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Efficiency of Trapezoidal Labyrinth Shaped stepped spillways
Journal Pre-proof Efficiency of Trapezoidal Labyrinth Shaped Stepped Spillways Amir Ghaderi, Saeed Abbasi, John Abraham, Hazi Mohammad Azamathulla PII:
S0955-5986(20)30059-5
DOI:
https://doi.org/10.1016/j.flowmeasinst.2020.101711
Reference:
JFMI 101711
To appear in:
Flow Measurement and Instrumentation
Received Date: 13 October 2019 Revised Date:
19 January 2020
Accepted Date: 11 February 2020
Please cite this article as: A. Ghaderi, S. Abbasi, J. Abraham, H.M. Azamathulla, Efficiency of Trapezoidal Labyrinth Shaped Stepped Spillways, Flow Measurement and Instrumentation, https:// doi.org/10.1016/j.flowmeasinst.2020.101711. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Elsevier Ltd. All rights reserved.
Efficiency of Trapezoidal Labyrinth Shaped Stepped Spillways Amir Ghaderi1, Saeed Abbasi2*, John Abraham3, Hazi Mohammad Azamathulla4 1- Ph.D. Student, Department of Civil Engineering, University of Zanjan, Zanjan, Iran. [email protected] 2*- Assistant Professor, Department of Civil Engineering, University of Zanjan, Zanjan, Iran. [email protected] 3- Professor, School of Engineering, University of St. Thomas, St Paul, MN, USA. E-mail: [email protected] 4- Professor, Department of Civil and Environmental Engineering, University of the West Indies, St. Augustine, Trinidad. [email protected] Abstract: A spillway with Trapezoidal Labyrinth Shaped steps (TLS) is proposed and investigated numerically. The results show that this spillway has better performance as it increases the magnification ratio LT/Wt (LT is the total edge length and Wt is the spillway width). A TLS stepped spillway is more effective for magnification ratios of 1.25, 1.45, 1.65 and 1.85 for energy dissipation compared to stepped spillways by 5.6, 8.5, 12.6 and 17% respectively. TLS stepped spillway has a lower residual head and a larger friction factor. The residual head ratio (Hres/dc) is ~2.89 in a TLS stepped spillway, while it is ~4.32 for a flat stepped spillway. Also, the friction factor varies from 0.79 to 1.33 for different magnification ratios, while this value is approximately 0.66 for the flat stepped spillway.
Keywords: TLS stepped spillway, Trapezoidal Labyrinth Shape, energy dissipation, residual energy, friction factor, FLOW-3D.
1. Introduction: Spillways are constructed to transfer high-energy water. Because the crest and the downstream portion of a spillway have different elevations, the water velocity will increase in the flow direction [1]. Recently, researchers have tried to increase the efficiency of these spillways and have made significant advancement in our understanding. Researchers generally attempt to reduce the velocity downstream by increasing the efficiency of energy dissipation in the stepped spillway. Based on the mentioned studies, the regimes of the flow on stepped spillways are categorized in skimming flow, transition flow and nappe flow (Fig. 1). Fig. 1
Extreme hydrodynamic oscillations and continuous mixing of air and water are of the main characteristics of the skimming flow regime. For this flow, the water on the steps travels as a continuous layer and the steps play the role of a large roughness. A large part of energy dissipation in the skimming flow will be achieved by the formation of vortices in the recirculation region (Fig. 1-a) [2]. Ohtsu and Yasuda (1997) were probably the first who introduced the concept of “transition flow” regime, but they did not elaborate on its properties. For a given geometry of stepped channel chute, a range of flow discharges results in a flow regime between a nappe flow at low discharges and skimming flow at large flow rates (Fig. 1-b)[3]. For the nappe flow regime, the flow drops and hits the lower step as a jet (Fig. 1-c) [2]. Various laboratory investigations have been carried out on stepped spillways by [4-6] Sorensen (1985), Peyaras et al. (1992) and Baylar et al. (2011) and the effects of flow patterns,
step geometry and spillway slope on energy dissipation have been investigated. Sorensen (1985) and Chanson (2001) performed many laboratory tests to diagnose flow and energy dissipation on stepped spillways and provided relationships to determine the type of flow and the amount of energy loss on these spillways [4,1]. Gonzalez and Chanson (2007) empirically investigated the characteristics of flow on a stepped spillway for different chute slopes for embankment dams; they presented some new design charts for stepped spillways of various geometries [7]. Felder and Chanson (2009) studied stepped spillways with a slope of 21.8° and step heights of 5 cm and 10 cm. Comparison of their results with previous studies indicated that the residual head at the end of the spillway was smaller than that of a stepped spillway with a slope of 15.9° [8]. Felder et al. (2012a, b) investigated the flow patterns and the residual head on stepped spillways of different configurations. Their results indicated that the slope of the spillway is one of the most important factors for energy dissipation. The performance of a pooled stepped spillway with a slope of 8.9° for reducing the residual head was better than that of a flat stepped spillway, while a contrary result was obtained for the slope of 26.6°[9-10]. Zhang and Chanson (2018) studied the air–water flow system and the energy dissipated by studying the effects of step shape in a 45° steep chute. The results showed flow patterns, aeration, and energy dissipation changes due to changing in step shape [11]. Parsaie and Haghiabi (2019) investigated the stage-discharge relation, discharge coefficient, and energy dissipation on the circular crested stepped spillways (CCSS). Their results indicated that The CCSS can dissipate the energy of flow between 90 and 40% and compare to their common type (smooth chute) [12].
Computational fluid dynamics (CFD) have long been utilized to simulate the flow in hydraulic structures. Tongratoke et al. (2009) attained reliable and integrated results by simulating the flow on stepped spillways. They used different turbulence models including two-equation models [13]. Nikseresht et al. (2013) simulated two-phase flows and compared the VOF scheme with mixture methods and investigated the energy dissipation in stepped spillways of different step slopes [14].
Morovati et al. (2016) investigated the influence of a vertical obstacle on the edge of steps on energy dissipation. Their results showed that decreasing the number of steps in pooled stepped spillways reduces the velocity and increases the relative energy dissipation [15].
Kaouachi et al. (2019) and Wan et al. (2019) numerically assessment of the inception point Location in stepped spillways. They results indicate that the numerical simulations were in accordance with experimental data, which allowed the analysis of inception point location that fits experimental results. The inception point location of the pooled stepped spillway is closer to the spillway crest than that of the flat stepped spillway [16-17].
Other researchers such as [18-22] performed numerical investigations on stepped spillways. The cost of physical models includes the construction costs of the initial model, laboratory tests and modifications; a numerical simulation is a tool that can be used to reduce laboratory tests in order to save time and money and is applicable to different hydraulic phenomena [23].
In this paper a new kind of step shapes proposes Trapezoidal Labyrinth Shaped steps for stepped spillway (TLS stepped spillway) and investigates the hydraulic performance this type
spillway. In particular, the energy dissipation efficiency of this type of spillway is evaluated using CFD techniques.
2. Model Development 2.1. Theoretical relations of flow regime detection The factors that affect the generation of various types of flow on stepped spillways include the geometry of the steps (length and height of the steps, l and h) and the flow rate Q. However, some researchers have proposed other criteria. The studies of many researchers such as [5, 24, 25, 26] have shown that the type of flow regime depends on the normalized critical depth, dc/h (dc is the critical depth of the flow) and a dimensionless variable of steps, h/l. According to Equations (1) to (3), criteria are presented for which a skimming flow regime occurs: h ≤ 0.9 l
Rajaratnam (1990)
dc ≥ 0.8 h
Peyras et al. (1992)
dc h > 1.01 − 0.37( ) h l
Chanson (1994)
dc h > 1.05 − 0.465( ) h l
0.4 ≤
(1)
(2) 0.2 ≤
h ≤ 1.25 l
(3)
Based on the Equations (1) to (3), the type of flow regime on the stepped spillway which is modeled in the current study is presented in Table 1 and Figure 2. Table 1 Fig. 2
2.2. Dimensional analysis The most important flow variables on the stepped spillways are: •
The acceleration of gravity (g) and the properties of the fluid; i.e. dynamic viscosity (µ) and mass density (ρ)
•
The hydraulic properties of flow including depth (y) and velocity (V)
•
The geometry of the spillway including step height (h), step length (l), number of steps (N0) and the spillways total height (Hdam).
In this study, the geometric shape of the steps is changed into a trapezoidal labyrinth shape; the geometric variables should be defined as follows: Nc: The number of labyrinth shape cycles, LT: Total length of the labyrinth shape ( LT = N c × (4a + 2lc ) ) and Wt: The width of the channel (Figure. 3). According to the notations in Figure. 4, the total head loss is expressed by ∆H = H0 – H where: H is the head on the considered step, H=ycosθ + αV2/2g; H0 is the upstream head of the spillway H0 =∆z+Y+α(V0)2/2g; V is local mean velocity (V=Q/yWt) and V0 is the approach velocity. Fig. 3 Fig. 4
Considering Wt>>y, the head loss should primarily depend on the number of steps (N0), the geometry of the steps (h, l) and the discharge (Q). The last one may be expressed by the critical depth dc=[(Q/Wt)2/g] 1/3 (Christodoulou, 1993). Therefore, in the TLS stepped spillway, the function through which the relation of variables can be expressed is: f (Hdam, N0, Nc, l, h, V, y, LT, Wt, g, ρ, µ)=0.
(4)
Using Buckingham’s π theory (the variables ρ, V and y are repeating variables) one can
obtain: d d L ∆H = f ( c , c , T , Re , Fr , N 0 , N c ) H0 h l Wt
(5)
As in this study, the number of steps and the number of labyrinth shape cycles are held constant; it is assumed that the force of fluid viscosity is insignificant compared to the inertia force for open channel flow [27]; so three dimensionless variables of Reynolds number Re, N0 and Nc could be eliminated.
2.3. The FLOW-3D model Computational Fluid Dynamics (CFD) is increasingly used to study a wide variety of complex Environmental Fluid Mechanics (EFM) processes; such as water flow and turbulent mixing of contaminants in rivers [28]. FLOW-3D® is a general purpose computational fluid dynamics (CFD) program for modeling a wide variety of fluid flow and heat transfer phenomena [29] which basically solves the Navier-Stokes equations [30-31]. The following equations describe the unsteady continuity and momentum equations in the Cartesian coordinates: ∂U i =0 ∂xi
ρ
(6)
∂U i ∂U i ∂U i ∂P ∂ + ρU j =− + (µ − ρ ui′u′j ) + ρ gi ∂ti ∂xi ∂xi ∂x j ∂x j
(7)
In which Ui and u i′ are average velocity and fluctuating velocities in the xi direction respectively, xi = (x,y,z), Ui = (U,V,W), and ui’ =(u’, v’, w’). The symbols ρ, µ, P, and gi are density,
dynamic
viscosity,
pressure,
and
gravitational
acceleration,
respectively.
Instantaneous velocity is defined as ui = U i + ui′ for the three directions. A turbulence model is necessary for accounting for the nonlinear Reynolds stresses. Despite there are more advance k-epsilon turbulence modelling studies such as Pu et al. (2014) and
Pu (2015) [32-33], For this purpose and in this study, the k-ε (RNG)1 turbulence model is used to simulate the turbulent flow characteristics. According to the previous researches in which flow on hydraulic structures such as stepped spillway, in spillway’s flip bucket and cte is simulated [34-38], the k-ε (RNG) model was utilized to model the domain. The RNG equations are presented by Yakhot et al. (1992) [39]. The FLOW-3D® software uses the Volume of Fluid (VOF) scheme for free surface modeling. This method is described as follows: ∂F 1 ∂ ∂ ∂ + ( FuA x ) + ( FvA y ) + ( FwA z ) = 0 ∂t V f ∂x ∂y ∂z
(8)
Here, F is the fraction function; F=0 when a cell is empty and F=1 when the cell is full [40].
3. Methods and materials 3.1. Geometric model In this study, results of the numerical solution are validated with experimental data of Felder et al. (2012a, b) [9-10]. They used a stepped spillway with ten steps of 52 cm length; the steps dimensions were 20 ×10 cm. Fig. 5 shows the schematic design of that stepped spillway. In this study, the geometry of the stepped spillway was adjusted to improve its hydraulic performance (energy dissipation rate). The changes to step geometry were obtained by employing a Trapezoidal Labyrinth Shape for each step. The models are classified into four configurations for TLS stepped spillway with four different magnification length ratios (LT/Wt) equal to 1.2, 1.4, 1.6 and 1.8; and one model with a conventional shape which was tested by Felder et al. (2012a, b) and is used for validation [9-10]. Fig. 6 shows the three-dimensional design and geometry of the TLS stepped spillway. 1
Re-Normalization Group
Fig. 5 Fig. 6
3.2. Computational grid and boundary conditions Four different meshes sizes were used in the vicinity of the weir. As listed in Table 2, comparison was made of the ratio of V/Vc at y/dc=0.645 obtained from numerical solution on step No. 9 with experimental results. Based on this mesh-refinement study, a computational mesh with 2,204,893 elements was selected for further calculations, with the selected appropriate mesh results in a relative error and maximum aspect ratio of 6.07%and 1.29, respectively. Table 2
The larger element size ds were set to be 1.8 cm with an element count of 868,271. The smaller element size was 0.8 cm and resulted in 1,336,622 elements. Finally, the flow domain was discretized by cubic elements with a total of 2,204,893 elements and the number of meshes was considered the same in all models (Figure. 7). According to the experimental condition all boundary conditions has been employed. In experimental research, predetermined discharge (Q) had been used at the beginning of spillway channel that specific discharge was used in the present paper subsequently. Furthermore the experimental results and flow condition hadjust been presented above steps and we couldn’t choose other boundary condition including specific pressure or specific velocity for output. Hence outflow boundary condition was used at the output to prevent the downstream boundary effects onlast steps results. Also a no-slip wall boundary condition was also used for thewalls and channel bed. The symmetry boundary condition was used for
the upper boundary which was located on top of the air phase. Table 3 lists the boundary conditions for the model. Table 3 Fig. 7
4. Results and discussion 4.1. Verification of the numerical model using laboratory results After the numerical simulation of steady-state flow on the stepped spillway, the results of the flat stepped spillway and interfacial velocity distributions on steps 8, 9 and 10 were validated with experimental data of Felder et al. (2012a, b) [9-10]. The error limits between the data obtained from FLOW-3D® and the experimental data were calculated according to Eqaution (9): E=
((V /V c )N − (V /V c )E ) ×100 (V /V c )N
(9)
In which E is the relative error, (V/Vc)N is the dimensionless velocity distribution from the numerical solution and (V/Vc)E is a dimensionless form of the measured interfacial velocity distributions in the laboratory. Figure 8 displays the interfacial velocity distributions and relative error for the numerical data. Regarding the relative error, it can be seen that the maximum difference is 8.55% and the maximum level of consistency between the numerical and experimental model is in the middle area of the depth up to the flow surface. One can conclude that there is good agreement between the numerical and experimental results. Fig. 8
4.2. Water surface profiles Water surface profiles on stepped spillway for Q=0.105 m3/s are presented for the skimming flow regime (Figure 9). Skimming flow is observed when there is a coherent stream between step edges and the development of recirculation vortices. It also creates a semi labyrinth water profile because of the interaction of the flow direction and the flow interference (Fig. 10). Therefore, with the utilization of this kind of spillway more energy is expected to. Fig. 9 Fig 10
4.3. Residual energy and turbulent energy dissipation Transferring the water downstream using spillways will produce a large amount of kinetic energy. In the case of stepped spillways, one of the important concerns of engineers is to know the amount of energy dissipation and the amount of residual energy downstream. In the present study, the energy dissipation ratio, ∆H/Hmax, and residual energy, Hres, are calculated at the edge of the final step of the TLS spillway models. The energy dissipation ratio,
∆H/Hmax, expresses the percentage of total energy loss along the stepped spillway relative to the upstream total head, Hmax. The total head can be evaluated as: H max =
3 × d c + H dam 2
(10)
In which Hdam is the height of the dam and dc is the critical flow depth. The total head loss ∆H can be estimated as ∆Η = Hmax - Hres in which the residual head, Hres, is H res = d × cos 2 θ +
U w2 2× g
(11)
The term Uw=q/y represents the mean flow velocity along the central axis of the spillway. The symbols θ, d, and g are the spillway slope, the equivalent clear water flow depth and the
gravitational acceleration, respectively. Figure 11 shows a comparison of flat and TLS stepped spillway energy dissipation under different flow conditions. It is evident that the trapezoidal labyrinth shape of the steps has dissipated more energy than the flat shape. As the flow rate increases, the rate of energy dissipation is reduced for all models. The rate of energy dissipation on the last step for four different magnification ratios (LT/Wt) of 1.2, 1.4, 1.6 and 1.8, is illustrated in Figure 12 as a function of the dimensionless variable of ∆z0/dc. The variable ∆z0 defines the vertical distance between the spillways crest and the last step (Figure 5). Comparing the performance of the flat stepped spillway with TLS shows that the energy loss in the TLS stepped spillway is greater. Also, increasing the magnification ratio (LT/Wt) results in increases to the energy dissipation. For similar flow conditions, the TLS stepped spillway is more effective at energy dissipation than a flat stepped spillway. For magnification ratios (LT/Wt) of 1.25, 1.45, 1.65 and 1.85, the percentages of efficiency improvements are 5.6%, 8.5%, 12.6%, and 17%, respectively. The flow interference on the TLS stepped spillway and the step shapes cause this efficiency improvement. Fig. 11 Fig. 12
It can be clearly seen in Figure 13 that changing the step shape to labyrinth will increase the streamlines interference. As a result, the energy dissipation will increase, especially for the downstream steps compared to the flat stepped spillway. It should be noted that the contour color scaling for the images in Figure 13 are not the same. Fig. 13
In Figure 14, the dimensionless residual head (Hres/dc) is depicted as a function of dimensionless discharge (dc/h). On the flat stepped spillway, the dimensionless residual head is the largest (~4.32) and for the same flow conditions (dimensionless discharge = 0.84); the average dimensionless residual head on the TLS stepped spillway is ~2.89. As the magnification ratio increases, the residual head decreases. Fig. 14
4.4. Flow resistance The flow resistance is primarily a form of drag in skimming flows [41] and is commonly expressed in the form of Darcy-Weisbach friction factor fe [22, 39]. The Darcy–Weisbach friction factor is estimated from the measured air-water flow properties and is calculated as Equation (12): fe =
8× g × S f × d 8 ×τ 0 = 2 ρ w ×U w U w2
(12)
Here, τ0 is the average shear stress, Sf is the friction slope and is equal to − ∂H ∂x (where H is the total head and x is the distance in the flow direction), d is the equivalent clear water flow depth and Uw = q/y is the flow mean velocity [42-43]. The friction factor results are illustrated in Figure 15 as a function of the dimensionless step cavity roughness height (hcosθ)/DH in which DH is the hydraulic diameter.
It is defined by DH =4By/(B+2y) for a
rectangular channel). It is observed that the friction factor (fe) in the flat stepped spillway is lower compared to the TLS-stepped spillway. This value is 0.66 in the flat stepped spillway and 0.79, 0.90, 0.91, and 1.33 in the TLS-stepped spillway for magnification ratios of 1.25, 1.45, 1.65 and 1.85, respectively. Also, it has been shown that the largest friction factor for the flat stepped spillway is obtained for the highest discharges.
Also, in the TLS stepped spillway, the highest friction factor is achieved for a low discharge and in a magnification ratio of 1.85. In other words, for low discharge, as the ratio of LT/Wt increases the friction factor for increases. The reason is from flow interference through the edges of the trapezoidal labyrinth part of the steps and the extension of the recirculation region (Figure 16). According to Chanson et al. (2002), flow resistance caused by recirculating rotating vortices and the energy dissipation changes by the motion exchange between the cavity flow and the main flow [40]. Also, in TLS stepped spillways, the interference of streamlines increases the resistance against the flow and increases the energy dissipation. Fig. 15 Fig. 16
5. Summary and conclusions Stepped spillways are designed to improve the hydraulic conditions of flow and the energy dissipation. Construction of these types of spillways will reduce the dimensions and cost of the stilling basin. In this research, a new type of stepped spillway; called a Trapezoidal Labyrinth Shaped stepped spillway (TLS stepped spillway) is proposed and its hydraulic performance is evaluated using FLOW-3D®. The summary and results of this research are as follows: • The highest error between the experimental and numerical results occurs at the 8th step by
8.55%. There is good agreement between the experimental and numerical results. • In TLS stepped spillways, the labyrinth shape of the steps causes interference of the
streamlines and improves the energy dissipation of the flow.
• Labyrinth-shaped steps increase the energy dissipation compared to flat steps under the same
flow conditions and as the magnification ratio (LT/Wt) increases, more energy dissipation occurs. For a TLS stepped spillway with a magnification ratio of 1.25, 1.45, 1.65 or 1.85, the energy dissipates more than that of the flat stepped spillway by 5.6%, 8.5%, 12.6%, or 17%, respectively. • Residual head calculations show that the flat stepped spillway under the same flow conditions
has a dimensionless residual head of ~4.32 and the TLS stepped spillway has an average residual head of ~2.89. Also increasing the magnification ratio results in decreasing the residual head. • It is observed that the friction factor (fe) for the flat stepped spillway is lower than that of the
TLS stepped spillway. This value equals 0.79, 0.90, 0.91, and 1.33 in the TLS stepped spillway for magnification ratios of 1.25, 1.45, 1.65 and 1.85, respectively; while it is 0.66 for the flat stepped spillway. The friction factors in TLS stepped spillways increase because of the flow interference through the edges of the labyrinth shapes and the recirculation region in this new type of stepped spillways.
Nomenclature q(Q/B)
(l/s)/m
Water discharge per unit width
B
cm
Width of the flume
h
cm
The step’s height
l
cm
The length of step
Ld
cm
The length of the stilling basin
Hs
cm
Height of stepped spillway
α
-
Chute angle
hc
cm
The critical depth
ht
cm
The tail water depth
ρs
kg/m3
Mass density of sediments
ρw
kg/m3
Mass density of the water
µ
(N-s)/m2
Re
-
The Reynolds number
g
m/s2
Acceleration of gravity
S
-
d50
mm
Bed material diameter of which is finer by weight
d10
mm
Grain size for which 10% of material is finer
d30
mm
Grain size for which 30% of material is finer
d60
mm
Grain size for which 60% of material is finer
Cu
-
Coefficient of uniformity
Cc
-
Coefficient of curvature
Frd
-
Particle Froude number
σg
-
Geometric standard deviation
Dynamic viscosity of water
The relative density of sediments
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[29] Flow Science, 2008. Incorporated: FLOW-3D users manual. Version 9.3, Santa Fe, New Mexico. [30] Zahabi, H., Torabi, M., Alamatian, E., Bahiraei, M., and Goodarzi, M. 2018. Effects of Geometry
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on
Sediment
Entrapment. Water, 10(12), 1725. [31] Ghaderi, A., and Abbasi, S. 2019. CFD simulation of local scouring around airfoil-shaped bridge piers with and without collar. Sādhanā, 44(10), 216. [32] Pu, J.H., Shao, S., and Huang, Y. 2014. Numerical and experimental turbulence studies on shallow open channel flows. Journal of Hydro-Environment Research, 8(1), 9-19. [33] Pu, J.H. 2015. Turbulence Modelling of Shallow Water Flows using Kolmogorov Approach, Computers and Fluids, 115, 66-74. [34] Shahheydari, H., Nodoshan, E. J., Barati, R., and Moghadam, M. A. 2015. Discharge coefficient and energy dissipation over stepped spillway under skimming flow regime. KSCE Journal of Civil Engineering, Vol. 19, No. 4, pp. 1174-1182. [35] Tabari, M. M. R., and Tavakoli, S. 2016. Effects of stepped spillway geometry on flow pattern and energy dissipation. Arabian Journal for Science and Engineering, Vol. 41, No. 4, pp. 1215-1224. [36] Daneshfaraz, R., Ghahramanzadeh, A., Ghaderi, A., Joudi, AR., and Abraham, J. 2016. Investigation of the Effect of Edge Shape on Characteristics of Flow under Vertical Gates. Journal‐American Water Works Association, 108(8): 425-432. [37] Daneshfaraz, R., and Ghaderi, A. 2017. Numerical Investigation of Inverse Curvature Ogee Spillway. Civil Engineering Journal, 3(11): 1146-1156.
[38] Daneshfaraz, R., Minaei, O., Abraham, J., Dadashi, S., and Ghaderi A. 2019. 3-D Numerical simulation of water flow over a broad-crested weir with openings. ISH Journal of Hydraulic Engineering, 1-9. [39] Yakhot, V., Orszag, S. A., Thangam, S., Gatski, T. B., and Speziale, C. G. 1992. Development of turbulence models for shear flows by a double expansion technique. Physics of Fluids A: Fluid Dynamics, Vol. 4, No. 7, pp. 1510-1520. [40] Ghaderi, A., Dasineh, M., Abbasi, S., and Abraham, J. 2020. Investigation of trapezoidal sharp-crested side weir discharge coefficients under subcritical flow regimes using CFD. Applied Water Science, Vol. 10, No. 1, p. 31. [41] Carosi, G., and Chanson, H. 2008. Turbulence characteristics in skimming flows on stepped spillways. Canadian Journal of Civil Engineering, Vol. 35, No. 9, pp. 865-880. [42] Chanson, H. 2000. Forum article, Hydraulics of Stepped Spillways: Current Status. Journal of hydraulic engineering, Vol. 126, No. 9, pp. 636-637. [43] Chanson, H., Yasuda, Y., and Ohtsu, I. 2002. Flow resistance in skimming flows in stepped spillways and its modeling. Canadian Journal of Civil Engineering, Vol. 29, No. 6, pp. 809-819.
Table 1. Hydraulic characteristics and the type of flow regime on the stepped spillway for the present study dc=(q2/g)1/3
Step height
Step length
(m)
(m)
(m)
0.04
0.08
0.1
0.065
0.12
0.08
Q(m3/s)
h/l
dc/h
Flow Regime
0.2
0.5
0.84
Skimming flow
0.1
0.2
0.5
1.17
Skimming flow
0.13
0.1
0.2
0.5
1.34
Skimming flow
0.095
0.15
0.1
0.2
0.5
1.50
Skimming flow
0.105
0.16
0.1
0.2
0.5
1.61
Skimming flow
0.113
0.17
0.1
0.2
0.5
1.69
Skimming flow
Table 2. Mesh Sensitivity analysis MAPE* (%) Int.
Ext.
Total
Max
V/Vc, y/dc=
V/Vc, y/dc= 0.645 –
mesh
mesh
number of
Aspect
0.645- step 9 in
step 9 in
(cm)
(cm)
cells
ratio
FLOW-3D
Experimental result
N1
2.5
1.5
1046729
1.79
2.132
2.938
27.43
N2
2.2
1.1
1648157
1.52
2.429
2.938
17.32
N3
1.8
0.8
2204893
1.29
2.741
2.938
6.07
N4
1.6
0.65
2876956
1.14
2.780
2.938
5.37
Test NO.
Finer meshes
100 ×
1 n X exp − X num ∑ X n 1 exp
The solution needs higher computational storage.
*Mean Absolute Percentage Error (MAPE), Xexp: Experimental value of X, Xnum: Numerical value of X, n: Count of data
Table 3. Applied boundary conditions Software FLOW-3D
Upstream
Downstream
Free surface
Floor
Lateral
boundary
boundary
boundary
boundary
boundaries
Volume flow rate
Outflow
Symmetry
Wall
Wall
Figure 1. Flow conditions on stepped channels: (a) skimming flow; (b) transition flow; (c) nappe flow (Ohtsu et al., 1997)
Figure 2. Flow regime conditions in the skimming flow region
Figure 3. Geometrical characteristics of the TLS stepped spillway
Figure 4. Side view with dimensional annotation
Figure 5. Schematic design of the stepped spillway used by Felder et al. (2012a, b)
Figure 6. Three-dimensional design of the Trapezoidal Labyrinth Shaped (TLS) stepped spillway
Figure 7. Meshed model in FLOW-3D
Figure 8. Dimensionless interfacial velocity distributions on steps 8, 9 and 10 of the flat stepped spillways; numerical simulation vs. experimental data (Q = 0.113 m3/s)
Figure 9. Skimming flow water surface profile on the stepped spillway falls (Q=0.105 m3/s)
Figure 10. Flow on the flat stepped spillway and its interference on transverse sides of the TLS stepped spillway
Figure 11. Percentage of energy dissipation on flat and TLS stepped spillways versus the dimensionless discharge dc/h
Figure 12. The rate of energy dissipation on flat and TLS stepped spillways versus the dimensionless elevation change
Figure 13. Comparison of energy dissipation on TLS spillway and flat spillways
Figure 14. Dimensionless residual head on flat and TLS stepped spillways versus dimensionless discharge
Figure 15. Darcy friction factors on the flat and the TLS stepped spillways
Figure 16. Recirculation regions in a skimming flow through a TLS stepped spillway
Stepped spillways are designed to improve the hydraulic conditions of flow and the energy dissipation. Construction of these types of spillways will reduce the dimensions and cost of the stilling basin. In this research, a new type of stepped spillway; called a Trapezoidal Labyrinth Shaped stepped spillway (TLS stepped spillway) is proposed and its hydraulic performance is evaluated using FLOW-3D®. The purpose of this research is to propose and investigate a new type of stepped spillways to achieve the highest dissipation level of energy of the flow conducted to the downstream utilizing FLOW3D model (CFD models). One can find and utilize TLS stepped spillways as next generation of stepped spillways which increase the efficiency and improve the hydraulic performance. The CFD models can be useful for analyzing hydrodynamics of hydraulic structures that some of their results were validated by experimental or real observations. The summary and results of this research are as follows: •
The highest error between the experimental and numerical results occurs at the 8th step by 8.55%. There is good agreement between the experimental and numerical results.
•
In TLS stepped spillways, the labyrinth shape of the steps causes interference of the streamlines and improves the energy dissipation of the flow.
•
Labyrinth-shaped steps increase the energy dissipation compared to flat steps under the same flow conditions and as the magnification ratio (LT/Wt) increases, more energy dissipation occurs. For a TLS stepped spillway with a magnification ratio of 1.25, 1.45, 1.65 or 1.85, the energy dissipates more than that of the flat stepped spillway by 5.6%, 8.5%, 12.6%, or 17%, respectively.
•
Residual head calculations show that the flat stepped spillway under the same flow conditions has a dimensionless residual head of ~4.32 and the TLS stepped spillway has an average residual head of ~2.89. Also increasing the magnification ratio results in decreasing the residual head.
•
It is observed that the friction factor (fe) for the flat stepped spillway is lower than that of the TLS stepped spillway. This value equals 0.79, 0.90, 0.91, and 1.33 in the TLS stepped spillway for magnification ratios of 1.25, 1.45, 1.65 and 1.85, respectively; while it is 0.66 for the flat stepped spillway. The friction factors in TLS stepped spillways increase because of the flow interference through the edges of the labyrinth shapes and the recirculation region in this new type of stepped spillways.
Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
S0955-5986(20)30059-5
DOI:
https://doi.org/10.1016/j.flowmeasinst.2020.101711
Reference:
JFMI 101711
To appear in:
Flow Measurement and Instrumentation
Received Date: 13 October 2019 Revised Date:
19 January 2020
Accepted Date: 11 February 2020
Please cite this article as: A. Ghaderi, S. Abbasi, J. Abraham, H.M. Azamathulla, Efficiency of Trapezoidal Labyrinth Shaped Stepped Spillways, Flow Measurement and Instrumentation, https:// doi.org/10.1016/j.flowmeasinst.2020.101711. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Elsevier Ltd. All rights reserved.
Efficiency of Trapezoidal Labyrinth Shaped Stepped Spillways Amir Ghaderi1, Saeed Abbasi2*, John Abraham3, Hazi Mohammad Azamathulla4 1- Ph.D. Student, Department of Civil Engineering, University of Zanjan, Zanjan, Iran. [email protected] 2*- Assistant Professor, Department of Civil Engineering, University of Zanjan, Zanjan, Iran. [email protected] 3- Professor, School of Engineering, University of St. Thomas, St Paul, MN, USA. E-mail: [email protected] 4- Professor, Department of Civil and Environmental Engineering, University of the West Indies, St. Augustine, Trinidad. [email protected] Abstract: A spillway with Trapezoidal Labyrinth Shaped steps (TLS) is proposed and investigated numerically. The results show that this spillway has better performance as it increases the magnification ratio LT/Wt (LT is the total edge length and Wt is the spillway width). A TLS stepped spillway is more effective for magnification ratios of 1.25, 1.45, 1.65 and 1.85 for energy dissipation compared to stepped spillways by 5.6, 8.5, 12.6 and 17% respectively. TLS stepped spillway has a lower residual head and a larger friction factor. The residual head ratio (Hres/dc) is ~2.89 in a TLS stepped spillway, while it is ~4.32 for a flat stepped spillway. Also, the friction factor varies from 0.79 to 1.33 for different magnification ratios, while this value is approximately 0.66 for the flat stepped spillway.
Keywords: TLS stepped spillway, Trapezoidal Labyrinth Shape, energy dissipation, residual energy, friction factor, FLOW-3D.
1. Introduction: Spillways are constructed to transfer high-energy water. Because the crest and the downstream portion of a spillway have different elevations, the water velocity will increase in the flow direction [1]. Recently, researchers have tried to increase the efficiency of these spillways and have made significant advancement in our understanding. Researchers generally attempt to reduce the velocity downstream by increasing the efficiency of energy dissipation in the stepped spillway. Based on the mentioned studies, the regimes of the flow on stepped spillways are categorized in skimming flow, transition flow and nappe flow (Fig. 1). Fig. 1
Extreme hydrodynamic oscillations and continuous mixing of air and water are of the main characteristics of the skimming flow regime. For this flow, the water on the steps travels as a continuous layer and the steps play the role of a large roughness. A large part of energy dissipation in the skimming flow will be achieved by the formation of vortices in the recirculation region (Fig. 1-a) [2]. Ohtsu and Yasuda (1997) were probably the first who introduced the concept of “transition flow” regime, but they did not elaborate on its properties. For a given geometry of stepped channel chute, a range of flow discharges results in a flow regime between a nappe flow at low discharges and skimming flow at large flow rates (Fig. 1-b)[3]. For the nappe flow regime, the flow drops and hits the lower step as a jet (Fig. 1-c) [2]. Various laboratory investigations have been carried out on stepped spillways by [4-6] Sorensen (1985), Peyaras et al. (1992) and Baylar et al. (2011) and the effects of flow patterns,
step geometry and spillway slope on energy dissipation have been investigated. Sorensen (1985) and Chanson (2001) performed many laboratory tests to diagnose flow and energy dissipation on stepped spillways and provided relationships to determine the type of flow and the amount of energy loss on these spillways [4,1]. Gonzalez and Chanson (2007) empirically investigated the characteristics of flow on a stepped spillway for different chute slopes for embankment dams; they presented some new design charts for stepped spillways of various geometries [7]. Felder and Chanson (2009) studied stepped spillways with a slope of 21.8° and step heights of 5 cm and 10 cm. Comparison of their results with previous studies indicated that the residual head at the end of the spillway was smaller than that of a stepped spillway with a slope of 15.9° [8]. Felder et al. (2012a, b) investigated the flow patterns and the residual head on stepped spillways of different configurations. Their results indicated that the slope of the spillway is one of the most important factors for energy dissipation. The performance of a pooled stepped spillway with a slope of 8.9° for reducing the residual head was better than that of a flat stepped spillway, while a contrary result was obtained for the slope of 26.6°[9-10]. Zhang and Chanson (2018) studied the air–water flow system and the energy dissipated by studying the effects of step shape in a 45° steep chute. The results showed flow patterns, aeration, and energy dissipation changes due to changing in step shape [11]. Parsaie and Haghiabi (2019) investigated the stage-discharge relation, discharge coefficient, and energy dissipation on the circular crested stepped spillways (CCSS). Their results indicated that The CCSS can dissipate the energy of flow between 90 and 40% and compare to their common type (smooth chute) [12].
Computational fluid dynamics (CFD) have long been utilized to simulate the flow in hydraulic structures. Tongratoke et al. (2009) attained reliable and integrated results by simulating the flow on stepped spillways. They used different turbulence models including two-equation models [13]. Nikseresht et al. (2013) simulated two-phase flows and compared the VOF scheme with mixture methods and investigated the energy dissipation in stepped spillways of different step slopes [14].
Morovati et al. (2016) investigated the influence of a vertical obstacle on the edge of steps on energy dissipation. Their results showed that decreasing the number of steps in pooled stepped spillways reduces the velocity and increases the relative energy dissipation [15].
Kaouachi et al. (2019) and Wan et al. (2019) numerically assessment of the inception point Location in stepped spillways. They results indicate that the numerical simulations were in accordance with experimental data, which allowed the analysis of inception point location that fits experimental results. The inception point location of the pooled stepped spillway is closer to the spillway crest than that of the flat stepped spillway [16-17].
Other researchers such as [18-22] performed numerical investigations on stepped spillways. The cost of physical models includes the construction costs of the initial model, laboratory tests and modifications; a numerical simulation is a tool that can be used to reduce laboratory tests in order to save time and money and is applicable to different hydraulic phenomena [23].
In this paper a new kind of step shapes proposes Trapezoidal Labyrinth Shaped steps for stepped spillway (TLS stepped spillway) and investigates the hydraulic performance this type
spillway. In particular, the energy dissipation efficiency of this type of spillway is evaluated using CFD techniques.
2. Model Development 2.1. Theoretical relations of flow regime detection The factors that affect the generation of various types of flow on stepped spillways include the geometry of the steps (length and height of the steps, l and h) and the flow rate Q. However, some researchers have proposed other criteria. The studies of many researchers such as [5, 24, 25, 26] have shown that the type of flow regime depends on the normalized critical depth, dc/h (dc is the critical depth of the flow) and a dimensionless variable of steps, h/l. According to Equations (1) to (3), criteria are presented for which a skimming flow regime occurs: h ≤ 0.9 l
Rajaratnam (1990)
dc ≥ 0.8 h
Peyras et al. (1992)
dc h > 1.01 − 0.37( ) h l
Chanson (1994)
dc h > 1.05 − 0.465( ) h l
0.4 ≤
(1)
(2) 0.2 ≤
h ≤ 1.25 l
(3)
Based on the Equations (1) to (3), the type of flow regime on the stepped spillway which is modeled in the current study is presented in Table 1 and Figure 2. Table 1 Fig. 2
2.2. Dimensional analysis The most important flow variables on the stepped spillways are: •
The acceleration of gravity (g) and the properties of the fluid; i.e. dynamic viscosity (µ) and mass density (ρ)
•
The hydraulic properties of flow including depth (y) and velocity (V)
•
The geometry of the spillway including step height (h), step length (l), number of steps (N0) and the spillways total height (Hdam).
In this study, the geometric shape of the steps is changed into a trapezoidal labyrinth shape; the geometric variables should be defined as follows: Nc: The number of labyrinth shape cycles, LT: Total length of the labyrinth shape ( LT = N c × (4a + 2lc ) ) and Wt: The width of the channel (Figure. 3). According to the notations in Figure. 4, the total head loss is expressed by ∆H = H0 – H where: H is the head on the considered step, H=ycosθ + αV2/2g; H0 is the upstream head of the spillway H0 =∆z+Y+α(V0)2/2g; V is local mean velocity (V=Q/yWt) and V0 is the approach velocity. Fig. 3 Fig. 4
Considering Wt>>y, the head loss should primarily depend on the number of steps (N0), the geometry of the steps (h, l) and the discharge (Q). The last one may be expressed by the critical depth dc=[(Q/Wt)2/g] 1/3 (Christodoulou, 1993). Therefore, in the TLS stepped spillway, the function through which the relation of variables can be expressed is: f (Hdam, N0, Nc, l, h, V, y, LT, Wt, g, ρ, µ)=0.
(4)
Using Buckingham’s π theory (the variables ρ, V and y are repeating variables) one can
obtain: d d L ∆H = f ( c , c , T , Re , Fr , N 0 , N c ) H0 h l Wt
(5)
As in this study, the number of steps and the number of labyrinth shape cycles are held constant; it is assumed that the force of fluid viscosity is insignificant compared to the inertia force for open channel flow [27]; so three dimensionless variables of Reynolds number Re, N0 and Nc could be eliminated.
2.3. The FLOW-3D model Computational Fluid Dynamics (CFD) is increasingly used to study a wide variety of complex Environmental Fluid Mechanics (EFM) processes; such as water flow and turbulent mixing of contaminants in rivers [28]. FLOW-3D® is a general purpose computational fluid dynamics (CFD) program for modeling a wide variety of fluid flow and heat transfer phenomena [29] which basically solves the Navier-Stokes equations [30-31]. The following equations describe the unsteady continuity and momentum equations in the Cartesian coordinates: ∂U i =0 ∂xi
ρ
(6)
∂U i ∂U i ∂U i ∂P ∂ + ρU j =− + (µ − ρ ui′u′j ) + ρ gi ∂ti ∂xi ∂xi ∂x j ∂x j
(7)
In which Ui and u i′ are average velocity and fluctuating velocities in the xi direction respectively, xi = (x,y,z), Ui = (U,V,W), and ui’ =(u’, v’, w’). The symbols ρ, µ, P, and gi are density,
dynamic
viscosity,
pressure,
and
gravitational
acceleration,
respectively.
Instantaneous velocity is defined as ui = U i + ui′ for the three directions. A turbulence model is necessary for accounting for the nonlinear Reynolds stresses. Despite there are more advance k-epsilon turbulence modelling studies such as Pu et al. (2014) and
Pu (2015) [32-33], For this purpose and in this study, the k-ε (RNG)1 turbulence model is used to simulate the turbulent flow characteristics. According to the previous researches in which flow on hydraulic structures such as stepped spillway, in spillway’s flip bucket and cte is simulated [34-38], the k-ε (RNG) model was utilized to model the domain. The RNG equations are presented by Yakhot et al. (1992) [39]. The FLOW-3D® software uses the Volume of Fluid (VOF) scheme for free surface modeling. This method is described as follows: ∂F 1 ∂ ∂ ∂ + ( FuA x ) + ( FvA y ) + ( FwA z ) = 0 ∂t V f ∂x ∂y ∂z
(8)
Here, F is the fraction function; F=0 when a cell is empty and F=1 when the cell is full [40].
3. Methods and materials 3.1. Geometric model In this study, results of the numerical solution are validated with experimental data of Felder et al. (2012a, b) [9-10]. They used a stepped spillway with ten steps of 52 cm length; the steps dimensions were 20 ×10 cm. Fig. 5 shows the schematic design of that stepped spillway. In this study, the geometry of the stepped spillway was adjusted to improve its hydraulic performance (energy dissipation rate). The changes to step geometry were obtained by employing a Trapezoidal Labyrinth Shape for each step. The models are classified into four configurations for TLS stepped spillway with four different magnification length ratios (LT/Wt) equal to 1.2, 1.4, 1.6 and 1.8; and one model with a conventional shape which was tested by Felder et al. (2012a, b) and is used for validation [9-10]. Fig. 6 shows the three-dimensional design and geometry of the TLS stepped spillway. 1
Re-Normalization Group
Fig. 5 Fig. 6
3.2. Computational grid and boundary conditions Four different meshes sizes were used in the vicinity of the weir. As listed in Table 2, comparison was made of the ratio of V/Vc at y/dc=0.645 obtained from numerical solution on step No. 9 with experimental results. Based on this mesh-refinement study, a computational mesh with 2,204,893 elements was selected for further calculations, with the selected appropriate mesh results in a relative error and maximum aspect ratio of 6.07%and 1.29, respectively. Table 2
The larger element size ds were set to be 1.8 cm with an element count of 868,271. The smaller element size was 0.8 cm and resulted in 1,336,622 elements. Finally, the flow domain was discretized by cubic elements with a total of 2,204,893 elements and the number of meshes was considered the same in all models (Figure. 7). According to the experimental condition all boundary conditions has been employed. In experimental research, predetermined discharge (Q) had been used at the beginning of spillway channel that specific discharge was used in the present paper subsequently. Furthermore the experimental results and flow condition hadjust been presented above steps and we couldn’t choose other boundary condition including specific pressure or specific velocity for output. Hence outflow boundary condition was used at the output to prevent the downstream boundary effects onlast steps results. Also a no-slip wall boundary condition was also used for thewalls and channel bed. The symmetry boundary condition was used for
the upper boundary which was located on top of the air phase. Table 3 lists the boundary conditions for the model. Table 3 Fig. 7
4. Results and discussion 4.1. Verification of the numerical model using laboratory results After the numerical simulation of steady-state flow on the stepped spillway, the results of the flat stepped spillway and interfacial velocity distributions on steps 8, 9 and 10 were validated with experimental data of Felder et al. (2012a, b) [9-10]. The error limits between the data obtained from FLOW-3D® and the experimental data were calculated according to Eqaution (9): E=
((V /V c )N − (V /V c )E ) ×100 (V /V c )N
(9)
In which E is the relative error, (V/Vc)N is the dimensionless velocity distribution from the numerical solution and (V/Vc)E is a dimensionless form of the measured interfacial velocity distributions in the laboratory. Figure 8 displays the interfacial velocity distributions and relative error for the numerical data. Regarding the relative error, it can be seen that the maximum difference is 8.55% and the maximum level of consistency between the numerical and experimental model is in the middle area of the depth up to the flow surface. One can conclude that there is good agreement between the numerical and experimental results. Fig. 8
4.2. Water surface profiles Water surface profiles on stepped spillway for Q=0.105 m3/s are presented for the skimming flow regime (Figure 9). Skimming flow is observed when there is a coherent stream between step edges and the development of recirculation vortices. It also creates a semi labyrinth water profile because of the interaction of the flow direction and the flow interference (Fig. 10). Therefore, with the utilization of this kind of spillway more energy is expected to. Fig. 9 Fig 10
4.3. Residual energy and turbulent energy dissipation Transferring the water downstream using spillways will produce a large amount of kinetic energy. In the case of stepped spillways, one of the important concerns of engineers is to know the amount of energy dissipation and the amount of residual energy downstream. In the present study, the energy dissipation ratio, ∆H/Hmax, and residual energy, Hres, are calculated at the edge of the final step of the TLS spillway models. The energy dissipation ratio,
∆H/Hmax, expresses the percentage of total energy loss along the stepped spillway relative to the upstream total head, Hmax. The total head can be evaluated as: H max =
3 × d c + H dam 2
(10)
In which Hdam is the height of the dam and dc is the critical flow depth. The total head loss ∆H can be estimated as ∆Η = Hmax - Hres in which the residual head, Hres, is H res = d × cos 2 θ +
U w2 2× g
(11)
The term Uw=q/y represents the mean flow velocity along the central axis of the spillway. The symbols θ, d, and g are the spillway slope, the equivalent clear water flow depth and the
gravitational acceleration, respectively. Figure 11 shows a comparison of flat and TLS stepped spillway energy dissipation under different flow conditions. It is evident that the trapezoidal labyrinth shape of the steps has dissipated more energy than the flat shape. As the flow rate increases, the rate of energy dissipation is reduced for all models. The rate of energy dissipation on the last step for four different magnification ratios (LT/Wt) of 1.2, 1.4, 1.6 and 1.8, is illustrated in Figure 12 as a function of the dimensionless variable of ∆z0/dc. The variable ∆z0 defines the vertical distance between the spillways crest and the last step (Figure 5). Comparing the performance of the flat stepped spillway with TLS shows that the energy loss in the TLS stepped spillway is greater. Also, increasing the magnification ratio (LT/Wt) results in increases to the energy dissipation. For similar flow conditions, the TLS stepped spillway is more effective at energy dissipation than a flat stepped spillway. For magnification ratios (LT/Wt) of 1.25, 1.45, 1.65 and 1.85, the percentages of efficiency improvements are 5.6%, 8.5%, 12.6%, and 17%, respectively. The flow interference on the TLS stepped spillway and the step shapes cause this efficiency improvement. Fig. 11 Fig. 12
It can be clearly seen in Figure 13 that changing the step shape to labyrinth will increase the streamlines interference. As a result, the energy dissipation will increase, especially for the downstream steps compared to the flat stepped spillway. It should be noted that the contour color scaling for the images in Figure 13 are not the same. Fig. 13
In Figure 14, the dimensionless residual head (Hres/dc) is depicted as a function of dimensionless discharge (dc/h). On the flat stepped spillway, the dimensionless residual head is the largest (~4.32) and for the same flow conditions (dimensionless discharge = 0.84); the average dimensionless residual head on the TLS stepped spillway is ~2.89. As the magnification ratio increases, the residual head decreases. Fig. 14
4.4. Flow resistance The flow resistance is primarily a form of drag in skimming flows [41] and is commonly expressed in the form of Darcy-Weisbach friction factor fe [22, 39]. The Darcy–Weisbach friction factor is estimated from the measured air-water flow properties and is calculated as Equation (12): fe =
8× g × S f × d 8 ×τ 0 = 2 ρ w ×U w U w2
(12)
Here, τ0 is the average shear stress, Sf is the friction slope and is equal to − ∂H ∂x (where H is the total head and x is the distance in the flow direction), d is the equivalent clear water flow depth and Uw = q/y is the flow mean velocity [42-43]. The friction factor results are illustrated in Figure 15 as a function of the dimensionless step cavity roughness height (hcosθ)/DH in which DH is the hydraulic diameter.
It is defined by DH =4By/(B+2y) for a
rectangular channel). It is observed that the friction factor (fe) in the flat stepped spillway is lower compared to the TLS-stepped spillway. This value is 0.66 in the flat stepped spillway and 0.79, 0.90, 0.91, and 1.33 in the TLS-stepped spillway for magnification ratios of 1.25, 1.45, 1.65 and 1.85, respectively. Also, it has been shown that the largest friction factor for the flat stepped spillway is obtained for the highest discharges.
Also, in the TLS stepped spillway, the highest friction factor is achieved for a low discharge and in a magnification ratio of 1.85. In other words, for low discharge, as the ratio of LT/Wt increases the friction factor for increases. The reason is from flow interference through the edges of the trapezoidal labyrinth part of the steps and the extension of the recirculation region (Figure 16). According to Chanson et al. (2002), flow resistance caused by recirculating rotating vortices and the energy dissipation changes by the motion exchange between the cavity flow and the main flow [40]. Also, in TLS stepped spillways, the interference of streamlines increases the resistance against the flow and increases the energy dissipation. Fig. 15 Fig. 16
5. Summary and conclusions Stepped spillways are designed to improve the hydraulic conditions of flow and the energy dissipation. Construction of these types of spillways will reduce the dimensions and cost of the stilling basin. In this research, a new type of stepped spillway; called a Trapezoidal Labyrinth Shaped stepped spillway (TLS stepped spillway) is proposed and its hydraulic performance is evaluated using FLOW-3D®. The summary and results of this research are as follows: • The highest error between the experimental and numerical results occurs at the 8th step by
8.55%. There is good agreement between the experimental and numerical results. • In TLS stepped spillways, the labyrinth shape of the steps causes interference of the
streamlines and improves the energy dissipation of the flow.
• Labyrinth-shaped steps increase the energy dissipation compared to flat steps under the same
flow conditions and as the magnification ratio (LT/Wt) increases, more energy dissipation occurs. For a TLS stepped spillway with a magnification ratio of 1.25, 1.45, 1.65 or 1.85, the energy dissipates more than that of the flat stepped spillway by 5.6%, 8.5%, 12.6%, or 17%, respectively. • Residual head calculations show that the flat stepped spillway under the same flow conditions
has a dimensionless residual head of ~4.32 and the TLS stepped spillway has an average residual head of ~2.89. Also increasing the magnification ratio results in decreasing the residual head. • It is observed that the friction factor (fe) for the flat stepped spillway is lower than that of the
TLS stepped spillway. This value equals 0.79, 0.90, 0.91, and 1.33 in the TLS stepped spillway for magnification ratios of 1.25, 1.45, 1.65 and 1.85, respectively; while it is 0.66 for the flat stepped spillway. The friction factors in TLS stepped spillways increase because of the flow interference through the edges of the labyrinth shapes and the recirculation region in this new type of stepped spillways.
Nomenclature q(Q/B)
(l/s)/m
Water discharge per unit width
B
cm
Width of the flume
h
cm
The step’s height
l
cm
The length of step
Ld
cm
The length of the stilling basin
Hs
cm
Height of stepped spillway
α
-
Chute angle
hc
cm
The critical depth
ht
cm
The tail water depth
ρs
kg/m3
Mass density of sediments
ρw
kg/m3
Mass density of the water
µ
(N-s)/m2
Re
-
The Reynolds number
g
m/s2
Acceleration of gravity
S
-
d50
mm
Bed material diameter of which is finer by weight
d10
mm
Grain size for which 10% of material is finer
d30
mm
Grain size for which 30% of material is finer
d60
mm
Grain size for which 60% of material is finer
Cu
-
Coefficient of uniformity
Cc
-
Coefficient of curvature
Frd
-
Particle Froude number
σg
-
Geometric standard deviation
Dynamic viscosity of water
The relative density of sediments
References [1] Chanson, H. 2001. Hydraulic design of stepped spillway and downstream energy dissipaters. Dam Engineering, Vol. 11, No. 4, pp. 205-242. [2] Chanson, H. and Toombes, L. 1997. Flow aeration at stepped cascades. Research Report No. CE155. Dept. of Civil Engineering, University of Queensland, Australia, p. 110.
[3] Ohtsu, I., and Yasuda, Y. 1997. Characteristics of flow conditions on stepped channels. Proceedings of the 27th IAHR Biennial Congress, Theme D, San Francisco, Calif., 10–15 August 1997. Edited by F.M. Holly Jr. and A. Alsaffar. ASCE, New York, USA. pp. 583–588. [4] Sorensen, R.
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Hydraulic Engineering, Vol. 111, No. 12, pp. 1461-1472. [5] Peyras, L., Royet, P., and Degoutte, G. 1992. Flow and Energy Dissipation over Stepped Gabion Weirs. Journal of Hydraulic Engineering, Vol. 118, No.5, pp. 707-717. [6] Baylar, A., Unsal, M. and Ozkan, F. 2011. The effect of flow patterns and energy dissipation over stepped chutes on aeration efficiency. KSCE Journal of Civil Engineering, Vol. 15, No. 8, pp. 1329-1334, DOI: 10.1007/s12205-011-1360-0 [8] Felder, S., and Chanson, H. 2009. Energy dissipation, flow resistance and gas-liquid interfacial area in skimming flows on moderate slope stepped spillways. Environmental fluid mechanics, Vol. 9, No. 4, pp. 427-441, DOI:
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[9] Felder, S., Guenther, Ph., Chanson, H., 2012a. Air-water flow properties and energy dissipation on stepped spillways: A physical study of several pooled stepped configurations. School of Civil Engineering, the University of Queensland, Brisbane QLD4072, Australia. Report CH87/12. [10] Felder, S., Guenther, Ph. Chanson, H. 2012b. Air entrainment and energy dissipation on an 8◦/9◦ slope stepped spillway with flat and pooled steps. School of Civil Engineering, the University of Queensland Brisbane QLD 4072. Australia. Report CH86/12. [11] Zhang, G. and H. Chanson. 2018. Effects of step and cavity shapes on aeration and energy dissipation performances of stepped chutes, J. Hydraul. Eng. Vol. 144, No. 9, 04018060.
[12] Parsaie, A. and Haghiabi, A. H. 2019. The hydraulic investigation of circular crested stepped spillway. Flow Measurement and Instrumentation, Vol. 69, 101624. [13] Tongkratoke, A., Chinnarasri, C., Pornprommin, A., Dechaumphai, P. and Juntasaro, V. 2009. Non-linear turbulence models for multiphase recirculating free-surface flow over stepped spillways. International Journal of Computational Fluid Dynamics, Vol. 23, No. 5, pp. 401-409. [14] Nikseresht, A. H., N. Talebbeydokhti, and M. J. Rezaei. 2013. Numerical simulation of two-phase flow on step-pool spillways. Scientia Iranica, Vol. 20, No. 2, pp. 222-230. [15] Morovati, K., Eghbalzadeh, A., and Soori, S. 2016. Study of Energy Dissipation of Pooled Stepped Spillways. Civil Engineering Journal, Vol. 2, No. 5, pp. 208-220. [16] Kaouachi, A., Carvalho, R. F. Benmamar, S. and Gafsi, M. 2019. Numerical assessment of the inception point in different stepped spillway configurations. Arabian Journal of Geosciences, Vol. 12, No. 8, p. 564. [17] Wan, W., Raza, A., and Chen, X. 2019. Effect of Height and Geometry of Stepped Spillway on Inception Point Location. Applied Sciences, Vol. 9, No. 10 p. 2091. [18] Tabbara, M., Chatila, J., and Awwad, R. 2005. Computational simulation of flow over stepped spillways. Computers and structures, Vol. 83, No. 27, pp. 2215-2224. [19] Dong, Z. Y. 2006. Numerical simulation of skimming flow over mild stepped channel. Journal of Hydrodynamics, Ser. B, Vol. 18, No. 3, pp. 367-371. [20] Daneshfaraz, R. Joudi, A. R. Ghahramanzadeh, A. and Ghaderi, A. 2016. Investigation of flow pressure distribution over a stepped spillway. Advances and Applications in Fluid Mechanics. Vol. 19, No. 4, pp.811–822.
[21] Wan, H., Li, R., Gualtieri, C., Yang, H., and Feng, J. 2017. Numerical Simulation of Hydrodynamics and Reaeration over a Stepped Spillway by the SPH Method. Water, 9(8), 565. [22] Bayon, A., Toro, J. P., Bombardelli, F. A., Matos, J., and López-Jiménez, P. A. 2018. Influence of VOF technique, turbulence model and discretization scheme on the numerical simulation of the non-aerated, skimming flow in stepped spillways. Journal of Hydro-environment Research, 19, 137-149. [23] Choufu, L., Abbasi, S., Pourshahbaz, H., Taghvaei, P. and Tfwala, S., 2019. Investigation of Flow, Erosion, and Sedimentation Pattern around Varied Groynes under Different Hydraulic and Geometric Conditions: A Numerical Study. Water, 11(2), p.235. [24] Rajaratnam, N. 1990. Skimming flow in stepped spillway. Journal of Hydraulic Engineering, Vol. 116, No. 4, pp. 487-591. [25] Chanson, H., 1994. State of the art of the hydraulic design of stepped chute spillways. Hydropower and Dams Journal, pp. 33-42. [26] Boes, R. M., Minor, H. E. 2000. Guidelines for the Hydraulic Design of Stepped Spillways, Hydraulics of stepped spillways. pp. 163-170. [27] Chow, V. T. 1959. Open Channel Hydraulics. McGraw-Hill, New York, USA. Christodoulou, G. C. 1993. Energy dissipation on stepped spillways. Journal of Hydraulic Engineering, Vol. 119, No. 5, pp. 644-650. [28] Blocken, B., and Gualtieri, C. 2012. Ten iterative steps for model development and evaluation
applied
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Fluid
[29] Flow Science, 2008. Incorporated: FLOW-3D users manual. Version 9.3, Santa Fe, New Mexico. [30] Zahabi, H., Torabi, M., Alamatian, E., Bahiraei, M., and Goodarzi, M. 2018. Effects of Geometry
and
Hydraulic
Characteristics
of
Shallow
Reservoirs
on
Sediment
Entrapment. Water, 10(12), 1725. [31] Ghaderi, A., and Abbasi, S. 2019. CFD simulation of local scouring around airfoil-shaped bridge piers with and without collar. Sādhanā, 44(10), 216. [32] Pu, J.H., Shao, S., and Huang, Y. 2014. Numerical and experimental turbulence studies on shallow open channel flows. Journal of Hydro-Environment Research, 8(1), 9-19. [33] Pu, J.H. 2015. Turbulence Modelling of Shallow Water Flows using Kolmogorov Approach, Computers and Fluids, 115, 66-74. [34] Shahheydari, H., Nodoshan, E. J., Barati, R., and Moghadam, M. A. 2015. Discharge coefficient and energy dissipation over stepped spillway under skimming flow regime. KSCE Journal of Civil Engineering, Vol. 19, No. 4, pp. 1174-1182. [35] Tabari, M. M. R., and Tavakoli, S. 2016. Effects of stepped spillway geometry on flow pattern and energy dissipation. Arabian Journal for Science and Engineering, Vol. 41, No. 4, pp. 1215-1224. [36] Daneshfaraz, R., Ghahramanzadeh, A., Ghaderi, A., Joudi, AR., and Abraham, J. 2016. Investigation of the Effect of Edge Shape on Characteristics of Flow under Vertical Gates. Journal‐American Water Works Association, 108(8): 425-432. [37] Daneshfaraz, R., and Ghaderi, A. 2017. Numerical Investigation of Inverse Curvature Ogee Spillway. Civil Engineering Journal, 3(11): 1146-1156.
[38] Daneshfaraz, R., Minaei, O., Abraham, J., Dadashi, S., and Ghaderi A. 2019. 3-D Numerical simulation of water flow over a broad-crested weir with openings. ISH Journal of Hydraulic Engineering, 1-9. [39] Yakhot, V., Orszag, S. A., Thangam, S., Gatski, T. B., and Speziale, C. G. 1992. Development of turbulence models for shear flows by a double expansion technique. Physics of Fluids A: Fluid Dynamics, Vol. 4, No. 7, pp. 1510-1520. [40] Ghaderi, A., Dasineh, M., Abbasi, S., and Abraham, J. 2020. Investigation of trapezoidal sharp-crested side weir discharge coefficients under subcritical flow regimes using CFD. Applied Water Science, Vol. 10, No. 1, p. 31. [41] Carosi, G., and Chanson, H. 2008. Turbulence characteristics in skimming flows on stepped spillways. Canadian Journal of Civil Engineering, Vol. 35, No. 9, pp. 865-880. [42] Chanson, H. 2000. Forum article, Hydraulics of Stepped Spillways: Current Status. Journal of hydraulic engineering, Vol. 126, No. 9, pp. 636-637. [43] Chanson, H., Yasuda, Y., and Ohtsu, I. 2002. Flow resistance in skimming flows in stepped spillways and its modeling. Canadian Journal of Civil Engineering, Vol. 29, No. 6, pp. 809-819.
Table 1. Hydraulic characteristics and the type of flow regime on the stepped spillway for the present study dc=(q2/g)1/3
Step height
Step length
(m)
(m)
(m)
0.04
0.08
0.1
0.065
0.12
0.08
Q(m3/s)
h/l
dc/h
Flow Regime
0.2
0.5
0.84
Skimming flow
0.1
0.2
0.5
1.17
Skimming flow
0.13
0.1
0.2
0.5
1.34
Skimming flow
0.095
0.15
0.1
0.2
0.5
1.50
Skimming flow
0.105
0.16
0.1
0.2
0.5
1.61
Skimming flow
0.113
0.17
0.1
0.2
0.5
1.69
Skimming flow
Table 2. Mesh Sensitivity analysis MAPE* (%) Int.
Ext.
Total
Max
V/Vc, y/dc=
V/Vc, y/dc= 0.645 –
mesh
mesh
number of
Aspect
0.645- step 9 in
step 9 in
(cm)
(cm)
cells
ratio
FLOW-3D
Experimental result
N1
2.5
1.5
1046729
1.79
2.132
2.938
27.43
N2
2.2
1.1
1648157
1.52
2.429
2.938
17.32
N3
1.8
0.8
2204893
1.29
2.741
2.938
6.07
N4
1.6
0.65
2876956
1.14
2.780
2.938
5.37
Test NO.
Finer meshes
100 ×
1 n X exp − X num ∑ X n 1 exp
The solution needs higher computational storage.
*Mean Absolute Percentage Error (MAPE), Xexp: Experimental value of X, Xnum: Numerical value of X, n: Count of data
Table 3. Applied boundary conditions Software FLOW-3D
Upstream
Downstream
Free surface
Floor
Lateral
boundary
boundary
boundary
boundary
boundaries
Volume flow rate
Outflow
Symmetry
Wall
Wall
Figure 1. Flow conditions on stepped channels: (a) skimming flow; (b) transition flow; (c) nappe flow (Ohtsu et al., 1997)
Figure 2. Flow regime conditions in the skimming flow region
Figure 3. Geometrical characteristics of the TLS stepped spillway
Figure 4. Side view with dimensional annotation
Figure 5. Schematic design of the stepped spillway used by Felder et al. (2012a, b)
Figure 6. Three-dimensional design of the Trapezoidal Labyrinth Shaped (TLS) stepped spillway
Figure 7. Meshed model in FLOW-3D
Figure 8. Dimensionless interfacial velocity distributions on steps 8, 9 and 10 of the flat stepped spillways; numerical simulation vs. experimental data (Q = 0.113 m3/s)
Figure 9. Skimming flow water surface profile on the stepped spillway falls (Q=0.105 m3/s)
Figure 10. Flow on the flat stepped spillway and its interference on transverse sides of the TLS stepped spillway
Figure 11. Percentage of energy dissipation on flat and TLS stepped spillways versus the dimensionless discharge dc/h
Figure 12. The rate of energy dissipation on flat and TLS stepped spillways versus the dimensionless elevation change
Figure 13. Comparison of energy dissipation on TLS spillway and flat spillways
Figure 14. Dimensionless residual head on flat and TLS stepped spillways versus dimensionless discharge
Figure 15. Darcy friction factors on the flat and the TLS stepped spillways
Figure 16. Recirculation regions in a skimming flow through a TLS stepped spillway
Stepped spillways are designed to improve the hydraulic conditions of flow and the energy dissipation. Construction of these types of spillways will reduce the dimensions and cost of the stilling basin. In this research, a new type of stepped spillway; called a Trapezoidal Labyrinth Shaped stepped spillway (TLS stepped spillway) is proposed and its hydraulic performance is evaluated using FLOW-3D®. The purpose of this research is to propose and investigate a new type of stepped spillways to achieve the highest dissipation level of energy of the flow conducted to the downstream utilizing FLOW3D model (CFD models). One can find and utilize TLS stepped spillways as next generation of stepped spillways which increase the efficiency and improve the hydraulic performance. The CFD models can be useful for analyzing hydrodynamics of hydraulic structures that some of their results were validated by experimental or real observations. The summary and results of this research are as follows: •
The highest error between the experimental and numerical results occurs at the 8th step by 8.55%. There is good agreement between the experimental and numerical results.
•
In TLS stepped spillways, the labyrinth shape of the steps causes interference of the streamlines and improves the energy dissipation of the flow.
•
Labyrinth-shaped steps increase the energy dissipation compared to flat steps under the same flow conditions and as the magnification ratio (LT/Wt) increases, more energy dissipation occurs. For a TLS stepped spillway with a magnification ratio of 1.25, 1.45, 1.65 or 1.85, the energy dissipates more than that of the flat stepped spillway by 5.6%, 8.5%, 12.6%, or 17%, respectively.
•
Residual head calculations show that the flat stepped spillway under the same flow conditions has a dimensionless residual head of ~4.32 and the TLS stepped spillway has an average residual head of ~2.89. Also increasing the magnification ratio results in decreasing the residual head.
•
It is observed that the friction factor (fe) for the flat stepped spillway is lower than that of the TLS stepped spillway. This value equals 0.79, 0.90, 0.91, and 1.33 in the TLS stepped spillway for magnification ratios of 1.25, 1.45, 1.65 and 1.85, respectively; while it is 0.66 for the flat stepped spillway. The friction factors in TLS stepped spillways increase because of the flow interference through the edges of the labyrinth shapes and the recirculation region in this new type of stepped spillways.
Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: