Applied Mathematical Modelling 38 (2014) 4308–4322

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Applied Mathematical Modelling

journal homepage: www.elsevier .com/locate /apm

Application of Smoothed Particle Hydrodynamics for modellinggated spillway flows q

http://dx.doi.org/10.1016/j.apm.2014.05.0080307-904X/Crown Copyright � 2014 Published by Elsevier Inc. All rights reserved.

q This article belongs to the Special Issue: Topical Issues drawn from CFD2012 on CFD in the Minerals and Process Industries.⇑ Corresponding author. Tel.: +61 3 95458010; fax: +61 3 95458080.

E-mail address: mahesh.prakash@csiro.au (M. Prakash).

Kate Saunders a, Mahesh Prakash a,⇑, Paul W. Cleary a, Mark Cordell b

a CSIRO Computational Informatics, Private Bag 33, Clayton South, VIC 3169, Australiab SMEC, PO Box 6208, St. Kilda Road Central, VIC 8008, Australia

a r t i c l e i n f o a b s t r a c t

Article history:Received 27 August 2013Received in revised form 15 May 2014Accepted 15 May 2014Available online 27 May 2014

Keywords:Smoothed Particle HydrodynamicsSpillwayWeir flowGatedDischarge ratePhysical model

Computational models of spillways are important for evaluating and improving dam safety,optimising spillway design and updating operating conditions. Traditionally, scaled downphysical models have been used for validation and to collect hydraulic data. Computationalfluid dynamics (CFD) models however provide advantages in time, cost and resourcereduction. CFD models also provide greater efficiency when evaluating a range of spillwaydesigns or operating conditions. Within the present literature, most studies of computa-tional spillway models utilise a mesh-based method. In this work we use the particle basedmethod of Smoothed Particle Hydrodynamics (SPH) to model weir flow through a four bay,gated, spillway system. Advantages of SPH for such modelling include automatic represen-tation of the free surface flow behaviour due to the Lagrangian nature of the method, andthe ability to incorporate complex and dynamic boundary objects such as gate structures ordebris. To validate the SPH model, the reservoir water depth simulated is compared with arelated physical study. The effect of SPH resolution on the predicted water depth is evalu-ated. The change in reservoir water level with discharge rates for weir flow conditions isalso investigated, with the difference in simulated and experimental water depths foundto range from 0.16% to 11.48%. These results are the first quantitative validation of theSPH method to capture spillway flow in three dimensions. The agreement achieved dem-onstrates the capability of the SPH method for modelling spillway flows.

Crown Copyright � 2014 Published by Elsevier Inc. All rights reserved.

1. Introduction

The ability of a dam to safely pass extreme flood events is affected by the maximum discharge capacity of its spillwaysand the decisions made regarding the spillway operation. Traditionally to assess spillway operation, physical models wereused. Computational spillway models however provide the advantage of time and cost efficiency, as well as the ability tocollect vast, accurate amounts of hydraulic data with ease and reproducibility. The role of physical models has thereforechanged to supplementation and validation of the computational models.

Historically most computational model studies of spillways have utilised mesh-based techniques. Savage and Johnson [1]simulated flow over a simple ogee crested spillway using the mesh-based, finite difference method with volume of fluid rep-resentation, Flow-3D [2]. Simulations were performed by assuming unit thickness along the width direction since the flow

K. Saunders et al. / Applied Mathematical Modelling 38 (2014) 4308–4322 4309

was essentially 2D in nature. The study demonstrated the capability of computational fluid models to recreate spillway flow,with simulated pressures on the spillway and discharge rates comparing well with physical data. Gessler [3] also used themesh-based Flow-3D, simulating spillway flow in three-dimensions. In contrast to other studies, the discharge rate was usedas the control variable and the reservoir water elevation was measured.

Studies of spillway flow using mesh-based techniques have included gated spillway designs. Chanel and Doering [4] sim-ulated weir flow over three ogee crested spillways, each with a different design head to spillway height ratio. Only represen-tative portions of the spillway gates were considered by using reflective boundary conditions. The computational modelimplemented was Flow-3D, in combination with a renormalized group turbulence model. In the study, simulated dischargerates were compared at a range of fixed headwater levels to rates from a scaled physical model. Results ranged from an overprediction of 9% to an under prediction of 24.4% in the levels as compared to the physical study. The study attributed thelarger variations in the discharge predictions to the results being mesh dependent in some cases, as well as possible issueswith the turbulence model.

To resolve orifice flow through a gated spillway with a mesh-based method and to achieve accurate results a nested meshwas required by Chanel and Doering [5]. These authors simulated orifice flow through the same spillways with different gateopenings, again comparing simulated discharge to physical model discharge. For a 4 m gate opening with a 0.25 m mesh res-olution (16 elements across the opening) the predicted discharge over-estimated the physical model by as much as 15%. Theextent of over-estimation did not change significantly with a mesh resolution variation from 1.0 to 0.25 m. A nested meshapproach with a resolution of 0.5 m away from the gate to a fine mesh of 0.125 m very close to the gate resulted in a sig-nificant improvement in the prediction, with an over-estimate of less than 2% for all flow rates analysed.

Li et al. [6] demonstrated the possibility of using computational fluid models as an alternative to costly physical modelexperiments when designing spillways for peak maximum flood conditions. In this study, weir flow was simulated in 3Dthrough an auxiliary spillway consisting of six gates. Simulations were performed using the mesh-based volume of fluid sol-ver, Fluent, and with a j–e turbulence model. Five modifications to the spillway’s geometry were investigated to determinewhich geometry best alleviated areas of large flow separation and recirculation. The selected spillway geometry was eval-uated with a physical test and peak maximum flood conditions were validated.

When undertaking spillway modelling there is a compromise between computational efficiency and accuracy. Lv et al. [7]considered the impact of mesh resolution on simulated results in a convergence study. Weir flow was simulated in twodimensions over a broad-crested weir using a mesh-based hybrid level set and volume of fluid method. Simulations wereperformed using four meshes of increasing resolution ranging from 0.03 to 0.005 m. The study concluded that a resolutionof 0.01 m provided the best balance of accuracy and computational efficiency to resolve headwater levels of between 0.1 and0.8 m (10–80 elements). The percentage error for this optimal resolution ranged from an under prediction of 10.2% to anover-prediction of 2.3%.

An alternative to mesh-based modelling is the particle based method of Smoothed Particle Hydrodynamics (SPH). Devel-oped initially by Monaghan [8], SPH has been applied in studies of geophysical flows, such as tsunamis and volcanoes [9],dam break flows [10,11] and reactive transport and mineral precipitation [12]. Compared to mesh-based methods, potentialadvantages of SPH for spillway modelling include; automatic representation of free surface flow behaviour due to theLagrangian nature of the method, the ability to incorporate complex three dimensional geometries, and the ability to includedynamic boundary objects such as gate movements, debris or ice formations. SPH can also easily capture splashing and flowfragmentation which can affect pressure distributions lower down in the spillway, particularly at high discharge rates. Atthis stage, longer computation times remain a current perceived shortcoming of SPH relative to mesh-based methods. Referto De and Bathe [13] for a discussion on efficiency comparison between mesh based and mesh free methods. While activestrands of research in SPH typically include improving the efficiency of SPH solvers methods and the accuracy of the num-erics, it is possible that the inherent advantages associated with SPH are already able to more than compensate for any per-ceived shortcomings when it comes to solving such dynamic flows in dynamic geometries. This sort of issue requiresinvestigation, as well as the issue of turbulence modelling for high Reynolds number flows since these are relatively newconcepts in the SPH literature.

Investigations into the application of SPH for modelling spillway flow are limited. Ferrari [14] used the standard weaklycompressible SPH in 2D to simulate flow over a sharp-crested weir profile. The investigation was done for a single flow rate,and with limited comparison between the simulated free surface profile and experimental results of Scimeni [15]. An SPHparticle spacing of 0.0075 m was used to resolve a headwater level of 0.55 m with 74 particles, but no resolution studywas presented. Lee et al. [16] simulated flow over a simple ski-jump spillway in 3D using the standard weakly compressibleSPH formulation but results were only visually compared. An SPH particle size of 0.2 m was used to resolve a spillway widthof 4 m (20 particles).

Improved forms of the standard SPH method have been suggested for various applications. For the current study some ofthe relevant literature in this context is specifically related to improving the pressure signals from the SPH model. Molteniand Colagrossi [17] introduced a density diffusion term in the mass conservation equation to deal with a noisy pressure fieldin the fluid flow calculation. Such a modification may help with comparing the pressure signals from experiments to thenumerically obtained values for spillway design. Ferrari [18] proposed a new parameter free SPH scheme that is able toachieve a high degree of pressure field accuracy and at the same time is also able to accurately track the free surface behav-iour. Along similar lines Antuono et al. [19] proposed a modified form with diffusive terms in both the continuity and energyequations to deal with high frequency numerical acoustic noise and to smooth the pressure field. While the measures

4310 K. Saunders et al. / Applied Mathematical Modelling 38 (2014) 4308–4322

suggested here are plausible for improving solutions from standard SPH implementations, their ability to result in improvedsimulations of real spillway flows is not known to us as yet. In this paper, we investigate the use of the weakly compressible(WC) SPH method [10] to model flow in three dimensions (3D) through a four bay, radial gated, submerged spillway system.The computational model includes the spillway wall, full reservoir and surrounding topography. Whilst 2D models or peri-odic segments of the spillway gates have been used in other studies [14,4], a 3D model is essential for this application toconsider the transverse variation of the reservoirs free surface with proximity to the spillway gates, and will be requiredfor future work where the gates will be opened to varying extents.

The purpose of this study is to extend the applicability of SPH for spillway modelling through quantitative comparisonwith physical model measurements. The study is comprised of four components:

1. SPH’s ability to capture the complex free surface behaviour through the spillway gates is demonstrated. The flow of waterdown the ski-jump and projected away from the spillway base is also captured.

2. We examine the extent of variation across the reservoirs free surface. This is important to understand, given that thegates can open and close to varying extents. Understanding the variation also allows for more accurate interpretationof results as there is also uncertainty with the sensor location used in the related physical study.

3. The sensitivity of simulated results to increased particle resolution is investigated: Resolution is particularly importantgiven the influence of gated structures on the flow behaviour [5].

4. A range of flow rates that induce weir flow conditions at the spillway gates are also modelled to determine how SPH han-dles the relationship between the reservoir water level and discharge behaviour.

The results of the study through extensive validation with physical model results, aim to provide confidence in the stan-dard WC SPH method for modelling spillway flows. Depending on the outcomes established in this baseline study with thestandard WC SPH method, improvements to the SPH method to investigate the pressure field and comparison with exper-imentally obtained pressure signals may be formulated for a future study.

2. Computation method and model design

2.1. The SPH methodology

In this section, we present an overview of the SPH methodology used in simulation. For a comprehensive review of themethod, readers are referred to Monaghan [20,8] and Cleary et al. [21]. Within the SPH method, particles are used to repre-sent the volume of fluid. Each particle has properties of mass, m, density, q, velocity, v and position, r. An interpolation kernelis used to smooth the discrete information stored by particles, to give smooth continuous fields, such as density and pres-sure. The interpolated value of any field A, at a position r is calculated by

AðrÞ ¼X

b

mbAb

qbWðr� rb;hÞ; ð1Þ

where the sum is over all particles b within a radius 2h of r and h is the interpolation length. Here W(r, h) is a spline basedinterpolation or smoothing kernel with radius 2h, that approximates the shape of a Gaussian function with compact support.

The cubic kernel is used for all calculations in this paper where:

WðuÞ ¼ a23� u2 þ 1

2u3

� �; if 0 � u < 1;

WðuÞ ¼ bð2� uÞ3; if 1 � u < 2;WðuÞ ¼ 0; if u � 2;

ð2Þ

where the constants a and b are:

In 2D : a ¼ 3=2p b ¼ 1=4p;In 3D : a ¼ 15=7p b ¼ 15=42p;

The gradient of function A is given by

rAðrÞ ¼X

b

mbAb

qbrWðr� rb;hÞ: ð3Þ

Using these interpolation formulas, systems of partial differential equations can be reformulated as coupled ordinary dif-ferential equations and solved using suitable time integration methods. We denote the position vector from particle b to par-ticle a by rab ¼ ra � rb. For the distancejrabj, Wab ¼Wðrab;hÞ is the interpolation kernel evaluated with smoothing length h.The continuity equation used in the SPH formulation is therefore (Monaghan [20]):

dqa

dt¼X

b

mbðva � vbÞrWab: ð4Þ

K. Saunders et al. / Applied Mathematical Modelling 38 (2014) 4308–4322 4311

This form of the continuity equation is Galilean invariant, as the positions and velocities appear only as differences.The momentum equation used to predict particle motion is

dva

dt¼ g�

Xb

mbPb

q2b

þ Pa

q2a

� �� n

qaqb

4lalb

ðla þ lbÞvabrab

r2ab þ g2

� �raWab; ð5Þ

where Pa and la are pressure and viscosity of particle a, g is the gravity vector, and vab ¼ va � vb. The small parameter, g isused to smooth out the singularity at rab = 0 and the parameter, n, is associated with the viscous term (Cleary [22]).

To calculate the pressure used in the momentum equation (4), we use the following equation of state,

P ¼ P0qq0

� �c

� 1� �

; ð6Þ

where P0 is the magnitude of the pressure and q0 is the reference density. For water the exponent c = 7 is used. The pressurescale factor P0 is given by:

cP0

q0¼ 100V2 ¼ c2

s ; ð7Þ

where V is the characteristic or maximum fluid velocity and cs is the speed of sound. The speed of sound is chosen to be muchlarger than the maximum velocity in the flow to ensure that the density variation is less than 1% and the flow can beregarded as incompressible.

2.1.1. Boundary particlesIn SPH, solid boundaries are also modelled using particles. These ‘‘boundary particles’’ are assigned properties similar to

the moving ‘‘fluid’’ particles: mass, position, density, viscosity, etc. This is analogous to using grid points along boundaries ingrid-based methods. To avoid penetration of fluid particles through the solid walls, the boundary particles are prescribedwith boundary forces that are exerted on fluid particles in the normal direction. The boundary forces used in this paperare of a Lennard-Jones form,

f ðaÞ ¼Aðl=aÞs if 0 < a < l;

Dða� bÞ2 if l < a < b;

0 otherwise;

8><>: ð8Þ

where,

D ¼ Asþ 2

2b

� �2

l ¼ sbsþ 2

; ð9Þ

The exponent s determines the steepness of the boundary force, and is typically of value 8. The boundary force factor Adetermines the magnitude of the boundary force, and is generally 0:01c2

s . All boundaries modelled in this paper are of a fixednature.

2.1.2. Time stepping and integrationThe time stepping used in the solver is explicit and is limited by the Courant condition which is modified for the presence

of viscosity:

Dt ¼mina

0:5h cs þ2nla

hqa

� ��� �; ð10Þ

where cs is the local speed of sound.For the time integration, an Improved Euler scheme is used with a predictor and corrector step as below for the velocity, v,

displacement, x and density q:Predictor step:

v� ¼ vn þDt2

Fmomn xn�1=2; vn�1=2

;

x� ¼ xn þDt2

vn;

q� ¼ qn þDt2

Fdenn xn�1=2; vn�1=2

;

ð11Þ

where the subscript n refers to the current time step and the superscripts mom and den stand for momentum and densityfunctions.

4312 K. Saunders et al. / Applied Mathematical Modelling 38 (2014) 4308–4322

Corrector step:

vnþ1 ¼ vn þ DtFmomnþ1 xnþ1=2; vnþ1=2

;

xnþ1 ¼ xx þ Dtv�;

qnþ1 ¼ qn þ DtFdennþ1 xnþ1=2;vnþ1=2

:

ð12Þ

2.1.3. Inflow/outflow conditionsIn the work presented here fluid particles are introduced in the domain through an inflow condition. This is because set-

ting up the entire domain at a high SPH particle resolution is computationally expensive. The inflow boundary condition con-sists of three sets of equally spaced rectangles each filled with equally spaced particles. The particle spacing is maintained asclose to the fluid particle spacing as possible in all directions in this zone. The particles in the two rectangles that are notadjacent to dynamic particles in the simulation are held stationary. The dynamic particles follow the Navier–Stokes equa-tions of motion. Particles in the third rectangle closest to the dynamic particles are dynamically released into the simulationat a speed set to represent the spillway discharge rate (see Section 2.2). When these particles are released, the two rectanglescontaining stationary particles are moved forward toward particles in the simulation at the discharge rate and one of thesesets is then released. A new rectangle of stationary particles is then created. This approach of creating the inflow particles issimilar to the one described in Federico et al. [23]. The outflow condition in these simulations is relatively simple. Particlesare removed from simulation, with their masses and densities no longer considered when the particles flow down the terrainand reach the end of the computational domain.

2.1.4. Related considerationsThe SPH method used in this paper does not impose any explicit turbulence model. The standard SPH method has under-

lying dissipation that contributes implicit turbulence modelling (analogous to multiple mesh-based methods), which is ableto damp out the effect of turbulence thereby resulting in converged solutions even at higher Reynolds numbers. This doesnot mean that an explicit treatment of turbulence will not have a beneficiary effect on the solution. At this stage, the addi-tional imposition of a turbulence model without understanding the origin and mechanism of the dissipation in the standardSPH formulation leads to additional complexity without any benefit, and is hence unnecessary and not pursued.

The SPH method used here also features a constant particle resolution. The use of a variable resolution SPH algorithmwould have beneficial effects in solution speed as well as possibly resolving boundary layers especially close to the walls.However the complicated nature of the spillway flow would require SPH nodes to be able to combine and divide and alsopreserve mass at the same time in a variable resolution implementation. This implementation was therefore not exploredat this investigatory phase in this project.

2.2. Physical model design

The Pala Tiloth dam is a 74 m high concrete gravity arch dam, with a crest elevation of 1670 m and crest length of 141 m.For flood control and reservoir management, the dam has a four bay, radial gated, submerged spillway system located at anelevation of 1631.53 m. Each of the spillway bays have dimensions 10.5 m wide by 11.9 m tall, with a combined design dis-charge capacity of 9200 m3/s for peak maximum flood. The spillway design uses a ski-jump to project the water dischargedhorizontally away from the dam wall base. This design reduces erosion effects at the dam wall base and thereby improvesdam stability and safety.

Physical testing of the spillway design was conducted by SMEC International Pty Ltd [24] using a scaled model of ratio1:60. The four gated structures of the physical model are shown in Fig. 1(a) and the entire model domain is shown inFig. 1(b). Water discharging from the spillway gates, flowing down the ski-jump and projected horizontally away fromthe spillway base is shown for a discharge rate of approximately 1000 m3/s in Fig. 1(c). Measurements taken in the physicalexperiments included the reservoir water elevation for a range of discharge rates with the gates both fully and partiallyopened. Pressures were also recorded along the centreline of the ski-jump and sediment transfer in the downstream plungepool was considered.

Froude number similarity can be used to compare the results from the model to the full scale spillway as the spillway flowconditions are governed by the ratio of inertial to gravitational forces, i.e.,

VffiffiffiffiffigL

p !

m

¼ VffiffiffiffiffigL

p !

sp

; ð13Þ

where V is a characteristic velocity (m/s), g is acceleration due to gravity and L is the length scale. Subscripts m and sp areused to represent the scaled model and full scale spillway respectively.

Defining the length scale ratio as Lr = Lsp=Lm and given the full spillway to model ratio is 60:1, we obtain Lr = 60. Using Eq.(13), the following secondary ratios are then obtained:

Velocity : Vr ¼Vsp

Vm¼ 7:75 : 1; ð14Þ

Fig. 1. Photograph of the four bay radial gated spillway used in the physical model testing: (a) perspex spillway gates used in testing (b) entire physicalmodel domain, including surrounding terrain (c) flow projected from the ski-jumps for a scale flow rate corresponding to 1000 m3/s.

K. Saunders et al. / Applied Mathematical Modelling 38 (2014) 4308–4322 4313

Discharge : Q r ¼ VrL2r ¼

Q sp

Qm¼ 27885 : 1; and ð15Þ

Reynolds number : Rr ¼Rsp

Rm¼ 465 : 1: ð16Þ

These secondary ratios allow for interpretation of the physical results at the full length scale. For example a model dis-charge of 35.86 L/s is equivalent to a spillway discharge of 1000 m3/s.

2.3. Computational model design

The numerical model was constructed at the same dimensions as the physical model. This allows direct comparison ofsimulated results to physical model results. The numerical model is used here primarily for validating it against the physicalmodel results. Simulations have not been performed at the full scale. However the flow physics in the physical and full scalemodels are comparable such that the flow can be expected to be fully turbulent in both instances. The numerical modelincluded a 3D CAD geometry of the spillway constructed from schematics shown in Fig. 2. A digital terrain model (DTM)of the topography surrounding the spillway, including the reservoir terrain and downstream areas, was also included.

The spillway opening width and height on the physical scale model were 0.1750 m (corresponding to actual scale 10.5 m)and 0.198 m (actual scale 11.9 m) respectively. Six SPH resolutions ranging from 0.01 to 0.035 m were used (0.6–2.1 m actualscale) to test the dependence of results on resolution. The number of SPH particles in the simulation for representing thefluid, terrain and the spillway geometry are given in Table 1 for each resolution. The fluid viscosity and density in the sim-ulations were that of water. Discharge rates ranging from 8.96 L/s (250 m3/s full scale) to 62.76 L/s (1750 m3/s full scale),increasing in increments of 8.96 L/s, were analysed. For consistency, all quantities are reported below in the physical modelscale.

Fig. 2. Two dimensional profile of the spillway gate and ski-jump as provided by SMEC. This schematic and photographs of the physical model were used toproduce the 3D computational geometry of the spillway used in simulations.

Table 1Number of SPH particles in simulation for each resolution used.

SPH resolution (m) Fluid Terrain Spillway(�103) (�103) (�103)

0.010 5392 1937 2420.015 2110 878 1980.020 972 490 610.025 589 287 510.030 438 224 500.035 242 148 25

4314 K. Saunders et al. / Applied Mathematical Modelling 38 (2014) 4308–4322

Fig. 3 shows 3 orthographic views and 1 oblique view of the model configuration. The reservoir water surface shown inFig. 3 is the level prior to commencement of flow through the spillway gates, defined to be time zero seconds. Similar to thephysical model, the reservoir area was enclosed by a wall at the back as shown in the top view. The location of this wall was1.855 m from the spillway gates. A rectangular inflow was positioned along the back wall with the top of the inflow belowthe bottom of the spillway gates (shown in Fig. 3c). The inflow was positioned as far as possible from the spillway gates toensure negligible effects on the discharge through the spillway gates. The reservoir water level was used to compare simu-lations with physical measurements. The reservoir water level was measured when the spillway simulation reached steadystate, which was when the discharge rate through the spillway was equal to the specified inflow rate and the reservoir waterlevel became invariant.

Twenty-five sensors were used to measure the reservoir water level during simulation. Twenty sensors were positionedin a gridded layout, another four sensors were positioned at the spillway crest and one sensor was placed to measure thewater level in the centre of the reservoir (Fig. 3c). Increased sampling was required closer to the spillway gates where greaterfree surface variation was observed. To address this, five extra sensors for each grid column were randomly generated using auniform distribution. These sensors were added to each column, ranging up to 0.35 m behind the spillway wall (Fig. 3c). Tocompare sensor readings an appropriate datum was needed as the terrain is uneven. The height of the terrain at the point ofintersection with the middle of the spillway base, �0.0905 m, was used as the datum.

For the highest particle resolution used, the simulation took 28 days to reach steady state when running in parallel usinga dual Xeon 8-core E5-2650 machine. In contrast, the lowest resolution simulation took 17 h to reach steady state. Fastersimulation would occur if the position of the initial water level was closer to the steady state level, as opposed to abuttingthe base of the spillway gates.

Fig. 3. Front (a), back (b), top (c) and oblique (d) views of the computational model setup. The front and back views show the four spillway gate openings.The red box in the top view indicates the position of the inflow inset within the spillway wall. The coloured circles in the top view show the locations of thewater height sensors. Sensors are coloured the similarly if they are the same perpendicular distance to the spillway wall. (For interpretation of thereferences to colour in this figure legend, the reader is referred to the web version of this article.)

K. Saunders et al. / Applied Mathematical Modelling 38 (2014) 4308–4322 4315

4316 K. Saunders et al. / Applied Mathematical Modelling 38 (2014) 4308–4322

3. Flow visualisation

Fig. 4 shows a visualisation of the flow through the spillway gates and down the ski-jump at 10, 18, 26 and 65 s for aninflow rate of 35.86 L/s in the physical scale model (corresponding to 1000 m3/s full scale) and particle size of 0.01 m(0.6 m full scale).

At 10 s, the water height above the spillway crest is 0.079 m (4.74 m full scale). Water has just begun to exit throughthe spillway gates with only a thin layer of water present on the ski-jump. A modest amount of fragmentation of water isvisible on the ski-jump. The velocity of the water exiting the ski-jump is approximately 1.25 m/s. At 18 s, the water heighthas increased to 0.109 m and the ski-jump has filled leading to a continuous stream of water with no fragmentation. Thevelocity of the water on the ski-jump is no longer constant, with a velocity profile that changes proportionally to changesin the slope of the ski-jump. The water exit velocity has increased to around 1.8 m/s. By 26 s, the water height hasincreased to 0.124 m and the velocity profile of the water in the ski-jump is approaching a steady state. The maximumvelocity of water is close to 2.5 m/s. A continued widening of the flow exiting the ski-jump is also seen in comparisonto flow visualisations of 10 and 18 s. Unlike the flow behaviour at 10 s, where water falls away from the ski-jump withlow energy, the water exiting the ski-jump at 26 s is propelled upward and horizontally away from the dam base. Cap-turing the water projected from the ski-jump is essential as this spillway design feature minimises erosion effects closeto the dam base thereby improving dam stability and safety. Steady state flow behaviour is reached at 65 s and the waterheight above the spillway crest stabilises at approximately 0.139 m. The velocity profile of water has also reached a steadystate.

Given the importance of capturing the trajectory of water from the ski-jump, the steady state behaviour of the flow in sideprofile is shown in Fig. 5. This visualisation of the flow shows the free surface shape and the magnitude of velocity. Fig. 5 alsocaptures the trajectory of water from the ski-jump and the point of impact of the water on the terrain. The simulated flowdown the ski-jump, such as the water trajectory and the width of flow, shares good qualitative agreement with flow behav-iour as seen in the physical model, Fig. 2(c). Note when considering the point of impact on the terrain, downstream waterwas not included in the computational model but was in the physical model. Also visible in side profile, Fig. 5, is water pool-ing beneath the spillway which recirculates when visualised dynamically.

4. Variations in the reservoir free surface

In this section, we analyse the transverse variation of the reservoir’s free surface. The free surface will vary with proximityto the spillway gates, and this variation is important to understand as the exact sensor location used in the physical testing isunknown. Examining the free surface is also important for determining if flow behaviour through a singular bay is represen-tative of all bays and for later investigations when the gates will be opened to varying extents. A discharge rate through thespillway gates of 35.86 L/s (1000 m3/s full scale) was simulated using a particle size of 0.01 m (0.6 m full scale). The simu-lated water depth at the gridded sensor locations is given in Table 3 and the sensor locations relative to the spillway areshown in Fig. 3(c).

For sensors that are positioned inline and perpendicular to spillway wall (Z axes), the reservoir’s free surface changes byas much as 0.0532 m (3.192 m full scale) with proximity to the spillway gates., This variation highlights the need for theexact sensor location used in physical testing to accurately compare simulated with experimental results (Table 3).

Variation in the free surface measured at sensors parallel to the spillway wall is less severe than in the perpendiculardirection (Table 3). For sensors positioned on the spillway crest (black in Fig. 3c), the transverse variation between read-ings is 0.01 m (0.6 m full scale). For sensors 0.087 m behind the spillway crest (pink in Fig. 3c), the transverse variationbetween readings is 0.0112 m (0.672 m full scale). For sensors positioned in parallel 0.36 m or greater behind the spillwaycrest, the reservoir’s free surface exhibits less transverse variation, up to 0.0027 m (0.162 m full scale). This insensitivity tolocation is expected as the reservoir water level converges to a fixed value away from the spillway gates. The transversevariation of the reservoir’s free surface sampled at sensors parallel to the spillway wall and away from the spillway gatesis consistent with a similar study by Gessler [3], where variations of up to 0.3 m full scale were observed within the res-ervoir water level.

Cross-sectional profiles of the reservoir were considered perpendicular to the spillway wall to compare the variations ofthe reservoir’s free surface in more detail. The cross-sections were approximated using a line of best fit of the form,y = m(1/z) + c, where c is the y intercept and m is the gradient. Table 4 gives the equations of the lines of best fit for eachtwo-dimensional cross-section, and the corresponding R-squared values. From the R-squared values, we observe that foreach of the lines of best fit parallel to centrelines of the ski-jumps, the inverse transform of the Z coordinate (1/z) is ahighly accurate fit. The fit is less accurate for the fixed X coordinate of �0.017. This follows as the X coordinate doesnot correspond to a spillway gate, so the behaviour of the water free surface closer to the spillway wall is different. Eachof the lines of best fit are shown using solid lines in Fig. 7. In general the lines of cross-sectional profiles corresponding tothe central spillway gates are translated lower for flow approaching the spillway gates. It is important to note that theinclusion of sensors within the spillway gates in the calculation, such as those on the crest, alters the accuracy of thefit. This follows intuitively as the free surface behaviour is expected to change as the water flows through the spillwaygate.

Fig. 4. Flow development through the spillway for a discharge rate of 35.86 L/s, a particles size of 0.010 m and times 10, 18, 26 and 65 s. The shading on thewater surface represents speed, with blue being 0 m/s, green intermediate and red being 2.5 m/s. (For interpretation of the references to colour in this figurelegend, the reader is referred to the web version of this article.)

K. Saunders et al. / Applied Mathematical Modelling 38 (2014) 4308–4322 4317

Fig. 5. Visualisation of steady state flow in side profile showing the trajectory of water off the ski-jump and point of impact of water on the terrain.

Table 2Simulated water depth for different resolutions. Percentage difference is between simulated reservoir depth and physical data. Relative percentage differencecompares lower resolution simulations with the highest resolution of 0.01 m.

Particle size (m) Simulated water depth (m) Percentage difference in depth Relative percentage difference

0.010 0.610 7.02 n/a0.015 0.612 7.37 0.330.020 0.611 7.19 0.160.025 0.623 9.30 2.130.030 0.635 11.40 4.100.035 0.653 14.56 7.05

Table 3Simulated water depth at the gridded sensor locations (Fig. 3) for an inflow rate of 35.86 L/s and particle size of 0.010 m.

TransectX1 = �0.17

TransectX2 = 0.138

TransectX3 = 0.446

TransectX4 = 0.754

Transect X5 = 1.062 Variation X axis

(No gate) (Gate 1) (Gate 2) (Gate 3) (Gate 4)

Z1 = �0.03(crest)

NA 0.5515 0.5553 0.5598 0.5615 0.0100

Z2 = �0.09 0.5986 0.5874 0.5893 0.5908 0.5925 0.0112Z3 = �0.36 0.6042 0.6026 0.6016 0.6015 0.6025 0.0027Z4 = �0.63 0.605 0.6036 0.6031 0.6024 0.6032 0.0026Z5 = �0.9 0.6057 0.6047 0.6034 0.603 No data (sensor error) 0.0027Variation Z axis 0.0071 0.0532 0.0481 0.0432 0.0417

Table 4Line of best fit for the reservoir free surface profiles with a fixed X coordinate (Table 3). The lines of best fit take an inverse transform of the Z coordinate andinclude the 5 extra sensors randomly distributed between Z = �0.36 and the spillway wall.

Transect X1 = �0.170 Transect X2 = 0.138 Transect X3 = 0.446 Transect X4 = 0.754 Transect X5 = 1.062(No gate) (Gate 1) (Gate 2) (Gate 3) (Gate 4)

Y intercept, c 0.6079 0.6083 0.6067 0.6059 0.6067Gradient, m 5.88 � 104 1.70 � 103 1.46 � 103 1.23 � 103 1.14 � 103

R-squared 0.87 0.99 0.99 0.98 0.99

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Two dimensional profiles of the reservoir (Fig. 7) show minimal variation in the free surface behaviour for those profilesin line with the centreline of the ski-jump. A single bay can therefore be used to represent flow behaviour exhibited in allfour bays if one is only interested in the flow dynamics in the spillway zone. However, two dimensional profiles start showvariations away from the centreline of the spillway ski-jumps (Fig. 7). For profiles taken in alignment with the centreline ofthe ski-jump an inverse transform is a highly accurate fit. The inverse transform may therefore be used as good approxima-tion for reservoir free surface behaviour for these instances.

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5. Resolution study

The accuracy of the simulated spillway flows was found to be dependent on the particle size used to resolve the flowbehaviour. A resolution study was therefore conducted to understand the trade-off between cost and accuracy for spillwayflow prediction using SPH. A fixed inflow rate of 35.86 L/s (1000 m3/s full scale) was used for the analysis.

Since the inflow rate was controlled, the simulated reservoir water level is the key prediction to be compared to the phys-ical model results. The simulated reservoir water level was measured at the sensor furthest from the spillway (Fig. 3c) toensure comparisons were not affected by variation of the free surface close to the spillway gates. The water level recordedin the physical testing for a discharge rate of 35.86 L/s was 0.570 m (34.2 m full scale).

The simulated reservoir water depths are compared to physical model results for different resolutions in Table 2. Twoquantities are used for comparison, namely:

(a) Percentage difference in depth which compares the simulated reservoir depth with physical data, and(b) Relative percentage difference comparing the depth obtained from lower resolution simulations with the highest res-

olution of 0.01 m (0.6 m full scale).

Results for the particle size of 0.015 m are comparable to the highest resolution simulation of 0.01 m, with a relative dif-ference of only half a percent. The particle size of 0.015 m therefore provides the desired balance of computational efficiencyand accuracy. Fig. 6 shows the percentage difference between the simulated water depth and the physical model depth forthe range of SPH resolutions.

At the highest resolution simulated; the percentage difference between the simulated and physically measured reservoirwater level is 7%. It is important to acknowledge that there is uncertainty with the sensor location used in the physical test-ing. The simulated percentage difference of 7% should therefore be interpreted as a worst case percentage difference, as thesampling location used for comparison is in the centre of the reservoir, as opposed to closer to the spillway wall where thewater surface profile decreases with proximity to the gates. Moreover, the disparity between simulation and experimentalwater level is reasonable given that experimental errors can be up to 5%, as stated by Gessler [3].

A possible simulation factor contributing to the 7% percentage difference could be the resolution is still too coarse. Toocoarse a resolution will affect the accuracy of the model since in SPH turbulence is accounted for inherently at a particlescale. In other spillway studies a much higher resolution has been used to resolve the headwater level (the water heightabove the spillway crest). For example, in a two-dimensional SPH model of a simple overflow spillway by Ferrari [14],the headwater level was resolved with 74 particles; however, a resolution study was not included, so it is unclear whetherthis amount of particles was necessary. In our study, at the highest resolution simulated, 20 particles were used to resolvethe headwater level.

Boundary layer effects could be compounded by gated structures, such as in the current spillway design. If this is indeed afactor, a very high resolution will be needed to more accurately resolve flow behaviours through the spillway gates. This canonly be achieved by using a variable resolution SPH model with fine particles closer to the gates. Omidvar et al. [25] used avariable particle mass SPH model for investigating wave body interactions. However this study was restricted to using onlystatic and discrete variations in resolutions and with no particle splitting/joining. A more recent study by Vacondio et al. [26]

Fig. 6. Percentage difference in simulated reservoir water depth for different particle resolutions for a discharge rate of 35.86 L/s.

Fig. 7. Lines of best fit for the reservoir free surface for fixed X coordinates and sensors in Table 2. These lines of best fit each include the 5 extra sensorsrandomly distributed between Z = �0.36 and the spillway wall. The discharge flow rate used was 35.86 L/s and particle resolution was 0.01 m.

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demonstrates the use of a variable particle splitting/joining algorithm in SPH that also conserves mass and momentum. Thistype of approach is suited to the current problem and will be the subject of a future investigation specifically targeted atspillway modelling.

We also refer to a mesh based study by Chanel and Doering [5], in which gated flows were modelled using a finite dif-ference based method. In this work, a uniform mesh approach, that was invariant to resolution, did not provide accurateagreement. A nested mesh with a higher resolution for the gated structures and lower resolution in the reservoir was neededto accurately resolve the gated flow. Despite the issues described above, the results in Fig. 6 suggest that resolution conver-gence occurs for a particle size of less than 0.02 m for the range of flow rates investigated here.

6. Weir flow

Weir flow occurs when the water flowing through the spillway does not contact the top of the gate opening. In this sec-tion, simulations of weir flow conditions at the spillway are presented using a particle size of 0.015 m. When the water con-tacts the top of the gate opening, the spillway operates under orifice flow conditions.

Fig. 8. Change in reservoir water depth with discharge rate at different distances from the spillway. The simulated reservoir water heights are shown inblue, orange, magenta and black, and the physical measurements are shown in red. (For interpretation of the references to colour in this figure legend, thereader is referred to the web version of this article.)

Table 5Comparison between simulated reservoir depth and physical model reservoir depth at the spillway crest and 1.2 m away from the spillway crest.

Rate (L/s) Model water depth (m) Simulated water depth (m)At crest/1.2 m from crest

Percentage difference in depthAt crest/1.2 m from crest

8.96 0.512 0.500/0.522 �2.34/+1.9517.93 0.534 0.523/0.558 �2.06/+4.4926.89 0.553 0.542/0.586 �1.99/+5.9735.86 0.570 0.561/0.612 �1.58/+7.3744.83 0.585 0.579/0.637 �1.03/+8.8953.79 0.598 0.595/0.658 �0.50/+10.0362.76 0.610 0.609/0.680 �0.16/+11.48

K. Saunders et al. / Applied Mathematical Modelling 38 (2014) 4308–4322 4321

Fig. 8 compares the water depths as a function of discharge rate at different distances from the spillway. The simulateddischarge rates are shown in blue, orange, magenta and black, while the physical model discharge rate is shown in red. Read-ings of sensors positioned behind the spillway gates were averaged across the X axis, with the colour of each line indicatingthe corresponding row of sensors in Fig. 3(c). In the above section; Variation of the Reservoir’s Free Surface, flow through onebay was considered representative of all four bays, so the average value of these sensors is a representative approximation ofthe reservoir water level. Fragmentation of flow occurred for water flowing through the spillway gates for the lowest flowrate simulated, which results from a lack of resolution. The irregular flow resulted in the sensors positioned on the spillwaycrest returning no data for some time-steps. The average value of readings for sensors in the crest and the lowest flow ratewas therefore not an accurate reflection of the height of the water at these points. Given that for the other flow rates andsensors the median and the mean of the sensor readings were comparable, the median of the sensor readings was usedto represent the height of the water on the spillway crest for the lowest flow rate simulated.

The predicted reservoir level measured at the spillway crest, as compared to other locations throughout the reservoir,provided the best agreement to the physical measurements. For all discharge rates simulated, the maximum difference insimulated reservoir level at the crest, as compared with experimental results, was only 2.4%, occurring at a flow rate of8.96 L/s. The predicted reservoir level measured in the centre of the reservoir, 1.2 m from the crest (darker blue in Fig. 8),provided the greatest difference to experimental results. At this location the maximum difference between the simulatedand experimental values is 11.50% and occurred at the highest discharge rate simulated of 62.76 L/s.

Table 5 compares the simulated water depth at the spillway crest and 1.2 m from the crest with the physical modelresults. At the spillway crest the simulated values are always below the physical measurements with the smallest differenceof 0.16% occurring at a discharge rate of 62.76 L/s and a maximum of 2.34% at 8.96 L/s. At 1.2 m from the crest the predictionsare always above the measurements with the difference ranging from 1.95% at 8.96 L/s to 11.48% at 62.76 L/s.

Uncertainty in physical sensor location impacts upon our ability to interpret the accuracy of the simulated results. If thephysical sensor was located close to the spillway crest, the SPH model agreement with the physical study is excellent. If thephysical sensor location was more central to the reservoir area, the agreement with physical model results is good at lowflow rates, but decreases as the flow rate increases. If the latter is the case, as mentioned in the section, Resolution Study,factors of turbulence and boundary layer effects may be influencing simulated results.

7. Conclusions

The ability of the particle based method, SPH, to model spillway flow through a four, bay gated spillway was investigated.The full spillway was modelled, demonstrating SPH’s capability to capture spillway flow in three dimensions, and to resolvethe complex free surface behaviour near the gated structures. Flow behaviour of water down the spillway ski-jump and pro-jected away from the spillway base was qualitatively compared to the physical model study showing good agreement. Accu-rate simulation of ski-jump flow is important for assessing dam stability, and for investigations of efforts aimed at reducingerosion effects at the spillway wall base and in the downstream plunge pool.

The extent of variation in the reservoir’s free surface was investigated to understand how the free surface changes withproximity to the spillway gates. Fifty sensors were used across the reservoir domain to sample the water level for a flow rateof 35.86 L/s and a particle size of 0.01 m. The maximum variation in reservoir water height at sensor locations parallel to thespillway wall was 0.0112 m (0.672 m full scale). The maximum variation observed perpendicular to the spillway wall was0.0532 m (3.192 m full scale). For sensors positioned parallel with the centreline of the ski-jump, an inverse transform ofthe perpendicular distance from the spillway wall (1/z) provided an accurate fit for the reservoir’s free surface profile intwo-dimensions.

A resolution study was conducted to determine the effect of particle resolution on simulated results. The percentage dif-ference in the simulated and experimental water depths decreased from 14.56% to 7.02% for resolutions increasing from0.035 to 0.01 m (5–17 particles across the width of the spillway opening). The solution became insensitive to resolutionfor an SPH particle size of less than 0.02 m. A particle size of 0.015 m was found to have the optimal balance of computationalefficiency and accuracy.

The relationship between discharge rate and reservoir water level was considered by modelling weir flow conditionsthrough the spillway. The optimal particle size of 0.015 m determined in the resolution study was used. For the weir flow

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rates simulated, the maximum difference between the simulated and experimental water depth was 11.5% for a dischargerate of 62.76 L/s. This depth was recorded in the centre of the reservoir area. At the spillway crest, the maximum differencewas only 2.4% for a discharge rate of 8.96 L/s. Given the sensor location used in the physical experiment is not accuratelyknown, and that our results show that the reservoir water level varies substantially with proximity to the spillway gates,agreement between simulation and experiment may be improved with better knowledge of physical model sensor location.

The results of this study form the most in depth quantitative validation of the SPH method for spillway flows to date. Infuture work, these results will be extended to consider pressures on the ski-jump, forces on the gate structures and the effectof dynamically moving gate structures. With further model calibration, this study demonstrates that the SPH method couldbe used to inform decisions of spillway design and operation.

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