water Article
SPHysics Simulation of Experimental Spillway Hydraulics Shenglong Gu 1,2,3 , Liqun Ren 2 , Xing Wang 2 , Hongwei Xie 1,2 , Yuefei Huang 1,3 , Jiahua Wei 1,2,3, * and Songdong Shao 4,5 1 2 3 4 5
*
State Key Laboratory of Plateau Ecology and Agriculture, Qinghai University, Xining 810016, China;
[email protected] (S.G.);
[email protected] (H.X.);
[email protected] (Y.H.) School of Water Resources and Electric Power, Qinghai University, Xinning 810016, China;
[email protected] (L.R.);
[email protected] (X.W.) State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing 100084, China Department of Civil and Structural Engineering, University of Sheffield, Sheffield S1 3JD, UK;
[email protected] State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University, Chengdu 610065, China Correspondence:
[email protected]; Tel.: +86-(0)10-6279-6325
Received: 28 October 2017; Accepted: 11 December 2017; Published: 14 December 2017
Abstract: In this paper, we use the parallel open source code parallelSPHysics based on the weakly compressible Smoothed Particle Hydrodynamics (WCSPH) approach to study a spillway flow over stepped stairs. SPH is a robust mesh-free particle modelling technique and has great potential in treating the free surfaces in spillway hydraulics. A laboratory experiment is carried out for the different flow discharges and spillway step geometries. The physical model is constructed from a prototype reservoir dam in the practical field. During the experiment, flow discharge over the weir crest, free surface, velocity and pressure profiles along the spillway are measured. In the present SPH study, a straightforward push-paddle model is used to generate the steady inflow discharge in front of the weir. The parallelSPHysics model is first validated by a documented benchmark case of skimming flow over a stepped spillway. Subsequently, it is used to reproduce a laboratory experiment based on a prototype hydraulic dam project located in Qinghai Province, China. The detailed comparisons are made on the pressure profiles on the steps between the SPH results and experimental data. The energy dissipation features of the flows under different flow conditions are also discussed. It is shown that the pressure on the horizontal face of the steps demonstrates an S-shape, while on the vertical face it is negative on the upper part and positive on the lower part. The energy dissipation efficiency of the spillway could reach nearly 80%. Keywords: Dahua hydraulic dam; parallelSPHysics; push-paddle inflow; skimming flow; spillway hydraulics; step geometry
1. Introduction The spillway is designed to prevent the overtopping of a dam and provide a diversion channel for the discharging flows. Spillway flow is a benchmark issue in hydraulic dam engineering and has gained considerable attention over the decades. Previous studies heavily focused on the laboratory experiments, field observations and mostly were based on the semi-empirical relationships which constitute the basis of spillway hydraulics. Spillway flows are usually related to the energy dissipations since quite a few hydraulic processes that occur over the spillway, such as the friction along the chute, consume a significant part of the flow energy through flow interactions and turbulences [1].
Water 2017, 9, 973; doi:10.3390/w9120973
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In the last two decades, significant progresses have been made in the field of experimental spillway hydraulics. For example, Boes and Hager [2] carried out a benchmark experimental study in a large flume using the fiber-optical instruments for the onset of skimming flows and they discussed the energy dissipation features of the stepped spillways and the design of training walls. The study aimed to develop a practical design principle that therefore lessens the needs of each individual physical model test. Chanson and Toombes [3] investigated the turbulent features of a spillway flow in a large-size experimental facility, where the flow was characterized by very energetic turbulence and free-surface aeration. State-of-the-art velocity and turbulence intensity measurements were taken in the study. They also explored the scales of the interfacial aeration and entrained bubble with relation to the entrained particles. In a more recent work, Bung [4] studied the air-water surface roughness in a self-aerated stepped chute flow and compared with the smooth invert model by using a high-speed camera and ultrasonic sensor. It was found that the surface wave height was lower on the smooth invert chute than on the stepped one, where with a decrease in the step height the entrapped air became more relevant. The Smoothed Particle Hydrodynamics (SPH) method is a mesh-free numerical modelling approach exclusively using the free-moving particles. Therefore, it is equipped with unique advantages of tracking various free surfaces with large deformation and it provides a very promising technique to study the spillway hydraulics. Since its early application in free surface flows [5], SPH has been widely used in the coastal hydraulics [6,7], fluid-structure interactions [8–10], multi-phase flows [11,12], dam-break flows [13–15], porous flows [16–18], and river ice dynamics [19–21]. It was found that the SPH method meets with some limitations for the flow with high Reynolds numbers in dam break application but there is no information available on the threshold Re value. By using an appropriate turbulence model, it is expected that SPH should be able to provide reasonably accurate results in such a field. In comparison, the SPH applications in hydraulic engineering, such as spillway flow, are generally under-reported and the reasons could be attributed to the unaffordable computational expenses arising from the domain size and simulation time. Besides, it is much more challenging to develop a two-phase SPH model that could accurately address the water-air interactions commonly found in the spillway flow. Different from the coastal wave breaking or dam-break in which the flow behaviour is transient, the hydraulic simulations involving the open channel or weir flows require gradual establishment of the stable flow conditions, which adds additional challenges to the particle nature of the SPH. The pioneering study of mesh-free numerical modelling on the hydraulics could be attributed to Gotoh et al. [22], who used the Moving Particle Semi-implicit (MPS) method to compute a turbulent jet. Although the model was not quantitatively validated by the experimental data, the proposed sub-particle scale (SPS) turbulence model and the soluble inflow wall boundary laid a milestone foundation for the numerous follow-on works in SPH turbulence [23,24] and SPH open channel flows [25,26]. Another category of SPH models commonly used in the flooding hydraulics is based on the solution of Shallow Water Equations (SWEs), i.e., SWE-SPH, which solves the nonlinear SWEs by using the SPH interpolation principles and benchmark works in this field were documented by [27,28]. The SPH simulations in spillway hydraulics, although being under-reported as compared with its popularity in other application fields, are gaining increasing attention due to their robustness in providing the accurate free surface variations and associated velocity and pressure information inside the complex flows. The modelling results could provide invaluable flow and spillway parameters to the engineering design practice. For example, Chem and Syamsuri [29] studied a hydraulic jump over the complex beds by using SPH and they also discussed the energy dissipations between the corrugated and smooth bed. Jonsson et al. [30] used the SPH to study hydraulic jumps in a rectangular channel for a wide range of Froude numbers with an improved periodical open boundary, in which they found that the jump toes oscillated with a frequency in good agreement with the experimental findings and the oscillation amplitudes increased with the Froude number. Husain et al. [31] used the 2D open source code SPHysics to investigate the pressure variations in the non-aerated flow
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region of a stepped spillway under skimming flow condition. Their simulations clearly disclosed the positive and negative pressure distributions on different parts of the vertical step face. In contrast, the present work quantifies the accuracy of pressure calculations by using the measured laboratory data and also the practical energy dissipation issues are explored. For more practical SPH applications, Lee et al. [32] used a parallel WCSPH code for 3D spillway flows for a site dam of Goulours Valley in Ariége, Midi-Pyrénées (France). Although the validations were only carried out at a qualitative level and the spillway surface was assumed to be smooth, the initiatives showed that SPH can handle the real-life scenarios. Their SPH simulations used about 1 M particles for a 16-s physical time and this cost 5 days of CPU time with 1024 processors on BlueGene/L. Saunders et al. [33] also applied the 3D SPH to a physical model experiment constructed from the Pala Tiloth dam, which was designed for flood control and reservoir management. For the highest particle resolution used, the simulations took 28 days to reach the steady state when running in parallel using a dual Xeon 8-core E5-2650 machine (PassMark Software, Surry Hills, Australia), while 17 h was needed for the three-time lower particle resolution. In their study, the water surface variations within the reservoir area close to the spillway were compared with both the experimental data and SPH results and the spillway surface was assumed to be smooth without any step configurations. In this paper, we use the 2D parallelSPHysics code (http://www.sphysics.org) for a laboratory spillway experiment carried out at Qinghai University, China. The physical model was built from a 1:40 ratio of the prototype engineering project, i.e., Dahua Hydraulic Dam located on Lara River of Qinghai Province. The reservoir has a catchment area of 83 km2 and storage capacity of 426 million m3 . It mainly serves the domestic water supply and agricultural irrigation for the downstream regions and the total project cost nearly 0.2 billion CNY. We plan to use the parallelSPH model to investigate the water surface, and velocity and pressure variations over the stepped spillway aiming to achieve the following purposes: (i) to show how SPH could be used as a practical engineering tool in real industry; (ii) to provide energy dissipation indicators for the different spillway geometries. One distinct feature of the present work is the use of mesh-free numerical model in the spillway hydraulics. This study field has been extensively explored by the pioneering researchers such as Chanson [34,35] due to its practical engineering interest. The paper is structured as follows. Section 2 presents a brief review of the SPH fundamentals and parallelSPHysics [36]. Section 3 presents the engineering background of Dahua Hydraulic Dam and the designs of spillway experiment. Section 4 is dedicated to the validation of the numerical model by using a documented benchmark spillway skimming flow. Section 5 is the SPH modelling of our laboratory experiment with detailed discussions on the simulation results. Finally, Section 6 concludes the whole work. 2. paralleSPHysics 2.1. Governing Equations and SPH Formulations The parallelSPHysics [36] is based on the WCSPH numerical scheme, which solves the following mass and momentum equations of a compressible flow Dρ + ρ∇ · u = 0 Dt
(1)
⇒ Du 1 1 = − ∇ P + g + υ0 ∇ 2 u + ∇ · τ Dt ρ ρ
(2)
where ρ is the density; t is the time; u is the velocity vector; P is the pressure; g is the ⇒ gravitational acceleration vector; υ0 is the kinematic viscosity; and τ is the turbulent shear stress tensor, which is modelled through an eddy-viscosity based Sub-Particle Scale (SPS) model [22] in parallelSPHysics. For more detailed SPS implementation in the code, the reader can refer to the manual of parallelSPHysics (http://www.sphysics.org). Equations (1) and (2) are presented in the form of single-phase formulation, and therefore the model cannot account for the effect of air entrainment.
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After the velocity is found, the fluid particles are translated by the following equation of motion Dr =u Dt
(3)
where r is the position vector. In parallelSPHysics, several time integration schemes are optioned. We use the predictor-corrector solution scheme since it is fully explicit in time and second-order accurate. Density filtering operations are also used to reduce the particle noises. To close the mass and momentum Equations (1) and (2) for a slightly compressible fluid, the following equation of state is employed to compute the fluid pressures
ρ P = B ( )γ − 1 ρ0
(4)
where B = c20 ρ0 /γ is the coefficient, in which c0 is the speed of sound at reference density; ρ0 = 1000 kg/m3 is the reference density and γ = 7 is the polytrophic constant. It should be noted that using the real value of sound speed in water could lead to a very small time step arising from the constraint of Courant-Fredrich-Levy (CFL) condition. In SPH practice, the minimum speed of sound should be taken at least 10 times larger than the maximum bulk flow velocity. This keeps the density variation at less than 1%. In SPH fundamentals, a reference particle a interacts with its neighbouring particles b within a kernel influence domain through the weighting function W. A Wendland kernel is used in this paper. It has a compact support with simple functional form and it is very stable against the particle clustering in SPH computations with good accuracy. Besides, the Wendland kernel is also more computationally efficient than the other high-order Spline based kernels. In SPH approximations, the value of any quantity A of a desired particle a and its gradient ∇ A can be estimated by using the following discretized summation equations for all the particles b located within the kernel influence range as m A(ra ) = ∑ b A(rb )Wab (5) ρb b
∇ A (r a ) = ∑ b
mb [ A(ra ) − A(rb )]∇ a Wab ρb
(6)
where mb and ρb are the mass and density of the neighbouring particle b, respectively; A(ra ) and A(rb ) represent the values of the quantity A at point ra and rb , respectively; ∇ A(ra ) is the gradient of quantity; and ∇ a Wab is the gradient of kernel function between the two particles. Here Wab = W (|rab |, h), where |rab | = |ra − rb | is the distance between particle a and b, and h is the kernel smoothing length. In this study, h is simply taken as 1.3 times of the particle spacing dx, by following the guidelines of SPHysics. By applying these SPH rules, all terms in the momentum Equation (2) can be transformed into the SPH algebraic forms. For example, the following anti-symmetric form is used for the pressure gradient
1 − ∇P ρ
a
= −∑ mb b
Pb Pa + 2 2 ρa ρb
!
∇ a Wab
(7)
The above principle is also applicable to the SPH velocity divergence operator. By mixing both, the following hybrid formulation is proposed for the viscosity term
υ0 ∇ 2 u
a
= ∑ mb b
4υ0 rab .∇ a Wab
(ρ a + ρb )|rab |2 ⇒
! uab
(8)
where uab = ua − ub is defined. The SPS turbulent shear stress τ in Equation (2) can also be formulated in a similar way.
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In practical SPH computations, contrary to Equation (3), the fluid particles are actually moved by using the XSPH variant proposed by [2] as follows Dra u = ua + ε∑ mb ba Wab Dt ρ ab b
(9)
where ε is a constant between 0 and 1; and the average density ρ ab = (ρ a + ρb )/2 is defined. XSPH makes the fluid particle move with a velocity that is closer to the averaged velocity of its neighbouring particles b depending on the coefficient ε. This can effectively prevent the fluid particles from overlapping with each other and stabilize the SPH simulations in spillway flows. Besides, a Sheperd filter is used every 30 time steps in the computation. The Sheperd filter is designed to smooth and re-assign the particle densities during the SPH computations, aiming to reduce the large pressure oscillation related to the density accumulation errors. It is one of the most straightforward and computationally least expensive correction techniques used in SPH. To address the kernel deficiency near the free surface or solid boundary, a constant correction technique as proposed by [37] is adopted. A predictor-corrector time marching scheme is used to solve the governing equations. For the turbulence modeling, an eddy-viscosity based Sub-Particle Scale (SPS) model as suggested by Gotoh et al. [22] is used. 2.2. Free Surface and Boundary Conditions In WCSPH computations, the free surface conditions are automatically satisfied due to the mesh-free nature of the method. The solid wall boundaries are treated to ensure the fluid particles of non-penetration of the walls, and the relevant velocity conditions to be satisfied. Various wall treatments have been proposed such as the repulsive boundary [5] and dynamic boundary [38]. The Dalrymple boundary, i.e., dynamic boundary, is used which is available in the SPHysics open source code. In this method, the boundary particles follow the same mass and momentum equations as the fluid particles but they do not move but rather remain fixed in the position. They are placed in a staggered manner on the solid boundary. The pressure Equation (4) is also solved on the solid boundary particles. The upstream boundary is provided by a solid vertical push-paddle with a constant horizontal displacement. Therefore, a prescribed flow depth and discharge can be imposed in front of the spillway weir. Further details on this push-paddle can be found in the following model validations and applications. The downstream boundary is an open boundary, where the fluid particles are allowed to freely move out of the computational domain, such as plunging into a stilling pool. 2.3. Brief Introduction of parallelSPHysics parallelSPHysics is a free open-source SPH code (http://www.sphysics.org) that was released in 2007. It was jointly developed by the researchers at Johns Hopkins University (Baltimore, MD, USA), University of Vigo (Vigo, Spain), University of Manchester (Manchester, UK) and University of Roma La Sapienza (Roma, Italy). The code is programmed in FORTRAN language and is specifically suitable for the free-surface hydrodynamics [39]. In this paper, we use a parallel version of SPHysics, i.e., parallelSPHyphysics [36], to carry out the simulations of spillway flow over a stepped surface. The code parallelSPHysics has the same SPH numerical schemes as the serial SPHysics code but has been designed to perform simulations of millions of particles. The code is parallelised through an MPI formalism, and thus requires the MPICH or OpenMPI to be installed on the parallel machine. More detailed documentations are available from the SPHysics website (http://www.sphysics.org). 3. Engineering Background and Laboratory Experiment Dahua Reservoir is located along the Lara River of Qinghai Province, China and it is 69 km away from the provincial capital Xining city. The reservoir has a total storage capacity of 4.26 million m3 with
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a flood storage of 0.53 million m3 . The reservoir operates at a normal water level of ∇3138.8 m and 6 of 19 a designed flood level of ∇3141.0 m, respectively. To discharge the redundant flood water during the rainy seasons, stepped spillways used onare theused downstream side of the dam flow energy. during the rainy seasons, steppedare spillways on the downstream sideto ofdissipate the dam to dissipate In order to fully evaluate the complex flow conditions over the step surface as well as to provide useful flow energy. In order to fully evaluate the complex flow conditions over the step surface as well as information for design purposes, a thorough understanding of the spillway hydraulics is required. to provide useful information for design purposes, a thorough understanding of the spillway For such a purpose, a physical laboratory model is thus built. A schematic prototype Dahua hydraulics is required. For such a purpose, a physical laboratory model layout is thusof built. A schematic Dam and location of the spillway is shown in Figure 1. As shown in the figure, the spillway includes layout of prototype Dahua Dam and location of the spillway is shown in Figure 1. As shown in the the flow energy dissipater and outflow channel, with a total of figure, theentrance, spillwaydischarge includes flume, the flow entrance, discharge flume, energy dissipater andlength outflow 216.97 m. The crest elevation of the weir is the same as that of the hydraulic dam at ∇ 3138.8 m. channel, with a total length of 216.97 m. The crest elevation of the weir is the same as that of the The spillway ogee shape has takes a smoothed connection with the main dam. hydraulic damtakes at an3138.8 m. Thewhich spillway an ogeestreamlined shape which has a smoothed streamlined 3 The designed flood the spillway 48.43discharge m . Thisof is the equivalent flood m recurrence 3. This is connection with the discharge main dam.ofThe designed isflood spillwaytoisa 48.43 of 50 years. The effective breadth of the spillway is 21 m and the operating water head above the equivalent to a flood recurrence of 50 years. The effective breadth of the spillway is 21 m and the weir crest is 2.2 m. On the safety side, our study considers a 1000-year recurrence flood with a flow operating water head above the weir crest is 2.2 m. On the safety side, our study considers a 1000discharge of 120.21 m3with /s. a flow discharge of 120.21 m3/s. year recurrence flood Water 2017, 9, 973
Figure 1. Schematic layout of Dahua Dam and its spillway. Figure 1. Schematic layout of Dahua Dam and its spillway.
A physical laboratory experiment is carried out at School of Water Resources and Electric Power, A physical laboratory experiment is carried out at School ofand Water Resourcesprovide and Electric Power, Qinghai University, to study the spillway flow characteristics, meanwhile, useful data Qinghai University, to study the spillway flow characteristics, and meanwhile, provide useful data to validate the numerical SPH model. The length dimension ratio of model to prototype dams to is validate to thebenumerical SPH the model. The length dimension ratiobyofconsidering model to prototype damsrestrictions is selected selected 1/40 to satisfy similarity laws of gravity also the lab space to becapacity 1/40 to of satisfy the similarity laws of gravity also by considering lab space restrictions and and the water supply facilities. An experimental site photo the is shown in Figure 2a,b for a capacity of the water supply facilities. An experimental site photo is shown in Figure 2a,b for a global global view of the facility and enlarged view of the spillway with pressure measurement, view of the facility and enlarged view of the spillway with pressure measurement, respectively. respectively. The experimental experimental spillway spillway flows flows follow follow aa 1000-year 1000-year recurrence recurrence of of prototype prototype flood flood as as mentioned mentioned The before. By By using using the experimental flow rate of before. the length length ratio ratioof of40, 40,this thiscorresponds correspondstotoananequivalence equivalenceofof experimental flow rate 3 /s. A kinematic similarity law, i.e., Froude together with Manning’s similarity, has been 0.01188 m 3 of 0.01188 m /s. A kinematic similarity law, i.e., Froude together with Manning’s similarity, has been observed. The Thewater waterhead headabove abovethe the model weir experiment is 5.5 model spillway observed. model weir in in thethe experiment is 5.5 cm.cm. TheThe model spillway has a breadth of 0.525 m and total length is 159.83 withananaveraged averagedbed bedslope slopeangle angleof of33.69°. 33.69◦ . ahas breadth of 0.525 m and thethe total length is 159.83 cmcm with Different sizes sizes of of the the spillway spillway steps steps are are used used in in the the experiment, experiment, leading leading to to the the number number of of steps steps at at31, 31, Different 45 and 62, respectively. The dimension of the steps is provided by their width a multiplied with height 45 and 62, respectively. The dimension of the steps is provided by their width a multiplied with height h, as as 4.5 4.5 cm cm ××3.0 3.0cm, cm,3.1 3.1cm cm× × 2.07 and 2.25 × cm, 1.5 cm, for above the above different numbers. h, 2.07 cmcm and 2.25 cmcm × 1.5 for the different stepstep numbers. All All types of the flow are observed in the range of skimming flows. types of the flow are observed in the range of skimming flows. In the detailed flowflow depth, velocity and pressure profilesprofiles along the spillway In the laboratory laboratorymeasurement, measurement, detailed depth, velocity and pressure along the steps have been recorded for validation purposes. The flow depth was measured by a depth ruler and spillway steps have been recorded for validation purposes. The flow depth was measured by a depth the velocity was measured a portable (Motorola Solutions, Chicago, IL, USA) with ruler and the velocity was by measured byLS300-velocimetry a portable LS300-velocimetry (Motorola Solutions, Chicago, a range 0.01–4.0 m/s. the measurements were taken after several minutes when the waterwhen level IL, USA)ofwith a range ofAll 0.01–4.0 m/s. All the measurements were taken after several minutes in the reservoir became stable. The flow depth and velocity were recorded at the centerline of the the water level in the reservoir became stable. The flow depth and velocity were recorded at the centerline of the spillway as well as two other points near the lateral wall at a distance of 5 cm. These are located along the spillway at a number of locations, and the velocity was recorded at a distance of 8 mm from the step corner in the normal direction to the pseudo bottom (as shown in Figure 9a). The flow depth and velocity data were sampled for 20–30 s and then time-averaged to eliminate the
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spillway as well as two other points near the lateral wall at a distance of 5 cm. These are located along the spillway at a number of locations, and the velocity was recorded at a distance of 8 mm from the step corner in the normal direction to the pseudo bottom (as shown in Figure 9a). The flow depth and velocity data were sampled for 20–30 s and then time-averaged to eliminate the fluctuations Water 2017, 9, 973 7 of 19 (the depth data were read 3–4 times during this period to take the average). With regard to the pressure measurement, the standard manometers in Figure 2btowere used to record the pressures at fluctuations (the depth data were read as 3–4shown times during this period take the average). With regard to of thethe pressure measurement, the the standard manometers as vertical shown in Figure 2b were used to record the centerline spillway, on both horizontal and step faces. the pressures at the centerline of the spillway, on both the horizontal and vertical step faces.
(a)
(b) Figure 2. Site photo of laboratory spillway experiment: (a) Global view of facility; (b) Enlarged view Siteofphoto laboratory spillway experiment: (a) Global view of facility; (b) Enlarged spillwayofwith pressure measurement.
Figure 2. spillway with pressure measurement.
view of
4. Model Validations through Benchmark Spillway Flow
For the SPH simulation of spillway flow with an open free surface, an accurate treatment of the 4. Model Validations through Benchmark Spillway Flow
inflow and outflow boundary is of great importance. Some useful techniques such as the periodic
model [32] are not applicable to the situation where the flow depth is different on the For the boundary SPH simulation of spillway flow with an open free surface, an accurate treatment of the upstream and downstream side, while a more advanced real inflow model [26] could have complex inflow and outflow boundary is of great importance. useful[31] techniques such as the periodic programming. Some researchers adopted a large water Some tank approach so the domain is large enough[32] and the outflow is not significantly constrained by the storage of water, but this boundary model are not process applicable to the situation where the flow depth iscould different on the cause excessive CPU load. In this work, we propose a simple but effective push-paddle approach to upstream and downstream side, while a more advanced real inflow model [26] could have complex realize an accurate inflow process. programming. Some researchers a large watersolid tankpaddle approach [31] so the domain is large The model is based on adopted the principle that a vertical is placed on the upstream boundary with aprocess constant is horizontal moving speed.constrained By giving a prescribed to the enough and the outflow not significantly by the unilateral storage motion of water, but this could paddle, an inflow process can be generated. As long as the speed of the paddle is carefully set to cause excessive CPU load. In this work, we propose a simple but effective push-paddle approach to match the desired discharge, a continuous and stable inflow can be achieved. The feasibility of using realize an accurate inflow process. such a push-paddle approach has been demonstrated in a validation work carried out for the weir flow in Here on we use inflow technique reproduce solid the numerical simulation of [31] The model is[40]. based thethis principle that atovertical paddle is placed onandthe upstream experimental study [41] by using the open source code parallelSPHysics. boundary with a constant horizontal moving speed. By giving a prescribed unilateral motion to the Meireles and Matos [41] conducted experimental studies on a stepped spillway that was 0.5 m paddle, an inflow process canThe be base generated. As long with as the speed ofm.the paddle iseach carefully set to high with 1V:2H slope. weir is broad-crested a length of 0.5 It has 10 steps, match the desired discharge, a continuous and stable inflow can be achieved. The feasibility of using such a push-paddle approach has been demonstrated in a validation work carried out for the weir flow in [40]. Here we use this inflow technique to reproduce the numerical simulation of [31] and experimental study [41] by using the open source code parallelSPHysics. Meireles and Matos [41] conducted experimental studies on a stepped spillway that was 0.5 m high with 1V:2H slope. The base weir is broad-crested with a length of 0.5 m. It has 10 steps,
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Water 2017, 973in height and 0.1 m in width. Husain et al. [31] first reproduced this experiment 8 of 19 each with 0.059,m by using a series version of SPHysics open source code. In their numerical model, a large water with 0.05 m in height and 0.1 m in width. Husain et al. [31] first reproduced this experiment by using tank that was 7.5 m long was used in front of the weir to achieve a stable inflow process throughout a series version of SPHysics open source code. In their numerical model, a large water tank that was the simulation time of interest. Due to the use of proposed push-paddle inflow model, the present 7.5 m long was used in front of the weir to achieve a stable inflow process throughout the simulation computational domain can be efficiently reduced. A schematic setup of the computational domain is time of interest. Due to the use of proposed push-paddle inflow model, the present computational shown in Figure 3.efficiently As shown in the A figure, the setup upstream area is filled with the fluid particles domain can be reduced. schematic of thewater computational domain is shown in Figure only between x = 3.5 m and 7.0 m with a vertical solid paddle being located at x = 3.5 m. This 3. As shown in the figure, the upstream water area is filled with the fluid particles only between leads x = to a significant reduction of athe particle numbers by four times the same resolution 3.5 m and 7.0 m with vertical solid paddle being located at assuming x = 3.5 m. This leads particle to a significant reduction the particle numbers by four times assuming the same particle with [31]. In theofpresent parallelSPHysics simulations, a particle spacing dx = resolution 0.004 m is with used.[31]. ThisInleads the present parallelSPHysics simulations, a particle dx = 0.004domain. m is used. This leads to roughlystudy to roughly a total number of 162,098 particles in the spacing computational Detailed sensitivity total number of 162,098 in the computational sensitivity study on the on thea particle resolution hasparticles been discussed by Husain etdomain. al. [31]Detailed and therefore their recommended particle resolution has been discussed by Husain et al. [31] and therefore their recommended value value is simply used in the present computation. The whole simulation time of spillway flow is is simply used in the present computation. The whole simulation−time of spillway flow is 12 s. The 12 s. The computational time step was initially chosen to be 10 5 s and then dynamically adjusted computational time step was initially chosen to be 10−5 s and then dynamically adjusted in the in thecomputations computations to fulfill the CFL condition. The program is run on a 16-core computer cluster to fulfill the CFL condition. The program is run on a 16-core computer cluster (CPU 2 (CPUGHz 2 GHz and RAM 32 Gb) (School of Water Resources and Power, ElectricQinghai Power,University, Qinghai China) University, and RAM 32 Gb) (School of Water Resources and Electric China) and it cost approximately four of days oftime. CPU time. and it cost approximately four days CPU
Figure 3. Schematic setup of computational domain for model validation.
Figure 3. Schematic setup of computational domain for model validation.
In the SPH study of [31], the flow depths, velocity profiles and coordinates of the inception point
In the flow depths, velocity profiles andfor coordinates of of thecritical inception onthe the SPH stepsstudy locatedofin[31], the non-aerated flow regions are examined a wide range depthpoint skimming flow condition. hflow c is the critical are flowexamined depth on the the weir on thewithin stepsthe located in the non-aerated regions forcritical a widesection rangeover of critical depth crest where the Froude number equals one. In the current parallelSPHysics simulations, the test within the skimming flow condition. hc is the critical flow depth on the critical section over the weir of Froude hc = 0.07935 m, 0.0716 m and 0.0635 shown inparallelSPHysics Table 1 in [31] is used. Based on the crest condition where the number equals one. In m theascurrent simulations, the test simulation results, Table 1 presents the experimental [31,41] and parallelSPHysics values of the condition of hc = 0.07935 m, 0.0716 m and 0.0635 m as shown in Table 1 in [31] is used. Based on discharge per unit width (q) for the different critical flow depths (hc) over the weir crest. It seems a the simulation results, Table 1 presents the experimental [31,41] and parallelSPHysics values of the good agreement has been reached. discharge per unit width (q) for the different critical flow depths (hc ) over the weir crest. It seems a good agreement has been reached. Table 1. Comparisons between experimental and parallelSPHysics discharge per unit width. hc (m) qEXP (m2/s) qSPH (m2/s) Relative Error (%) Table 1. Comparisons between experimental and parallelSPHysics discharge per unit width. 0.07935 0.07 0.0683 −2.43 0.0716 0.06 2 0.0578 2 −3.67 hc (m) Relative Error (%) qEXP (m /s) qSPH (m /s) 0.0635 0.05 0.0523 +4.60 0.07935 0.07 0.0683 −2.43 0.0716 0.06 0.0578 −3.67 over the stepped For the case of critical depth hc = 0.0716 m, the computed particle snapshots 0.0635 0.05 0.0523 +4.60 spillway are shown in Figure 4a–d, for the different time instants. It shows that under such a skimming flow condition, the flow passes over the weir crest like a jet and hits the horizontal face of all the steps. One hmajor part of the flow coherently skims over the pseudo bottom For thedownstream case of critical depth c = 0.0716 m, the computed particle snapshots over the stepped formed by the step outer edges. Another minor part is entrapped inshows the recirculation zone under the spillway are shown in Figure 4a–d, for the different time instants. It that under such a skimming main stream and inside the step cavity delimited by the step faces. In this area there exists an flow condition, the flow passes over the weir crest like a jet and hits the horizontal face of all the extensive exchange with the main flows and the water particles move around a triangular path. These downstream steps. One major part of the flow coherently skims over the pseudo bottom formed by simulation patterns are very similar to those observed in [31]. However, as the computations proceed the step edges. Another part is shown entrapped in the4d, recirculation zone under the main stream to aouter relatively longer time minor at t = 12 s as in Figure the boundary of two different flows and inside the step cavity delimited by the step faces. In this area there exists an extensive exchange becomes unclear.
with the main flows and the water particles move around a triangular path. These simulation patterns
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are very similar to those observed in [31]. However, as the computations proceed to a relatively longer time Water at t =2017, 12 9, s as 973 shown in Figure 4d, the boundary of two different flows becomes unclear. 9 of 19
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(a)
(b)
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Figure 4. SPH particle snapshots of flow over the stepped spillway at different time t: (a) 1.2 s; (b) 2 s;
Figure 4. SPH particle snapshots of flow over the stepped spillway at different time t: (a) 1.2 s; (b) 2 s; (c) 8 s; (d) 12 s. (c) (d) (c) 8 s; (d) 12 s. Figure 4. SPH particle snapshots of flow over the stepped spillway at different time t: (a) 1.2 s; (b) 2 s;
In their experiment, Meireles and Matos [41] also measured the streamwise flow velocity parallel (c) 8 s; (d) 12 s. In the their experiment, Meirelessections and Matos [41]various also measured the streamwise flow velocity parallel to chute slope at different under flow discharges. To quantify the accuracy of parallelSPHysics as well as the proposed push-paddle inflow model, Figure 5a–c shows the to the chute slope at different sections under[41] various flow discharges. To quantify the parallel accuracy of In their experiment, Meireles and Matos also measured the streamwise flow velocity numerical and experimental velocity profiles at section x = 8.1 m, which is downstream of the weir, parallelSPHysics as well as the proposed push-paddle inflow model, Figure 5a–c shows the numerical to the chute slope at different sections under various flow discharges. To quantify the accuracy of for the three different critical flow depths h c as shown in Table 1. The details of velocity measurement and experimental velocity profiles at section x = 8.1 m, which is downstream of the weir, for the parallelSPHysics as well as the proposed push-paddle inflow model, Figure 5a–c shows thethree and the instruments used could found in of seen from the figure, spite the numerical and experimental profiles at original section xwork = 8.1 m,[41]. which is downstream of the in weir, different critical flow depths hcvelocity asbeshown inthe Table 1. The details ofAs velocity measurement and of some slight deviations between the numerical and experimental velocities, the general agreement for the three critical flow depths hc as shown 1. The details velocity measurement instruments useddifferent could be found in the original work in of Table [41]. As seen fromofthe figure, in spite of some between the two is quite satisfactory. It in is shown thatwork the velocity values do not change much the instruments used be found original of [41]. As seen from the figure,too in spite slightand deviations between thecould numerical and the experimental velocities, the general agreement between along the flow depth. According to [31,41], this is the region close to the weir crest, where the velocity of some slight deviations between the numerical and experimental velocities, the general agreement the two is quite satisfactory. It is shown that the velocity values do not change too much along the flow is relatively also nearly constant the that entire flow depthvalues due todo thenot factchange that the thickness between thelow twoand is quite satisfactory. It isover shown the velocity too much depth. According to [31,41], this isHowever, the region close to the weirincreases crest, where the the velocity is relatively of boundary layer is quite small. as the flow velocity towards downstream along the flow depth. According to [31,41], this is the region close to the weir crest, where the velocityof low and also nearly constant over thewill entire flow depth due to the fact that the thickness of aboundary the spillway, theand boundary layer fully develop and then it would expected have more is relatively low also nearly constant over the entire flow depth due tobethe fact thatto the thickness layervaried is quite small. However, as the flow velocity increases towards the downstream of the spillway, velocity profile such as that follows the power law. of boundary layer is quite small. However, as the flow velocity increases towards the downstream of
0.030 0.035 0.025 0.030 0.020 0.025 0.015 0.020 0.010 0.015 0.005 0.010 0.000 0.0051
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the boundary layer fully develop then it would be expected a more varied velocity the spillway, thewill boundary layer willand fully develop and then it would to be have expected to have a more Exp Exp profile suchvelocity as that profile follows theaspower law. the power law. Exp varied such that follows
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Figure 5. Comparisons between experimental and parallelSPHysics velocity profiles at x = 8.1 m for (a) (b) (c) the critical flow depth hc: (a) 0.0635 m; (b) 0.0716 m; (c) 0.07935 m. Figure 5. Comparisons between experimental and parallelSPHysics velocity profiles at x = 8.1 m for Figure 5. Comparisons between experimental and parallelSPHysics velocity profiles at x = 8.1 m for the critical flow depth hc: (a) 0.0635 m; (b) 0.0716 m; (c) 0.07935 m.
the critical flow depth hc : (a) 0.0635 m; (b) 0.0716 m; (c) 0.07935 m.
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5. 5. Model Applications through SpillwayFlow Flow Model Applications throughExperimental Experimental Spillway Water 2017, 9, 973
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InInthis model isis used used toto reproduce reproducethe theprevious previouslaboratory laboratory thissection, section, the the parallelSPHysics parallelSPHysics model 5. Model Applications through Experimental Spillway Flow experiment of prototype Dahua Hydraulic spillway As mentioned before, the experiment of prototype Dahua Hydraulic Dam Dam spillway project.project. As mentioned before, the experiment In this section, the parallelSPHysics model is used to reproduce the previous laboratory experiment was carried out by a 1:40 ratio model. In the numerical simulations, it was very difficult was carried out by a 1:40 ratio model. In the numerical simulations, it was very difficult to reproduce the experiment of experimental prototype Dahua Hydraulic spillway project. As mentioned before,Therefore, the to reproduce the conditions due toDam the very shallow water above the crest. spillway weir experimental conditions due to the very shallow water head above the head spillway weir experiment was carried out by a 1:40 ratio model. In the numerical simulations, it was very difficult crest. Therefore, a relatively largerwhich numerical model, is based on1:10 a similarity of 1:10and a relatively larger numerical model, is based on a which similarity ratio of between ratio the model to reproduce the experimental conditions dueisto the very shallow water head above the SPH spillway weir between the model and prototype built. For aof better interpretation ofresults, the simulation prototype spillways, is built. For a spillways, better interpretation the SPH simulation the numerical crest. Therefore, a relatively larger numerical model, which is based on a similarity ratio of results, the numerical and experimental values areprototype both converted totheir the prototype ones for1:10 their and experimental values are both converted to the ones for comparisons. between the model and prototype spillways, is built. For a better interpretation of the SPH simulation comparisons. results, the numerical and experimental values are both converted to the prototype ones for their 5.1. Computational Settings 5.1.comparisons. Computational Settings Three different spillway conditions, represented by the step numbers as shown in Section 3, Three different spillway conditions, stepvalues numbers as shown in Sectionfrom 3, arethe 5.1. Computational are simulated, i.e., theSettings step number beingrepresented 62, 45 and by 31.the These have been proposed simulated, i.e., the step number being 62, 45 and 31. These values have been proposed from the different conditions, represented by the diagram step numbers as shown in Section domain 3, are feasibilityThree study of the spillway real spillway design. A schematic of the computational is feasibility study of the real spillway design. schematic diagram the been computational domain is simulated, i.e., step number being 62,water 45Aand 31. These values of have proposed from the shown in Figure 6. the It includes an upstream tank, a push-paddle for inflow generation, an ogee shown in Figure 6. It includes an upstream water tank, a push-paddle for computational inflow generation, an ogee study thestepped real spillway design. A schematic diagram of the domain weir feasibility connected withofthe spillways and a downstream energy dissipation tank. An is inflow weir connected with the stepped spillways and a downstream energy dissipation tank. An inflow shown in Figure 6. It includes an upstream water tank, a push-paddle for inflow generation, an ogee 2 discharge per unit width of 0.181 m /s is generated, and the water head above the weir crest is 0.22 m. 2/s is generated, and the water head above the weir crest is 0.22 discharge per unitwith width 0.181 mspillways weir connected theofstepped and a downstream energy dissipation tank. An inflow The stepped spillway is 6.4 m long. The water head looks relatively large compared to the length of m. discharge The stepped is of 6.40.181 m long. water head relatively to the per spillway unit width m2/sThe is generated, andlooks the water headlarge abovecompared the weir crest is length 0.22 theofspillway, since asince morea safety design was considered in the study based onbased a 1000-year recurrence the spillway, more safety design was considered in the study on a 1000-year m. The stepped spillway is 6.4 m long. The water head looks relatively large compared to the length of recurrence prototype flood assince mentioned Section 3.in Section of prototype flood asin mentioned 3. of the spillway, a more safety design was considered in the study based on a 1000-year In SPH simulations, a particle spacing dx 0.004 is used leads a total number recurrence of prototype aflood as mentioned indx Section 3.m m In SPH simulations, particle spacing ofof = =0.004 is used andand thisthis leads to a to total number of − 5 −5 s and of nearly nearly300,000 particles. The initial time wasmas chosen as this 10 s and automatically In300,000 SPH simulations, a particle spacing ofwas dxstep =chosen 0.004 is10 used to a later total number of in particles. The initial time step and laterleads automatically changed particles. initial step CFL was chosen as 10−5 time s andwas later20 automatically changed changed in300,000 the computation the condition. The simulation was 20 s inand thenearly computation followingThe thefollowing CFL time condition. The simulation s andtime it cost roughly 20– it the computation following the CFL condition. The simulation time was 20 s and it cost roughly 20– cost roughly 20–26 days CPU time for the different step numbers on the same computer cluster 26 days CPU time for the different step numbers on the same computer cluster (but with 48-core) as 26 days CPU time for theindifferent numbers the same computer cluster (but 48-core) (but with 48-core) asvalidation used the validation test in 4.ofTo show thewith stability ofas inflow used in the model testmodel instep Section 4. Toon show theSection stability inflow generations, Figure 7 used in the model validation test in Section 4. To show the stability of inflow generations, Figure 7 shows the time-dependent discharge hydrograph for the case of 62-stepfor spillway. The point generations, Figure 7 shows the time-dependent discharge hydrograph the case of gauging 62-step spillway. shows the time-dependent dischargeentrance hydrograph case 62-stepthe spillway. The gauging point is gauging located just upstream of the at xfor = the 7.9entrance m. Itofshows flow The point is located justweir upstream of the weir at x = 7.9increasing m. It shows thedischarge increasing is located just upstream of the weir entrance at x = 7.9 m. It shows the increasing flow discharge following thefollowing forward motion of themotion push-paddle. After some very the flow flow discharge the forward of the push-paddle. Afterslight someoscillations, very slight oscillations, following the forward motion of the push-paddle. After some very slight oscillations, the flow stabilizes at time tat= time 8 s. thedischarge flow discharge stabilizes discharge stabilizes at time t = 8 s. t = 8 s.
Figure6.6.SPH SPHcomputational computational domain domain of stepped spillway flow. Figure Figure 6. SPH computational domainof ofstepped steppedspillway spillwayflow. flow.
Figure 7. Discharge hydrograph upstream of weir entrance.
Figure7.7.Discharge Discharge hydrograph hydrograph upstream Figure upstreamofofweir weirentrance. entrance.
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5.2. Flow Patterns over Stepped Spillways According to the SPH simulation results, the time history of particle snapshots are shown in Water 2017, 9, 973 11 of 19 Figure 8a–d, at different time instants based on the 62-step spillway. It shows the importance of steps in dissipating flowover energy and thus reducing the need of a large basin at the end of the spillway. 5.2. Flowthe Patterns Stepped Spillways The steps can increase the surface roughness of flow path and then accelerate the growth of turbulent According to the SPH simulation results, the time history of particle snapshots are shown in boundary layer. This facilitates the initiation of the self-aeration in reality thatthecould protect Figure 8a–d, at different time instants based on 62-step spillway. It shows importance of the stepsspillway surfaceinfree of cavitation [31,41]. Figure the 8 shows skimming dissipating the flowdamage energy and thus reducing need ofaa typical large basin at the endflow of thepattern spillway.over the steps can increase the surface roughness of contracts flow path and then accelerate thestart growth turbulent steppedThe spillways. The water surface initially somewhat at the of of the spillway path boundary layer. This facilitates initiation of Then self-aeration in reality thatup could protectdownstream the spillway due to due to the sudden acceleration fromthethe gravity. it gradually rises towards surface free of cavitation damage [31,41]. Figure 8 shows a typical skimming flow pattern over the the frictional influence of the steps and the accumulation of incoming flows. With the development of stepped spillways. The water surface initially contracts somewhat at the start of the spillway path boundary gravitational boundary drag. the flow reaches quasi-stable duelayer, to the the sudden accelerationforce from balances the gravity.the Then it gradually risesThen up towards downstream due state with little variations in the flow depth. Eventually theof skimming flow over spillway resembles to the frictional influence of the steps and the accumulation incoming flows. With the the development boundary layer, uniform the gravitational force balances boundary drag. Then thethe flow reaches quasi-into the a classicofopen channel flow. At the end ofthe the stepped spillway, flow plunges stablefor state with little variations in the flow depth. Eventually the skimming flow over theThe spillway tailing pool energy dissipation, when a large scale hydraulic jump is observed. violent flow resembles a classic open channel uniform flow. At the end of the stepped spillway, the flow plunges interaction as demonstrated by the breaking free surface is expected to dissipate remaining flow energy. into the tailing pool for energy dissipation, when a large scale hydraulic jump is observed. The violent Large scale flow circulation is also found in the region enclosed by the step and bottom floor upstream flow interaction as demonstrated by the breaking free surface is expected to dissipate remaining flow of the hydraulic jump. energy. Large scale flow circulation is also found in the region enclosed by the step and bottom floor Besides, in ofFigure 8a–d, tjump. = 2 s corresponds to the onset of the overflow, while times t = 8, 12 upstream the hydraulic Besides, in Figure t =flow 2 s corresponds to the onset of thedischarge overflow, while t =head 8, 12 and and 15 s almost indicate a 8a–d, stable state in which the flow and times water in front of 15 s almost indicate a stable flow state in which the flow discharge and water head in front of the the spillway remain more or less unchanged. The existence of consistent cavity flow patterns over spillway remain more or less unchanged. The existence of consistent cavity flow patterns over the the spillway steps could be clearly found in the zoomed portions of the figure. However, these cavity spillway steps could be clearly found in the zoomed portions of the figure. However, these cavity flows demonstrate somewhat variations andspace space due to influence the influence flow turbulence flows demonstrate somewhat variationsininthe the time time and due to the of flowofturbulence that is highly unstable. that is highly unstable.
(a)
(b)
(c)
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Figure 8. Computed particle snapshots of 62-step spillway flow at different time instants at: (a) t = 2
Figure 8. Computed particle snapshots of 62-step spillway flow at different time instants at: (a) t = 2 s; s; (b) t = 8 s; (c) t = 12 s; (d) t = 15 s. (b) t = 8 s; (c) t = 12 s; (d) t = 15 s.
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5.3. Model Model Validation Validation of of Pressure Pressure on on Spillway Spillway Ftep Ftep faces faces 5.3. A good good knowledge knowledge of faces of of a spillway helps to understand the A of the thepressure pressurefeatures featureson onthe thestep step faces a spillway helps to understand spillway hydraulics and improve the structure reliability. The cavitation formation and evolution in the the spillway hydraulics and improve the structure reliability. The cavitation formation and evolution non-aerated flow region under the skimming flow condition is closelyisrelated the pressure in the non-aerated flow region under the skimming flow condition closelytorelated to thevariations pressure when they are below the atmospheric level. Husain et al. [31] investigated the pressure flowpressure field on variations when they are below the atmospheric level. Husain et al. [31] investigated the the horizontal and vertical faces of the steps by using a series version of SPHysics for the experiment flow field on the horizontal and vertical faces of the steps by using a series version of SPHysics for of [41], but there was but a lack of was pressure for the data model Besides, Husain al. [31] the experiment of [41], there a lackdata of pressure for validation. the model validation. Besides,etHusain also suggested that no detailed information may have been available in the literature relating to the et al. [31] also suggested that no detailed information may have been available in the literature pressure measurement in the non-aerated flow region over moderate slope stepped spillways under relating to the pressure measurement in the non-aerated flow region over moderate slope stepped the skimming flow Toflow address such a practical issuesuch in the this section spillways under thecondition. skimming condition. To address a spillway practical hydraulics, issue in the spillway provides quantitative pressure measurement on the horizontal and vertical faces of the steps from hydraulics, this section provides quantitative pressure measurement on the horizontal and vertical our laboratory In addition, the parallelSPHysics simulation results are compared and faces of the stepsexperiment. from our laboratory experiment. In addition, the parallelSPHysics simulation results validated by the measured data. are compared and validated by the measured data. In the the numerical numerical simulations, simulations, both both the the velocity velocity and and pressure pressure values values of of spillway spillway flows flows were were In extracted. The solved from the the momentum Equation (2) so (2) it is so a total extracted. The pressure pressurewas was solved from momentum Equation it ispressure a total including pressure both the hydrostatic and dynamic components. The velocity sampling point is located the including both the hydrostatic and dynamic components. The velocity sampling point isalong located spillway at a number of locations at a distance of 0.008 m from the step corner in the normal direction along the spillway at a number of locations at a distance of 0.008 m from the step corner in the normal to the pseudo shownas in shown Figure in 9a.Figure The pressure point is shown direction to thebottom, pseudoasbottom, 9a. The sampling pressure sampling point in is Figure shown 9b, in for the 31-step spillway case. The scales presented here correspond to the experimental dimensions. Figure 9b, for the 31-step spillway case. The scales presented here correspond to the experimental Both the velocity proposed hereafter are thehereafter weight average neighboring dimensions. Bothand the pressure velocity values and pressure values proposed are the of weight averageSPH of particles within the kernel support range for a true representation of the flow field. neighboring SPH particles within the kernel support range for a true representation of the flow field.
(a)
(b)
Figure Figure 9. 9.Experimental Experimentaldata datagauging gaugingpoints pointsalong alongthe thespillway spillwayfor forthe the31-step 31-stepcase: case:(a) (a)Velocity Velocitypoint; point; (b) Pressure point. (b) Pressure point.
Figures 10 and 11a,b show the pressure distributions on the horizontal and vertical step faces of Figures 10 and 11a,b show the pressure distributions on the horizontal and vertical step faces of the spillway, respectively, at two different stream-wise step locations, for the 31 and 45 step number the spillway, respectively, at two different stream-wise step locations, for the 31 and 45 step number cases. The results are presented for both the SPH simulations and experimental data converted to the cases. The results are presented for both the SPH simulations and experimental data converted prototype values. It should be noted that only two measurement points are available for the pressure to the prototype values. It should be noted that only two measurement points are available for profile on the vertical face of 45-step spillway, as shown in Figure 11b, while three measuring points the pressure profile on the vertical face of 45-step spillway, as shown in Figure 11b, while three are available in other conditions. First a quite satisfactory agreement has been found at all of these measuring points are available in other conditions. First a quite satisfactory agreement has been found locations. An error analysis showed that the normalized average errors between the numerical and at all of these locations. An error analysis showed that the normalized average errors between the experimental results are 0.8–4.3% in the horizontal pressures and 0.7–4.7% in the vertical pressures. numerical and experimental results are 0.8–4.3% in the horizontal pressures and 0.7–4.7% in the vertical Besides, a consistent trend of the curves is shown in the two pressure profiles either on the horizontal pressures. Besides, a consistent trend of the curves is shown in the two pressure profiles either on the or on the vertical faces. However, there exist some kinds of differences in their magnitudes, which horizontal or on the vertical faces. However, there exist some kinds of differences in their magnitudes, could be attributed to the unsteadiness of the vortices formed inside the step cavities according to the which could be attributed to the unsteadiness of the vortices formed inside the step cavities according study in [31]. to the study in [31]. By examining these pressure profiles, it is shown that all of those on the horizontal face of the By examining these pressure profiles, it is shown that all of those on the horizontal face of the steps steps demonstrate a half S-shape. The pressure on the horizontal boundary points is positive with a demonstrate a half S-shape. The pressure on the horizontal boundary points is positive with a peak peak value, and then gradually decreases towards the step corner with a minimum value. According value, and then gradually decreases towards the step corner with a minimum value. According to [31], to [31], the positive pressure can be attributed to the reverse flow inside the cavity, while the peak pressure value occurs when the main stream hits the horizontal step face. This pattern of the pressure on the horizontal step surface is very close to that observed by Husain et al. [31] in their SPH results
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theWater positive pressure can be attributed to the reverse flow inside the cavity, while the peak pressure 2017, 9, 973 13 of 19 value occurs when the main stream hits the horizontal step face. This pattern of the pressure on the horizontal step validations. surface is very close that observed byalong Husain al. [31]step in their with with no data As for thetopressure profiles theet vertical faces,SPH theyresults demonstrate nomore data variations. validations.Generally, As for theitpressure profiles along the vertical step faces, they demonstrate more is shown that the negative pressure area appears on the upper part of variations. Generally, it is positive shown that the negative pressure areapart appears theface upper of theitsstep the step face, while the pressure occurs on the lower of theon step andpart reaches peak face, while thestep positive pressure occurs on thethe lower part of the steparises face and its peak value at value at the corner. Following [31,41], negative pressure fromreaches the combinations of high theflow stepvelocities corner. Following [31,41], the negative pressure arises from the combinations of high flow close to the step outer edge and the stepped surface. On the other hand, the positive velocities close to the stepformation outer edge theeddies stepped surface. otherflows hand,inside the positive pressure pressure is due to the ofand small caused by On the the reverse the step cavity. is due to the of validation small eddies bycomputations, the reverse flows the stepdata cavity. For a formation more robust of caused the SPH the inside experimental of Amador et al. For a more robust validation thethe SPH computations, the experimental data of L/k Amador et al. [42] L [42] were added to Figure 10a,bofon horizontal and vertical faces at different s values, where were 10a,b on the and to vertical facesstep at different L/kcharacteristic L and s values, where andadded ks are to theFigure distance from the horizontal spillway crest the outer edge and roughness ks height, are the distance from the spillway crest to the outer step edge and characteristic roughness height, respectively. There is a good trend of the data consistency between the two results, but some respectively. There is a good trendinofthe thepressure data consistency between the two results, some obvious obvious discrepancies do exist values due to the difference in the but flow/step condition discrepancies do exist point in theused pressure values due to the difference inthe thepotentials flow/stepofcondition data and data sampling in the two studies. Nonetheless, pressure and prediction sampling point used in the two studies. Nonetheless, the potentials of pressure prediction by the SPH by the SPH model on the step faces can be well demonstrated. model on the step faces can be well demonstrated. 3.5
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(b) Figure 10. Pressure distributions on: (a) horizontal; (b) vertical step faces, for the 31-step spillway Figure 10. Pressure distributions on: (a) Horizontal; (b) Vertical step faces, for the 31-step spillway case. case.
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(b) Figure Pressure distributions on:Horizontal; (a) Horizontal; (b) Vertical step for faces, the 45-step Figure 11. 11. Pressure distributions on: (a) (b) Vertical step faces, the for 45-step spillwayspillway case. case.
5.4. Analysis of Energy Dissipation Efficiency of Spillway with Different Geometries 5.4. Analysis of Energy Dissipation Efficiency of Spillway with Different Geometries One significant practical value of the stepped spillways is their highly efficient flow energy One significant practical value of the stepped spillways is their highly efficient flow energy dissipation capacities. This could prevent the downstream channel from severe erosion and stabilize dissipation capacities. This could prevent the downstream channel from severe erosion and stabilize the main flow stream in the river management. Severe erosions at the downstream channel may harm the main flow stream in the river management. Severe erosions at the downstream channel may harm the stability of the hydraulic structures as documented by Dodaro et al. [43,44]. More practical issues the stability of the hydraulic structures as documented by Dodaro et al. [43,44]. More practical issues on the stepped spillway design and energy dissipation have been thoroughly studied in the benchmark on the stepped spillway design and energy dissipation have been thoroughly studied in the of [45–47]. Therefore, a thorough understanding of the energy dissipation features can help to better benchmark of [45–47]. Therefore, a thorough understanding of the energy dissipation features can evaluate the design of stepped spillways and other energy dissipaters such as the tailing basin. In this help to better evaluate the design of stepped spillways and other energy dissipaters such as the tailing section, a comparison is made between the SPH and experimental energy dissipation efficiencies along basin. In this section, a comparison is made between the SPH and experimental energy dissipation the streamwise spillway direction for the 62, 45 and 31 step cases, respectively. These step numbers efficiencies along the streamwise spillway direction for the 62, 45 and 31 step cases, respectively. correspond to the step dimensions of 9 cm × 6 cm, 12.4 cm × 8.28 cm and 18 cm × 12 cm in the These step numbers correspond to the step dimensions of 9 cm × 6 cm, 12.4 cm × 8.28 cm and 18 cm numerical simulations and 2.25 cm × 1.5 cm, 3.1 cm × 2.07 cm and 4.5 cm × 3.0 cm in the laboratory × 12 cm in the numerical simulations and 2.25 cm × 1.5 cm, 3.1 cm × 2.07 cm and 4.5 cm × 3.0 cm in experiment. Another purpose is to use this comparison to validate the flow velocity computations as the laboratory experiment. Another purpose is to use this comparison to validate the flow velocity well, since the energy is calculated from the potentials related to the height of fluids and the kinematics computations as well, since the energy is calculated from the potentials related to the height of fluids related to the flow velocity. and the kinematics related to the flow velocity. Figure 12a–c shows the energy dissipation curves of the three different step numbers. Figure 12a–c shows the energy dissipation curves of the three different step numbers. The The horizontal coordinate is indicated by the number of the pile, which is essentially a common term horizontal coordinate is indicated by the number of the pile, which is essentially a common term to to represent the horizontal distance measured from the start of the spillway presented in prototype represent the horizontal distance measured from the start of the spillway presented in prototype values. The results show that the energy dissipation increases along the downstream-wise directions. values. The results show that the energy dissipation increases along the downstream-wise directions.
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Water 2017, 9, 973 of the spillway, the gradient of the energy curve is steep and it becomes 15 of 19 At the beginning milder after the pile number reaches around 0 + 40.0 m for all the three cases, until a quasi-terminal value At the beginning of the spillway, the gradient of the energy curve is steep and it becomes milder after is approached. The maximum energy efficiency experimentally 75.17% forvalue the 62-step the pile number reaches around 0 + dissipation 40.0 m for all the threeiscases, until a quasi-terminal is case, approached. as comparedThe with the SPH value of 80.98% with an averaged error of 8.20%. On the other maximum energy dissipation efficiency is experimentally 75.17% for the 62-stephand, for the 45-step case thewith maximum is error experimentally as hand, compared case, as compared the SPH energy value ofdissipation 80.98% with efficiency an averaged of 8.20%. On80.96% the other with for 84.98% by SPH with an averaged error of 3.17%.efficiency With regard to the 31-step case,asthese values are the 45-step case the maximum energy dissipation is experimentally 80.96% compared with 84.98%and by SPH an averaged error ofbe 3.17%. With to the dissipation 31-step case,of these values flow 76.04%, 81.77% 5.8%,with respectively. It should noted thatregard the energy a uniform are 76.04%, and 5.8%, shouldhas be noted that the energy dissipation of characteristics a uniform in some energy81.77% dissipaters, likerespectively. corrugatedItpipes, a constant rate for certain flow flow in some energy dissipaters, like corrugated pipes, has a constant rate for certain flow and does not vary as the flow conveys, according to the study of Calomino et al. [48]. characteristics and does not vary as the flow conveys, according to the study of Calomino et al. [48]. Here it is also interesting to note the 45-step spillway dissipated slightly more flow energy as Here it is also interesting to note the 45-step spillway dissipated slightly more flow energy as compared with the other two cases. This could be understood as follows: for the 31-step spillway, compared with the other two cases. This could be understood as follows: for the 31-step spillway, therethere is anisinsufficient bottom theincoming incomingflow flow velocity, the flow an insufficient bottomfriction frictionto to dampen dampen the velocity, andand the flow couldcould jumpjump on the step surface and regain their energy before reaching the next step. On the other on the step surface and regain their energy before reaching the next step. On the other hand,hand, too many steps maymay notnot bebe beneficial the spillway spillwaybed bed could become similar torough the rough too many steps beneficialeither, either,since since the could become similar to the bed of channel under such Therefore, energy dissipation bedanofopen an open channel under sucha adense densestep step configuration. configuration. Therefore, thethe energy dissipation efficiency of the spillway could decreaseaccordingly. accordingly. ItItmight thatthat there should exist exist efficiency of the spillway could decrease mightbebeconcluded concluded there should an optimum step dimension and number for a particular spillway design in view of achieving the the an optimum step dimension and number for a particular spillway design in view of achieving maximum energy dissipation, which depends on the inflow conditions, bed slope and spillway maximum energy dissipation, which depends on the inflow conditions, bed slope and spillway length, length, etc. A large series of experimental or numerical trails would be required to find out the most etc. A large series of experimental or numerical trails would be required to find out the most desirable desirable spillway configuration. spillway configuration.
Exp SPH
100
energy dissipation rate (%)
80 60 40 20 0 -20 -40
0+070.0
0+060.0
0+050.0
0+040.0
0+030.0
0+020.0
0 . 0 0 0 + 0
0+010.0
-60
pile number (m)
(a) Exp SPH
100
60 40 20 0 -20 -40
pile number (m)
(b) Figure 12. Cont.
0+070.0
0+060.0
0+050.0
0+040.0
0+020.0
0+010.0
-80
0+030.0
-60
0+000.0
energy dissipation rate (%)
80
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Exp SPH
100
energy dissipation rate (%)
80 60 40 20 0 -20 -40
0+070.0
0+060.0
0+050.0
0+030.0
0+020.0
0+010.0
0+000.0
-80
0+040.0
-60
pile number (m)
(c) Figure 12. Energy dissipation curves along the downstream direction of spillway for Figure 12. Energy dissipation curves along the downstream direction of spillway for different step different step numbers: (a) 62-step; (b) 45-step; (c) 31-step. numbers: (a) 62-step; (b) 45-step; (c) 31-step.
6. Conclusions
6. Conclusions
This study demonstrates the great potentials of the SPH modeling technique in practical engineering of a stepped spillway flow. Byofusing open source code parallelSPHysics This studyapplications demonstrates the great potentials the the SPH modeling technique in practical to a documented benchmark and a self-implemented based on the prototype engineering applications of a stepped spillway flow.laboratory By using spillway the openflow source code parallelSPHysics hydraulic dambenchmark project, the and numerical accuracy of the laboratory model is fully evidenced. proposed to a documented a self-implemented spillway flowThe based on thepushprototype paddle inflow model can generate a quite quite stable inflow hydrograph, which is also easy to hydraulic dam project, the numerical accuracy of the model is fully evidenced. The proposed program into the SPH numerical scheme and effective in reducing the computational domain by a push-paddle inflow model can generate a quite quite stable inflow hydrograph, which is also easy large margin. The SPH computations are critically validated by the velocity and pressure data to program into the SPH numerical scheme and effective in reducing the computational domain by through either the literature or the laboratory measurement. An error analysis showed that the a large margin. SPH computations are critically validated velocity andhorizontal pressure data average errorsThe between the numerical and experimental results by arethe 0.8–4.3% in the through either the literature the laboratory measurement. Andistributions error analysis showed that the pressures and 0.7–4.7% in theorvertical pressures. Besides, the pressure on the horizontal average thespillway numerical experimental are 0.8–4.3% horizontal pressures and errors verticalbetween faces of the stepand show significantly results different patterns. Onin thethe former it is always with peak value towards the outer edge the of the step with a decreasing trend to horizontal the corner, and and positive 0.7–4.7% in athe vertical pressures. Besides, pressure distributions on the while on the latter it has alternatively positive and negative pressure values, which is highest vertical faces of the spillway step show significantly different patterns. On the former it near is always the step and lowest near thethe step edge.edge In the three different step configurations positive withcorner a peak value towards outer of study, the step with a decreasing trend to theare corner, i.e., it thehas step numbered atpositive 62, 45 and 31.negative The computed energy dissipations also agree near whileconsidered, on the latter alternatively and pressure values, which is highest with the experimental data within an error of 8.2%, 3.1% and 5.8%, respectively, for the above three the step corner and lowest near the step edge. In the study, three different step configurations are step conditions. One highlight of the present study is that the SPH pressure computations on the step considered, i.e., the step numbered at 62, 45 and 31. The computed energy dissipations also agree faces are quantitatively validated by the measured experimental data. It seems the parallelSPH could withprovide the experimental withintool an error 8.2%, 3.1% andflows 5.8%,inrespectively, for the above a promisingdata simulation for theofstepped spillway a laboratory scale with great three step potentials conditions. One highlight of the present study is that the SPH pressure computations on the step in real hydraulic engineering. faces are However, quantitatively validated by the measured experimental data. It seems the parallelSPH it should be realized that SPH still has quite a few limitations in its application to thecould spillway flows. Compared withtool the for gridthe modeling technique such as RANS model, SPHscale consumes provide a promising simulation stepped spillway flows in a laboratory with great more CPU time, especiallyengineering. for a lower water head over the spillway crest, when a very refined particle potentials in real hydraulic resolution be needed to achieve realistic simulation results. Besides, the stepped flowto the However,must it should be realized that SPH still has quite a few limitations in its spillway application is often associated with the water-air two phase interactions. A single-phase SPH model would not spillway flows. Compared with the grid modeling technique such as RANS model, SPH consumes be able to accurately simulate the real flow situations and a two-phase SPH model should provide a more CPU time, especially for a lower water head over the spillway crest, when a very refined particle more convincing simulation technique. On the other hand, it should also be realized that SPH has resolution must be needed to achieve realistic simulation results. Besides, the stepped spillway flow very superior advantages when dealing with the large deformation or broken free surfaces associated is often associated the water-air two phase interactions. A single-phase SPH model would with the flow jetwith or flow-step interactions over the spillway. Due to its mesh-free nature, SPH can not be able to accurately simulate the real flow situations and a two-phase SPH model should provide capture these complex free surface patterns much better than the RANS modeling techniques. a more convincing simulation technique. Onapplications the other hand, should alsohave be realized that SPH has Besides, the more recent promising SPH in theit water field been vigorously by [49,50]. very demonstrated superior advantages when dealing with the large deformation or broken free surfaces associated
with the flow jet or flow-step interactions over the spillway. Due to its mesh-free nature, SPH can capture these complex free surface patterns much better than the RANS modeling techniques. Besides, the more recent promising SPH applications in the water field have been vigorously demonstrated by [49,50].
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Acknowledgments: This research work is supported by the National Natural Science Foundation of China (No. 51479087), National Key R&D Program of China (No. 2017YFC0404303; No. 2017YFC0403600; No. 2017YFC0403603), Chinese Ministry of Water Resources Special Funds for Scientific Research on Public Causes (No. 201501028), Science and Technology Projects State Grid Corporation of China (No. 52283014000T), and Open Research Fund Program of State Key Laboratory of Hydroscience and Engineering, Tsinghua University (No. 397SKLHSE-2015-B-02) and Open Research Fund Program of State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University (No. SKHL1512). Author Contributions: S. Gu, L. Ren and X. Wang made the computations and data analysis; H. Xie guided the engineering project and provided the data; J. Wei and Y. Huang drafted the manuscript; S. Shao made the proof reading and editing. All authors contributed to the work. Conflicts of Interest: The authors declare no conflict of interest.
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