A Smooth Particle Hydrodynamics discretization forthe modelling of free surface flows and rigid body

dynamics

Ricardo B. Canelas1, Jose M. Domınguez2, Alejandro J. Crespo2, MonchoGomez-Gesteira2, Rui M.L. Ferreira1

Abstract

Unsteady hydrodynamic forces on an unrestricted rigid body are of considerablepractical importance, considering that the solid material advected by a fluidflow may contribute significantly to the momentum balance. It is also of greattheoretical interest since the motion of the solid mass may be difficult to modeldue to the complexity of the modes and range of scales involved in momentumtransfer by the fluid motion.

This work describes a unified discretization of rigid solids and fluids, allowingfor detailed and resolved simulations of fluid-solid flows. The model is basedon the fundamental conservation laws of hydrodynamics, namely the continuityand Navier-Stokes equations, and Newton’s equations for rigid body dynamics.The numerical solution, based on Smoothed Particle Hydrodynamics (SPH),resolves solid-fluid interactions in a broad range of scales. Such entails detailsof momentum transfer at solid boundaries to large scales typical of engineeringproblems, such as transport of debris or hydrodynamic actions on structures. Aδ-SPH term is added to the continuity equation, allowing for a correct interfacedescription.

A general overview of the method is presented, and a set of numerical ex-periments are carried out in order to compare the results with analitical andknown numerical solutions.

Keywords: Smooth Particle Hydrodynamics, Rigid Body Dynamics,

Multi-phase, Free-surface, Meshless methods

1CEHIDRO, Instituto Superior Tecnico, UL, Lisbon, Portugal, e-mail: ri-cardo.canelas@ist.utl.pt, ruimferreira@ist.utl.pt

2Environmental Physics Laboratory (EPHYSLAB), Universidade de Vigo, Ourense,Spain, e-mail: jmdominguez@uvigo.es; alexbexe@uvigo.es; mggesteira@uvigo.es, web:http://ephyslab.uvigo.es

Preprint submitted to Journal of Computational Physics February 4, 2014

1. INTRODUCTION

The interaction of solid material and a fluid flow is a common occurrenceand, at the engineering scales, it may be associated with highly unsteady events.A number of areas, from coastal, to offshore, maritime and fluvial hydraulicsprovide a large spectrum of problems whose solution can be approximated by5

considering the solid material perfectly rigid. A model that is capable of pro-viding meaningful solutions for all of these areas and is scalable, from the com-putational point of view, to be applied to real engineering cases is of paramountimportance, since it can provide immediate means for risk analysis and allowfor optimization of consequent mitigation measures.10

For many applications treating the flow as single phase, or a continuummedium, is clearly insufficient. A model that can shed light into the mechanismsof these flows must attempt to characterise all relevant interactions at theirproper scale. The main difficulties with modelling the events from the saidareas arise from the characteristics of the phenomena and scale. The latter15

poses a problem even for the simplest models, since modelling a single eventcan require remarkably large domains. This relates directly with the former,since the type of interaction and its relevant scales may require high resolutionadding to such large domains. Hence the need for high performance models andimplementations.20

Within the meshless framework, efforts have been made on unifying solid andfluid modelling. Koshizuka et al. (1998) modelled a rigid body as a collectionof Moving Particle Simulation (MPS) fluid particles, rigidified by default. Thishas become the standard approach due to its simplicity and elegance. Mon-aghan et al. (2003) and Rogers et al. (2010), employing the same principle,25

modelled the effects of wave interaction on rigid bodies resorting to SmoothedParticle Hydrodynamics (SPH) and special considerations for the particles thatbelonged to the solid body, effectively including a form of frictional behaviour.For normal interactions, continuum potential based forces were used, not basedin contact mechanics theories. Maruzewski et al. (2010) modelled the solid as a30

rigid boundary with imposed motion, using the ghost particle technique. Suchapproach encounters generalisation issues for arbitrary geometries and is usuallyused for simple cases, as spheres or other smooth surfaces.

This work relies on the DualSPHyics code (www.dual.sphysics.org), and rep-resents an effort to improve and validate the solid-fluid descriptions. It uses the35

same fundamental technique of Koshizuka et al. (1998): particles that consti-tute a rigid body have their relative position fixed and are regarded by thefluid particles as SPH particles. This allows for a simple coupling between fluidand solid descriptions, since no special treatment of the interaction with thesolid phase is needed. A δ-SPH (Molteni & Colagrossi, 2009) term is added to40

the continuity equation, controlling the density field fluctuations and contribut-ing for the mitigation of known solid-fluid interface deficiencies (Colagrossi &Landrini, 2003). The implementation has been already validated for interac-tion between fluid and fixed structures (Crespo et al., 2011; Gomez-Gesteiraet al., 2012). The DualSPHysics code enables simulation of millions of particles45

2

at a reasonable computation time by using GPU cards (Graphics ProcessingUnits) as the execution devices. This allows to somewhat alleviate the previ-ously expressed concerns about requirements of scale and resolution, since thecomputations are made up to two orders of magnitude faster than on normalCPU systems (Domınguez et al., 2013a). A Multi-GPU code was developed to50

further phase out the increased memory consumption by running large-scale,high-resolution simulations. Domınguez et al. (2013b) showed that very highefficiency was achieved for hundreds of GPUs using the Multi-GPU implemen-tation of DualSPHysics.

A set of numerical experiments and analytical solutions are recovered from55

the literature in order to provide a benchmark for the results achieved withthe revised DualSPHysics implementation. Previous efforts with the methodol-ogy focus mainly on practical applications (Rogers et al., 2010), not providinga more systematic study of the fundamental properties of these systems. Im-portant features that the model should respect include free stream consistency,60

simple dynamics of a buoyant body with various densities and the correct re-covery of equilibrium states. This work addresses these topics in an attempt tocharacterise the presented model with regards to the quality of its solutions andpossible limitations.

2. METHOD FORMULATION65

In SPH, the fluid domain is represented by a set of nodal points wherephysical quantities such as position, velocity, density and pressure are known.These points move with the fluid in a Lagrangian manner and their propertieschange with time due to the interactions with neighbouring nodes. The thermSmoothed Particle Hydrodynamics arises from the fact that the nodes, for allintended means, carry the mass of a portion of the medium, hence being eas-ily labelled as ”particles”, and their individual angular velocity is disregarded,hence ”smooth”. The method relies heavily on integral interpolant theory (Mon-aghan, 2005), that can be resumed to the exactness of

A (r) =

∫Ω

A (r′) δ (r − r′) dr′, (1)

for any continuous function A(r) defined in r′, where Ω is the domain, δ is theDirac delta function and r is a position in space. The nature of the Dirac deltafunction renders this identity numerically useless however, and an approxima-tion at r can be obtained by replacing it with a suitable weight function W ,called a kernel function. W should be an even function, defined on a compactsupport, i.e. if the radius is εh then W (r − r′, h) = 0 if |r − r′| ≥ εh, withlimh→0W = δ and

∫ΩW (r′, h) dr′ = 1, where h is the smoothing length and

defines the size of the kernel support (Liu, 2003). This leads to

A (r) =

∫Ω′A (r′) W (r − r′, h) dr′, (2)

3

known as the integral interpolant. An approximation to discrete Lagrangianpoints can be made, by a proper discretization of the integral

Ai ≈∑j

AjW (rij , h)Vj , (3)

called the summation interpolant, extended to all particles j, |rij | = |ri−rj | ≤εh, where Vj is the volume of particle j and Ai is the approximated variable atparticle i. The cost of such approximation is that particle first order consistency,i.e., the ability of the kernel approximation to reproduce exactly a first orderpolynomial function, may not be assured by the summation interpolant, since∑

j

VjW (rij , h) ≈ 1, (4)

which is especially understandable in situations were the kernel function doesnot verify compact support, for example near the free surface or other openboundaries in our cases. Mitigations may be considered, as the Shepard andMLS corrections. In the work of Colagrossi & Landrini (2003) spatial gradientesare computed using the gradient of the kernel function.70

3. DISCRETIZATION OF GOVERNING EQUATIONS

3.1. Equations of motion in SPH

The proposed SPH formulation relies on the discretization of the compress-ible Navier-Stokes system

dv

dt= −∇p

ρ+ µ∇2v + g (5)

dρ

dt= −ρ∇v, (6)

where v is the velocity field, p is the pressure, ρ is the density and µ and gare the kinematic viscosity and body forces, respectively. This is to avoid thenecessity of solving a Poisson equation, using p = f(ρ) (Lee et al., 2010). Thecontinuity equation is discretized as

dρidt

= −ρi∑j

(vi − vj)∇W (rij , h) + Φi, (7)

where mi is the mass of particle i and Φi is a diffusive term (Molteni & Cola-grossi, 2009), designed to stabilize the density field from high-frequency oscilla-tions, written as

Φi = 2δhc0∑j

(ρj − ρi)rij ·∇W (rij , h)

|rij |2 + η2

mj

ρj, (8)

4

where δ is a free parameter, c0 is the numerical sound velocity and η is a smallnumber to present singularities with very close particles. Equation (5) can bewritten as

dvidt

= −∑j

mj

(piρ2i

+pjρ2j

)∇W (rij , h)+

+∑j

mj

(4µrij∇W (rij , h)

(ρi + ρj)(|rij |2 + η2)

)vij +

∑j

mj

(τiρ2i

+τjρ2j

)∇W (rij , h) + g

(9)

The first term of the right side is a symmetrical, balanced form of the pressureterm (Monaghan, 2005). The second and third terms represent a laminar vis-cosity term (Morris et al., 1997) and a sub-partcile stress (SPS) (Dalrymple &Rogers, 2006), respectivley. The SPS term introduces the effects of turbulentmotion at smaller scales than the spatial discretization. Following the eddyviscosity assumption and using Favre-averaging, the SPS stress tensor for acompressible fluid can be written as

ταβρ

= 2νtSαβ −2

3kδαβ −

2

3CI∆

2δαβ |Sαβ |2, (10)

where ταβ is the sub-particle stress tensor, νt = (CS |rij |)2|Sαβ | is the eddyviscosity, CS is the Smagorinsky constant, k is the SPS turbulence kinetic en-ergy, CI = 6.6 × 10−3 and Sαβ is the local strain rate tensor, with |Sαβ | =75

(2SαβSαβ)1/2.Rigid bodies are groups of SPH particles whose variables are integrated

in time with a different set of equations. Newton’s equations for rigid bodydynamics are used, and the discretization consists of summing the contributionsfrom each SPH node, as

MIdV I

dt=∑k∈I

mkfk (11)

IIdΩI

dt=∑k∈I

mk(rk −RI)× fk, (12)

where body I possesses a mass MI , velocity V I , inertial tensor II , angularvelocity ΩI and center of gravity RI . fk is the force by unit mass appliedto particle k, belonging to body I. This force encompasses body forces, fluidresultants as well as the result of any rigid contact that might occur, a case80

not contemplated in this work. The fluid forces are computed with Equation9, where the viscous formulation should provide an adequate viscous drag. Themodel can be seen as an application of the Dynamic Bounday Conditions (Cre-spo et al., 2007), where the boundary is made to additionally follow Equations(11) and (12). No ad-hoc terms are added, since all the dynamics are a result of85

the fundamental, particle-wise, solution of Equations (11) and (12). A known

5

difficulty of this formulation is the overestimation of the density (Price, 2008;Saitoh & Makino, 2013), resulting from an entropy jump across the interface.This results in an increased distance of fluid-solid particles due to the addedforce from the pressure gradient, effectively disturbing the viscous forces com-90

puted at that interface (Colagrossi & Landrini, 2003). The inclusion of theδ-SPH diffusive term in Equation (7) allows for a correct density estimationacross the interface, thus mitigating the deficiency problem that affected theviscous forces.

3.2. Pressure field recovery, stability region and particle movement95

Following Monaghan (2005), the commonly used estimate relationship betweenpressure and density is Tait’s equation

pi =ρ0c

20

γ

[(ρiρ0

)γ− 1

](13)

where ρ0 is a reference density, c0 is a numerical speed of sound on the mediumand γ = 7 for a fluid like water. According to Equation (13), the compressibilityof the fluid depends on c0, in such a way that for a high enough sound celeritythe fluid is virtually incompressible. However the value of c0 in the model shouldnot be the actual speed of sound, as the stability region is defined by

∆t = C min

mini

(√h

|f i|

); min

i

h

c0 + maxj |hvijrij

r2ij|

, (14)

where C is a constant of the order of 10−1 (Gomez-Gesteira et al., 2010). Thefirst term results from the consideration of force magnitudes and the second isa version of the classical CFL condition. This expression takes into accountnumerical information celerities and a restriction arising from the viscous terms(Gomez-Gesteira et al., 2010). If the sound celerity in the simulation is too100

high, it will render ∆t very small and the computation more expensive. Follow-ing Monaghan (2005), c0 is kept to an artificial value of around 10 times themaximum flow speed, restricting the relative density fluctuations at less than1%. As a consequence, the estimated pressure field given by Equation (13) usu-ally shows some instabilities and may be subject to scattered distributions. The105

δ-SPH diffusive terms contribute to the density field and smooth most of thehigh frequency oscillations.

Particle positions are updated every time-step, but instead of integratingdri/dt = vi, the XSPH smoothed velocity variant (Monaghan, 2005) is used,with the recomended parameter values. This procedure results in a locally110

smoother velocity field, while preserving linear and angular momentum. Theparticles are moved more orderly and no dissipation but dispersion is introduced.

4. RESULTS

In this section, a set of numerical and analytical solutions are used to com-pare the results of the SPH model. These include free-stream studies, and115

6

buoyancy tests. These comparisons should demonstrate a set of properties thatare fundamental for modelling fluids and solids, such as that no spurious errorsare introduced at the interface of the mediums, forces are correctly reproducedand taken into account and that unstable equilibrium solutions are not favouredor tolerated by the discretization.120

A Quintic (Wendland, 1995) kernel is employed for all the results. It is as

W (rij , h) = αD

(1− q

2

)4

(2q + 1) , 0 ≤ q ≤ 2, (15)

where q = |rij/h|, αD = 7/(4πh2) in 2D and (21/16πh3) in 3D.Simulations in section 4.1 were ran in a 3D domain, with sections 4.2 and 4.3

showing results from 2D simulations in order to compare to the available bench-mark solutions. The smoothing length is defined as h = 1.2

√dx2 + dy2 + dz2.

The C parameter from the stability region condition, Equation (14), was set at125

0.20 for all simulations.

4.1. Free stream consistency

A simple property of a discretization method should be free stream com-pliance. For a given velocity of the fluid and the rigid body, given that initialrelative velocities are zero, they should remain zero as time evolves and the130

equations are integrated, showing that there are no errors in the integratorsand that these are robust regarding truncation errors. The first case consistsof a 6 × 6 × 6 m patch of fluid and a 2 × 2 × 2 m rigid square in the center ofthe fluid, with a dx = dy = dz = 0.05 m initial inter particle spacing. Both aregiven a v = 1 ms−1 initial velocity, gravity acceleration is zero and there are no135

solid boundaries. Figure 1 shows a slice of a detail of the velocity field, aroundthe lower right corner of the rigid square.

Figure 1: Velocity field detail for a corner of the square at any time step.

The field is constant in time, no deviations along the solid-fluid interfacewere introduced during the 30 s of simulation.

A more demanding case is to consider a non-zero acceleration, since numer-140

ical errors arising from the usage of two sets of equations (for fluid and rigidbodies) should be more noticeable. The same geometry as for the previous ex-ample is set up, but with zero initial velocity and non-zero constant acceleration.Figure 2 shows a slice of the velocity fields at three separate instants.

7

Figure 2: Velocity field detail for a corner of the square at t = 5 s, t = 10 s and t = 15 s.

The same conclusions as for the first case can be drawn: the velocity field145

evolves without any irregularities being developed at the fluid-solid interface,streamlines are not disturbed by numerical errors.

4.2. Unsteady motion of bodies through a free surface

This section deals with the motion of cylinders in a fluid, subjected exclu-sively to buoyant forces. Three sets of results are compiled and compared with150

the current results. Moyo & Greenhow (2000) studied the problem using a po-tential flow model and Fekken (2004) later compared the results with an addedmass model solution and a numerical model that employs a Volume of Fluid(VOF) discretization. The latter provides the most relevant comparison, sinceviscous forces are considered for both body and fluid and the used VOF method155

is compatible with topological changes of the free surface geometry without anyspecial treatment (unlike the used potential flow method). Cylinders with ar = 1 m radius of several relative densities are placed at specific depths and thesimulation is allowed to evolve. The 2D simulation has a domain of an 8 m longperiodic box with a fluid depth of 7 m and a rigid boundary at the bottom, with160

an initial inter particle spacing of dx = dz = 0.03 m. Velocities are consideredat the centroid of the cylinder and are non-dimensionalised as V = v/

√gr, were

v is the velocity and g is the gravity acceleration magnitude. Time is madenon-dimensional as T = t

√g/r.

Cylinders of ρ = 0.6ρw and ρ = 0.9ρw are placed at a depth of D = 5 m.165

Figure 3 shows the system for the ρ = 0.6ρw case at four instants and Figure 4shows the velocity of the cylinders for both cases.

Both the potential flow model and the added mass model show a linearevolution of the velocity, expectable since no drag is taken into account onthe first and the fluid is not affected by the drag computed on the second.170

The Fekken and SPH solutions however show a non-linear acceleration on theascending part of the solution. Given the lower buoyant force of the ρ = 0.9ρwcase, the acceleration is much less pronounced and so are the added mass effectsand viscous drag on the dynamics of the system. The results again follow thesame trends as for the ρ = 0.6ρw, but a oscillation is seen in the velocity signal.175

This is due to pressure waves travelling on the domain and reflecting at thebottom.

8

Figure 3: Rising cylinder with ρ = 0.6ρw. T = 0; T = 3.13; T = 6.26; T = 9.40.

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

1.2

1.4

T(−)

V(−

)

0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

T(−)

V(−

)

Figure 4: Non-dimensional vertical velocity for a cylinder. Left - ρ = 0.6ρw; Right - ρ = 0.9ρw.Added mass modelFekken (2004)(−·−); Moyo & Greenhow(Moyo & Greenhow, 2000)(−−−);FekkenFekken (2004)(· · · ); DualSPHysics(—).

The interface between fluid and solid particles is subjected to the same defi-ciencies as described for large density ratio cases (Colagrossi & Landrini, 2003),in this case generated by forcing the relative position of the solid phase particles.180

An overestimation of the density on the solid particles occurred, producing ahydrophobic effect that resulted in a drastic reorganization of the fluid particles.This happened in order to cope with the increased density gradient at the inter-face and the entropy jump that occurred, considering that the solid particles areordered very strictly, not necessarily generating a constant equilibrium distance185

for a fluid particle across the interface. The δ-SPH term effectively curbs thesecascade behaviours by not allowing an erroneous density field to be computed atthe interface locus. Across the simulations the fluid particles remain at a similardistance from the solid and other fluid particles. This enables the viscous termto be computed as intended.190

For a case of engulfment, a cylinder with ρ = 1.2ρw at D = 0m (halfsubmerged) is set. Figure 5 shows the state of the system at four instants and

9

Figure 6 plots the vertical displacement and velocity of the cylinder.

Figure 5: Sinking cylinder with ρ = 1.2ρw. T = 1.57; T = 3.13; T = 4.70; T = 6.27.

0 2 4 6 8−5

−4

−3

−2

−1

0

T (−)

Dis

plac

emen

t (m

)

0 2 4 6 8−0.8

−0.6

−0.4

−0.2

0

T (−)

V (

−)

Figure 6: Left - Displacement of cylinder; Right - Non-dimensional vertical velocity for acylinder of ρ = 1.2ρw. Moyo & Greenhow (2000)(−−−); Fekken (2004)(· · · ); DualSPHysics(—).

Again, one can see how the SPH results are close to the VOF solution butwith less pronounced drag. The inflections on the velocity plots are a result of195

kinetic energy transfer from the body to the fluid, in order to accommodate thefree surface deformations that occurred.

4.3. Equilibrium position of floating bodies

An important scenario is equilibrium position retrieval of floating bodies.It is demanding for a numerical discretization since the system is set in an200

unstable equilibrium position and is then allowed to evolve until it reaches astable equilibrium. Fekken (2004) introduces numerical results for a 0.10×0.05mrectangle, vertically placed in a tank, half submerged, with ρ = 0.5ρw. The SPHsimulations were set with dx = dz = 0.003 m. Figure 7 shows the evolutionof the system and the angle of the object with the horizontal (Ω) along time,205

comparing with the data from Fekken.The cylinder correctly finds the stable equilibrium position, at Θ = −90 de-

grees and the damped oscillatory rotation around that point closely matches theVOF results. This result is particularly important since it shows that the systemavoids unstable positions, correctly reproduces the dynamics of the event and210

10

0 1 2 3 4 5−120

−100

−80

−60

−40

−20

0

Time(s)

Θ(d

eg)

Figure 7: Left - Displacement of rectangle at t = 0.0s, t = 1.1s and t = 5.0s; Right - Anglehistory of the rectangle. Equilibrium(−−−); Fekken (2004)(· · · ); DualSPHysics(—).

is robust even with low resolution simulations. Unlike on the VOF simulations,no force was imposed in the first instants of the simulation in order to force thesystem to diverge from the initial position. This appears to be a result of thelarge amount of interactions involved in every time-step, together with machineprecision and non-repeatability of the thread order of the parallel implementa-215

tion. Very small numerical unbalances, that would otherwise be diffused, areamplified by the unstable configuration of the system.

5. CONCLUSIONS AND FUTURE WORK

A general SPH discretization, expanded to support the inclusion of arbitrar-ily shaped rigid solids, was presented and discussed. This allows for a unified220

description of the media, without using extra terms in the discretization to ac-count for coupling and geometrical effects. A δ-SPH term allows to stabilisethe density field, a known difficulty with the used formulation, and at the sametime manages to minimize another known issue: systematic density overestima-tion at interfaces. This term seemed to be the missing ingredient to the correct225

characterisation of the fluid-solid interface.The comparison of results from the SPH approach with known solutions is

presented. This allowed to characterise the behaviour of the model by testingthree main properties: (I) conservation of the velocity field in free stream cases;(II) buoyancy induced dynamics; (III) fate of unsteady equilibrium configura-230

tions.This work focuses exclusively on fluid-solid interaction, however, multi-body

systems with solid-solid contacts are also within the capabilities of the model,employing ideas from contact mechanics, as well as considering the solid de-formable. Work is being developed around these topics, that represent a relevant235

contribution to the SPH method.

11

Acknowledgements

This research was partially supported by project PTDC/ECM/117660/2010,funded by the Portuguese Foundation for Science and Technology (FCT). Firstauthor acknowledges FCT for his PhD grant, SFRH/BD/75478/2010.240

This work was partially financed by Xunta de Galicia under project Pro-grama de Consolidacion e Estructuracion de Unidades de Investigacion Com-petitivas (Grupos de Referencia Competitiva) co-funded by European RegionalDevelopment Fund (FEDER), and also financed by Ministerio de Economa yCompetitividad under Project BIA2012-38676-C03-03.245

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