Numerical Investigation of Surge Waves Generated by Submarine Debris Flows

water

Article Numerical Investigation of Surge Waves Generated by Submarine Debris Flows

Zili Dai, Jinwei Xie, Shiwei Qin * and Shuyang Chen

Department of Civil Engineering, Shanghai University, 99 Shangda Road, Shanghai 200444, China; [email protected] (Z.D.); [email protected] (J.X.); [email protected] (S.C.) * Correspondence: [email protected]; Tel.: +86-18817879593

Abstract: Submarine debris flows and their generated waves are common disasters in Nature that may destroy offshore infrastructure and cause fatalities. As the propagation of submarine debris flows is complex, involving granular material sliding and wave generation, it is difficult to simulate the process using conventional numerical models. In this study, a numerical model based on the smoothed particle hydrodynamics (SPH) algorithm is proposed to simulate the propagation of submarine debris flow and predict its generated waves. This model contains the Bingham fluid model for granular material, the Newtonian fluid model for the ambient water, and a multiphase granular flow algorithm. Moreover, a boundary treatment technique is applied to consider the repulsive force from the solid boundary. Underwater rigid block slide and underwater sand flow were simulated as numerical examples to verify the proposed SPH model. The computed wave profiles were compared with the observed results recorded in references. The good agreement between the numerical results and experimental data indicates the stability and accuracy of the proposed SPH model. Keywords: submarine debris flow; surge wave; smoothed particle hydrodynamics; water–soil Citation: Dai, Z.; Xie, J.; Qin, S.; interface; multiphase flow Chen, S. Numerical Investigation of Surge Waves Generated by Submarine Debris Flows. Water 2021, 13, 2276. https://doi.org/10.3390/ 1. Introduction w13162276 Submarine debris flows are widely distributed on continental shelves, continental slopes and in deepwater areas, where they pose a serious threat to offshore infrastructure, Academic Editor: Giuseppe Pezzinga such as submarine pipelines and cables, offshore oil and gas platforms, and offshore wind farms [1]. In addition, submarine debris flows can generate huge surge waves that Received: 4 July 2021 can seriously threaten the safety of infrastructure and people living in coastal areas. For Accepted: 18 August 2021 Published: 20 August 2021 example, the 1958 Lituya Bay debris flow-generated tsunami produced a run-up of over 400 m [2]. Another significant tsunami was generated by a submarine debris flow at the Nice

Publisher’s Note: MDPI stays neutral airport in France on 16 October 1979, and swept away 11 people [3]. The 1994 Skagway with regard to jurisdictional claims in debris flow-generated tsunami destroyed a railway dock, damaged the harbor, and killed a published maps and institutional affil- construction worker in Alaska, USA [4,5]. A submarine debris flow in Papua New Guinea iations. generated a wall of water measuring 15 m high that killed over 2100 people in July 1998 [6]. Furthermore, the 2006 Java tsunami, generated by a submarine debris flow off of Nusa Kambangan Island, produced a run-up in excess of 20 m at Permisan [7]. In 2010, the slope failures of river deltas along the Haitian coastline triggered tsunamis with a height of 3 m, which caused at least three fatalities and some damage to infrastructure [8]. In 2018, the Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. submarine debris flow induced by the Palu earthquake resulted in devastating tsunamis This article is an open access article and caused more than 2000 fatalities in Sulawesi, Indonesia [9–11]. These debris flow- distributed under the terms and generated tsunami disasters have therefore attracted a lot of attention from researchers. conditions of the Creative Commons Catastrophic tsunami waves generated by submarine debris flows are difficult to Attribution (CC BY) license (https:// observe and record because of their inaccessibility and unpredictability. Current approaches creativecommons.org/licenses/by/ to investigating the surge waves generated by submarine debris flows mainly focus on 4.0/). physical model tests and numerical simulations. For instance, physical model tests were

Water 2021, 13, 2276. https://doi.org/10.3390/w13162276 https://www.mdpi.com/journal/water Water 2021, 13, 2276 2 of 13

conducted based on the generalized Froude similarity to study the complex wave patterns generated by debris flows at Oregon State University [12]. Later, McFall and Fritz [13] conducted physical modeling using gravel and cobble materials in different topographic conditions, and analyzed offshore tsunami waves. Wang et al. [14] conducted physical model tests to analyze the effect of volume ratio on the maximum tsunami amplitude, and obtained an optimization model to predict the maximum tsunami amplitude. Miller et al. [15] designed a large flume model test to measure the shape and amplitude of the near-field waves and investigate the fraction of debris flow mass that activates the leading wave. Meng [16] conducted a series of model tests to investigate the waves generated by debris flow at laboratory scale, and the effect of the sliding mass on the wave features was analyzed. Takabatake et al. [17] conducted subaerial, partially submerged, and submarine debris flow-generated tsunamis to investigate the different characteristics of these three types of tsunamis. Although some promising results have been obtained, the physical model tests conducted in the abovementioned work require a great deal of manpower and material resources. In addition, the size effect is an inevitable problem in physical modeling. With the rapid development of computer techniques and numerical algorithms, numerical modeling of such events is critical to better understand the mechanism of surge waves and predict their propagation behavior, and become more prevalent in recent years. For example, Lynett and Liu [18] derived a numerical model to simulate the waves generated by a submarine debris flow and to predict the run-up. Yavari–Ramshe and Ataie–Ashtiani [19] presented a comprehensive review on past numerical investigations of debris flow-generated tsunami waves. In this review study, numerical models based on depth-averaged equations (DAEs) were tabulated according to their conceptual, mathemati- cal, and numerical approaches. Recently, a novel multiphase numerical model based on the moving particle semi-implicit method was proposed and applied to simulate submerged debris flows and tsunami waves. The model was validated and evaluated through comparison with the experimental data and other numerical models [20]. Sun et al. [21] solved the Navier–Stokes equations and incompressible flow continuity equation and developed a numerical model to investigate the propagation of a debris flow-generated tsunami under different initial submergence conditions. Rupali et al. [22] developed a coupled model based on the boundary element method and spectral element method to describe the characteristics of debris flow-generated waves. Baba et al. [23] adopted a two-layer flow model to simulate the submarine mass movement and surged wave propagation. In addition, computational fluid dynamics (CFD) has been widely applied to analyze tsunami wave propagation [24,25]. In the field of CFD, a mesh-free particle method named smoothed particle hydrodynamics (SPH) was proposed as an astrophysics application and has been widely used in various engineering fields [26]. For example, Iryanto [27] established an SPH model to simulate the waves induced by aerial and submarine landslides. Wang et al. [28] proposed a coupled DDA–SPH model to deal with the solid–fluid interaction problem, and applied it to simulate a block sliding along an underwater slope. However, the landslide mass was simulated as a rigid body, and its deformation was neglected in these two models. This work aimed to establish an SPH model to simulate surge waves generated by submarine debris flows, and to show the accuracy and stability of this model in the simulation of multiphase flow problems.

2. Materials and Methods 2.1. SPH Algorithm The SPH method was proposed in 1977 for astrophysical applications [29]. Recently, this method has been widely applied to a large variety of engineering ﬁelds [30–33]. Compared to the mesh-based methods, the major advantage of this method is that it bypasses the need for numerical meshes and avoids the mesh distortion issue and a great deal of computation for mesh repartition [34]. Water 2021, 13, 2276 3 of 13

In the SPH method, the governing equations can be transformed from partial dif- ferential equation form into SPH form by two approximation methods. The ﬁrst one is to produce the functions using integral representations, which is usually called kernel approximation. The second one is to discretize the problem domain into a set of particles, which is called particle approximation. Based on these two approximations, ﬁeld variables and their derivative in SPH form can be expressed by the following equations [35]:

N mj f xj f (xi) ≈ ∑ W xi − xj , h (1) j=1 ρj

N ∂ f (x ) mj f xj ∂W xi − xj , h i ≈ −∑ · (2) ∂xi j=1 ρj ∂xi where f is an arbitrary ﬁeld function, x is the coordinates of the SPH particles, m is the mass of particles, ρ is the density, W is the smoothing function, h is the smoothing length, and N is the total number of neighboring particles.

2.2. Governing Equations In the presented model, soil and water in a large deformation situation are assumed as two different ﬂuids that occupy the space by a certain volume fraction. They satisfy the conservation law of mass and momentum, respectively [36]:

1 D(ρesφs) + φs∇ · ves = 0 (3) ρes Dt

1 D(ρewφw) + φw∇ · vew = 0 (4) ρew Dt Dves 1 1 φs = ∇σes + φsg + fs (5) Dt ρes ρes Dvew 1 1 φw = ∇σew + φwg + fs (6) Dt ρew ρew where t is the time, g is the gravitational acceleration. v denotes the velocity vector, and σ denotes the stress tensor. The subscripts w and s represent water and soil, respectively. Moreover, ϕs and ϕw are the volume fractions of soil and water, and fs is the force exerting on the soil by the water, which can be determined by:

fs = −φs∇p + fd (7)

where f d is the Darcy penetrating force, and p is the pressure term of the ﬂuid, which is calculated by an equation of state [37]:

ρ c2 ρ γ p = 0 s − 1 (8) γ ρ0

where ρ0 is the reference density, which can be measured through laboratory tests; ρ is the density obtained through the mass conservation equation; cs is the sound speed, which can be set equal to ten times the maximum velocity [38]; and γ is the exponent of the equation of state, and is usually set to 7.0 [39].

2.3. Material Model In the presented SPH model, the submarine debris flow is assumed to be a Bingham viscous fluid, and the relationship between the shear stress and the shear strain rate is defined as: . τ = ηγ + τy (9) Water 2021, 13, x FOR PEER REVIEW 4 of 15

2.3. Material Model In the presented SPH model, the submarine debris flow is assumed to be a Bingham viscous fluid, and the relationship between the shear stress and the shear strain rate is defined as: Water 2021, 13, 2276 4 of 13 τηγ +τ = y (9) where τ denotes the shear stress, η represents the viscosity coefficient in fluid dynamics, γ . where meansτ denotes the shear the strain shear rate, stress, andη representsτy is the Bingham the viscosity yield stress. coefficient in fluid dynamics, γ meansThe the ambient shear strain water rate, is assumed and τy is as the a classical Bingham Newtonian yield stress. fluid, and the deformation behaviorThe can ambient be described water is as: assumed as a classical Newtonian fluid, and the deformation behavior can be described as: τη= .γ τ = ηγ (10) −3 TheThe viscosity viscosity of of water water ηη isis 1.7 1.7 ×× 1010− Pa·s.3 Pa ·s.

2.4.2.4. Approach Approach for for Multiphase Multiphase Granular Granular Flow Flow Modeling Modeling InIn this this SPH SPH model, model, the the approach approach for for multiphase multiphase granular granular flow flow proposed proposed by by Tajnesaie Tajnesaie etet al. [[20]20] isis adopted.adopted. AsAs shown shown in in Figure Figure1, the1, the debris debris flow flow is represented is represented by granular-type by granular- type(in brown (in brown color) color) particles, particles, and the and water the iswater represented is represented by fluid-type by fluid- (in bluetype color)(in blue particles. color) particles.As the granular As the flowgranular is quite flow fast, is quite and the fast, permeability and the permeability is low, it is is assumed low, it is in assumed this model in thisthat model the mass that exchange the mass betweenexchange the between ambient the and ambient pore and water pore is negligible.water is negligible. Each particle Each particlehas its own has physicalits own physical and mechanical and mechanical properties. properties. The fluid-type The fluid-type particles particles are regarded are re- as gardedNewtonian as Newtonian fluid, and thefluid, viscosity and the is viscosity constant. is In constant. contrast, theIn contrast, granular-type the granular-type particles are particlesregarded are as Binghamregarded viscousas Bingham fluid, viscous and the fluid, viscosity and the is variable viscosity and is variable determined and bydeter- the minedrheological by the model, rheological as described model, inas Section described 2.3 .in Section 2.3.

FigureFigure 1. 1. ParticleParticle representation representation of of mu multiphaseltiphase granular continuum.

2.5.2.5. Treatment Treatment on Water–Soil Interface AsAs shown shown in in Figure Figure 2a,2a, density density and and mass mass ar aree discontinuous discontinuous near near the the water–soil water–soil in- Water 2021, 13, x FOR PEER REVIEW terface,interface, reducing reducing the the computational computational accura accuracycy and and stability stability when when solving solving the the governing 5 of 15 equations.equations. Therefore, Therefore, boundary boundary treatments treatments on on the the water–soil water–soil interface interface should be made when solving the governing equations. As shown in Figure 2a, a is the central particle (in red) and b represents the neighboring particles of the other phase (in yellow). In this model, particle b is converted into particle b' (in blue) of the same phase with particle a, as shown in Figure 2b, which occupies the same position, velocity, and volume as the original particle b. Only the position, volume, velocity, and pressure of these neighboring particles are taken into account in particle approximation.

(a) (b) FigureFigure 2. 2. TreatmentTreatment on on the the interface interface of of different different fluids: ﬂuids: (a (a) )original original particles; particles; (b (b) )converted converted particles. particles.

2.6. BoundaryAs shown Condition in Figure 2a, a is the central particle (in red) and b represents the neighboring particles of the other phase (in yellow). In this model, particle b is converted into particle During the submarine debris flow propagation, the fluid particles face resistance b’ (in blue) of the same phase with particle a, as shown in Figure2b, which occupies the from the local seabed. Therefore, to truly simulate the debris flow motion, it is important same position, velocity, and volume as the original particle b. Only the position, volume, to accurately calculate the repulsive force from the solid boundary. In the SPH method, velocity, and pressure of these neighboring particles are taken into account in particle repulsive particles are widely used on the boundary contours to exert a force on the fluid approximation. particle in the direction of the line connecting both particles, as shown in Figure 3.

Figure 3. Schematic illustration of the solid boundary treatment.

The repulsive force, Fij, can be determined by Equation (11) [40]:

 nn12 rrX r  00−≥ij 0 Dif2 ,1  rr r  ij ijrij ij F=ij   (11)  r 01if 0 < r  ij

where D is a parameter that depends on the highest flow velocity, r0 is the cutoff distance, |rij| is the distance between a fluid particle and a boundary virtual particle, Xij is the differ- ence between the position vectors of a fluid particle and a boundary virtual particle, and n1 and n2 are parameters that are usually equal to 12 and 4, respectively.

3. Model Validation and Application 3.1. Waves Generated by Underwater Rigid Block Sliding To validate the developed SPH model, an underwater rigid block sliding experiment performed by Heinrich [41] was simulated, and the numerical results were compared with test data. Figure 4 shows the experimental setup. In the experiment, a block slid down a slope of 45° on the horizontal and generated water waves. The shore was simulated by another incline with a 15° slope. Water surface was level with the intersection of these two inclines, and the water depth in the channel was 1 m. The block was triangular in the cross-section with dimensions of 0.5 m × 0.5 m. The density was 2000 kg/m3. During the experiment, the block slid into the water under the effect of gravity, and then stopped

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Water 2021, 13, 2276 (a) (b) 5 of 13 Figure 2. Treatment on the interface of different fluids: (a) original particles; (b) converted particles.

2.6.2.6. BoundaryBoundary ConditionCondition DuringDuring thethe submarinesubmarine debrisdebris flowflow propagation,propagation, thethe fluidfluid particlesparticles faceface resistanceresistance fromfrom thethe locallocal seabed.seabed. Therefore,Therefore, toto trulytruly simulatesimulate thethe debrisdebris flowflow motion,motion, itit isis importantimportant toto accuratelyaccurately calculatecalculate thethe repulsiverepulsive forceforce fromfrom thethe solidsolid boundary.boundary. InIn thethe SPHSPH method,method, repulsiverepulsive particlesparticles areare widelywidely usedused onon thethe boundaryboundary contourscontours toto exertexert aa forceforce onon thethe fluidfluid particleparticle inin thethe directiondirection ofof thethe lineline connectingconnecting bothboth particles,particles, asas shownshown inin Figure Figure3 .3.

FigureFigure 3.3. SchematicSchematic illustrationillustration ofof thethe solidsolid boundaryboundary treatment.treatment.

TheThe repulsiverepulsive force, force,F Fijij,, cancan bebe determineddetermined byby EquationEquation (11)(11) [[40]:40]:

 n n 1 2 X r r0 nnr0 ij 0  D  − 12, i f ≥ 1 |r | rr|r | X 2 r|rij| Fij = ijDif00−≥ij |rij|ij ,1 0 (11)  r0 2   rrij ijr < 1 r ij  0 i f r ij F=ij  | ij| (11)  r 01if 0 < where D is a parameter that depends on ther highest flow velocity, r0 is the cutoff distance,  ij |rij| is the distance between a fluid particle and a boundary virtual particle, Xij is the differencewhere D is between a parameter the position that depends vectors on of the a fluidhighest particle flow andvelocity, a boundary r0 is the virtualcutoff distance, particle, and|rij| isn 1theand distancen2 are parametersbetween a fluid that particle are usually and equal a boundary to 12 and virtual 4, respectively. particle, Xij is the differ- ence between the position vectors of a fluid particle and a boundary virtual particle, and n1 3. Model Validation and Application and n2 are parameters that are usually equal to 12 and 4, respectively. 3.1. Waves Generated by Underwater Rigid Block Sliding 3. ModelTo validate Validation the developed and Application SPH model, an underwater rigid block sliding experiment performed3.1. Waves Generated by Heinrich by [Underwater41] was simulated, Rigid Block and Sliding the numerical results were compared with test data. Figure4 shows the experimental setup. In the experiment, a block slid down a To validate the developed SPH model, an underwater rigid block sliding experiment slope of 45◦ on the horizontal and generated water waves. The shore was simulated by performed by Heinrich [41] was simulated, and the numerical results were compared with another incline with a 15◦ slope. Water surface was level with the intersection of these Water 2021, 13, x FOR PEER REVIEW test data. Figure 4 shows the experimental setup. In the experiment, a block slid down6 of 15 a two inclines, and the water depth in the channel was 1 m. The block was triangular in the slope of 45° on the horizontal and generated water waves. The shore was simulated by cross-section with dimensions of 0.5 m × 0.5 m. The density was 2000 kg/m3. During the another incline with a 15° slope. Water surface was level with the intersection of these two experiment, the block slid into the water under the effect of gravity, and then stopped when inclines, and the water depth in the channel was 1 m. The block was triangular in the whenit ran it into ran a into small a small buffer buffer at the at channel the channel bottom. bottom. As shown As shown in Figure in Figure4, the 4, computational the computa- cross-section with dimensions of 0.5 m × 0.5 m. The density was 2000 kg/m3. During the tionaldomain domain was 5.0 was m × 5.01.3 m m ×in 1.3 the mx inand they directions.x and y directions. Table1 lists Table the calculating1 lists the parameterscalculating parametersusedexperiment, in the used SPH the in simulation.block the SPHslid simulation.into the water under the effect of gravity, and then stopped

FigureFigure 4. 4. ExperimentalExperimental setup setup of of an an underwater underwater rigi rigidd block block sliding sliding along along an an inclined inclined plane. plane.

Table 1. Parameters used in the SPH simulation of the rigid block sliding experiment.

Rigid block density ρs (kg/m3) 2000 Water density ρw (kg/m3) 1000 Water viscosity coefficient η (Pa·s) 1.7 × 10−3 Acceleration of gravity g (m/s2) 9.8

Figure 5 shows the SPH simulated results for the rigid block sliding experiment. When the block slid down, surge waves were generated because the kinetic energy of the sliding block was transferred to the ambient water body. The surge waves propagated out from the near field to the far field because of the height differences of the water surface. Energy conversion finished after the rigid block stopped, and then both the kinetic and potential energy of the water waves started to decay. The positions of the block and the wave profiles at different times were observed.

(a) t = 0.0 s

(b) t = 0.5 s

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when it ran into a small buffer at the channel bottom. As shown in Figure 4, the computational domain was 5.0 m × 1.3 m in the x and y directions. Table 1 lists the calculating parameters used in the SPH simulation.

Water 2021, 13, 2276 6 of 13 Figure 4. Experimental setup of an underwater rigid block sliding along an inclined plane.

Table 1. Parameters used in the SPH simulation of the rigid block sliding experiment. Table 1. Parameters used in the SPH simulation of the rigid block sliding experiment. Rigid block density ρs (kg/m3) 2000 3 Rigid blockWater density density ρs (kg/m ) ρw (kg/m3) 20001000 3 WaterWater densityviscosity coefficient ρw (kg/m ) η (Pa·s) 1000 1.7 × 10−3 Water viscosity coefficient η (Pa·s) 1.7 × 10−3 Acceleration of gravity g (m/s2) 9.8 Acceleration of gravity g (m/s2) 9.8 Figure 5 shows the SPH simulated results for the rigid block sliding experiment. When theFigure block5 shows slid down, the SPH surge simulated waves resultswere generated for the rigid because block the sliding kinetic experiment. energy of When the slidingthe block block slid was down, transferred surge wavesto the ambient were generated water body. because The the surge kinetic waves energy propagated of the sliding out fromblock the was near transferred field to the to far the field ambient because water of the body. height The surgedifferences waves of propagated the water surface. out from Energythe near conversion field to the finished far field after because the rigid of the bl heightock stopped, differences and ofthen the both water the surface. kinetic Energy and potentialconversion energy finished of the after water the waves rigid blockstarted stopped, to decay. and The then positions both the of kinetic the block and and potential the waveenergy profiles of the at water different waves times started were to observed. decay. The positions of the block and the wave profiles at different times were observed.

(a) t = 0.0 s

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(b) t = 0.5 s

(d) t = 1.5 s

FigureFigure 5. 5.SPHSPH simulated simulated results results for for the the rigid rigid block block sliding sliding experiment: experiment: ( a) t = 0.0 s; ( b) t = 0.5 s; (c) t = = 1.0 s; ( d)) tt = 1.5 1.5 s. s.

To verify the simulation accuracy of the SPH model, the numerical results were compared with the experimental data from Heinrich [41]. Figure 6 compares the numerical and experimental vertical displacement time history of the rigid block. The numerical results coincided with the experiment data well. After an acceleration phase at the beginning, the rigid block reached a constant velocity of 0.6 m/s. At approximately 1.0 s, the block reached the flume bottom in the SPH model as well as in the experiment. Figure 7 shows the comparison between the simulated and observed wave profiles at different times (t = 0.5 s, t = 1.0 s and t = 1.5 s). Although some discrepancy can be observed, there is close overall agreement between the experimental and numerical wave profiles. The maximum run-up of the wave during the simulation was about 0.10 m, which matches the experimental data well.

Figure 6. Comparison between the numerical and experimental vertical displacement time history of the rigid block.

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(d) t = 1.5 s

Water 2021, 13, 2276 Figure 5. SPH simulated results for the rigid block sliding experiment: (a) t = 0.0 s; (b) t = 0.5 s; (c7) oft 13 = 1.0 s; (d) t = 1.5 s.

To verify the simulation accuracy of the SPH model, the numerical results were com- paredTo with verify the theexperimental simulation data accuracy from ofHein therich SPH [41]. model, Figure the 6 numerical compares resultsthe numerical were com- andpared experimental with the experimental vertical displacement data from Heinrichtime history [41 ].of Figure the rigid6 compares block. The the numerical numerical re- and sultsexperimental coincided vertical with the displacement experiment timedata well. history After of thean acceleration rigid block. phase The numerical at the begin- results ning,coincided the rigid with block the experiment reached a constant data well. velocity After an of acceleration0.6 m/s. At approximately phase at the beginning, 1.0 s, the the blockrigid blockreached reached the flume a constant bottom velocity in the SPH of 0.6 model m/s. Atas approximatelywell as in the experiment. 1.0 s, the block Figure reached 7 showsthe flume the bottomcomparison in the between SPH model the as simulated well as in and the experiment.observed wave Figure profiles7 shows at thedifferent compari- timesson between (t = 0.5 s, the t = simulated 1.0 s and t and = 1.5 observed s). Although wave some profiles discrepancy at different can times be observed, (t = 0.5 s ,theret = 1.0 s isand closet = 1.5overall s). Although agreement some between discrepancy the experimental can be observed, and numerical there is close wave overall profiles. agreement The maximumbetween the run-up experimental of the wave and numericalduring the wavesimulation profiles. was The about maximum 0.10 m, run-up which ofmatches the wave theduring experimental the simulation data well. was about 0.10 m, which matches the experimental data well.

Water 2021, 13, x FOR PEER REVIEW 8 of 15 FigureFigure 6. 6. ComparisonComparison between between the the numerical numerical and and experime experimentalntal vertical vertical displacement displacement time time history history of ofthe the rigid rigid block. block.

(a) t = 0.5 s

(b) t = 1.0 s

(c) t = 1.5 s FigureFigure 7. Comparison 7. Comparison between between the simulated the simulated and observed and observed wave profiles wave proﬁlesat different at different times: (a)times: t = (a) t = 0.50.5 s; ( s;b) ( bt =) t1.0= 1.0s; (c s;) t (=c )1.5t = s. 1.5 s.

To evaluate the performance of the presented model furtherly, the numerical results are compared with those calculated by Tajnesaie et al. [20] using a MPS model, and Wang et al. [28] using a coupled DDA-SPH model. Figure 8 compares the water profiles calculated based on the above-mentioned numerical models at t = 0.5 s and t = 1.0 s. It shows that all the numerical models produce similar water profiles which are consistent with that observed in the experiments.

(a) t = 0.5 s

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(a) t = 0.5 s

(b) t = 1.0 s

(c) t = 1.5 s Water 2021, 13, 2276 Figure 7. Comparison between the simulated and observed wave profiles at different times: (a8) oft = 13 0.5 s; (b) t = 1.0 s; (c) t = 1.5 s.

To evaluate the performance of the presented model furtherly, the numerical results To evaluate the performance of the presented model furtherly, the numerical results are compared with those calculated by Tajnesaie et al. [20] using a MPS model, and Wang are compared with those calculated by Tajnesaie et al. [20] using a MPS model, and Wang et al. [28] using a coupled DDA-SPH model. Figure 8 compares the water profiles calcu- et al. [28] using a coupled DDA-SPH model. Figure8 compares the water proﬁles calculated lated based on the above-mentioned numerical models at t = 0.5 s and t = 1.0 s. It shows based on the above-mentioned numerical models at t = 0.5 s and t = 1.0 s. It shows that that all the numerical models produce similar water profiles which are consistent with all the numerical models produce similar water proﬁles which are consistent with that that observed in the experiments. observed in the experiments.

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(a) t = 0.5 s

(b) t = 1.0 s

FigureFigure 8. 8.ComparisonComparison of ofthe the experimental experimental and and numeri numericalcal (from (from the the models models of of current current study study and and otherother numerical numerical studies) studies) water water profiles proﬁles at at different different times: times: (a ()a t) =t =0.5 0.5 s; s;(b ()b t) =t 1.0= 1.0 s. s.

ToTo quantitatively quantitatively compare compare the the results, results, the the Pearson Pearson correlation correlation coefficient coefficient of ofthe the ex- ex- perimentalperimental and and numerical numerical water water profiles profiles in in Figure Figure 88 are are calculated. calculated. In In this this case, case, 40 40 points points withwith an an equal equal horizontal horizontal distance distance on on the the water water profile profile are are selected selected as as the the analysis analysis objects. objects. AssumingAssuming that that the the vector vector X (Xx1, (xx21, ,xx3,2 ……,, x3, ...... x40) represents, x40) represents the calculated the calculated elevations elevations of these points,of these and points, Y (y1, andy2, yY3, (……,y1, y 2y, 40y)3 ,represents...... , y40 the) represents measured the on measuredes, the Pearson ones, correlation the Pearson coefficientcorrelation of coefficientthe two vectors of the can two be vectors calculated can be as: calculated as:

nn ∑ (()x −−− x)(()y − y)  xxyyiii i i==1 CorrCorr(X,()YXY),= = i 1 (12) s nnnn (12) ()−×222 () − 2 ∑(xxiii−xyyx) × ∑ (yi − y) i=ii==111i=1

wherewhere xx and andy arey the are averagethe average values values of the of twothe two data data sets, sets, respectively. respectively. AccordingAccording to to Equation Equation (12), (12), the the correlation correlation coefficients coefficients between between the the experimental experimental and and numericalnumerical results results are are listed listed in in Table 2 2.. It It shows shows that that all all the the correlation correlation coefficients coefficients calculated calcu- latedare closeare close to 1, to which 1, which means means that that the the numerical numerical results results have have a strong a strong correlation correlation with with the theexperimental experimental profiles. profiles. Therefore, Therefore, the the presented presented SPH SPH model model can can successfully successfully simulate simulate such suchcomplex complex flows. flows.

TableTable 2. 2.CorrelationCorrelation coefficients coefﬁcients between between the experime the experimentalntal and numerical and numerical results results on the on water the pro- water files.proﬁles. Case Current Study Tajnesaie et al. [20] Wang et al. [28] Case Current Study Tajnesaie et al. [20] Wang et al. [28] t = 0.5 0.9336 0.9389 0.9297 t = 0.5 0.9336 0.9389 0.9297 t = 1.0 t = 1.0 0.9132 0.9132 0.9206 0.9206 0.9040 0.9040

3.2. Waves Generated by the Underwater Sand Flow In Section 3.1, the block was assumed to be rigid, which is not consistent with the actual situation of submarine debris flow. The large deformation of the solid phase should be taken into account in the simulation of the submarine debris flow propagation. There- fore, an experiment of the surge waves generated by underwater sand flow, conducted by Rzadkiewicz et al. [42], was simulated. Figure 9 shows the experimental setup. In this experiment, the maximum water depth was 1.6 m. Under the influence of gravity, a mass of sand (0.65 m × 0.65 m in cross section) slid down along a slope of 45°, and generated waves propagated outward.

Water 2021, 13, 2276 9 of 13

3.2. Waves Generated by the Underwater Sand Flow In Section 3.1, the block was assumed to be rigid, which is not consistent with the actual situation of submarine debris flow. The large deformation of the solid phase should be taken into account in the simulation of the submarine debris flow propagation. Therefore, an experiment of the surge waves generated by underwater sand flow, conducted by Rzadkiewicz et al. [42], was simulated. Figure9 shows the experimental setup. In this experiment, the maximum water depth was 1.6 m. Under the influence of gravity, a mass Water 2021, 13, x FOR PEER REVIEW 10 of 15 of sand (0.65 m × 0.65 m in cross section) slid down along a slope of 45◦, and generated waves propagated outward.

FigureFigure 9. 9. ExperimentalExperimental setup setup of of the the underwater underwater sand sand flow. ﬂow.

3 InIn the the underwater underwater sand sand flow flow experiment experiment,, the the density density of the of the sand sand was was 1950 1950 kg/m kg/m3, and, theand grain the graindiameter diameter ranged ranged from 2from to 7 mm. 2 to In 7 mm.the numerical In the numerical simulation, simulation, the sand flow the sandwas assumedflow was to assumed be a Bingham to be a fluid. Bingham The fluid.parameters The parametersare listed in are Table listed 3. inIn Tablethe absence3. In the of absence of measurements, the viscosity coefficient and yield stress of Bingham model can measurements, the viscosity coefficient and yield stress of Bingham model can be deter- be determined by trial and error. According to Ataie–Ashtiani and Shobeyri [43], under mined by trial and error. According to Ataie–Ashtiani and Shobeyri [43], under the con- the conditions of viscosity coefficient ηg = 0.15 Pa·s and Bingham yield stress τy = 750 Pa, ditions of viscosity coefficient ηg = 0.15 Pa·s and Bingham yield stress τy = 750 Pa, the nu- the numerical results match the experimental data well. Therefore, the same parameter merical results match the experimental data well. Therefore, the same parameter values values were used in this study. were used in this study.

Table 3. Parameters used in the SPH simulation of underwater sand flow. Table 3. Parameters used in the SPH simulation of underwater sand flow. Density of sand ρ (kg/m3) 1950 Density of sand s ρs (kg/m3) 1950 Viscosity coefficient of sand flow ηg (Pa·s) 0.15 Viscosity coefficient of sand flow ηg (Pa·s) 0.15 Bingham yield stress of sand flow τy (Pa) 750 3 y BinghamDensity yield of water stress of sand flow ρw (kg/m ) τ (Pa) 1000 750 3 −3 Viscosity coefficientDensity of of water water ηw (Pa·s) ρw (kg/m ) 1.7 × 10 1000 2 AccelerationViscosity ofcoefficient gravity of water g (m/s ) ηw (Pa·s) 9.8 1.7 × 10−3 Acceleration of gravity g (m/s2) 9.8 Figure 10 shows the SPH simulated results for the underwater sand flow. Unlike the previousFigure 10 numerical shows the example,SPH simulated the solid results phase for no the longer underwater behaved sand as aflow. rigid Unlike block butthe previousdeformed numerical significantly example, when the sliding solid downphase alongno longer the slope.behaved The as profilesa rigid block of water but andde- formedsand slope significantly at different when times sliding are presented. down along The the sand slope. slope The keeps profiles its initialof water shape and atsand the slopebeginning at different of slide times motion, are presented. as shown The in Figure sand 9slopea,b, while keeps at itst initial= 0.8 s,shape most at of the the begin- sand ningparticles of slide are motion, concentrated as show at then in flow Figure front, 9a,b, as while shown at in t = Figure 0.8 s,9 mostd. of the sand particles are concentrated at the flow front, as shown in Figure 9d. Figure 11 compares the simulated slope configurations with experimental data [42] and those calculated by the VOF model proposed in [42] and the ISPH model proposed in [43]. It shows that the simulated slope configurations are coincident with experimental results. Table 4 lists the Pearson correlation coefficients of the numerical and experimental slope configurations shown in Figure 11. The comparison shows that the SPH model proposed in this study has the equivalent simulation accuracy with the existing numerical models.

WaterWater2021 2021, ,13 13,, 2276 x FOR PEER REVIEW 11 of 15 10 of 13

(a) t = 0.2 s

(b) t = 0.4 s

Water 2021, 13, x FOR PEER REVIEW 12 of 15

(d) t = 0.8 s FigureFigure 10. 10. SPHSPH simulated simulated results results for the for underwater the underwater sand flow: sand ( ﬂow:a) t = 0.2 (a) s;t (=b) 0.2 t = s;0.4 (b s;) t(c=) t 0.4 = 0.6 s; (s;c ) t = 0.6 s; (d()d t) =t 0.8= 0.8 s. s.

Table 4. Correlation coefficients between the experimental and numerical results on the water profiles.

Case Current Study VOF Results [42] ISPH Results [43] t = 0.4 0.9092 0.8641 0.9197 t = 0.8 0.8761 0.8318 0.8804

(a) t = 0.4 s

(b) t = 0.8 s Figure 11. Comparison of the experimental and numerical (from the models of current study and other numerical studies) slope configurations at different times: (a) t = 0.4 s; (b) t = 0.8 s.

Water 2021, 13, x FOR PEER REVIEW 12 of 15

(d) t = 0.8 s Water 2021, 13, 2276 11 of 13 Figure 10. SPH simulated results for the underwater sand flow: (a) t = 0.2 s; (b) t = 0.4 s; (c) t = 0.6 s; (d) t = 0.8 s.

Table 4.Figure Correlation 11 compares coefficients the between simulated the slopeexperime configurationsntal and numerical with experimental results on the datawater [ 42pro-] and files.those calculated by the VOF model proposed in [42] and the ISPH model proposed in [43]. ItCase shows that the Current simulated Study slope configurations VOF Results are coincident[42] with ISPH experimental Results [43] results. Tablet = 0.44 lists the Pearson 0.9092 correlation coefficients 0.8641 of the numerical and experimental 0.9197 slope configurations shown in Figure 11. The comparison shows that the SPH model proposed in t = 0.8 0.8761 0.8318 0.8804 this study has the equivalent simulation accuracy with the existing numerical models.

(a) t = 0.4 s

(b) t = 0.8 s

FigureFigure 11. 11. ComparisonComparison of ofthe the experimental experimental and and numeri numericalcal (from (from the themodels models of current of current study study and and otherother numerical numerical studies) studies) slope slope conf conﬁgurationsigurations at different at different times: times: (a) (ta =) t0.4= 0.4s; (b s;) (tb =) 0.8t = s. 0.8 s.

Table 4. Correlation coefficients between the experimental and numerical results on the water profiles.

Case Current Study VOF Results [42] ISPH Results [43] t = 0.4 0.9092 0.8641 0.9197 t = 0.8 0.8761 0.8318 0.8804

4. Conclusions Submarine debris flow-generated water waves are natural phenomena that occur under certain conditions and can result in damage of infrastructure and loss of life in coastal areas. In this study, an SPH model with a multiphase granular flow algorithm is presented for numerical simulation of surge waves generated by submarine debris flows. The Bingham fluid model and Newton fluid model are used to describe the motion behavior of submarine debris flow and ambient water, respectively. A simple treatment is proposed to ensure the continuity of the density and pressure near the interface of these two fluids and to avoid numerical instability. This model was first used to simulate an experiment of underwater rigid block sliding. The simulated wave profiles were compared with the observed results to verify the stability and accuracy of the SPH model. Water 2021, 13, 2276 12 of 13

Then, this model was applied to simulate submarine debris flow. Surged waves generated by the underwater sand flow were computed and compared with the tested results. The good agreement between the numerical and tested results proves that the proposed SPH model is suitable to simulate such complex flows. In fact, submarine debris flows move across 3D submarine topography and may change direction and split or join in response to the complex topography. Therefore, the 2D model presented in this study cannot truly reproduce the complex dynamic process, and a 3D model with parallel computing techniques is necessary to improve the calculation accuracy and efficiency.

Author Contributions: Conceptualization, Z.D. and S.Q.; methodology, Z.D. and J.X.; writing— original draft preparation, Z.D. and J.X.; writing—review and editing, S.Q. and S.C.; visualization, J.X. and S.C.; supervision, S.Q.; funding acquisition, Z.D. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the National Natural Science Foundation of China (grant No. 42102318) and the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data presented in this study are available on request from the corresponding author. Acknowledgments: The authors are grateful for the support from the Department of Civil Engineer- ing, Shanghai University. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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