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Three-dimensional numerical simulations of subaerial landslide generated waves Clohan, William Daley 2015

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Three-Dimensional Numerical Simulations of Subaerial LandslideGenerated WavesbyWilliam Daley ClohanB. Engineering, McGill University, 2008A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFMaster of Applied ScienceinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Civil Engineering)The University of British Columbia(Vancouver)January 2015© William Daley Clohan, 2015AbstractThis research aims to advance the continuing effort of general purpose computational fluid dynam-ics model validation of subaerial landslide generated wave (SLGW) simulations. Specifically, usingthe open source program weakly compressible Smooth Particle Hydrodynamics model, DualSPHysics,three-dimensional simulations are quantitatively compared against a combination of physical model dataand traditional general-purpose computational fluid dynamics, Flow-3DTM, data.Many simulations were conducted to determine the effect of both numerical parametrization andnumerical scheme prescriptions on SLGW accuracy. A systematic approach was taken to parse outinsignificant physical processes using Flow-3DTM - specifically surface tension - and to determine theoptimal numerical scheme settings that yield the most accurate results for both Flow-3DTM and Dual-SPHysics.From this research, it is found that DualSPHysics is able to accurately simulate both wave generationand wave propagation, but tends to over-predict the maximum wave run-up by about 70%. In contrast,Flow-3DTM was able to accurately simulate wave propagation, but under predicted wave generation byabout 25% and over predicted the maximum wave run-up by about 40%.The question as to why both DualSPHysics and Flow-3DTM both over predict the maximum waverun-up during a SLGW simulation is still open. However, it is speculated that this due to a lack of eitherenergy dissipation through air entrainment or eigenfrequency consideration’s.iiPrefaceThis thesis is an original and unpublished work by W.D. Clohan.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.1 Wave generation research . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 Wave propagation research . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.1.3 Wave run-up research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.1.4 Integrated subaerial landslide generated wave research . . . . . . . . . . . . . 101.2 Aim of present work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.3 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Computational Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1 Background information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 Flow-3DTM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 Smoothed particle hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3.1 Integral representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3.2 Derivative representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.3.3 Particle approximation of continuity . . . . . . . . . . . . . . . . . . . . . . . 182.3.4 Particle approximation of Navier-Stokes . . . . . . . . . . . . . . . . . . . . . 192.3.5 Moving the particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3.6 Equation of state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21iv2.3.7 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3.8 Time integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.3.9 Computer hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 Physical Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.1 Model setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2 Model results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2.1 Slide motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 Numerical Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.1 Baseline modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.1.1 Eulerian computational fluid dynamics . . . . . . . . . . . . . . . . . . . . . 304.1.2 Smoothed particle hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . 374.2 Developmental modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.2.1 Eulerian computational fluid dynamics . . . . . . . . . . . . . . . . . . . . . 434.2.2 Smoothed particle hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . 474.3 Final results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.3.1 Eulerian computational fluid dynamics . . . . . . . . . . . . . . . . . . . . . 624.3.2 Smoothed particle hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . 645 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.1 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74vList of TablesTable 2.1 Differential form of continuity and Navier-Stokes equations . . . . . . . . . . . . . 13Table 3.1 Accelerometer technical specifications . . . . . . . . . . . . . . . . . . . . . . . . 27Table 4.1 Baseline Flow-3DTM error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Table 4.2 Baseline DualSPHysics default settings . . . . . . . . . . . . . . . . . . . . . . . . 37Table 4.3 Baseline DualSPHysics error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Table 4.4 Developmental Flow-3DTM Error . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Table 4.5 Developmental Flow-3DTM parameter settings . . . . . . . . . . . . . . . . . . . . 46Table 4.6 Developmental DualSPHysics parameter settings . . . . . . . . . . . . . . . . . . . 50Table 4.7 Developmental DualSPHysics Error . . . . . . . . . . . . . . . . . . . . . . . . . 51viList of FiguresFigure 1.1 Phases of landslide generated waves . . . . . . . . . . . . . . . . . . . . . . . . . 1Figure 2.1 Smoothed particle hydrodynamics domain and kernel support . . . . . . . . . . . 16Figure 3.1 Subaerial landslide generated wave test stand . . . . . . . . . . . . . . . . . . . . 26Figure 3.2 Results of physical experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Figure 3.3 Slide motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Figure 4.1 Baseline Flow-3DTM point free surface results . . . . . . . . . . . . . . . . . . . . 32Figure 4.2 Baseline Flow-3DTM elevation view frames . . . . . . . . . . . . . . . . . . . . . 33Figure 4.3 Baseline Flow-3DTM plan view frames . . . . . . . . . . . . . . . . . . . . . . . . 34Figure 4.4 Baseline Flow-3DTM centreline free surface overlaid onto physical experiment images 35Figure 4.5 Baseline DualSPHysics point free surface results . . . . . . . . . . . . . . . . . . 38Figure 4.6 Baseline DualSPHysics elevation view frames . . . . . . . . . . . . . . . . . . . . 39Figure 4.7 Baseline DualSPHysics plan view frames . . . . . . . . . . . . . . . . . . . . . . 40Figure 4.8 Baseline DualSPHysics centreline free surface overlaid onto physical experimentimages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Figure 4.9 Developmental wave height comparison between DualSPHyscis & Physical Data -Baseline simulation with a new viscosity coefficient of 0.20 and 0.15 . . . . . . . 49Figure 4.10 Developmental wave height comparison between DualSPHyscis & Physical Data -Baseline simulation with a new viscosity coefficient of 0.10 and 0.05 . . . . . . . 52Figure 4.11 Developmental wave height comparison between DualSPHyscis & Physical Data-Baseline simulation with a coefficient of sound of 20 and 30 . . . . . . . . . . . . 53Figure 4.12 Developmental wave height comparison between DualSPHyscis & Physical Data- Baseline simulation with a Wendland kernel, a new smoothing length coefficient(1.23 and 1.5), and a new viscosity coefficient . . . . . . . . . . . . . . . . . . . 54Figure 4.13 Developmental wave height comparison between DualSPHyscis & Physical Data -Baseline simulation with a Wendland kernel, a new smoothing length coefficient of1.23, a delta of 0.1, a new coefficient of sound of 40, and a new viscosity coefficient0.15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55viiFigure 4.14 Developmental wave height comparison between DualSPHyscis & Physical Data -Baseline simulation with a Wendland kernel, a new smoothing length coefficient of1.23, a new viscosity coefficient 0.15, and a symplectic time integration scheme . . 56Figure 4.15 Developmental wave height comparison between DualSPHyscis & Physical Data -Baseline simulation with a Wendland kernel, a new smoothing length coefficient of1.23, a new viscosity coefficient 0.15, a smaller slide, and periodic North and Southwall boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Figure 4.16 Developmental wave height comparison between DualSPHyscis & Physical Data- Baseline simulation with a Wendland kernel, a new smoothing length coefficientof 1.23 and 2.02, a new viscosity coefficient 0.15, a smaller slide, and an epsiloncoefficient of zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Figure 4.17 Developmental wave height comparison between DualSPHyscis & Physical Data -Baseline simulation with a Wendland kernel, a new smoothing length coefficient of1.23, a new viscosity coefficient of 0.05 and 0.01, and a smaller slide . . . . . . . 59Figure 4.18 Developmental wave height comparison between DualSPHyscis & Physical Data -Baseline simulation with a Wendland kernel, a new smoothing length coefficient of1.23, a new viscosity coefficient of 0.02 and 0.03, and a smaller slide . . . . . . . 60Figure 4.19 Developmental wave height comparison between DualSPHyscis & Physical Data -Baseline simulation with a Wendland kernel, a new smoothing length coefficient of1.23, a new viscosity coefficient of 0.05, a smaller slide, a coefficient of sound of20, and a complex slide description . . . . . . . . . . . . . . . . . . . . . . . . . 61Figure 4.20 Finalized Flow-3DTM point free surface results . . . . . . . . . . . . . . . . . . . 63Figure 4.21 Finalized DualSPHysics point free surface results . . . . . . . . . . . . . . . . . . 65Figure 4.22 Finalized DualSPHysics elevation view frames . . . . . . . . . . . . . . . . . . . 66Figure 4.23 Finalized DualSPHysics plan view frames . . . . . . . . . . . . . . . . . . . . . . 67Figure 4.24 Final DualSPHysics centreline free surface overlaid onto physical experiment images 68Figure 5.1 Finalized and baseline DualSPHysics and Flow-3DTM point free surface results . . 72viiiAcknowledgmentsI would like to express my gratitude to my supervisor, Professor Bernard Laval. His encouragement toexplore my interests made this research both enjoyable and enriching.I wish to also thank the Natural Sciences and Engineering Research Council of Canada for myIndustrial Postgraduate Scholarship. In addition, I would like to express my gratitude to the Universityof British Columbia, as the UBC Civil Engineering Flood Fund provided generous contributions to myresearch.Finally, I would like to acknowledge my wife Jarleen. Without her support, confidence, and patiencethis thesis would not be what it is.ixChapter 1IntroductionLandslide generated waves (LWG) are gravity dominated waves that are generated by a sudden distur-bance, or impulsive force, in a body of water.As a landslide rapidly interacts with a body of water, a large volume of water is displaced andthen forms a large, fast-moving wave. The generated wave can then travel vast distances with minimalattenuation until it collides with and runs-up a shoreline. This phenomenon is extremely complex anddifficult to evaluate. One way to analyze a LGW problem is by breaking it into a wave generation phase,a wave propagation phase, and a wave run-up phase, as shown in Figure 1.1Physically, the generation phase is a transfer of potential energy (from a slide) to kinetic energy (mo-tion of water) through either a landslide that originates entirely on land (subaerial landslide), originatesentirely underwater (subaqueous landslide), or originates partially on land and partially underwater (par-tially submerged landslide). The amplitude of the generated wave is dependent on the slide’s volume,velocity, density, porosity, shape, and hill slope angle[59].Subaqueous and partially submerged landslides are infamously known for generating destructiveTsunami waves in excess of 20-m[59] high, either in the open ocean (e.g. the 1883 Krakatau Tsunamiwhere 36,000 people died or the 1998 Papua New Guinea Tsunami where 2,200 people died) or inland,on man-made reservoirs (e.g. the 1960 Vajont partially submerged landslide that killed roughly 2,000Figure 1.1: Phases of landslide generated waves1people). In contrast, subaerial landslides tend to only be known for their destruction when occurringinland, in natural or man-made reservoirs. For example, of the roughly 15,600[67] people who havedied because of subaerial landslide generated waves, only two were killed in a non-inland event inLituya Bay, Alaska (July 9th, 1958). The remaining causalities all perished in one of seven inlandsubaerial landslide generated wave (SLGW) events: Five in Norway - one in Rammerfjell (17 perishedon January 8th, 1741), one in Tjelle (32 perished February 22nd, 1756), two in Ravenfjell (61 perishedJanuary 15th, 1905 and 72 more on September 13th, 1936), and one in Tafjord (41 perished on April7th, 1934); one in Shimabara, Japan (roughly 15,000 perished on May 21st, 1792); and, one in YanahuinLake, Peru (400-600 perished March 18th, 1971).Despite the fact that subaerial, subaqueous, and partially submerged landslides all generate waveswhen they interact with a body of water, the underlying physics of each is significantly different. Thisthesis will focus solely on inland subaerial landslide generated waves - herein referred to as SLGW.The wave propagation phase encompasses wave energy dispersion1 that characteristically changesduring times of shoaling, diffracting, and refracting. The wave run-up phase looks at when a wave hitsa boundary (like a shoreline), and tends to run-up the boundary to an elevation several times that of itsoriginal approaching height.The three phase framework has worked well for analyzing SLGW and there has been a wide breadthof research into subaerial landslide wave generation, wave propagation, and wave run-up. However,there is no “one” way of integrating findings from all the relevant research. Instead, the conscientiousengineer needs to use either judgement in applying analytical/empirical relationships between subaeriallandslides and a body of water’s response, or use scaled physical models or computational fluid dynam-ics (CFD) models for case specific situations.Again, this research is limited to waves generated by inland subaerial landslides, and the objectiveis to correlate the characteristics of a subaerial landslide to the characteristics of the water wave itgenerates upon impact. More specifically, freely dropping or sliding a rigid or deformable mass (hereinrefereed to as the forcing mechanism) down a 0-90 slope into still water of finite volume and developingan empirical relation between the motion of the forcing mechanism and the height of the wave(s) itgenerates - most notably the maximum wave height.1.1 Literature reviewThe following literature review is broken down into wave generation research, wave propagation re-search, wave run-up research, and SLGW research.1.1.1 Wave generation researchThe birth of subaerial landslide wave generation research started in 1845 when John Scott Russell pub-lished his “Report on Waves”[65]. Here, Russell discussed his findings on solitary wave generation1The reader is referred to a text (like Sorensen’s Basic Coastal Engineering book[69]) on linear, Stokes, Cnoidal, Solitary,and bore wave theory2when a free falling vertical box was dropped into a flume of water2, and he concluded that a wave’svolume is identical to the initial water displacement by the vertical box, which was the first reportedrelationship between a forcing mechanism and the waves it generates.Following Russell’s 1845 paper, there was a 104-year void in subaerial landslide wave generationresearch. It was not until 1949, when Johnson and Bermel[35] dropped circular steel discs into a flumeof shallow water, did SLGW research pick up again3. That said, their research was not overly fruitful.Aside from using data from five tests to show a linear relationship between the fall height of a 472-kg1.2-m diameter steel disc and the wave length of the first resulting wave, not much can be inferred fromtheir work.The first comprehensive study of SLGW was undertaken by Kamphuis and Bowering in 1970[36].Using a two-dimensional (2D) 45-m long and 1-m wide physical model, they released a rigid sled ofvariable weight from a variety of locations down a roller ramp of variable slope into water depths rangingfrom 0.23-0.46-m. During each test run, the sled velocity was recorded as well as wave heights at threelocations. Analysis allowed them to construct a dimensionless equation (Equation 1.1) that relates thedimensionless maximum wave height at a point, hmax,dim = hmax/h (where hmax is the maximum waveheight and h is the still water depth) , to slide Froude number, F = vs/pgh (where vs is the slide impactvelocity), dimensionless slide volume, 8 = Vs/wh2 (where Vs is the slide volume and w is the slidewidth), specific gravity of the slide, G = rs/rw (where rs is the bulk density of the slide and rw is thedensity of water), dimensionless propagation distance, X = x/h (where x is the distance from impact),dimensionless time slide spends underwater, Ts = tspg/h (where ts is the underwater motion time),landslide front slope, F (in radians), slope angle, a (in radians), and landslide porosity, n:h(x)max,dim =FF,8,G,X ,Ts,F,a,n(1.1)They found the most influential parameters to be the slide Froude number, F, and the dimensionlessslide volume, 8.In the same year as Kamphuis and Bowering, Noda[57] and Wiegel[84] performed their own exper-iments using a Scott Russell wave generator (SRWG). They also found that the maximum wave heightgenerated was highly influenced by the slide Froude number. Furthermore, Noda[57] suggested thatdepending on the box width and velocity, the waves generated from the SRWG could be analyzed usingeither linear or solitary wave theory (see Section 1.1.2)Expanding on Kamphuis and Bowering’s work[36], Huber[33] conducted one of history’s mostcomprehensive SLGW studies. He used granular material to represent a subaerial slide, and carried outover 1000 2D and 3D test runs. During which, he used a slide mass, ms, ranging from 5-50-kg, a slideimpact velocity ranging from 1-5-m/s, a water depth ranging from 0.12-0.36-m, and a slope angle of28 to 60. Rounded river rock with a mean diameter of 0.002-m was used for the slide material.Analysis of Huber’s (1980) work by Huber and Hager[34] generated the first relationship between2This method of generating waves has since been coined the “Scott Russell wave generator” and is used by many SLGWresearchers.3Johnson and Bermel’s motivations were not actually provoked by SLGW research, but rather assisting with the planningof atomic bomb testing near the Bikini Atoll.3a granular subaerial slide and the maximum wave height at a specific point. However, unlike Kamphuisand Bowering[36], Noda[57], and Wiegel[84], they did not take into account the slide Froude number.Instead they suggested the maximum generated wave height at a point was mainly dependent on thenon-dimensional landslide volume, and could be described in 2D as:h(x)max = 0.88 ·h481/2G1/4X1/4sina (1.2)or in 3D (note, r is the radial distance away from the slide impact):h(x)max = 1.76 · sina · cos2"2g81/2G1/43✓rh◆2/3#(1.3)Slingerland and Voight[68], used the slide’s dimensionless kinetic energy to relate the maximumwave generated at a location r/h⇡ 4 from the slide impact as:log✓h(x = r/h⇡ 4)maxh◆=1.25+0.71 · log✓0.5rsrwVsh3v2sgh◆(1.4)orlog✓h(x = r/h⇡ 4)maxh◆=1.25+0.71 · log✓0.5wF28Gh◆(1.5)where they arrived at Equation 1.5 through empirical of data generated by a scaled physical model studyof subaerial landslides into the Mica dam reservoir in British Columbia, Canada[81]. They then usedEquation 1.5 to reasonably reproduce the Lituya Bay maximum generated wave at r/h⇡ 4.Walder et al.[80], felt that the maximum wave generated may be influenced by a slide’s Froude num-ber (Kamphuis and Bowering[36], Noda[57], and Wiegel et al.[84]), dimensionless volume(Kamphuisand Bowering[36],Huber and Hager[34]), or dimensionless kinetic energy(Slingerland and Voight[68]),but these were not the dominating relationships. Instead Walder et al.[80] argue the key relationshipis the ratio of the dimensionless submerged slide travel time to the dimensionless slide volume, Ts/8.Where the basis for their argument is founded on 2D results from a physical experiment they carried outin a 1-m long, 0.285-m wide flume with 0.051-m to 0.13-m of water, and a rigid variably weighted boxthat slid down a concave slide. They also strengthened their argument by “reasonably” using this ratioto approximate the Lituya Bay event[47].Panizzo[60] also advocated that the dominate parameter for the maximum SLGW was the dimen-sionless submerged slide travel time, but he also valued the dimensionless slide volume. Having saidthat, his experiments were not 2D, but actually 3D experiments in a 12-m long, 6-m wide, and 0.8-mdeep tank, and his conclusion differs from Atai-Ashtiani and Nik-Khah’s[2] 3D experimental conclu-sions.Atai-Ashtiani and Nik-Khah’s[2] used a 25-m long, 2.5-m wide, and 1.8-m deep tank with six uni-formly placed wave measuring devices along the tank centreline. They then slid a variety of rigid shapes(i.e. square, short rectangle, tall rectangle, triangle, short/narrow hemisphere, etc.) and deformable4shapes (i.e. granular material in fabric bags) down a variable sloped ramp into water. According tothem, the maximum generated 3D wave is “strongly affected” by the slide Froude number, ramp slope,and slide rigidity, but “weakly affected” by ramp slope. For slide rigidity, they found a 35% decrease inthe maximum generated wave when using a deformable shape, as opposed to a rigid object. They alsosuggest the following dimensionless relationship:hmaxh=0.405+0.078(8F2)1.28✓Ts8◆✓ls◆0.12✓ xh◆0.48(1.6)Where l is the slide length and s is the slide thickness and Equation 1.6 is based on the aggregate datafrom Kamphuis and Bowering in 1970[36], Huber[33], Walder et al.[80], as well as Atai-Ashtiani andNik-Khah[2], which greatly improves the reliability of the above equation. That said, what they - and themany other researcher’s before them - neglected to focus on, was the underlying physics of a subaeriallandslide impacting water. That is, SLGW research was considered a black box type of research. Inthat, researcher’s only focused on a slides pre-impact velocity, shape, volume, kinetic energy, etc. (i.e.“black box” input) and the generated wave(s) occurring post-impact (i.e. “black box” output). However,the PhD. work of Fritz[19] attempted to fill that void.Fritz[19] constructed an ingenious flexible 2D physical experiment to pragmatically evaluate granu-lar SLGW events, the foundation of which was based on his pneumatic wave maker. He recognized thatchanges in the slide Froude number would not always generate a 2D wave in his model. That is, assum-ing a constant volume, the higher the material is placed up on his slide, the more it will tend to thin outand generate a “tongue” like structure where the central portion would impact the water much earlierthen the sides. This implies, waves would also be generated radial and inevitably reflect off the modelwalls. Not only that, for each variation of slide Froude number, the shape of the slide would alwayschange. Using his pneumatic wave maker, Fritz was able to accelerate granular material at differentrates while maintaining the front shape of the material4. Using particle image velocimetry, Fritz[19]was able to quantify the impact zone physics and show once again that the slide Froude number is oneof the dominating parameters, which he showed by reproducing the Lituya Bay event using dynamicsimilitude [20, 21, 47].As an extension of Fritz’s work[19], Zweifel[85] and Heller[29, 29, 30] used Fritz’s experimentalsetup to conduct their own research. Zweifel showed that the higher a slide’s density, the higher themaximum generated wave would be, also that the slide Froude number was the dominant parameterfor “slow” slides whereas the water depth and slide thickness were the dominant parameters for “fast”slides.Heller[27, 28, 30] used Fritz’s[19] 223 test runs and his 211 runs to generate (what he calls) auniversal impact parameter, P, to describe the maximum generated SLGW within ±30%:P = FS1/2M1/4 ·pcos[(6/7)a] (1.7)4The granular material was placed in a rigid box that was pneumatically accelerated, then at a given location, the front ofthe box would open and the material would be shot out[19]5With Heller’s impact parameter taking into account the slide Froude number, the slide dimensionlessmass, M = ms/(rwwh2), and the slide dimensionless thickness, S = s/h, he argues this parameter is themost valid parameter for estimating the maximum generated wave height caused by subaerial landslides.However, he does indicate that this parameter is only valid for 2D and in his own words states ’the futurein terms of an efficient and accurate prediction of the wave features for a specific bathymetry lies innumerical simulations’.Following Heller’s advice (albeit fourteen years before Heller said it!), Monaghan and Kos[53]used a robust 2D weakly-compressible Smoothed Particle Hydrodynamics (SPH) numerical model[22,44, 51] (discussed in detail in Section 2.3) to investigate waves generated using both a numerical andphysical SRWG. The results indicated their SPH model overshot their physical model data by 3-18%;however, they did find that the amplitude of the leading wave, A1, could be described as:A1 = 3h1/3✓ms40rww◆2/3(1.8)Later in 2008, Ataie-Ashtiani[3] conducted the same type of experiment with a SRWG, but this timethey used a 2D incompressible model. Unfortunately, they did not compare their physical experimentaldata with their numerical data, but rather their analytical solution to the SRWG problem (i.e. solitarywave theory - see Section 1.1.2).Monaghan et al.[55] conducted a 2D physical experiment where a weighted rigid box slid downa concave slope into water. They then ran 2D SPH simulations and compared the two results. Theyfound an error between the two to be between 15-50%. This led them to concluded the 2D SPH modelqualitatively compared well with the physical experiment, instead of quantitatively.Das et al.[14] qualitatively looked at a 2D numerical simulation of a triangular wedge sliding intowater of 1-m in depth. They used both Flow-3DTM and an open source SPH code to conclude that Flow-3DTM “appears to better predict” experimental free surface elevations. Qiu[61] and Viroulet et al.[79]also evaluated a 2D SPH numerical simulation of a triangular wedge sliding into water, and found theSPH code to be reasonable.Finally, using an experiment similar to Atai-Ashtiani and Nik-Khah[2], Liu et al.[43] constructed a3D 104-m long, 3.7-m wide, and 4.6-m deep flume and ran either a triangular or hemispherical sled intothe water. They then used the physical data to “validate” a numerical code of theirs, the results of whichappeared qualitatively accurate.Despite much research into SLGW, there appears to be no real consensus on what the dominat-ing parameters are. Nevertheless, most researches agree that the important parameters are the slideFroude number, F, dimensionless volume, 8, and dimensionless mass, M, which suggests the likelymost suitable parameter is Heller’s universal impact parameter[30], P. That said, even Heller cautionsthis parameter is really only suitable for 2D situations and that 3D numerical site specific research needsto be carried out.For the CFD community, this is an invitation to numerically investigate 3D SLWG events. However,only a handful of researchers have conducted qualitative research, and even less have done quantitative6research. As such, there is a void that needs to be filled.1.1.2 Wave propagation researchThe field of wave propagation research is one in which an attempt is made to mathematically describe(either analytically or empirically) how waves move through water of varying depths, h. Specificallyit focuses on quantifying the speed at which waves move through water, c, and the energy/power theycarry with them.Generally speaking, there are many formulations based on relating wave height, H, length, L, andperiod, T, to wave celerity and energy; however, the main theories used for SLGW research are lin-ear wave theory, solitary wave theory, Stokes wave theory, and cnoidal wave theory. The followingdiscussion will focus on these four theories.In 1841, George Biddell Airy published the first mathematical construct to evaluate waves propagat-ing through water (known today as the linear wave theory) in his highly influential paper titled “Tidesand Waves”[1, 9]. Airy assumed pressure is hydrostatic and flow is irrotational, inviscid, and incom-pressible. And he suggests that the temporal free surface, h(x, t), of a linear periodic wave be expressedas a function of wave height, length, and period as it propagates through water. That is:h(x, t) = H2cos(2pLx2pTt) (1.9)Although Airy’s linear wave theory was a giant leap forward in the field of fluid dynamics, thetheory had many limitations and caveats. Namely, the theory is a sinusoidal first-order accurate smallamplitude wave theory that is only valid, according to Dean[15], for H/h < 0.3, H/L < 0.006, andHL2/h3 << 1[27]. Furthermore, the theory is linear, meaning it negates higher order non-linear termswhich become important when a wave height relative to the flow depth and wavelength increases past0.3 and 0.006, respectively.Joseph Valentin Boussinesq, however, did account for non-linear effects in his 1871 mathematicalconstruct of solitary waves[5]. Here, Boussinesq assumed a non-hydrostatic5 rotational flow conditionto mathematically confirm John Scott Russell’s solitary wave generation observations made in 1845[65].Boussinesq’s theory implies that the non-linear temporal free surface of a solitary wave could beexpressed as a function of wave amplitude, a, and speed in the following manner:h(x, t) = asech2 r3a4h3(x ct)!(1.10)Where c is defined by Russell[65] as:c =pg(h+a) (1.11)The major caveat to the solitary wave theory is that it is based on the assumption that there existsonly one wave peak and no wave trough, which implies an infinite wave length. This of course is not5Implying pressure is not simply the specific gravity of water times its depth (or height above a given point).7practical for engineering applications and arguably has no real use for real world applications.In 1847, George Gabriel Stokes developed his own theory on non-linear oscillatory waves[71],which today is called “Stokes wave theory”. Fundamentally, this theory is based on irrotational flowand non-hydrostatic pressure distributions, and is generally only appropriate for deep water waves -specifically for H/h < 0.1[69], H/h < 1, and HL2/h3 ⇡ 1 [74]. A second-order formulation, accordingto Dean[16], of the free surface for a Stokes wave is (using the above noted notation):h(x, t) = H2cos✓2pLx2pTt◆+pH28Lcosh2pL hsinh32pL h✓2+ cosh✓4pLh◆◆cos✓2✓2pLx2pTt◆◆(1.12)In 1895, Diederik Johannes Korteweg and Gustav de Vries published their work “On the change ofform of long waves advancing in a rectangular canal, and on a new type of long stationary waves”[38].Here they developed a non-linear wave theory that allows for periodic waves to exist in shallow-water[16]. Korteweg and de Vries coined their theory the ”Cnoidal Wave Theory” due to their inclusionof a Jacobian elliptical function, cn.The Korteweg and de Vries’ Cnoidal Wave Theory is most useful for shallow water wave investi-gations - specifically when HL2/h3 > 26. But due to it’s complexity, most authors choose not to usethis theory, which this author has chosen to do as well, and the reader is directed to Wiegel’s paper on“A presentation of condial wave theory for practical application”[83] for a comprehensive cnodial wavetheory discussion.Following the works of Airy, Boussinesq, Russell, Stokes, and Korteweg most research in the fieldof wave probation focused on refining these original works by increasing their order of accuracy[16, 69],or by evaluating when one theory breaks down and another takes over[16, 46]. For example, as brieflymentioned above, Stokes theory is appropriate for deep water conditions, while Cnoidal Waver Theoryis better for shallow water conditions. However, the underlying principals remain the same.From inspection of the linear wave, solitary wave, Stokes wave, and Cnoidal wave theories, it isapparent that these all assume the initial wave height is known, wave trains are infinity long and periodic,and water depths are constant. Sound judgment is required when applying them. As such, for SLGWresearch these theories can be problematic, namely because SLGW are highly non-linear, finite wavetrains that may or may not be periodic, and they can rapidly transition from deep to intermediate toshallow depths (though most are shallow water) during their travels. Therefore, it is extremely difficultto soundly use an analytical theory without suitable expertise. Lastly, as discussed above (Section 1.1.1),predicting the exact wave height due to a subaerial landslide impact is not feasible.1.1.3 Wave run-up researchWave run-up is defined as the vertical height above the still water level an incident wave reaches whenit collides with a boundary. For coastal waters, this could either be a typical wind-generated ocean wavebreaking and running a meter or so up a beach, or a tsunami wave running 20-m[59] up a denselypopulated shoreline. For inland subaerial landslide generated waves (SLGW), this could be a wave8running up and overtopping a dam face.Although the focus for this research is on inland SLGW, the physical concept of wave run-up issimilar for coastal or inland waters. Rather it is dependent on the shape and roughness of the boundarythe wave strikes, the depth of water and bed slope adjacent to the boundary toe, the permeability of theboundary, and the speed, length, and period of the incident wave[76]. As such, because of the breadthof variables, there is no complete analytical description of this phenomenon, and research is limited toempirical relations for linear monochromatic6 wave run-up.One of the first comprehensive studies on wave run-up was carried out by the US Army Corpsof Engineers (USACE) from 1973-1984[75, 76]. Here the USACE presents graphical relationshipsbetween wave run-up on a variety of sloped surfaces (e.g. impermeable rigid flat/stepped/curved wallsand permeable rubble slopes) and incident wave heights and periods, with different still water depths atthe toe of a variably sloped boundary. The results indicated that the larger the wave height, the smallerthe waver period, and/or the steeper the sloped boundary is, the higher the run-up will be. However, it isnoted that this work is limited to linear monochromatic waves and does not account for complex slopedsurfaces, non-linear effects, or 2D/3D waves.During the same time as the USACE study, Chue[6] developed a “universal” wave run-up formula(Equation 1.13) that is useful for nonbreaking wave run-up. Here, Chue relates the run-up, R, to thestill water depth, h, the incident wave period, L, and the angle of the sloped boundary the incident waveimpacts, q , as follows:Rh= 1.8✓13.111hL◆✓ p2q◆1/2(1.13)Costas Synolokis[72], on the other hand, used a series of laboratory experiments to validate a sim-plified analytical solution he developed for non-breaking solitary wave run-up (Equation 1.14). Here,Synolokis relates wave run-up, R, to wave height, h , still water depth, h, and the angle of the slopedboundary the wave impacts, q , as follows:Rh= 2.831pcotq✓hh◆ 54(1.14)The simple form of Synolokis equation makes it attractive for general engineering applications;however, because it is only valid when (h/h) 12  0.288tan(q), and does not account for non-linearterms, its use is limited for complex situations.To combat the need for estimating wave run-up on breaking waves, Hajime Mase[45] suggestsrelating wave run-up, R, to wave height, h , and wave period, L, as follows:Rh = 2.32✓tanqph/L◆0.77(1.15)However, Mase’s equation is only valid for 2  q  11 and 0.007 h/L, meaning it is only valid for6Waves having only one wave length (i.e. only one wave signal)9boundaries with gentle slopes and waves with heights that are relatively much smaller then their periods.Consequently, Mase’s equation is not overly practical for SLGW, as these wave’s have heights that arerelatively much larger then their periods.In fact, inspecting the equations developed by the USACE, Synolokis, and Mase, it is noted thatnon of these are valid for inland SLGW problems. Instead, only scaled physical models or validatedcomputational fluid dynamics (CFD) models should be used.1.1.4 Integrated subaerial landslide generated wave researchAlthough individual research into wave generation, propagation, and run-up phases has been fruitful, itis still necessary to look at subaerial landslide generated wave (SLGW) events as a whole, and integratethese three phases into one investigation. This is easily justified by simply looking at the difficultyassociated with coupling the analytical/empirical relations between the three phases. For example, howis the maximum SLGW best described? By Kamphuis and Bowering[36], Noda[57], Wiegel et al.[84],Slingerland and Voight[68], Huber and Hager[34], Atai-Ashtiani and Nik-Khah’s[2], or Heller[27, 28,30]? Then, what is the best method to use for wave propagation? Is the wave a linear wave, a Stokeswave, a Cnoidal wave, a solitary wave, or an amalgamation of these four wave types? Lastly, once awave hits a boundary, how can the maximum wave run-up be estimated? Here, likely the situation issuch that the wave is non-linear and the boundary it is running up is much more complex then a gentlyslopped flat impermeable wall.Therefore, because of the vast number of variables and alternative methods for evaluating eachphase, an integrated approach using either scaled physical models (SPM) or computational fluid dy-namics (CFD) models are used. The preferred approach is to use a SPM due to its ability to nearlyreproduce prototype conditions. But due to their cost, both in terms of money and time, and scaleeffects, CFD models are increasingly being used.However, the main issue with CFD models, is that they need to be properly validated before gen-eral application to SLGW work is accepted within the hydrotechnical engineering community. Wherevalidation ultimately stems from the “exact” reproduction of real world SLGW events or high fidelityphysical experimentation data, which is unfortunately rare to come by.One of the first attempts to validate a CFD model for SLGW was carried out by Harbitz, Pedersen,and Gjevik[26]. They tried to reproduce a real world SLGW event (the Tafjord, Norway slide mentionedabove) using their own CFD7 model, but they under predicted the run-up by over -100%.Liu et. al.[43] developed their own CFD model8 and attempted to validate it against physical exper-iments they conducted. Here, they were specifically investigating the run-up generated behind the slideas it entered a body of water. The results of which reproduced the general trend of the temporal run-up,but did not accurately capture run-up magnitudes.One of the first commercially available software packages used for SLGW validation was Flow-3DTM by Basu[4]. Here a 2D Flow-3DTM model was used to numerically re-create Fritz’s Lituya bay7linearized hydrostatic shallow water approximation discretized using finite differences8Three-dimensional “filtered momentum and continuity” equations discretized using a finite volume scheme with a vol-ume of fluid free surface tracker and LES turbulence model.10physical model experiments[20], which included a deformable slide material. To do this, Basu employedFlow-3D’sTM two-phase drift-flux model to represent a sediment-water slurry that he approximated asthe slide9. The results of this simulation were then looked at from a wave generation point of view anda wave run-up point of view.Basu’s Flow-3DTM model qualitatively reproduces the impact crater Fritz describes in his physicalexperiments, but tends to over predict the maximum wave run-up of the actual Lituya bay event by28-56%. This, according to Basu, could be due to his choice of using a 2D approximation, and/or thephysical approximation of the slide as a sediment-water slurry.Using the same two-phase drift-flux model in Flow-3DTM, Kim[37] also shows that the Flow-3DTMover predicts the maximum wave run-up by 0-78% (depending on location). However, because of lackof physical model data, he could not parse out what was causing the discrepancy. That is, was it a wavegeneration, propagation, or run-up issue. Consequently, as with Basu’s results, it is very difficult to saythat Flow-3DTM has been adequately validated for SLGW simulation.Lastly, it is noted that there exists no quantitative 3D numerical simulation/validation of Dual-SPHysics for SLGW.1.2 Aim of present workThe aim of this research is to conduct the first DualSPHysics simulations for three-dimensional inlandsubaerial landslide generated wave (SLGW) events, and to validate these simulations using high fidelityphysical model data. In addition, this research aids in the continuing effort of validating the generalpurpose computational fluid dynamics model Flow-3DTM for SLGW events.1.3 Thesis outlineThis thesis is has been broken into five chapters. Chapter 1 presents a comprehensive literature review ofall relevant subaerial landslide generated wave research conducted to date. Chapter 2 discusses the nu-merical models used in this research, but mainly focuses on the mathematical backbone of the SmoothedParticle Hydrodynamics model called DualSPHysics. Chapter 3 goes over the physical experiment usedto evaluate both Flow-3DTM and DualSPHysics. Chapter 4 quantitatively compares the Flow-3DTM andDualSPHysics simulations to the data generated by the physical experiment. Lastly, Chapter 5 com-pletes this thesis with a detailed summary of this work, some concluding remarks, and a brief outlookfor future numerical subaerial landslide generated wave research.9Basu himself notes this as not being an exact representation of Fritz’s slide, but it was/is the only option provided byFlow-3DTM .11Chapter 2Computational Methods2.1 Background informationDescribing subaerial landslide generated waves (SLGW) using a computational fluid dynamics (CFD)program is no trivial task. Not only does the CFD code have to account for moving (and possiblydeforming) boundaries (i.e. the slide), but it also must capture the free surface, which for SLGW is ahighly complex and multi-faceted problem. Wave generation during impact causes a turbulent crashingwave that is difficult to capture numerically, while wave run-up involves a rapid decrease in flow depth.Fortunately, there are several general purpose/specialized CFD packages available that can performSLGW simulations. Flow-3DTM in particular has been shown to yield reasonable results[4, 14, 37, 56],which makes it a suitable tool to compare against new numerical schemes.In the case of this research, a relatively new numerical scheme, Smoothed Particle Hydrodynamics,will be evaluated against both physical experimental data and Flow-3DTM simulations of SLGW.This section will provide a description of both Flow-3DTM and DualSPHysics , and how their un-derling numerics make use of the governing equations of motion.2.1.1 Governing equationsIn the field of hydrotechnical computational fluid dynamics, two of the many relevant equations are thecontinuity and the Navier-Stokes equations. When coupled together, they describe the physical velocityand pressure fields of a fluid. These are collectively called the equations of motion. Table 2.1 presentsthem in their differential form for both Eulerian and Lagrangian descriptions.Despite their compact form, these equations are wrought with mathematical richness and may there-fore be discussed in great detail. Such a large discussion, however, is outside the context of this thesisand the reader is directed to comprehensive texts by Landau and Lifshitz[41], Tritton[73], Currie[12],and Kundu et. al[39, 40] for further information. That said, relevant details about these equations arediscussed in the following sections.12Table 2.1: Differential form of continuity and Navier-Stokes equationsEulerian Form Lagrangian FormContinuity ∂r∂ t +u ·—r =r(— ·u)DrDt =r(— ·u)Navier-Stokes r( ∂u∂ t +u ·—u) =—P+rg+Gr(DuDt ) =—P+rg+GNote: r=density; t=time; p=pressure; u=velocity vector; g=gravity; G=dissipative terms2.2 Flow-3DTMFlow-3DTM is a commercially available software produced by FlowScience. The program was developby Dr. C.W. Hirt in 1980 after he left the Los Alamos National Laboratory. The method uses a rectilinearmesh to discretize a computational domain using the Finite Volume Method, and solves the Eulerianthree-dimensional Navier-Stokes and Continuity equations.At the core of the Flow-3DTM program is what FlowScience has coined the “True Volume of Fluid”,a modified version of Hirts original construct[32]. This allows for efficient tracking of complex free-surface’s and the reproduction of solid boundary interfaces. As such, many consulting engineering firmsand research teams use Flow-3DTM to investigate hydrotechncial problems (eg. spillway performance,hydraulic jumps, weir design, etc.).The intent of this section is not to re-construct the Finite Volume Method or present the underlyingnumerics of Flow-3DTM1, but rather to qualitatively focus on some of the relevant features that arespecific to this research and give the reader a feel for how Eulerian CFD works. Specifically, this sectionwill focus on spatial discretization, free surface tracking, and momentum advection and viscosity.Spatial discretization: In the field of CFD, the computational domain can be spatial discretized usingeither a unstructured or structured mesh. The mesh is a discrete set of connected nodes that constructthousands to hundreds of millions of tetrahedral or hexahedral cells used to solve the equations ofmotion.Unstructured meshes, are a random construct of cells that range in both size and shape and haveno direct connectivity2 - this connectivity comes from clever bookkeeping. A structured mesh, on theother hand, uses a logical construct to generate cells that can be either uniform in size, or non-uniform;however, the general shape remains constant throughout the domain.Both meshes are fixed in space - and hence called a Eularian frame of reference. Computations aresubsequently evaluated at fixed points using a Finite Volume method.In general terms, the Finite Volume method uses a flux integral to approximate the rate of “things”(i.e. mass, momentum, velocity, etc.) travelling through a given volume (i.e. a cell). A more detaileddescription can be found in Versteeg[78].1Which can be difficult when working with proprietary algorithms2Cell one is not necessary connected to cell two, etc..13Free surface tracking: Of particular interest to hydrotechnical engineers is the time evolution of thefree surface. Unfortunately, this is rather difficult to do in Eularian CFD and requires special treatment.Flow-3DTM uses it’s proprietary Volume of Fluid method to track the free surface. It does this by findingthe volume of water occupied in each cell and prescribing the free surface as the cell with the smallestfractional volume that is above zero.The Volume of Fluid method has been shown to be quite robust and computationally efficient. How-ever, it does lack the ability to completely track complex non-linear free surfaces, like for example awave.Momentum advection and viscosity: Viscosity is implemented in Flow-3DTM through the Navier-Stokes equations. However, additional dissipation (i.e. numerical viscosity) is encountered when mo-mentum is advected between cells. Here, momentum is carried from one cell to another and the resultis a combination of the current and previous values. This process smooths out momentum gradientsbetween cells, and when performed over many steps a diffusion of momentum is observed[31]Flow-3DTM uses a first-order momentum advection scheme as the default. However, when theirsecond-order scheme is used the simulation is less dissipative and performs well for free surface waves[18].Turbulence: Flow-3DTM allows a user to select from one of the following five turbulence models[18]:1. Prandtl mixing length model: Turbulence is taken into account by spatially “enhancing” the kine-matic viscosity in such a way viscous losses resemble TKE dissipation.2. One-equation turbulence model: In this model, the kinetic energy associated with turbulence (i.e.turbulent kinetic energy - TKE) is approximated as the half the summation of the squared compo-nent velocity fluctuations (u0,v0, and w0)TKE = 0.5(u02 + v02 +w02) (2.1)Equation 2.1 is then dispersed using Flow-3D’sTM “FAVORized” equations of motion, and therate of TKE dissipation is approximated as:e = 0.09r32✓TKE1.50.07⇤Lmin◆(2.2)where Lmin is the smallest domain dimension.3. Two-equation (k-e) turbulence model: The two-equation (k-e) model is essentially a more so-phisticated extension of the one-equation model. Instead of computing the rate of TKE dissipa-tion based on a purely empirical relationship, it is instead computed using a empirical-numerical(i.e. using Flow-3D’sTM “FAVORized” equations of motion with some empirical coefficients)approach, but is limited to a minimum value defined by Equation 2.2.144. Renormalization-Group (RNG) model: ”The RNG model uses equations similar to the equationsfor the k-e model. However, equation constants that are found empirically in the standard k-emode are derived explicitly.”5. Large Eddy Simulation (LES) model: The LES method models all turbulent flow structures thatcan be resolved by the numerical mesh and only approximates features that are smaller than thegrid.From inspection, the simplest of methods is the Prandtl method. Here it is noted that it is rathersimple to implement and requires the least amount of CPU resources. That said, one must use cationwhen using this method, as it is only really suitable for fully developed steady flows[18].The one-, two-equation and RNG models are all derivatives of the same fundamental assumptions;however, they vary in sophistication. The one-equation being the least, while the RNG being the mostsophisticated. As such, they also each influence the time to complete a simulation by varying degrees.Where the-one equation is the quickest, while the RNG model is the slowest.Finally the LES model is considered the most advance and accurate model of them all. However, itis noted that not only is it the slowest, it is also plagued by numerical instability issues that are highlydependent on the users mesh building skills.From a practical point of view, the most widely used model is the RNG model. It yields a suitablebalance been both simulation accuracy and practical simulation run times.Computational hardware: With this research residing in the field of CFD, some description of thehardware Flow-3DTM uses is needed.For the standard3 version of Flow-3DTM , the hardware used is simply a computer’s central pro-cessing unit (CPU). The CPU may include several cores that can be used for computations. Intuitively,the speed at which a Flow-3DTM simulation comes to completion would linearly increase with eachadditional CPU core; however, this is not the case, as the more cores used for a simulation, the moreoverhead (in the form of communication) between cores is required. In fact, it is observed that Flow-3DTM simulation speed plateaus at 16-cores. The significance of this may not initially be visible, butwhen factored in with decreased rates of CPU speed growth (i.e. the industry no longer follows Moore’sLaw), one is forced to ask: has the speed of CFD modelling reached its limit with traditional CPUs?2.3 Smoothed particle hydrodynamicsSmoothed particle hydrodynamics is a mathematical construct that approximates spatial derivativeswithout the need for a computational mesh. It does this by prescribing a Lagrangian description ofmotion to a discrete set of randomly spaced computational points (or particles) and interpolating be-tween neighbouring particles using an appropriate function.This, however, is a vastly different approach from traditional CFD constructs; as such, the followingsections outline the fundamental mathematics of SPH and will focus on how DualSPHysics[11, 24, 25]3There is also a parallel version of Flow-3DTM that allows simulations to be solved on a Linux cluster15takes advantage of this construct.2.3.1 Integral representationAt the core of SPH, lies the integral interpolation method (or smoothing kernel, or kernel). Here acontinuous field function, f, that depends on a three-dimensional position vector, x, within a givendomain, W, that is bounded by the surface, ∂W, (see Figure 2.1) can be exactly represented as theconvolution product with the Dirac delta function, d :f (x) =Zf (x0)d (x-x0)dx0 (2.3)where x0 is a neighbouring position vector, dx0 is an elemental volume that is located within W, andd (x-x0) is given by:d (x-x0) =( 1 f or x = x00 f or x 6= x0 (2.4)However, Equation 2.3 only holds for continuous functions as the Dirac delta function is infinitelynarrow. To accommodate discrete functions, the Dirac delta function can be replaced with a suitableapproximation, say W (x-x0,h), that is of a finite width, h. In the context of SPH, such functions aretypically called kernels (or smoothing kernel) and are discussed below. Equation 2.3 can be written asfollows (note, angular brackets, hi, are used to define the approximated representation of f ):f(x)⇡ hf(x)i= Z f(x0)W (x-x0,h)dx0 (2.5)!"#δΩΩωδωFigure 2.1: Smoothed particle hydrodynamics domain with three types of kernel support. Thecentre kernel has complete support. The right kernel has a truncated support, but is withinthe domain. And the left kernel has truncated that extends past the main domain.16Smoothing kernelsThe kernel function is a “distribution” function that is evaluated using neighbouring particles. Thecloser the particle is, the more influence it has on the the kernel function. Theoretically, this influencetends to zero as the distances between two particles approach infinity. However, this is impractical fornumerical schemes. As such, the kernel function is forced to be zero at a prescribed distance h, calledthe smoothing length.The smoothing length defines the radial distance away from a particle that other particles will in-fluence that particle. In three-dimensions, this would be a volumetric construct, and is typically calledthe kernel support, w , that is bounded by the surface ∂w . The larger the smoothing length, the moreparticles will influence a given particle and yield a more accurate results. However, with more parti-cles supporting a kernel more computational time is required. Oger[58] suggests using a normalizedsmoothing length (i.e. hDx , where Dx is the particle size) of 3 to limit errors in spatial derivatives, butadmits this yields excess computational time. As such, he indicates a normalized value of 1.2-1.5 is anacceptable balance between time and error.Kernel functions can theoretically be any distribution function; however, the following caveats limitwhat kernel functions can be used[42]:1. The kernel must be normalized for convergence[17]:RwW (x-x0,h) = 12. The kernel must approach the Dirac delta function as the smoothing length tends to zero, or elsethe integral interplant will not converge[17] : limh!0W (x-x0,h) = d (x-x0)3. The kernel must be symmetric, so that two particles an equal distance away will have the sameinfluence: W ((x-x0),h) =W (x-x0,h)4. The kernel must have compact support: W (x-x0,h) = 0 for |x-x0|>hWith this in mind, the so called “golden” kernel is a Gaussian kernel. It is stable and accurate for higherorder spatial derivatives[42]. However, a Gaussian kernel does not have compact support because itasymptotes to zero with increasing x. Consequently, several alternative kernel functions have beendeveloped. For example,• Modified Gaussian: W (x-x0,h) = 1p3/2h3 exp(q2)• Quadratic: W (x-x0,h) = 54ph3 [ 316q2 34q+ 34 ] for  q 2• Cubic spline[54]: W (x-x0,h) =8><>:1 32q2 + 34q3 f or 0 q 114(2q)3 f or 1 q 20 f or q 2• Wendland[82]: W (x-x0,h) = 2116(1 q2)4(2q+1) for 0 q 217(note: q is the non-dimensional distance between particles)Generally speaking, any of these can be implemented in SPH; however, in DualSPHysics only theCubic spline and Wendland kernels are implemented, and of these, the Cubic spline is the most widelyused since it most closely resembles a Gaussian distribution. That said, numerical stability[42] andparticle clumping[63] issues have been noted. The Wendland kernel, is said to be more accurate thanthe cubic spline, but increases computational cost[25].2.3.2 Derivative representationUsing Equation 2.5, a spatial derivative, ∂ f (x)∂x , can now be written as:⌧∂ f (x)∂x=Zw∂ f (x0)∂x W (x-x0,h)dx0 (2.6)Applying the product rule, Equation 2.6 is re-cast as:⌧∂ f (x)∂x=Zw∂∂x [ f (x0)W (x-x0,h)dx0]Zwf (x0)∂W (x-x0,h)∂x0 dx0 (2.7)If f is a vector or scalar field, then applying Gauss’s theorem[39, 66] to the first term on the right-handside of Equation 2.7 yields:⌧∂ f (x)∂x=Z∂wf(x0)W (x-x0,h) ·ndSZwf (x0)∂W (x-x0,h)∂x0 dx0 (2.8)Equation 2.8 is the complete SPH expression to represent a spatial derivative in either a vector orscalar field, respectively. However, to enforce point 4 above (i.e. the kernel must have compact support),the first term on the right-hand side in Equation 2.8 must be zero. Therefore, a spatial derivative in eithera vector or scalar field can be approximated as:⌧∂ f (x)∂x=Zwf (x0)∂W (x-x0,h)∂x0 dx0 (2.9)For the case where the kernel has a truncated support (i.e. at a boundary), the approximation of thespatial derivative falls apart. That said, special techniques (as discussed in Section 2.3.7) can be used torectify this problem.Equation 2.9 now allows the spatial derivative of a function to be determined using values of thefunction and spatial derivatives of the smoothing kernel, rather than from spatial derivatives of thefunction itself[42]. This approach will now be used to transform the differential from of the equationsof motion (see Table 2.1) into the discrete Smoothed Particle Hydrodynamics approximation of theequations of motion.2.3.3 Particle approximation of continuityReferring back to the SPH integral representation of a functions derivative (Equation 2.9), when thekernel is symmetric and the elemental volume, dx0, is replaced by the mass of a particle over its density18(i.e. a discrete volumetric construct), it can be re-written in discrete form as:⌧∂ f (xa)∂xa⇡Âbmbrbf (xb)∂W (xa-xb,h)∂xb(2.10)where b denotes the particle label up to the number of particles in support of particle a; and mb, rb, andxb are the mass, density, and position of particle b, respectively.However, it is noted that this derivative does not vanish if f (x) is a constant. To rectify this,Monaghan[52] suggests using the product rule of differentiation as follows:∂ f (x)∂x =1F"∂ (F f (x))∂x  f (x)∂F∂x#(2.11)where F is any differentiable function. Re-casting in SPH form with F= r , we get the following:∂ f (x)∂x =1ra"Âbmbrbrb f (xb)∂W (xa-xb,h)∂xb f (xa)∂W (xa-xb,h)∂xb#(2.12)or, if re-arranged and simplified∂ f (xa)∂x =1raÂb"mb( f (xb) f (xa))∂W (xa-xb,h)∂xb#(2.13)Equation 2.13 now vanishes when f (x) is a constant.Now, replacing — ·u with Equation 2.13 in the Lagrangian description of continuity (Table 2.1), thediscrete SPH continuity equation can be constructed as:DraDt=Âbmb(uaub)∂W (xa-xb,h)∂xb(2.14)Equation 2.14 is how DualSPHysics generally exploits the continuity equation using SPH; however,alternative codes may set F, in Equation 2.11, equal to one[52] or some other differentiable function.2.3.4 Particle approximation of Navier-StokesIn Lagrangian form, the Navier-Stokes equation can be written as (Table 2.1)r✓DuDt◆=—P+rg+G (2.15)If an inviscid fluid is assumed, Equation 2.15 can be written as:r✓DuDt◆=—P+rg (2.16)Using the discrete SPH equation for first derivatives (Equation 2.13), Equation 2.16 can now beexpressed as:19ra✓DuaDt◆=Âbmbra(PbPa)∂W (xa-xb,h)∂xb+rg (2.17)However, this does not conserve neither linear or angular momentum since the the force on particlea from b is not the same as the force on b from a (i.e. Pa 6= Pb). Instead, Monaghan[50] suggests tosymmetrize the pressure gradient term as follows:—Pr = —✓Pr◆+Pr2—r (2.18)Casting Equation 2.18 into SPH form (using Equation 2.10), we get the following:—Pr =Âbmbrb✓Pbrb◆∂W (xa-xb,h)∂xb+Para2Âbmbrbrb∂W (xa-xb,h)∂xb(2.19)Simplifying Equation 2.19 and placing it into into Equation 2.16, we get:DuaDt=Âbmb✓Para2+Pbrb2◆∂W (xaxb,h)∂xb+g (2.20)Equation 2.20 is now spatially discretized for an inviscid fluid. To include viscous effects, Monaghan[50]suggests using an artificial viscosity term, P, in Equation 2.20:DuaDt=Âbmb✓Para2+Pbrb2+Pab◆∂W (xaxb,h)∂xb+g (2.21)Where Pab is equivalent to the dissipative term, G in Table 2.1 and can be any suitable construct. How-ever, in DualSPHysics, Pab is defined as:Pab =8<:a 0.5(CaCa)h(uaub)·(xaxb)(rarb)[(xaxb)2+0.01h2]f or (uaub) · (xaxb) < 00 f or (uaub) · (xaxb) > 0(2.22)Where a is the coefficient of viscosity; u is the particle velocity; x is the particle position vector; r isthe particle density; and C is the particle speed of sound.From Equation 2.22 it is noted that a and h are user defined variables, where a has the mostinfluence on the viscous term.2.3.5 Moving the particlesAs the SPHmethod is a Lagrangian description, the computational points must move with the fluid. Thiscan be achieved by simply saying dxadt = ua or by Monaghan’s[49] suggestion of an “XSPH variant”:dxadt= ua + eÂbmb✓ubua0.5(ra +rb)◆W (xa-xb,h) (2.23)In this case, the XSPH variant moves a particle with the average velocity of its neighbours.DualSPHysics uses both methods, where e is a definable parameter ranging from zero to one.202.3.6 Equation of stateWith three equations (Equation 2.14, Equation 2.21, Equation 2.23) and four unknowns (velocity vector,position vector, pressure, and density) there is a closure problem.Analytically, this closure problem can be solved by assuming that water is an incompressible fluid,and the continuity equation reduces to — · u = 0 and the pressure can be expressed as a non-linearfunction of velocity that can be solved using a traditional Poisson solver. However, this can becomerather complex to solve numerically, so a “weakly compressible” assumption can be made. This allowsfor an efficient closure scheme.DualSPHysics uses the following equation of state to relate pressure to density:P(r) = roco27✓ rro◆71(2.24)Where ro is the reference density (usually 1000kg/m3 for water) and co is the speed of sound.Of particular note, Equation 2.24 indicates that a small change in density, r , will change the pres-sure. For weakly compressible fluids, this change does not occur instantaneously. Rather, it takes timefor a density change to effect pressure elsewhere in the SPH domain. The time it takes to make thischange is governed by the speed of sound. The faster the speed of sound, the “stiffer” the fluid, and themore dissipative the fluid will be[63].In DualSPHysics co is defined as the “coefficient of sound” multiplied by the maximum particlevelocity. The significance of which was just discussed, but has other implications with time integration(Section 2.3.8).Furthermore, it is well known in the SPH community that pressure oscillations about a particlecan occur. This may not be significant for certain cases where pressure or force terms are not explicitlyneeded, but it can cause significant heartache when coupled with a structural solver. Molteni[48] suggestsmoothing out these pressure oscillations using what he calls a “Delta-SPH” term. Essentially, this termfilters out high frequency density oscilations prior to solving for pressure. This research did not requirethe use of the Delta-SPH term.2.3.7 Boundary conditionsUntil now, it has been noted that the smoothing kernel requires full support. However, when a par-ticle is near a boundary, the support becomes truncated and the kernel falls apart. For astrophysicalcomputations where the problem has no boundaries, this poses no problem. But in hydrotechnical en-gineering problems boundaries are extremely important. As such, the kernel support is required to besupplemented so that it is no longer truncated.Several attempts have been made to implement suitable boundary conditions. Monaghan[51] used aset of particles along a boundary that used a repulsive force to keep fluid particles at bay (analogous torepulsive forces encountered between molecules). Randles[62] completed the kernel support for parti-cles that encroached on a boundary by taking the mirror image of truncated support along the boundaryline - this they called ghost particles. Crespo[10] on the other hand, used two layers of staggered fixed21particles to implement what he called a “dynamic boundary particle”. All three boundary conditionshave their pros and cons and can be discussed in detail; however, as DualSPHysics exclusively usesdynamic boundary particles, this research will focus on them.Physically, dynamic particles work by generating a density gradient along a boundary, which in turngenerates a pressure gradient that creates a repulsion mechanism along the boundary interface.Numerically, dynamics boundary particles are implemented by using an alternative SPH momentumequation:duadt=✓2c2W (xaxb,h)(W (xaxb,h)+W (xaxa,h))2+mbPab◆∂W (xaxb,h)∂xa+g (2.25)It is noted that boundary particles are easy to implement “since they can be calculated inside thesame loop as fluid particles” and hence “save considerable computational time”[10]. But like mostthings, this easy of implementation comes at a cost. It has been noted by Crespo and observed in thisresearch that using this method can cause an unphysical repulsion mechanism along a boundary with athin layer of fluid. As such, a gap between the flow and the boundary can be seen. In the case of thisresearch, this causes a moving boundary to be artificially larger in volume (see Section 4.2.2).For the case where the kernel support is truncated due to a free surface, there is no cause for alarm.This is actually where SPH excels, because a lack of support at the free surface simply means there areno particles influencing that particle, which is a natural way of tracking the free surface. Consequently,SPH is well poised for tracking complex non-linear free surfaces, and why many coastal hydrodynamicresearchers have used it[7, 13, 23, 64, 70].In addition to dynamic boundary particles, DualSPHysics can also prescribe periodic boundaries.That is, should the kernel support of a particle extend past a boundary, it can complete its support usingparticles from the opposite boundary. For example, when simulating flow through a rectangular flume,the outflow particles would support the inflow particles.2.3.8 Time integrationWith the Navier-Stokes equations reduced to a set of ordinary differential equations, any stable timeintegration scheme may be applied[52]. In DualSPHysics , the two available schemes are the Verlet andSymplectic schemes[25]:Verlet scheme: The Verlet scheme is a second-order integrator, that Verlet[77] discusses in detail; how-ever, in essence, it uses data from the previous time (i.e. time equals n-1) and the current time (i.e. timeequals n) to find the density, r , velocity, u, and position of a particle at the next time step (i.e. time equaln+1):228>>>>>>>>>>><>>>>>>>>>>>:(a) rn+1a = rn1a +2Dt✓Âbmbrb(uaub)∂W (xa-xb,h)∂xa◆n(b) un+1a = un1a +2Dt✓Âbmb✓Para2+ Pbrb2+Pab◆∂W (xaxb,h)∂xa +g◆n(c) xn+1a = xna +Dtuna +0.5Dt2✓ua + eÂbmb✓ubua0.5(ra+rb)◆W (xa-xb,h)◆n 9>>>>>>>>>>>=>>>>>>>>>>>; (2.26)However, because these equations are not coupled, this scheme tends to diverge if every 50th time stepis not updated in the following manner[25]:8>>>>>>>>>>><>>>>>>>>>>>:(a) rn+1a = rna +Dt✓Âbmbrb(uaub)∂W (xa-xb,h)∂xa◆n(b) un+1a = una +Dt✓Âbmb✓Para2+ Pbrb2+Pab◆∂W (xaxb,h)∂xa +g◆n(c) xn+1a = xna +Dtuna +0.5Dt2✓ua + eÂbmb✓ubua0.5(ra+rb)◆W (xa-xb,h)◆n 9>>>>>>>>>>>=>>>>>>>>>>>; (2.27)Symplectic scheme: The symplectic scheme is a second-order accurate integrator that is a slightlymore complex scheme than the Verlet scheme. Particle positions (Equation 2.28a) and densities (Equa-tion 2.28b) are computed mid-step. After which, momentum (Equation 2.29a) and particle positions areupdated at end of the step (Equation 2.29b) using mid-step values, while the drn+1dt is updated using un+1and xn+1[52].8>>>><>>>>:(a) rn+1/2a = rna + Dt2✓Âbmbrb(uaub)∂W (xa-xb,h)∂xa◆(b) xn+1/2a = xna +Dt2✓ua + eÂbmb✓ubua0.5(ra+rb)◆W (xa-xb,h)◆ 9>>>>=>>>>; (2.28)8><>:(a) xn+1a = xn+1/2a + Dt2 un+1(b) (Vru)n+1a = (Vru)n+1/2a + Dt2 (d(Vru)dt )n+1a9>=>;(2.29)Where V is the volume of a particle.Both schemes are dependent on variables already discussed, but they are also dependent on anincremental time step, Dt. In DualSPHysics, the time step is evaluated using the minimum time stepcomputed by either a forcing condition (so that particles do not get too close to each other over a time-step) or a combined viscous diffusion and Courant-Friedrichs-Lewy[8] (CFL) condition[49]:23Dt = min8>>>><>>>>:min[0.3(ph/ | fa|)]min"✓0.3hca+max h((uaub)·(xaxb))(xaxb)2◆ (2.30)Where fa is the force per unit mass on a particle, ca is the speed of sound evaluated for each particle, uis a particle’s velocity, and x is a particle’s position.From inspection of the second constraint in Equation 2.30, it is noted that the dominating variableis the speed of sound. The smoothing length is O(0.01), the difference in velocity between particles isassumed O(1), the difference in particle position is related to particle size, O(0.001-10), and the speedof sound in water is O(1000). As such, Dt will be impractically small for numerical simulations, so anapproximation to this is to assume the speed of sound is at least ten times the maximum velocity. Thisassumption limits the density fluctuations on DualSPHysics to within 1%. For example, if a particle’spressure is proportional to a reference density multiplied by the square of a particle’s velocity (p⇠ rou2a),then Equation 2.24 can be re-written as:raro=✓7u2ac2o+1◆1/7⇡ 1+u2ac2o+O✓7u2ac2o2◆(2.31)therefore,Draro⇠u2ac2o(2.32)and if co = 10ua, Equation 2.32 goes to:Draro⇠u2a100u2a= 1% (2.33)Equation 2.33 validates approximating the speed of sound as ten time the particle velocity.2.3.9 Computer hardwareInstead of using CPUs to execute CFD code in serial or parallel with several CPU cores, DualSPHysicsuses graphical processing units (GPUs)[11], to compute things in parallel using hundreds of low costGPU cores.For example, simulations performed in this thesis were carried out using a NVIDIATM Tesla-C2075GPU with 448 cores, where the decrease in simulation run times were orders of magnitude lower whencompared with a serial execution.Unfortunately, not all CFD codes are well suited for GPU execution4. Hence, not many commercialcodes use GPUs. That said, because of the open question stated earlier (i.e. has the speed of CFDmodelling reached its limit with traditional CPUs), software developers are now trying to utilize GPUsfor cheaper and faster simulations.4This is mainly due to how matrices are solved24Chapter 3Physical Experiment3.1 Model setupThe physical experiment, Figure 3.1, was constructed and tested at Northwest Hydraulic Consultants,North Vancouver, Canada laboratory. It consisted of a 0.61-m wide flume with a 30-deg sloped westwall, a 45-deg slopped east wall, two vertical walls on the north and south sides of the model, and a1.02-m horizontal section connecting the toe of both the east and west slopes.Subaerial land slide generated waves (SLGW)were created by releasing a 0.18x0.31x0.31-mweightedacrylic box with a 45-deg sloped nose down the west slope of the model. When released, the slide trav-elled (by gravity) 0.77-m before impacting the 0.61-m deep water; after which, it travelled another1.05-m, through the water, before hitting a stop block.Using National Instruments (NI) LabVIEWTM 2012 with two NI USB-6008 four channel 12-bitdata acquisition boards, slide acceleration and displacement, as well as wave heights were recorded fora period of six-seconds at a frequency of 100Hz. The six instruments used to measure this data are asfollow:Slide acceleration: The acceleration of the slide as it travelled down the west slope was measuredusing a Memsic CXL04GP1 single axis accelerometer. Manufacture specifications are outlined in tableTable 3.1. The overall accuracy, E, of this device can be inferred as E =pe1 + e2 + ...+ e6 = 3.2%,where e is the individual error of each component in the accelerometer.Slide displacement: A Celesco SG1-120-3 potentiometer was used to measure the slide displacementover time as it travelled down the west slope. Accuracy of this device is stated by the manufacture as0.01-m over 3.05-m. Or ± 0.35% full scale (FS).Wave probes 1, 2, & 3: RBR WG-50 capacitance wave-probes, which work by measuring the changein capacitance of the probe as the water level changes with time. They have an accuracy (stated by themanufacture) of ± 0.4%, and react instantaneously.The probes were installed vertically in the model so that half their length was located within the water,while the other half was outside the water (still water conditions). Output data consisted of temporal25Figure 3.1: Subaerial Landslide Generated Wave Test Stand. Elevation view shown in panel (a) and plan view shown in panel (b).(Note: all units are in meters)26Table 3.1: Accelerometer technical specificationsSpecification Value UnitsInput Range (e1) ± 4 G†Zero Drift (e2) ± 0.1 G†Sensitivity (e3) 500 ± 15 mV/G†Transverse Sensitivity (e4) ± 5 % of SensitivityNon-Linearity (e5) ± 0.2 % FS‡Alignment Error (e6) ± 2 Degrees† Multiples of Gravity; ‡ Full Scalevoltages that were converted to wave heights above the still water level.Wave probe 4: A Milone PN-6573TC-24 continuous fluid level probe was used. It reacts instanta-neously and has a resolution of ± 0.25-mm with an accuracy of 0.006-m over its 0.607-m length ( 1%FS). The probe laid flat along the east slope with half of its length residing in the water, and the otherhalf outside the water. Output data consisted of temporal voltages that were converted to wave heightsabove the still water level.In addition to the above mentioned instrumentation, three Allied Manta G-125 video cameras withTheia MY110M machine vision lenses (i.e. no optical distortion) were used to record the water’s freesurface for each test. Theoretically, this setup could capture 30 frames per second; however, in reality,it only achieved 15-18 frames per second.During each test run, the model would operate in darkness while a thin sheet of laser light was castalong a plane offset 0.18-m from the centreline of the flume. This, combined with using luminescentdye, allowed for enhanced visualization of the water’s free surface.3.2 Model resultsThe physical experiment, discussed above, consisted of releasing the slide from a single fixed locationon the west ramp and recording data for a period of 6-sec. Experimental random error was reduced byrepeating this five times.The data recovered from all six instruments during each of the five test runs can be seen in Figure 3.2.Here, it is observed that this data is highly repeatable. The maximum coefficient of variation (standarddeviation over the mean value) is 0.33, 0.02, 3.81, 1.02, 0.21, and 0.16 for the slide acceleration, slidedisplacement, wave probe 1, wave probe 2, wave probe 3, and wave probe 4, respectively.From Figure 3.2, it is found that after an instantaneous release, the slide travelled down the westslope with an average acceleration of 3.8m/s2 until it impacted the water, thereafter decelerating at aaverage rate of -3.5m/s2.The maximum observed generated wave was the wave caused by the slide upon initial impact withthe water. This wave travelled the length of the flume and rode up the east wall to a height of 0.08-27−3.5−2.5−1.5−0.50.5Slide Accelerationx Gravity [m/s2]Slide Displacement  −10−5051015Wave Probe 1Wave Height [cm]Wave Probe 20 1 2 3 4 5 6−10−5051015Wave Probe 3Wave Height [cm]Time [sec]0 1 2 3 4 5 6Wave Probe 4Time [sec]Run 1Run 2Run 3COVFigure 3.2: Results of the five physical experiment test runs. Note, slide displacement y-axis ticksare at 0 0.5 1.0 1.5 2.0-meters.m above the still water elevation. However, this was not the observed maximum wave run-up. Thisoccurred during the next run-up cycle, when the wave reached 0.086-m above the still water elevation.At this point, the wave running up the east wall was not simply a result of east-west reflection of thefirst wave, but also a combination of additional waves being generated post impact behind the slide, andnorth-south wave reflections. After the slide came to a rest, the waves in the flume gradually dissipateddue to boundary friction, and viscosity.3.2.1 Slide motionAlthough slide acceleration and displacement were recorded in the physical experiment, they are notuseful input parameters for the numerical models. Both Flow-3DTM and DualSPHysics require velocityprofiles for prescription of motion. As such, the acceleration data was integrated and the displacement28data was differentiated to get two separate velocity profiles for all five tests.Figure 3.3 shows the ensemble average of the slide velocity using both acceleration and displace-ment data during a six second period. Here, it is noted that between time = 0 and time = 1.25-secthe computed slide motion using either the potentiometer or accelerometer data yields similar results(root-mean-square-difference for this period is 0.05-m/s). However, after time=1.25-sec the slide mo-tion computed using the potentiometer and accelerometer differs considerably. The potentiometer dataindicates that at 1.25-sec, the slide rapidly changes from travelling down the west slope (positive veloc-ity) to travelling up the west slope (negative velocity) for a brief period of time before again travellingdown the west slope and ultimately stopping. This data matches visual observations, as it was noted theslide did not completely stop when it hit a stop block (located at the toe of the slide), but rather bouncedoff the stop block once before coming to a complete stop. The accelerometer data, however, does not tellthe same story. It suggests something occurred at time=1.25-sec and the slide velocity changes by firstdecreasing, then increasing before stabilizing at a constant velocity of approximately 0.75-m/s. Thisof course is not physical, and suggests something was wrong with the data after 1.25-sec. As such, theslide motion was taken as the average of the computed motion from the potentiometer and accelerometerdata sets between time = 0 to time = 1.25-sec, and then set to 0-m/s, thereafter. This motion was thenused as an input file for numerical simulations and classified as a ’simple’ description of slide motion.Subsequently, the entire potentiometer data set was used as a ’complex’ slide description of motion.0 1 2 3 4 5 6−0.500.511.522.53Time [sec]Velocity [m/s]  Average Accelerometer VelocityAverage Potentiometer Velocity (Complex)Slide Velocity Used for Numerical Simulations (Simple)Figure 3.3: Slide motion computed from the average of five accelerometer and potentiometer datasets. A simple slide motion used for numerical simulations is the average between the aver-age accelerometer and average potentiometer data from Time=0 to Time=1.25 sec, and thenset to zero thereafter. A complex slide motion was also used, and is defined as the entirepotentiometer data set.29Chapter 4Numerical ExperimentRegardless of the robustness and maturity of a numerical scheme, there will always be a need to evaluatethe scheme’s default settings. This is especially true for general purpose schemes where the defaultsettings are the appropriate settings for the majority of scenarios, not necessarily for the desired one.As such, the physical data discussed earlier (see Chapter 3) is used to validate both the Flow-3DTMand DualSPHysics models using their respective default settings, thereafter referred to baseline mod-elling. Subsequently, the physical model data is used to calibrate both models through a series ofdevelopmental modelling until an optimal set of parameters is found.In this chapter, the sections on baseline modelling, developmental modelling, and final results rep-resent the typical path of a simulation sequence used to find the optimal settings for a given scenario (inthis case, aerial landslide generated waves). These sections have further been split into two subsectionsto accommodate both Flow-3DTM and DualSPHysics models.4.1 Baseline modellingBaseline modelling consisted of running both Flow-3DTM and DualSPHysics models with their defaultsettings, and comparing their output with the physical experiment data described in the previous chapter(see Chapter 3). For both numerical models this is described in detail below.4.1.1 Eulerian computational fluid dynamicsUsing the same dimensions as the physical model, a computerized three-dimensional model was con-structed and imported into Flow-3DTM as a sterolithography (STL) file. Once in Flow-3DTM , 5.8million 0.010-m sized square cells in a 4.39(L)x0.60(W)x2.20(H)-m rectangular mesh were used todiscretize the 3D-model.The slide was prescribed a simple velocity profile based on data collected from the physical model(as discussed above, Section 3.2.1). Surface roughness for the east and west sloped faces was set to 0-m(i.e. hydraulically smooth). The modelled fluid was water at 293k, and a dynamic Renormalization-Group (RNG) turbulence model with the default settings were used in Flow-3DTM v10.1[18].Similar to the physical model, Flow-3DTM simulated a period of six seconds and recorded wave30heights at wave probe locations 1-3 (Figure 4.1). For wave probe 4, Flow-3DTM does not explicitlycompute the temporal wave run-up heights. Instead, it outputs a text file containing a longitudinalcentreline slice of the free surface. This output however, is not the free surface between fluid and air,but rather between fluid/solid and air. Flow-3DTM does not differentiate between fluid or solid whencomputing the free surface. As such, two different post processing schemes were used to capture thewave run-up height.1. Threshold operator: Using a rotated frame of reference that aligns with the east wall (i.e. theeast wall is the new zero reference line), the first free surface elevation above a given threshold(0.00001-m) was considered the location at which the free surface intersected the east wall.2. Maximum second-derivative: Here, the maximum second-derivative of the free surface profilenear the east wall is considered the location at which the free surface intersected the east wall.Both methods appear to yield reasonable results (Figure 4.1), but they also have their issues. Thethreshold operator tends to work very well when fluid is travelling up the east wall; but when travellingdown, this method under predicts the free surface due to significant noise in the data. The maximumsecond-derivative generally works better then the threshold operator method, but its output is noisy dueto sudden changes in the location of maximum second-derivative. For example, at one time step themaximum second-derivative may be at the free surface/east wall interface, but then it may jump to arandom section where the slope of the free surface is quite large. To alleviate these issues, results fromboth methods are presented.Frames of the Flow-3DTM simulation can be seen in Figure 4.2 and Figure 4.3. Here, the slide can beseen impacting the fluid at about 0.8-sec and generating the initial impact wave. Figure 4.4b shows thatduring impact, Flow-3DTM does not capture the splash nor the non-linear shape of this impact wave.As such, Flow-3DTM under predicts the impact wave height (at time equal to about 1.2sec) by 36%,Figure 4.1a.At 1.0sec, the fluid starts to wrap around behind the slide before hitting itself at 1.2-sec (Figure 4.3).After this, the initial impact wave is observed to travel from west to east (Figure 4.4c) until time equal toabout 2.20-sec, where it runs 0.092-m (15% higher than the physical experiment) up the east slope beforereflecting back. During which, the secondary wave generated behind the slide at 1.2-sec is observed totravel west to east and reflect off the north and south walls, which can seen in Figure 4.3.After the initial impact wave reflects off the east slope, it starts to interact with the wave generatedbehind the slide and the waves reflected off the north and south walls before running back up the eastslope to a height of 0.172-m at 3.8-sec (Figure 4.2 and Figure 4.4d), or 96% higher than the physicalexperiment.Following the maximum wave run-up, subsequent run-ups were again over-predicted and Flow-3DTM wave probe 1-3 ( Figure 4.1) data became less accurate. Table 4.1 presents the associated errorwith the baseline testing. The relative maximum wave amplitude error is the relative error between thephysical and Flow-3DTM data pertaining to the maximum amplitude in each wave probe data set.31Based on the above noted observations, a series of developmental tests were carried out to try andreduce the error between Flow-3DTM and the physical experiment - specifically the maximum waverun-up. The description of these tests and their results can be found in Section 4.2.1.−10−505101520(a) Wave Probe 1 (F1)(F3D−RMSE=1.70)Wave Height [cm]−10−505101520(b) Wave Probe 2 (F1)(F3D−RMSE=1.40)Wave Height [cm]−10−505101520(c) Wave Probe 3 (F1)(F3D−RMSE=1.07)Wave Height [cm]0 1 2 3 4 5 6−10−505101520(d) Wave Probe 4 (F1)(F3D−RMSE=3.62)Wave Height [cm]Time [sec]  Flow−3DPhysicalFigure 4.1: Baseline Flow-3DTM and physical model data of the free surface measured at waveprobe 1-4 locations. Panel d represents the evaluated wave run-up using the maximum secondderivative method (green line) and the threshold operator method (magenta line).32Time: 0.20sec Time: 0.40sec Time: 0.60secTime: 0.80sec Time: 1.00sec Time: 1.20secTime: 1.40sec Time: 1.60sec Time: 1.80secTime: 2.00sec Time: 2.20sec Time: 2.40secTime: 2.60sec Time: 2.80sec Time: 3.00secTime: 3.20sec Time: 3.40sec Time: 3.60secTime: 3.80sec Time: 4.00sec Time: 4.20secTime: 4.40sec Time: 4.60sec Time: 4.80secTime: 5.00sec Time: 5.20sec Time: 5.40secTime: 5.60sec Time: 5.80sec Time: 6.00secFigure 4.2: Baseline elevation view of Flow-3DTM frames (Test- #F1). Note, colour gradient rep-resent velocity magnitude, where blue is ’slow’ and red is ’fast’.33Time: 0.20sec Time: 0.40sec Time: 0.60secTime: 0.80sec Time: 1.00sec Time: 1.20secTime: 1.40sec Time: 1.60sec Time: 1.80secTime: 2.00sec Time: 2.20sec Time: 2.40secTime: 2.60sec Time: 2.80sec Time: 3.00secTime: 3.20sec Time: 3.40sec Time: 3.60secTime: 3.80sec Time: 4.00sec Time: 4.20secTime: 4.40sec Time: 4.60sec Time: 4.80secTime: 5.00sec Time: 5.20sec Time: 5.40secTime: 5.60sec Time: 5.80sec Time: 6.00secFigure 4.3: Baseline plan view of F3 Flow-3DTM frames (Test- #F1)34(a)(b)(c)(d)(e)(f)Figure 4.4: Baseline Flow-3DTM centreline free surface overlaid onto physical experiment images.Panels a, b, c, d, e, and f represent time at 0.00, 0.55, 0.96, 1.58, 3.66, and 5.82-sec, respec-tively.35Table 4.1: Baseline Flow-3DTM errorWave probeRelative max. waveamplitude errorRoot mean square errorWS1 -36% 1.70-cmWS2 -19% 1.40-cmWS3 -18% 1.07-cmWS4a† 96% 3.62-cmWS4b‡ 100% 2.82-cm† Maximum second derivative; ‡ Threshold operator364.1.2 Smoothed particle hydrodynamicsA copy of the STL file used in the Flow-3DTM model was imported into DualSPHysics and the domainwas discretized into 700,000 boundary particles and 1.3 million fluid particles, for a total of 2.0 million0.010-m sized particles.The slide was prescribed a simple velocity profile based off data collected from the physical model(as discussed above, Section 3.2.1), and default settings presented in Table 4.2 were used to initiate asimulation:Table 4.2: Baseline DualSPHysics default settingsParameter ValueCFL 0.2Coefficient of sound 10Smoothing length coefficient 0.866025Reference density 1000kg/m3Time stepping verletKernel Cubic splineViscosity coefficient 0.24As with the physical experiment and Flow-3DTM, DualSPHysics simulated a period of six seconds,during which it computed the free surface height at the wave probe 1-41 locations at a frequency of50Hz. The data from this test can be seen in Figure 4.5. Frames of the DualSPHysics simulation can beseen in Figure 4.6 and Figure 4.7. Here, like the Flow-3DTM simulation, the slide can be seen impacting(Figure 4.8b) the fluid at 0.8-sec and generating an initial impact wave that is 1% higher then the physicalexperiment. At 1.0-sec, the fluid starts to wrap around behind the slide and then hits its self at 1.2sec(Figure 4.7). Then, the initial impact wave is observed to travel from west to east (Figure 4.8c), but itsmotion is ’sluggish’ after running up 0.12-m (or 50% more than the physical experiment) up the eastwall (Figure 4.8d), where the wave dissipates quickly (Figure 4.8e and f).Comparing Figure 4.6 and Figure 4.7 against Figure 4.2 and Figure 4.3 shows DualSPHysics ismore dissipative than Flow-3DTM . Table 4.3 quantifiably presents the associated error with the baselinetesting. Furthermore, looking at the width of the impact crater 1.2-sec for both the DualSPHysics andFlow-3DTM simulations suggests that the DualSPHysics slide is much wider. In fact, when measuredit is 13.25% wider than the Flow-3DTM impact crater. Also, in Flow-3DTM the initial impact waveappeared radial, whereas DualSPHysics it clearly is not radial.1Unlike Flow-3DTM , DualSPHysics is well-poised to capture maximum wave run-up heights. This is primarily due to itsability to explicitly distinguish between the fluid and solids. As such, post-processing of the output data was trivial.37−10−505101520(a) Wave Probe 1 (S1)(SPH−RMSE=2.64)Wave Height [cm]−10−505101520(b) Wave Probe 2 (S1)(SPH−RMSE=2.24)Wave Height [cm]−10−505101520(c) Wave Probe 3 (S1)(SPH−RMSE=1.82)Wave Height [cm]0 1 2 3 4 5 6−10−505101520(d) Wave Probe 4 (S1)(SPH−RMSE=2.40)Wave Height [cm]Time [sec]  DualSPHysicsPhysicalFigure 4.5: Baseline DualSPHysics and physical model data of the free surface measured at waveprobe 1-4 locations.38Time: 0.20sec Time: 0.40sec Time: 0.60secTime: 0.80sec Time: 1.00sec Time: 1.20secTime: 1.40sec Time: 1.60sec Time: 1.80secTime: 2.00sec Time: 2.20sec Time: 2.40secTime: 2.60sec Time: 2.80sec Time: 3.00secTime: 3.20sec Time: 3.40sec Time: 3.60secTime: 3.80sec Time: 4.00sec Time: 4.20secTime: 4.40sec Time: 4.60sec Time: 4.80secTime: 5.00sec Time: 5.20sec Time: 5.40secTime: 5.60sec Time: 5.80sec Time: 6.00secFigure 4.6: Baseline elevation view of DualSPHysics frames (Test- #S1). Note, colour gradientrepresent velocity magnitude, where blue is ’slow’ and red is ’fast’.39Time: 0.20sec Time: 0.40sec Time: 0.60secTime: 0.80sec Time: 1.00sec Time: 1.20secTime: 1.40sec Time: 1.60sec Time: 1.80secTime: 2.00sec Time: 2.20sec Time: 2.40secTime: 2.60sec Time: 2.80sec Time: 3.00secTime: 3.20sec Time: 3.40sec Time: 3.60secTime: 3.80sec Time: 4.00sec Time: 4.20secTime: 4.40sec Time: 4.60sec Time: 4.80secTime: 5.00sec Time: 5.20sec Time: 5.40secTime: 5.60sec Time: 5.80sec Time: 6.00secFigure 4.7: Baseline plan view of DualSPHysics frames (Test- #S1). Note, colour gradient repre-sent velocity magnitude, where blue is ’slow’ and red is ’fast’.40Therefore, with such poor agreement with the physical experiment, the following questions must beasked:1. Is DualSPHysics accurately generating the same waves as the physical experiment?2. Is DualSPHysics overly dampend due to inaccurate numerical dissipation?3. Are waves accurately reflected at boundaries?To answer these questions, a series of developmental tests were carried out to parse out the issue(s).A description of these tests and their results can be found in Section 4.2.2.Table 4.3: Baseline DualSPHysics errorWave probeRelative max. waveamplitude errorRoot mean square errorDualSPHysics1 1% 2.64-cm2 23% 2.24-cm3 27% 1.82-cm4 39% 2.40-cmFlow-3DTM1 -36% 1.70-cm2 -19% 1.40-cm3 -18% 1.07-cm4a† 96% 3.62-cm4b‡ 100% 2.82-cm† Maximum second derivative; ‡ Threshold operator41(a)(b)(c)(d)(e)(f)Figure 4.8: Baseline DualSPHysics centreline free surface overlaid onto physical experiment im-ages. Panels a, b, c, d, e, and f represent time at 0.00, 0.55, 0.96, 1.58, 3.66, and 5.82-sec,respectively.424.2 Developmental modellingThe purpose of this section is to evaluate appropriate user-controlled variables, and to discover how theyinfluence Flow-3DTM and DualSPHysics solutions, an objective of which is to find an optimal series ofvariable settings that can be used for similar simulations.4.2.1 Eulerian computational fluid dynamicsFlow-3DTM allows a user to select which physics need to be solved for a specific problem, the type ofsolver (i.e. explicit or implicit), and in some cases, what degree of accuracy (i.e. first or second-order)the solver may have.The hydrotechnical physics packages that Flow-3DTM includes are as follows:• Air entrainment• Bubble and phase change• Cavitation• Density evaluation• Dissolving objects• Drift-flux• Elasto-visco-plasticity• Fluid sources• Granular flow• Heat transfer• Moving and simple deforming objects• Porous media• Sediment scour• Shallow water• Surface tension• Viscosity and turbulenceFor this work, the “moving and simple deforming objects” as well as the “viscosity and turbulence”physics packages were primarily used. However, “air entrainment”, “density evaluation”, “drift-flux”,and “surface tension” physics packages were also used on selected simulations. Flow-3DTM suggestsusing an explicit scheme when possible, as it usually produces a more accurate result[18]. The onlyexception to this is solving the pressure field. In this case, an implicit scheme is used to maintainsimulation stability[18].As for degree of accuracy, the only possible user selection is the accuracy of the momentum advec-tion solver. The default is set to a first-order approximation; however, there is an option of a second-ordersolver.To evaluate all these options, a series of seventeen development tests were conducted. Parameterchanges included altering cell size (see Section 2.2), surface roughness, slide description (see Sec-tion 3.2.1), momentum advection accuracy, and which physics to include. A complete list of all thesimulations (including the baseline and final test) are presented in Table 4.5. Images of these simula-tions are not shown because changes in the results are not easily observed through visual inspection,instead a description of observations is provided.43Changing the surface roughness of the east and west sloped walls (north and south stayed at 0-m)from 0-m (F1) to 0.00025-m (F3) did very little to change the results. Further increasing the surfaceroughness to 0.001-m (F6) did reduce the maximum wave run-up error from 96% to 88% when usingthe maximum second-derivative method (MSD), but did little for the threshold operator (TO) method.Qualitatively, 0.001-m seems too high when looking at the physical experiment surfaces, and 0-m seemsinappropriate. As such, all other tests were carried out using a surface roughness of 0.00025-m.Including Flow-3D’sTM default surface tension package with the baseline settings (F4) reduced themaximum wave run-up error from 96% to 94% while using the MSD method, but did little for the TO.At first, it was thought that surface tension effects may be important near the leading run-up edge, andmay be responsible for energy loss in the physical experiment. However, these results suggest otherwise.Using a second-order momentum advection scheme instead of a first-order scheme (F5) not onlyreduces the root-mean-square-error of all the probes, but also reduces the maximum wave run-up errorfrom 96% to 64%. This seems appropriate, as the driving mechanism for this problem is the transfer andadvection of momentum. It is also noted that “this method performs well for free surface waves“[18]. Infact, on average, using a second-order scheme versus a first-order one reduced the error by about 50%.When reducing the baseline mesh from 0.01-m to 0.0075-m (F14), the maximum wave run-up erroractually became much worse. It went from 96% to 122% while using the MSD method, and from 99%to 124% for the TO method. This is most peculiar, as a reduction in cell size tends to be more accurate.However, when using a mesh cell size of 0.0075-m and a second-order momentum advection scheme(F9), the error is reduced to 59% and 54% for the MSD and TO methods, respectively.If the complex slide description is used (see Section 3.2.1) with the baseline settings (F12), there isno notable changes in to the results. But, if the mesh cell size is reduced to 0.0075-m (F11), then themaximum wave run-up error balloons to 130% and 137% for the MSD and TO methods, respectively.If the second-order momentum advection scheme is used with the complex slide description and amesh cell size of 0.0075-m (F10), the error drops to 43% and 44% for both the MSD and TO methods,respectively. However, if the mesh cell size is further reduced to 0.005-m (F15), then the maximumwave run-up error increases to 80% and 85% for the MSD and TO methods, respectively. Note, besidesthe change in cell size, the 0.005-m simulation (F15) was executed using Flow-3DTM MP v5.02.15.Theoretically, this version of Flow-3DTM is identical to Flow-3DTM v10.1, except for the added benefitof mesh decomposition and multi-node computations; however, there are some minor discrepancies. Todetermine whether or not the increase in error in F15 (when compared to F10) was due to a versionissue, F10 was re-run on Flow-3DTM MP v5.02.15. This test did yield different results, and the RMSEand REMA error’s did change, but only by a small percentage. As such, a version issue is not considereda problem for this large increase in error.Finally, knowing that the baseline test neglected air entrainment, the Flow-3D’sTM “air entrainment”and “drift-flux” package2 was activated while using a complex slide description and a second-ordermomentum advection scheme. The maximum wave run-up error for this simulation is 54% and 53% forthe MSD and TO methods, respectively. This simulation was also attempted with a mesh cell size of2Both the “air entrainment” and “drift-flux” packages are required for both entrainment and release of air440.0075-m, but failed due to an excessively small time step requirement.In summary, surface tension and surface roughness have little impact on Flow-3DTM maximumwaverun-up predictions, while the accuracy of the momentum advection scheme is the single most importantparameter. Combinations of surface roughness, surface tension, mesh cell sizes, slide descriptions, andmomentum advection accuracy do not always yield better results. In some cases, the error is magni-fied. That said, using a second-order momentum advection scheme, with a complex slide description,and a mesh cell size of 0.0075-m provided the most accurate Flow-3DTM results for prediction of themaximum wave run-up during SLGW simulations.Table 4.4: Developmental Flow-3DTM ErrorTest NumberRMSE1 REMA2WS1 WS2 WS3 WS4 WS1 WS2 WS3 WS4a3 WS4b4F15 1.7 1.4 1.1 3.6 -36 -18 -17 96 99F3 1.7 1.4 1.0 3.6 -36 -19 -17 96 100F4 1.7 1.4 1.1 3.5 -36 -17 -17 94 99F5 1.4 1.2 1.0 3.1 -44 -25 -30 64 65F6 1.7 1.4 1.0 3.5 -37 -22 -17 88 98F7 1.4 1.3 1.1 2.9 -48 -30 -23 59 63F8 1.4 1.2 1.0 3.0 -45 -25 -30 59 63F9 1.5 1.2 1.0 2.8 -47 -24 -27 59 54F106 1.5 1.1 1.0 2.7 -47 -20 -30 43 44F11 1.6 1.4 1.3 3.5 -15 -10 5 130 137F12 1.6 1.4 1.2 3.4 -30 -16 -15 96 99F13 1.5 1.2 1.2 2.9 -48 -22 -26 55 55F14 1.6 1.3 1.2 3.6 -26 -21 -6 122 124F15 1.2 0.9 0.7 3.1 -38 -27 -13 80 85F16 1.5 1.1 1.1 2.5 -47 -26 -30 48 43F17 1.6 1.1 1.0 3.1 -47 -19 -32 54 531 Root-mean-square-error [cm]2 Relative error of maximum amplitude [%]3 Maximum second-derivative4 Threshold operator5 Baseline test (Section 4.1.1)6 Final test45Table 4.5: Developmental Flow-3DTM parameter settingsTest Number1Flow-3DVersion1SlideVelocity Cell Size2SurfaceTension3SurfaceRoughness2MomentumAdvection4 Run Time5F16 10.1 Simple 0.010 No 0.00000 1st Order 6F3 10.1 Simple 0.010 No 0.00025 1st Order 6F4 10.1 Simple 0.010 Yes 0.00025 1st Order -F5 10.1 Simple 0.010 No 0.00025 2nd Order -F6 10.1 Simple 0.010 No 0.00100 1st Order -F7 10.1 Simple 0.010 No 0.00100 2nd Order -F8 10.1 Simple 0.010 Yes 0.00100 2nd Order -F9 10.1 Simple 0.008 No 0.00025 2nd Order -F107 10.1 Complex 0.008 No 0.00025 2nd Order 26F11 10.1 Complex 0.008 No 0.00025 1st Order 27F12 10.1 Complex 0.010 No 0.00025 1st Order 6F13 10.1 Complex 0.010 No 0.00025 2nd Order 11F14 10.1 Simple 0.008 No 0.00025 1st Order 22F15 MP 5.02.15 Complex 0.005 No 0.00025 2nd Order 33F16 MP 5.02.15 Complex 0.008 No 0.00025 2nd Order 5F17 10.1 Complex 0.010 No 0.00025 2nd Order 91 All simulations used an implicit pressure solver, explicit free surface pressure & advection solvers2 Units are meters3 Activation of Flow-3D’s default surface tension module4 Order of momentum advection scheme (i.e. accuracy)5 Run time is measured in hours of CPU time on an 8 core AMD FX-8320 3.5Hz processor6 Baseline test7 Final test464.2.2 Smoothed particle hydrodynamicsUnlike Flow-3DTM, which has an intuitive physics packages and well researched user variables, Dual-SPHysics is a bit less mature and requires both a strong knowledge of its underlying numerics and bruteforce to find the appropriate combination of variables for a given problem. Consequently, twenty-twodevelopment tests were required to find a suitable solution. Table 4.6 presents a complete list of all thetests and their settings. Table 4.7 presents the root-mean-square-error and relative error of the maximumamplitude. A description of observations follows.Reducing the viscosity coefficient from 0.25 (S1) to 0.05 (S2-S5) produces less dissipative results,and the DualSPHysics results approach the physical experiment results (Figure 4.9 and Figure 4.10).However, as viscosity was reduced, the maximum wave run-up increased. This physically makes sense,because if less energy is dissipated, then there is more energy driving wave run-up.DualSPHysics results are even more dissipative when the coefficient of sound is increased (S1 toS6 & S7-Figure 4.11; and S13 to S14). This agrees with Robinson’s observation that “increasing thesound speed results in a significant increase in numerical dissipation”[63]). Additionally, increasing thecoefficient of sound from 10 (S1/S13) to 20 (S6/S11) doubled the computational time. While goingfrom 10 (S1) to 30 (S7) tripled the computational time.Switching from a Cubic kernel with a smoothing length coefficient of 0.87 to a Wendland kernelwith a smoothing length coefficient of 1.5 (from S2 to S8) increases the impact wave error to 113%(panel a Figure 4.12) and the maximum wave run-up error to 59% (Figure 4.12).Reducing the particle size from 0.01-m to 0.005-m while using a smoothing length coefficient of1.23 (see Section 2.3.1) appears to do little to enhance the solution. Aside from the spike in the initialimpact wave3 (Figure 4.12), this simulation qualitatively appears very similar to the previous simulation(S8), and does not notably change with a smaller particle size (S9 to S13). Therefore, combiningthis with a required 49-hours of GPU time and 450-GB of hard dive space, it was decide to keep theDualSPHysics particle sizes at 0.01-m. However, the smoothing length coefficient was kept at 1.23.For all developmental simulations, the Delta-SPH coefficient (see Section 2.3.6) had been set tozero. That said, for test S10 (Figure 4.13) it was set to 0.1 and the simulation ’leaked’ particles andfailed. Consequently, the Delta-SPH coefficient was kept at zero for the reminder of the study.In an attempt to evaluate the implications of different time integration schemes, test S12 used thesame parameters as S11, but used a Symplectic scheme rather than a Verlet scheme. The differencebetween the two RMSEs and REMAs is marginal, however the main difference is that the Symplecticscheme took twice as long to complete and there appears to be a bug in the DualSPHysics code, as theoutput for wave probe 4 was all zeros. This occurred when running S21 with a Symplectic scheme (S24- results not presented).For test S1-S13 it is noted that the first wave generated upon impact notably differs from the physicalexperiment. It appears to occur before the physical wave. Also, looking at the width of the impact craterat 1.0sec (in Figure 4.7) it is apparent that its width is much larger than the Flow-3DTM impact crater. Infact, when measured it is 13.25% wider than the Flow-3DTM impact crafter - which is an artifact of the3Likely caused by a rogue particle47Dynamic boundary particles (see Section 2.3.7) used in DualSPHysics. Consequently, scaling the slideby 86.75% about the centreline at the slide’s leading edge, yields an almost perfect computed impulsewave. As such, all other simulations used a slide that was 13.25% smaller.Above, it was asked if DualSPHysics accurately reflects waves at boundaries? To evaluate this, theNorth and South walls were converted from Dynamic boundary particles to open periodic boundaries.The justification for this is if two identical waves hit each other, then this would be the equivalent of asingle wave hitting a vertical wall. If the dynamic boundary particles were preventing waves from beingreflected, then this approach would parse that out. However, it appears from test S15 ( Figure 4.15)that this is not the case. The RMSE of all the probes did decrease, but only marginally (average ofRMSE from S14 was 1.95-cm, while for S15 it was 1.80). Furthermore, this setup is not physical, sosimulations using this type of boundaries were no longer carried out.In Section 2.3.5, it is explained how the particles are moved using an XSPH variant. For all simula-tions this was set to a default value of 0.5. However, for this test, it was normally set to zero. Meaningthe particles were moved according to their individual velocity and not some fraction of their groupvelocity. The results of this indicate that it marginally influences the RMSE, but did reduce the relativemaximum error in each probe. That said, the results are still highly dissipative.Oger[58] states “errors in the gradient approximations are non-negligible” unless a high value of hDxis used. Therefore, this value was increased from 2.13 (S14) to 3.5 (S17). By doing this, the numberof particles in each kernel support went from 77 to 343 and dissipation marginally decreased, while thecomputational time tripled. As such, hDx was reverted back to 2.13.With the kernel type, particle size, hDx , Delta-SPH coefficient, XSPH variant, and time steppingmethod optimally selected as Wendland, 0.01-m, 2.13, 0, 0.5, and Verlet, respectively, viscosity wasagain looked at. For tests S18-S21 (Figure 4.17 and Figure 4.18) the viscosity coefficient was adjustedbetween 0.05, 0.01, 0.02, and 0.03, respectfully. Here it can be seen that DualSPHysics is now ap-proaching the physical experimental results. The data is slightly out of phase, but the wave amplitudesare resolved much better. Test S21 specifically has the lowest RMSE and REMA.For tests S1-S22 a simple velocity prescription (see, Section 3.2.1) was used. Test S23, on the otherhand, used S21’s settings and a complex description. The average RMSE went from 2.15 (S21) to 2.35(S23) while the REMA only marginally changed.In summary, DualSPHysics has many user definable settings that all play a significant role in thefinal solution. For this SLGW case, one of the main issues was finding the right combination that limitedexcess numerical dissipation and allowed higher-order waves to be generated. The optimal combination(test run S21) was found to be using a simple slide velocity, a particle size of 0.01, a Wendland kernel,a smoothing length of 1.23, a viscosity coefficient of 0.03, a coefficient of sound of 10, an XSPHcoefficient of 0.5, and Verlet time integration scheme with a CFL of 0.5, which can be used for futureALGW testing.The more pressing issue is the artificial increase in slide volume caused by the dynamic boundaryparticles. For this work, this issue was fixed by simply reducing the slide volume by 13.25%. However,this value is somewhat arbitrary and it is not known if all subsequent SLGW simulations should use the48same reduction or if they will have their own unique reduction value.−10−505101520(a) Wave Probe 1(S2)(SPH−RMSE=2.57)Wave Height [cm](b) Wave Probe 1(S3)(SPH−RMSE=2.47)  −10−505101520(c) Wave Probe 2(S2)(SPH−RMSE=2.15)Wave Height [cm](d) Wave Probe 2(S3)(SPH−RMSE=2.07)−10−505101520(e) Wave Probe 3(S2)(SPH−RMSE=1.79)Wave Height [cm](f) Wave Probe 3(S3)(SPH−RMSE=1.81)0 1 2 3 4 5 6−10−505101520(g) Wave Probe 4(S2)(SPH−RMSE=2.54)Wave Height [cm]Time [sec]0 1 2 3 4 5 6(h) Wave Probe 4(S3)(SPH−RMSE=2.74)Time [sec]DualSPHysics PhysicalFigure 4.9: Developmental wave height comparison between DualSPHyscis & Physical Data.Panels a, c, e, and g represent the baseline simulation with a new viscosity coefficient of0.20 (test number S2). Panels b, e, f, and h represent the baseline simulation with a newviscosity coefficient of 0.15 (test number S3)49Table 4.6: Developmental DualSPHysics parameter settingsTestNumber1SlideVelocityParticlesizeKernelTypeS. LengthCoeff.2 h/Dx3ViscosityCoeff.Coeff. ofSound e4 CFL5TimeIntegrationRunTime6S18 Simple 0.01 Cubic 0.87 1.50 0.25 10.00 0.50 0.20 Verlet 3S2 Simple 0.01 Cubic 0.87 1.50 0.20 10.00 0.50 0.20 Verlet 3S3 Simple 0.01 Cubic 0.87 1.50 0.15 10.00 0.50 0.20 Verlet 3S4 Simple 0.01 Cubic 0.87 1.50 0.10 10.00 0.50 0.20 Verlet 3S5 Simple 0.01 Cubic 0.87 1.50 0.05 10.00 0.50 0.20 Verlet 4S6 Simple 0.01 Cubic 0.87 1.50 0.25 20.00 0.50 0.20 Verlet 6S7 Simple 0.01 Cubic 0.87 1.50 0.25 30.00 0.50 0.20 Verlet 9S8 Simple 0.01 Wendland 1.50 2.60 0.15 10.00 0.50 0.20 Verlet 7S9 Simple 0.005 Wendland 1.23 2.13 0.15 10.00 0.50 0.50 Verlet 49S107 Simple 0.01 Wendland 1.23 2.13 0.15 10.00 0.50 0.50 Verlet 1S11 Simple 0.01 Wendland 1.23 2.13 0.15 40.00 0.50 0.50 Verlet 7S12 Simple 0.01 Wendland 1.23 2.13 0.15 40.00 0.50 0.50 Symplectic 13S13 Simple 0.01 Wendland 1.23 2.13 0.15 10.00 0.50 0.50 Verlet 3S14 Simple 0.01 Wendland 1.23 2.13 0.15 10.00 0.50 0.50 Verlet 2S15 Simple 0.01 Wendland 1.23 2.13 0.15 10.00 0.50 0.50 Verlet 2S16 Simple 0.01 Wendland 1.23 2.13 0.15 10.00 0.00 0.50 Verlet 2S17 Simple 0.01 Wendland 2.02 3.50 0.15 10.00 0.50 0.50 Verlet 6S18 Simple 0.01 Wendland 1.23 2.13 0.05 10.00 0.50 0.50 Verlet 3S19 Simple 0.01 Wendland 1.23 2.13 0.01 10.00 0.50 0.50 Verlet 4S20 Simple 0.01 Wendland 1.23 2.13 0.02 10.00 0.50 0.50 Verlet 3S219 Simple 0.01 Wendland 1.23 2.13 0.03 10.00 0.50 0.50 Verlet 3S22 Simple 0.01 Wendland 1.23 2.13 0.05 20.00 0.50 0.50 Verlet 5S23 Complex 0.01 Wendland 1.23 2.13 0.03 10.00 0.50 0.50 Verlet 31 All simulations had a particle spacing of 0.010-m (except S10, which used a 0.005-m spacing), and a Delta-SPH formulation coefficient equal to zero;2 Smoothing Length Coefficient; 3 Smoothing length (h) over particle spacing (Dx); 4 XSPH variant coefficient; 5 Courant-Friedrichs-Lewy Number;6 Run time is measured in hours of GPU time on a 448 core Nvidia Tesla C2075 card7 Test used a Delta-SPH coefficient equal to 0.1. Simulation also failed to complete due to particle leakage; 8 Baseline test; 9 Final test50Table 4.7: Developmental DualSPHysics ErrorTest NumberRMSE1 REMA2WS1 WS2 WS3 WS4 WS1 WS2 WS3 WS4S13 2.6 2.2 1.8 2.4 1 23 27 39S2 2.6 2.1 1.8 2.5 6 22 29 42S3 2.5 2.1 1.8 2.7 17 20 30 44S4 2.3 2.0 1.8 3.2 31 18 37 49S5 2.5 2.0 1.9 4.1 75 18 52 88S6 3.0 2.5 2.1 2.4 17 9 30 27S7 3.1 2.6 2.3 2.7 17 6 30 15S8 2.7 2.3 2.0 2.9 19 37 43 59S9 2.6 1.9 1.7 2.4 113 6 26 39S10 16.2 16.1 16.0 2.4 -17 -72 -15 39S11 3.2 2.8 2.5 2.8 29 17 41 36S12 3.3 2.9 2.7 2.9 33 23 54 43S13 2.6 2.2 2.0 2.8 8 42 40 57S14 2.3 1.9 1.5 2.1 4 -31 -12 1S15 2.0 1.5 1.3 2.4 -1 -32 -11 36S16 2.1 1.6 1.3 2.4 -5 -24 0 21S17 2.3 1.7 1.4 2.6 -0 -21 0 21S18 2.1 1.6 1.4 3.2 -10 -28 -12 57S19 2.4 1.8 1.5 4.3 -13 4 -7 85S20 2.2 1.7 1.4 3.8 -24 -17 -6 78S213 2.2 1.6 1.4 3.4 -3 -19 -9 72S22 2.2 1.8 1.5 2.4 9 -25 -6 27S23 2.3 1.8 1.6 3.7 -7 -14 0 721 Root-mean-square-error [cm] 2 Relative error of maximum amplitude [%]3 Baseline test 4 Final test51−10−505101520(a) Wave Probe 1(S4)(SPH−RMSE=2.33)Wave Height [cm](b) Wave Probe 1(S5)(SPH−RMSE=2.52)  −10−505101520(c) Wave Probe 2(S4)(SPH−RMSE=2.01)Wave Height [cm](d) Wave Probe 2(S5)(SPH−RMSE=2.03)−10−505101520(e) Wave Probe 3(S4)(SPH−RMSE=1.84)Wave Height [cm](f) Wave Probe 3(S5)(SPH−RMSE=1.93)0 1 2 3 4 5 6−10−505101520(g) Wave Probe 4(S4)(SPH−RMSE=3.20)Wave Height [cm]Time [sec]0 1 2 3 4 5 6(h) Wave Probe 4(S5)(SPH−RMSE=4.10)Time [sec]DualSPHysics PhysicalFigure 4.10: Developmental wave height comparison between DualSPHyscis & Physical Data.Panels a, c, e, and g represent the baseline simulation with a new viscosity coefficient of0.10 (test number S4). Panels b, e, f, and h represent the baseline simulation with a newviscosity coefficient of 0.05 (test number S5)52−10−505101520(a) Wave Probe 1(S6)(SPH−RMSE=2.98)Wave Height [cm](b) Wave Probe 1(S7)(SPH−RMSE=3.08)  −10−505101520(c) Wave Probe 2(S6)(SPH−RMSE=2.51)Wave Height [cm](d) Wave Probe 2(S7)(SPH−RMSE=2.62)−10−505101520(e) Wave Probe 3(S6)(SPH−RMSE=2.11)Wave Height [cm](f) Wave Probe 3(S7)(SPH−RMSE=2.31)0 1 2 3 4 5 6−10−505101520(g) Wave Probe 4(S6)(SPH−RMSE=2.43)Wave Height [cm]Time [sec]0 1 2 3 4 5 6(h) Wave Probe 4(S7)(SPH−RMSE=2.74)Time [sec]DualSPHysics PhysicalFigure 4.11: Developmental wave height comparison between DualSPHyscis & Physical Data.Panels a, c, e, and g represent the baseline simulation with a new coefficient of sound of20 (test number S6). Panels b, e, f, and h represent the baseline simulation with a newcoefficient of sound of 30 (test number S7)53−10−505101520(a) Wave Probe 1(S8)(SPH−RMSE=2.68)Wave Height [cm](b) Wave Probe 1(S9)(SPH−RMSE=2.61)  −10−505101520(c) Wave Probe 2(S8)(SPH−RMSE=2.34)Wave Height [cm](d) Wave Probe 2(S9)(SPH−RMSE=1.92)−10−505101520(e) Wave Probe 3(S8)(SPH−RMSE=2.03)Wave Height [cm](f) Wave Probe 3(S9)(SPH−RMSE=1.68)0 1 2 3 4 5 6−10−505101520(g) Wave Probe 4(S8)(SPH−RMSE=2.89)Wave Height [cm]Time [sec]0 1 2 3 4 5 6(h) Wave Probe 4(S9)(SPH−RMSE=2.40)Time [sec]DualSPHysics PhysicalFigure 4.12: Developmental wave height comparison between DualSPHyscis & Physical Data.Panels a, c, e, and g represent the baseline simulation using a Wendland kernel, a newsmoothing length coefficient of 1.5, and a new viscosity coefficient of 0.15 (test numberS8). Panels b, e, f, and h represent the baseline simulation using a Wendland kernel, asmaller particle size of 0.005-m, a new smoothing length coefficient of 1.23, and a newviscosity of 0.15 (test number S9 - note, no wave run-up data is available).54−10−505101520(a) Wave Probe 1(S10)(SPH−RMSE=16.17)Wave Height [cm](b) Wave Probe 1(S11)(SPH−RMSE=3.21)  −10−505101520(c) Wave Probe 2(S10)(SPH−RMSE=16.13)Wave Height [cm](d) Wave Probe 2(S11)(SPH−RMSE=2.77)−10−505101520(e) Wave Probe 3(S10)(SPH−RMSE=16.02)Wave Height [cm](f) Wave Probe 3(S11)(SPH−RMSE=2.45)0 1 2 3 4 5 6−10−505101520(g) Wave Probe 4(S10)(SPH−RMSE=2.40)Wave Height [cm]Time [sec]0 1 2 3 4 5 6(h) Wave Probe 4(S11)(SPH−RMSE=2.79)Time [sec]DualSPHysics PhysicalFigure 4.13: Developmental wave height comparison between DualSPHyscis & Physical Data.Panels a, c, e, and g represent the baseline simulation using a Wendland kernel, a newsmoothing length coefficient of 1.23, a delta coefficient of 0.1, and a new viscosity coef-ficient of 0.15 (test number S10 - note due to particle leakage, simulation failed). Panelsb, e, f, and h represent the baseline simulation using a Wendland kernel, a new smoothinglength coefficient of 1.23, a new coefficient of sound of 40, and a new viscosity of 0.15 (testnumber S11).55−10−505101520(a) Wave Probe 1(S12)(SPH−RMSE=3.32)Wave Height [cm](b) Wave Probe 1(S13)(SPH−RMSE=2.58)  −10−505101520(c) Wave Probe 2(S12)(SPH−RMSE=2.89)Wave Height [cm](d) Wave Probe 2(S13)(SPH−RMSE=2.21)−10−505101520(e) Wave Probe 3(S12)(SPH−RMSE=2.65)Wave Height [cm](f) Wave Probe 3(S13)(SPH−RMSE=1.95)0 1 2 3 4 5 6−10−505101520(g) Wave Probe 4(S12)(SPH−RMSE=2.87)Wave Height [cm]Time [sec]0 1 2 3 4 5 6(h) Wave Probe 4(S13)(SPH−RMSE=2.79)Time [sec]DualSPHysics PhysicalFigure 4.14: Developmental wave height comparison between DualSPHyscis & Physical Data.Panels a, c, e, and g represent the baseline simulation using a Wendland kernel, a newsmoothing length coefficient of 1.23, a new coefficient of sound of 40, a new viscosity of0.15, and a symplectic time integration scheme (test number S12). Panels b, e, f, and h rep-resent the baseline simulation using a Wendland kernel, a new smoothing length coefficientof 1.23, and a new viscosity coefficient of 0.15 (test number S13).56−10−505101520(a) Wave Probe 1(S14)(SPH−RMSE=2.29)Wave Height [cm](b) Wave Probe 1(S15)(SPH−RMSE=1.97)  −10−505101520(c) Wave Probe 2(S14)(SPH−RMSE=1.88)Wave Height [cm](d) Wave Probe 2(S15)(SPH−RMSE=1.53)−10−505101520(e) Wave Probe 3(S14)(SPH−RMSE=1.51)Wave Height [cm](f) Wave Probe 3(S15)(SPH−RMSE=1.26)0 1 2 3 4 5 6−10−505101520(g) Wave Probe 4(S14)(SPH−RMSE=2.08)Wave Height [cm]Time [sec]0 1 2 3 4 5 6(h) Wave Probe 4(S15)(SPH−RMSE=2.45)Time [sec]DualSPHysics PhysicalFigure 4.15: Developmental wave height comparison between DualSPHyscis & Physical Data.Panels a, c, e, and g represent the baseline simulation using a Wendland kernel, a newsmoothing length coefficient of 1.23, a new viscosity of 0.15, and a slide that is 13.25%smaller (test number S16). Panels b, e, f, and h represent the baseline simulation using aWendland kernel, a new smoothing length coefficient of 1.23, a new viscosity of 0.15, aslide that is 13.25%, and periodic boundary conditions imposed on the North and Southwall (test number S15).57−10−505101520(a) Wave Probe 1(S16)(SPH−RMSE=2.07)Wave Height [cm](b) Wave Probe 1(S17)(SPH−RMSE=2.30)  −10−505101520(c) Wave Probe 2(S16)(SPH−RMSE=1.56)Wave Height [cm](d) Wave Probe 2(S17)(SPH−RMSE=1.74)−10−505101520(e) Wave Probe 3(S16)(SPH−RMSE=1.33)Wave Height [cm](f) Wave Probe 3(S17)(SPH−RMSE=1.41)0 1 2 3 4 5 6−10−505101520(g) Wave Probe 4(S16)(SPH−RMSE=2.37)Wave Height [cm]Time [sec]0 1 2 3 4 5 6(h) Wave Probe 4(S17)(SPH−RMSE=2.64)Time [sec]DualSPHysics PhysicalFigure 4.16: Developmental wave height comparison between DualSPHyscis & Physical Data.Panels a, c, e, and g represent the baseline simulation using a Wendland kernel, a newsmoothing length coefficient of 1.23, a new viscosity of 0.15, a slide that is 13.25% smaller,and an epsilon of zero (test number S16). Panels b, e, f, and h represent the baseline simu-lation using a Wendland kernel, a new smoothing length coefficient of 2.02, a new viscosityof 0.15, and a slide that is 13.25% smaller (test number S17).58−10−505101520(a) Wave Probe 1(S18)(SPH−RMSE=2.07)Wave Height [cm](b) Wave Probe 1(S19)(SPH−RMSE=2.36)  −10−505101520(c) Wave Probe 2(S18)(SPH−RMSE=1.58)Wave Height [cm](d) Wave Probe 2(S19)(SPH−RMSE=1.77)−10−505101520(e) Wave Probe 3(S18)(SPH−RMSE=1.37)Wave Height [cm](f) Wave Probe 3(S19)(SPH−RMSE=1.50)0 1 2 3 4 5 6−10−505101520(g) Wave Probe 4(S18)(SPH−RMSE=3.21)Wave Height [cm]Time [sec]0 1 2 3 4 5 6(h) Wave Probe 4(S19)(SPH−RMSE=4.29)Time [sec]DualSPHysics PhysicalFigure 4.17: Developmental wave height comparison between DualSPHyscis & Physical Data.Panels a, c, e, and g represent the baseline simulation using a Wendland kernel, a newsmoothing length coefficient of 1.23, a new viscosity of 0.05, and a slide that is 13.25%smaller (test number S18). Panels b, e, f, and h represent the baseline simulation using aWendland kernel, a new smoothing length coefficient of 1.23, a new viscosity of 0.01, anda slide that is 13.25% smaller (test number S19).59−10−505101520(a) Wave Probe 1(S20)(SPH−RMSE=2.21)Wave Height [cm](b) Wave Probe 1(S21)(SPH−RMSE=2.19)  −10−505101520(c) Wave Probe 2(S20)(SPH−RMSE=1.68)Wave Height [cm](d) Wave Probe 2(S21)(SPH−RMSE=1.60)−10−505101520(e) Wave Probe 3(S20)(SPH−RMSE=1.45)Wave Height [cm](f) Wave Probe 3(S21)(SPH−RMSE=1.39)0 1 2 3 4 5 6−10−505101520(g) Wave Probe 4(S20)(SPH−RMSE=3.84)Wave Height [cm]Time [sec]0 1 2 3 4 5 6(h) Wave Probe 4(S21)(SPH−RMSE=3.41)Time [sec]DualSPHysics PhysicalFigure 4.18: Developmental wave height comparison between DualSPHyscis & Physical Data.Panels a, c, e, and g represent the baseline simulation using a Wendland kernel, a newsmoothing length coefficient of 1.23, a new viscosity of 0.02, and a slide that is 13.25%smaller (test number S20). Panels b, e, f, and h represent the baseline simulation using aWendland kernel, a new smoothing length coefficient of 1.23, a new viscosity of 0.03, anda slide that is 13.25% smaller (test number S21).60−10−505101520(a) Wave Probe 1(S22)(SPH−RMSE=2.24)Wave Height [cm](b) Wave Probe 1(S23)(SPH−RMSE=2.35)  −10−505101520(c) Wave Probe 2(S22)(SPH−RMSE=1.85)Wave Height [cm](d) Wave Probe 2(S23)(SPH−RMSE=1.76)−10−505101520(e) Wave Probe 3(S22)(SPH−RMSE=1.48)Wave Height [cm](f) Wave Probe 3(S23)(SPH−RMSE=1.62)0 1 2 3 4 5 6−10−505101520(g) Wave Probe 4(S22)(SPH−RMSE=2.37)Wave Height [cm]Time [sec]0 1 2 3 4 5 6(h) Wave Probe 4(S23)(SPH−RMSE=3.65)Time [sec]DualSPHysics PhysicalFigure 4.19: Developmental wave height comparison between DualSPHyscis & Physical Data.Panels a, c, e, and g represent the baseline simulation using a Wendland kernel, a newsmoothing length coefficient of 1.23, a new viscosity of 0.05, a slide that is 13.25% smaller,and a new coefficient of sound of 20 (test number S22). Panels b, e, f, and h represent thebaseline simulation using a Wendland kernel, a new smoothing length coefficient of 1.23, anew viscosity of 0.05, a slide that is 13.25% smaller, and a complex slide description (testnumber S23).614.3 Final resultsInvestigations of the developmental tests carried out above indicate the Flow-3DTM test F10 and Du-alSPHysics S21 yield results that most closely represent the physical experiment results. This sectionprovides a detailed discussion of these results.4.3.1 Eulerian computational fluid dynamicsOf the seventeen developmental tests, test F104 yielded the most accurate results. Here, the initialimpulse wave (Figure 4.20a at ⇠1.2-sec) is in phase but under predicted by -46.5%, after which thewave signal improves in amplitude but becomes out of phase.For wave probes 2 and 3, Flow-3DTM reproduces the first two waves well; however, once the re-flected waves start to interact with the main waves, the results become slightly less accurate. The firstwave in wave probe 2 occurs in the physical experiment at 1.51-sec while it occurred in Flow-3DTM at1.56-sec and was under predicted by -34.4%. The second wave occurred in the physical model at 2.59-sec while it occurred in Flow-3DTM at 2.64-sec and was under predicted by -25.9%. The maximumwave in wave probe 2 occurred at 4.31-sec with a height of 8.52-cm for the physical experiment, and4.26-sec with a height of 6.81-cm for Flow-3DTM .The first wave in wave probe 3 occurs in the physical experiment at 1.90-sec while it occurred inFlow-3DTM at 1.80-sec and was under predicted by -25.62%. The second wave occurred in the physicalmodel at time equal to 3.00-sec while it occurred in Flow-3DTM at 3.00-sec and was under predictedby -45.5%. The maximum wave in wave probe 3 occurred at time equal to 3.00-sec with a height of6.37-cm for the physical experiment, and 1.80-sec with a height of 4.46-cm for Flow-3DTM .The first run-up on wave probe 4 occurs in the physical experiment at 2.35-sec while it occurred inFlow-3DTM at either 2.26 or 2.24-sec and was under predicted by either -4.75% or -5.27%, dependingon which method was used to capture the run-up (see Section 4.1). The second run-up occurred in thephysical model at 3.49-sec while it occurred in Flow-3DTM at 3.56 or 3.62-sec and was over predictedby either 43.0% or 44.3%. The maximum run-up in wave probe 4 occurred at time equal to 3.49-secwith a height of 8.6-cm for the physical experiment, and 3.56 or 3.62-sec with a height of 12.32 or12.43-cm for Flow-3DTM .4Test F16 also generated a suitable result; however, it was executed using Flow-3DTM MP v5.02.15 and is not overly usedin the hydrotechnical community62−10−505101520(a) Wave Probe 1(F1)(F3D−RMSE=1.70)Wave Height [cm](b) Wave Probe 1(F10)(F3D−RMSE=1.52)  −10−505101520(c) Wave Probe 2(F1)(F3D−RMSE=1.40)Wave Height [cm](d) Wave Probe 2(F10)(F3D−RMSE=1.09)−10−505101520(e) Wave Probe 3(F1)(F3D−RMSE=1.07)Wave Height [cm](f) Wave Probe 3(F10)(F3D−RMSE=1.03)0 1 2 3 4 5 6−10−505101520(g) Wave Probe 4(F1)(F3D−RMSE=3.62)Wave Height [cm]Time [sec]0 1 2 3 4 5 6(h) Wave Probe 4(F10)(F3D−RMSE=2.67)Time [sec]Flow−3D PhysicalFigure 4.20: Finalized (F10) and baseline (F1) Flow-3DTM and physical model data of the freesurface measured at wave probe 1-4 locations. Panels g and h represent the evaluated waverun-up using the maximum second derivative method.634.3.2 Smoothed particle hydrodynamicsOf the twenty-three DualSPHysics simulations, the twenty-first (S21) yielded the most acceptable re-sults.Comparing Figure 4.5 and Figure 4.21 (see also Figure 5.1) it is evident that the final test (S21) ismuch less dissipative than the baseline test (S1). Improvement in fluidity can be seen in Figure 4.22 andFigure 4.23.The slide impact (Figure 4.21a at ⇠1.2-sec and Figure 4.24b) generates an impulse wave that is outof phase by 0.06-sec and under predicted by -3%, after which the wave signal gets more out of phase andunder predicts amplitudes. This seems appropriate as the reflected waves (Figure 4.21d are incorrectlyreproduced.For wave probes 2 and 3, DualSPHysics reproduces the first two waves well; however, once thereflected waves start to interact with the main waves, they become less accurate. The first wave in waveprobe 2 occurs in the physical experiment at 1.51-sec while it occurred in DualSPHysics at 1.62-secand was under predicted by -8.4%. The second wave occurred in the physical model at 2.59-sec whileit occurred in DualSPHysics at 2.50-sec and was under predicted by -12.5%. The maximum wave inwave probe 2 occurred at time equal to 4.31-sec with a height of 8.52-cm for the physical experiment,and 2.50-sec with a height of 6.74-cm for DualSPHysics .The first wave in wave probe 3 occurs in the physical experiment at 1.90-sec (Figure 4.21) while itoccurred in DualSPHysics at 1.84-sec and was under predicted by -4.9%. The second wave occurred inthe physical model at 3.00-sec while it occurred in DualSPHysics at 2.94-sec and was under predictedby -9.0%. The maximum wave in wave probe 3 occurred at 3.00-sec with a height of 6.37-cm for thephysical experiment, and 2.94-sec with a height of 5.80-cm for DualSPHysics.While for wave probes 1-3, DualSPHysics tended to under predict wave heights, wave probe 4was over predicted. The first run-up occurs in the physical experiment at 2.35-sec while it occurred inDualSPHysics at 2.18-sec and was over predicted by 21%. The second run-up occurred in the physicalmodel at 3.49-sec while it occurred in DualSPHysics at 3.54-sec and was over predicted by 71.8%.The maximum run-up in wave probe 4 occurred at 3.49-sec with a height of 8.6-cm for the physicalexperiment, and 3.54-sec with a height of 14.8-cm for DualSPHysics. That said, it is interesting to notethat although the maximum wave run-up is over predicted, DualSPHysics actually locates the horizontalextent of the run-up almost perfectly (Figure 4.24d).64−10−505101520(a) Wave Probe 1(S1)(SPH−RMSE=2.64)Wave Height [cm](b) Wave Probe 1(S21)(SPH−RMSE=2.19)  −10−505101520(c) Wave Probe 2(S1)(SPH−RMSE=2.24)Wave Height [cm](d) Wave Probe 2(S21)(SPH−RMSE=1.60)−10−505101520(e) Wave Probe 3(S1)(SPH−RMSE=1.82)Wave Height [cm](f) Wave Probe 3(S21)(SPH−RMSE=1.39)0 1 2 3 4 5 6−10−505101520(g) Wave Probe 4(S1)(SPH−RMSE=2.40)Wave Height [cm]Time [sec]0 1 2 3 4 5 6(h) Wave Probe 4(S21)(SPH−RMSE=3.41)Time [sec]DualSPHysics PhysicalFigure 4.21: Finalized (S21) and baseline (S1) DualSPHysics and physical model data of the freesurface measured at wave probe 1-4 locations.65Time: 0.20sec Time: 0.40sec Time: 0.60secTime: 0.80sec Time: 1.00sec Time: 1.20secTime: 1.40sec Time: 1.60sec Time: 1.80secTime: 2.00sec Time: 2.20sec Time: 2.40secTime: 2.60sec Time: 2.80sec Time: 3.00secTime: 3.20sec Time: 3.40sec Time: 3.60secTime: 3.80sec Time: 4.00sec Time: 4.20secTime: 4.40sec Time: 4.60sec Time: 4.80secTime: 5.00sec Time: 5.20sec Time: 5.40secTime: 5.60sec Time: 5.80sec Time: 6.00secFigure 4.22: Finalized elevation view of DualSPHysics frames (test S21). Note, colour gradientrepresent velocity magnitude, where blue is ’slow’ and red is ’fast’.66Time: 0.20sec Time: 0.40sec Time: 0.60secTime: 0.80sec Time: 1.00sec Time: 1.20secTime: 1.40sec Time: 1.60sec Time: 1.80secTime: 2.00sec Time: 2.20sec Time: 2.40secTime: 2.60sec Time: 2.80sec Time: 3.00secTime: 3.20sec Time: 3.40sec Time: 3.60secTime: 3.80sec Time: 4.00sec Time: 4.20secTime: 4.40sec Time: 4.60sec Time: 4.80secTime: 5.00sec Time: 5.20sec Time: 5.40secTime: 5.60sec Time: 5.80sec Time: 6.00secFigure 4.23: Finalized plan view of DualSPHysics frames (test S21). Note, colour gradient repre-sent velocity magnitude, where blue is ’slow’ and red is ’fast’.67(a)(b)(c)(d)(e)(f)Figure 4.24: Final (test S21) DualSPHysics centreline free surface overlaid onto physical experi-ment images. Panel’s a, b, c, d, e, and f represent time at 0,00, 0.55, 0.96, 1.58, 3.66, and5.82-sec, respectively.68Chapter 5Discussion and ConclusionsThe aim of this research was to conduct the very first DualSPHysics simulations for three-dimensionalinland subaerial landslide generated wave (SLGW) events, and to validate these simulations using highfidelity physical model data. In addition, this research aids in the continuing effort of validating thegeneral purpose computational fluid dynamics model Flow-3DTM for SLGW events.From the physical model experiment, the maximum wave amplitude was 0.117-m above the stillwater level and occurred at 1.2-sec. This wave then traveled down the flume where it runs-up theopposing slope to a height of 0.080-m above the still water level at 2.35-sec. The second run-up occurred1.14-sec later at 3.49-sec, and went up to a height of 0.086-m above the still water level, which was themaximum observed run-up. The reason for the second run-up height being higher then the first isspeculated as either the result of constructive interference between reflected waves, and/or the increasein volume associated with slide now being submerged.Using Flow-3DTM, it was found that with the default settings, Flow-3DTM achieved a good compar-ison with physical experiment results. However, changing the default first-order momentum advectionscheme to a second-order scheme decreased the maximum wave run-up error by 50%. Here, it wasable capture the phase of the first generated wave at 1.2-sec, but under predicted its height by -34.4%.As such, it under predicted the first run-up wave by about 5%, but surprisingly over predicted the sec-ond run-up wave (i.e. the maximum wave run-up) by about 43% and was out of phase by 0.07-sec to0.13-sec.As discussed in Section 1.1.4, Basu[4] and Kim[37] also indicate that Flow-3DTM over predicts themaximum wave run-up of SLGW by 28-56% and 0-78%, respectively — which is consistent with thisresearch. However, unlike Basu and Kim, this research explored, in detail, the effect of both numericalparametrization (i.e. numerical representation of specific parameters) and numerical scheme prescrip-tion. Specifically, this research investigated the influence of including Flow-3D’sTM air entrainmentand surface tension modules, as well as looking at the influence of first- or second-order momentumadvection schemes. The results of which suggest that while neither air entrainment or surface tensionsignificantly influence the maximum wave run-up, selection of a second-order momentum advectiondoes play a significant role. As such, it turns out by simply applying a second-order momentum ad-vection scheme to the baseline Flow-3DTM simulation (F1), the maximum wave run-up error is reduced69from 99% to 44%, as shown in Figure 5.1.DualSPHysics, on the other hand, required significant adjustments to reproduce the physical ex-periment. Additionally, scaling down the slide to 86.75% of its original size was required to counterthe artificial increase in slide volume due to the presence of dynamic boundary particles. After which,DualSPHysics almost exactly reproduced the first generated wave (see Figure 5.1). It was out of phaseby 0.06-sec and under predicted the value by -3%. However, the first run-up wave was over predictedby 21% and the second (i.e. the maximum wave run-up) by 72%, which is 29% more error than Flow-3DTM. Despite that, DualSPHysics actually out-performed Flow-3DTM when predicting the maximumwave generated by the slide.From this work, wave propagation in DualSPHysics has been shown to be significantly influencedby SPH viscosity coefficient and the coefficient of sound. High values of each coefficient drasticallyincreases energy dissipation, and decreased the “fluidness” of the simulated fluid. For this research, theideal viscosity coefficient and coefficient of sound was 0.03 and 10, respectively. In addition, it has beenshown that using a Wendland smoothing kernel is better at limiting artificial energy dissipation in wavepropagation then a Cubic kernel is.Although the treatment of boundaries in DualSPHysics has been shown to be problematic, it doesnot appear to influence wave propagation nor wave reflection. In DualSPHysics, boundaries are treatedas static particles that do not move with the fluid particles. As such, their effect on the SPH integralinterplant is the development of a pressure gradient which repels neighbouring fluid particles, but doesnot induce a no-slip boundary (i.e. zero velocity of fluid particles at the boundary) condition, whichtherefore does not account for energy losses along a “rough” boundary. For wave reflection, this treat-ment works very well. By replacing the north and south solid boundaries with periodic boundaries (i.e.the north domain edge is influenced by the south domain edge, and visa versa), an idealized conditionof perfect reflection can be evaluated for impinging waves. Here, it is assumed that a wave impactinga solid wall is similar to a wave hitting its mirrored self. Therefore, if a DualSPHysics simulation withdynamic boundaries greatly differed from a simulation with periodic boundaries, it would be known thatdynamic boundaries are negatively influencing the fluid particles. However, from this research it seemsthis is not the case. Waves reflected off solid boundaries comprised of dynamic boundary particlesclosely match waves reflected due to impact with their mirrored selfs.The issue with boundary particles in DualSPHysics, is that they artificially increase the size ofthe boundary. For example, this research shows that the slide used to generated subaerial landslidegenerated waves was artificially larger in the DualSPHysics domain then what was prescribed. As such,the generated waves were significantly different then that of the physical model. To combat this issue,the slide was scaled down by 13.25%.Despite this problem being a boundary driven problem, and the issues with the boundary imple-mentation in DualSPHysics, DualSPHysics is much better at generating waves than Flow-3DTM. It hasbeen shown that DualSPHysics can generate the first impulse wave to within -3% of the peak ampli-tude, while Flow-3DTM generated it to within 46%. However, Flow-3DTM does much better at wavepropagation and wave run-up than DualSPHysics does.70The implementation of boundaries in Flow-3DTM is significantly more mature and robust than inDualSPHysics , which prevents issues with dubious energy losses and/or reflections along a bound-ary. However, Flow-3D’sTM implementation of momentum advection1, does influence artificial energylosses, and subsequently maximum wave run-up. For example, this research has shown that going froma first- to a second-order momentum scheme yields an average reduction in relative maximum waverun-up error by 50%.In conclusion, although DualSPHysics and Flow-3DTM can sufficiently reproduce a SLGW eventfor some applications, they may not yet be suitable for high precision applications. Furthermore, neithermodel is suitable for fully capturing all the wave generation, wave propagation, and wave run-up details.Rather, it has been shown that DualSPHysics is better at wave generation, while Flow-3DTM is betterat wave propagation and wave run-up. That said, DualSPHysics and Flow-3DTM both tended to overpredict the maximum wave run-up by 43-72%.1In a Eulerian frame of reference, mass, momentum, and velocity need to be advected throughout a computation domainto solve the Navier-Stokes equations. As such, the order of the numerical scheme used to advect momentum dictates howmuch numerical viscosity is introduced.71−10−505101520(a) S1F1−Wave Probe1(SPH−RMSE=2.64; F3D−RMSE=1.70)Wave Height [cm] (b) S21F10−Wave Probe1(SPH−RMSE=2.19; F3D−RMSE=1.52)  −10−505101520(c) S1F1−Wave Probe2(SPH−RMSE=2.24; F3D−RMSE=1.40)Wave Height [cm] (d) S21F10−Wave Probe2(SPH−RMSE=1.60; F3D−RMSE=1.09)−10−505101520(e) S1F1−Wave Probe3(SPH−RMSE=1.82; F3D−RMSE=1.07)Wave Height [cm] (f) S21F10−Wave Probe3(SPH−RMSE=1.39; F3D−RMSE=1.03)0 1 2 3 4 5 6−10−505101520(g) S1F1−Wave Probe4(SPH−RMSE=2.40; F3D−RMSE=3.62)Wave Height [cm]Time [sec]0 1 2 3 4 5 6(h) S21F10−Wave Probe4(SPH−RMSE=3.41; F3D−RMSE=2.67)Time [sec]DualSPHysics Flow−3D PhysicalFigure 5.1: Finalized and baseline DualSPHysics, Flow-3DTM, and physical model data of the freesurface measured at wave probe 1-4 locations. For the Flow-3D results in panels g and h, themaximum second derivative method is used to present the evaluated wave run-up.725.1 Future workDespite much time spent investigating and researching, it is not known why Flow-3DTM or Dual-SPHysics over predicts the maximum wave run-up. However, two theories are here proposed:1. Energy dissipation through air entrainment: When the slide hits the water it generates a wavethat travels the length of the flume and runs-up the east slope, which Flow-3DTM and Dual-SPHysics capture. As the slide travels completely through the free surface, an impact crater isformed and water starts to collapse behind the slide. The collapse of water generates a spectrumof waves that interacts with the initial impulse wave to generate the second run-up on the eastslope, which is the maximum wave run-up. If the collapse of water is not accurately simulated, orenergy loss due to air entrainment is not accurately represented, then the driving energy will notbe the same as the physical model, and the maximum wave run-up will differ. However, if this bethe case, then why are the signals in wave probes 1-3 so much closer to the physical experiment?It is speculated that these apparently minor discrepancies are compounded at the east wall due tothe ratio of wave lengths and flume length. That is, net constructive wave interference is likelyoccurring right at the east slope.2. Eigenfrequency: During impact of the slide with the water, a wide spectrum of waves could begenerated and subsequently propagate in all three axis. Over time, these waves interact with oneanother and generate an Eigen frequency that matches that of the flume. At this point, the waverun-up ’stalls’ and the free surface along the east wall no longer moves. 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