Geometry and boundary conditions The geometries and meshes were generated by using Gambit. The parameters of the created models and its meshing specifications were given in Table 1. The typical view of model geometry and boundary conditions are given in Fig. 2. The open channel boundary conditions were applied to the CFD model as the pressure inlet at the main channel inlet, the pressure outlets at the boundaries which open to atmosphere and labyrinth side weir outlet (Fig. 2). The pressure inlet boundary condition presents an option which ensures to easily describe the bottom and the surface levels of the main channel and also the velocity of the main channel flow. A water level of downstream weir is imposed initially by the pressure inlet boundary at the main channel outlet. It is assumed that the free surface levels of the inlet and outlet boundaries in the main channel are approximately equal initially. After the solution is converged, the water levels reach own natural level in the subcritical conditions. For subcritical outlet flows (F1 o1), if there are only two phases, then the pressure is taken from the pressure profile specified over the boundary, otherwise the pressure is taken from the neighboring cell. Based on the Froude number when F1 o1 the flow is known to be subcritical where disturbances can travel upstream as well as downstream. In this case, downstream conditions might affect the flow upstream [31].
3
aspects known as iterative convergence and grid convergence, in terms of time-dependent problems, iterative convergence at every time step should be checked. It must be ensured that iterative convergence is achieved with at least three orders-ofmagnitude decrease in the normalized residuals for each equation solved. Each time steps are maximum 40 iterations and all residuals drop under three orders every time steps. As shown in Fig. 3, iterative convergences were achieved for each Froude number of the each CFD model of labyrinth side weirs (for L¼0.25 m, y ¼601, and p ¼0.20 m). It is pointed out that this is quite time-consuming process despite of using the HPC system. A method for discretization error estimation is the Richardson extrapolation (RE) method which was first used by Richardson [33]. The Grid Convergence Index (GCI) method which is based on Richardson Extrapolation was used to estimate the discretization errors. The procedure of this method was outlined by Celik et al. [34] as following. For three dimensional calculations the representative cell, mesh or grid size l was estimated by "
l¼
N 1X ðDV i Þ Ni¼1
#1=3 ð5Þ
3.2. Solution convergence The numerical verification in this paper is partially based on the American Society of Mechanical Engineers (ASME) editorial policy statement, which provides a framework for CFD uncertainty analysis [32]. Convergence investigation involves two
Fig. 3. Iterative convergence for various Froude numbers. (a) Hydraulic profile of the triangular side weir, (b) Plane view of triangular side weir and (c) Front view of the triangular side weir.
Fig. 2. Model geometry and boundary conditions [18].
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M.C. Aydin, M.E. Emiroglu / Flow Measurement and Instrumentation 29 (2013) 1–8
4. Results and discussion
where DVi is the volume of ith cell and N is the number of cell. Three different sets of grid are selected and the apparent order is calculated using the equation below. P¼
1
ln j32 =j21 þ qðPÞ
lnðr 21 Þ
The discharge coefficients Cd of CFD models were calculated by Eq. (1) using the data obtained from CFD analyses as calculated by Emiroglu et al. [25] for their physical model. Emiroglu et al. [4] found that the discharge coefficient Cd for labyrinth side weir included angles of 451 and 601 is greater than the Cd coefficient for the other included angles (y ¼901, 1201 and 1501). They stated that the effect of L/b ratio, which plays an important role in the discharges capacity of labyrinth side weir, on Cd for labyrinth side weir included angles of 451 and 601 is very important due to the increase in the intensity of secondary flow created by lateral flow. Therefore, in the CFD analyses, it is only considered that the triangular labyrinth side weir included angles of 451 and 601. To show the effects of parameter (h1 p)/p on discharge coefficients, Cd values obtained from both the CFD and physical models are plotted against the dimensionless nappe height (h1 p)/p for L/b ¼0.5 and 1.0 in Fig. 5(a–d). These figures show that Cd decreases with the increase in (h1 p)/p. The effect of (h1 p)/p on Cd can be explained with the discontinuity region. This discontinuity region has a strong secondary motion next to the boundary of the weir side. Tullis et al. [3] also stated that the values of discharge coefficient decrease with the increasing nappe height. The decreasing trend of Cd with the increase of (h1 p)/p increases with reducing labyrinth side weir included angle. For this reason, higher nappe heights on labyrinth side weir are not selected by hydraulic designers. The discharge capacity of the labyrinth side weir increases with the reducing included angle (y). Thus, the effect of nappe height on discharge of labyrinth side weir is more significant in the small angles. Therefore, the designer must carefully be choosing the nappe height.
ð6Þ
where j21 ¼ f2 f1, j32 ¼f3 f2, r is the grid refinement factor, defined as r21 ¼ l2/l1 for l1 o l 2, f1, f2, f3 are the key variables which play an important role for the considered study. If the grid refinement (r) is constant, q(P) ¼0. Then, it is calculated following approximate relative errors:
f f
e21 ¼
1 2
f1
ð7Þ
Finally, the fine grid convergence index was calculated as [35] GCI21 f ine ¼
1:25e21 r P21 1
ð8Þ
Three various grid sizes which are fine-grid, base grid (10 mm) and coarse-grid were used to control grid convergence. The refinement factors between the coarse and fine grids (r ¼ l coarse/ l fine 41) is 1.11 according to Roache [36] who recommended a minimum 10% change in the grid refinement factor. The values of key variable (f1, f2, f3) were considered as the overflow discharges of side weir for three grid sizes. The numerical errors for L¼0.25 m, y ¼601, p¼0.12 m were estimated in Table 2. According to this table, the fine-grid evaluation of GCI can be given as between 0.77% and 7.88% which corresponds to the weir discharges of 0.12 and 0.52 L/s approximately. The maximum discretization error with the averaged apparent order was also obtained as 2.13% in terms of velocity magnitude in the middle of side-weir by Aydin [18].
3.3. Turbulence model In order to determine the sensitivity of results to turbulence, the six various turbulence models (RSM, standard k–e, RNG k–e, Realizible k–e, k–o, Spalart–Allmaras, where k is the turbulence kinetic energy, e is the turbulence dissipation rate, and o is the specific dissipation rate of turbulence) were compared in Fig. 4 for L¼0.25 m, y ¼601, p ¼0.12 m and F1 ¼0.43. Fig. 4 shows that the variations of dimensionless velocity ratios (V/Vm) at the middle of the weir entrance with the turbulence models are quite slight especially for RSM, Standard k–e, Realizable k–e and Spalart– Allmaras turbulence models besides other two models. Additionally, the RSM is probably the best competent model among all the models to simulate the surface shapes especially the surface fluctuations and vortex occurrence in the side-weir for subcritical condition [18].
Fig. 4. Comparison of different turbulence models.
Table 2 Discretization errors. (h1-p)/h1
0.25 0.30 0.35 0.40 0.45
Discharges (L/s) f1
f2
f3
7.69 9.86 12.52 15.66 19.29
6.68 9.37 12.30 15.46 18.86
6.23 9.34 12.39 15.38 18.30
e21
e32
r21
r32
pave
e21
e32
GCI21 fine (%)
GCI32 fine (%)
1.0066 0.4858 0.2175 0.2018 0.4385
0.4487 0.0283 0.0935 0.0833 0.5587
1.11 1.11 1.11 1.11 1.11
1.11 1.11 1.11 1.11 1.11
11.26 11.26 11.26 11.26 11.26
0.131 0.049 0.017 0.013 0.023
0.067 0.003 0.008 0.005 0.030
7.88 2.96 1.05 0.77 1.37
4.04 0.18 0.46 0.32 1.78
M.C. Aydin, M.E. Emiroglu / Flow Measurement and Instrumentation 29 (2013) 1–8
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Fig. 5. (a–d) Comparison of Cd coefficients against the dimensionless nappe height.
Fig. 6 (a–d) shows the comparison of Cd values obtained from CFD and experimental studies against the Froude number. The results generally indicated that discharge coefficient increases ¨ with increasing Froude number. Agaccioglu and Yuksel [13] also found the similar tendency. But, the variation of Cd against F1 is not significant for the small width of the side weir as L/b¼ 0.5. Moreover, there has been a little reduction of Cd with increasing Froude number (Fig. 6a) like sharp-crested side weirs. The main reason of this is the low intensity of secondary motion created by lateral flow for small value of L/b. However for higher L/b¼1.0 the increasing of Cd values against F1 is more significant, as shown in Fig. 6(c–d). The strength of the secondary flow was affected by the length of the side weir and Froude number. An increase on the secondary flow causes the growth of the deviation angle and kinetic energy towards the side weir when the relative side-weir length increases. Therefore, an increase of F1 values also increases Cd values. El-Khashab and Smith [9] also mentioned that the secondary flow condition due to lateral flow is dominant when a side weir is relatively long (i.e., L/b41). The development of secondary flow towards the downstream direction inside the labyrinth side weir is presented in Fig. 7. The intensity of secondary flow created by lateral flow is defined as the ratio of the mean kinetic energy of the lateral motion to the total kinetic energy of main flow at a given cross section. The intensity of secondary flow is probably created by lateral flow shown in Fig. 8. Turbulence and velocity streamlines oriented towards the side weir. By orientation of the velocity streamlines towards side weir, the occurrence of the stagnation zone and the vortex plays an important role in flow interactions (see Fig. 8). The strength of the secondary
flow created by the lateral flow was affected by the length of the side weir, crest height of side weir and Froude number. An increase in the secondary flow causes the growth of the deviation angle and kinetic energy towards the side weir when the relative length of the side weir increases. The occurrence of the vortex near to middle of labyrinth side weir in XY-plane gets complicated the secondary flow phenomenon in YZ-plane. Due to the vortex occurrence while the secondary flow moves counter-clockwise at the upstream of middle of the weir, the secondary flow moves clockwise at the downstream of middle of the weir (see Fig. 7). As it can be seen from Figs. 5 and 6, the discharge coefficients from CFD analyses are quietly compatible with the experimental data observed. In Fig. 9, the weir discharges obtained from CFD analyses and computed from Eq. (1) with Cd values of Emiroglu et al. [4] were compared. As seen in this figure, the CFD and the calculated values are in reasonable agreement with 711% errors. Additionally, to evaluate the accuracy between experimental and CFD results, the root mean square error (RMSE) and the average percent error (APE) criteria were used. The RMSE and APE are given as 2.016% and 10.64% respectively for the side weir discharges. These values present a satisfactory agreement between experimental and CFD results. vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N u1 X 2 RMSE ¼ t ð9Þ ðQ ðobservedÞQ s ðCFDÞÞ Ni¼1 s
APE ¼
N
100 X
Q s ðobservedÞQ s ðCFDÞ
N i¼1
Q s ðobservedÞ
ð10Þ
6
M.C. Aydin, M.E. Emiroglu / Flow Measurement and Instrumentation 29 (2013) 1–8
Fig. 6. (a–d) Comparison of Cd coefficients against the Froude number.
Fig. 7. Secondary flows in the different cross sections. (a) Plan, (b) Section II-II, (c) Section III-III, (d) Section IV-IV and (e) Section V-V.
M.C. Aydin, M.E. Emiroglu / Flow Measurement and Instrumentation 29 (2013) 1–8
7
Fig. 8. Velocity vectors on free surface at junction location. (a) XY-plan, (b) X-component and (c) Y-component.
in which N is the number of data set and Qs is the discharge of weir.
5. Conclusions In the present study, the CFD (Fluent) analyses of labyrinth side weir located on a straight channel were performed to investigate the effects of some dimensionless parameters as F1 and (h1 p)/p at the certain values of y and L/b. The open channel boundary conditions used in the CFD models provide an efficient approach for simulation of the flow over the labyrinth side weir. The GCI analysis performed with the fine, base (10 mm) and coarse grid to determine the numerical sensitivity to the grid of
model. It is reported that the fine-grid evaluation of GCI as between 0.77% and 7.88% in terms of overflow discharges. Both the results of CFD and physical model showed that while Cd coefficient decreases with increasing values of (h1 p)/p, it increases with increasing F1 due to the effects of secondary flow inside the labyrinth side weir. Only for small values of L/b as 0.5, Cd values decreases with increase of Froude number, like sharpcrested side weir because of low intensity of secondary flow. For all results, it is concluded that a reasonable agreement was achieved between the CFD results and the experimental observations. Additionally, an interval error of 711%, RMSE¼2.016 and APE¼10.64% respect to the side weir discharges between CFD and experimental results is reported. The presented results in this study can encourage further the researchers in making new different designs of labyrinth side weir by using CFD.
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Fig. 9. Comparison of the discharges values.
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