A Smooth Particle Hydrodynamics discretization forthe modelling of free surface ﬂows and rigid bodydynamics
Ricardo B. Canelas
1
, Jose M. Dom´ınguez
2
, Alejandro J. Crespo
2
, MonchoG´omezGesteira
2
, Rui M.L. Ferreira
1
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 ﬂuidsolid ﬂows. The model is basedon the fundamental conservation laws of hydrodynamics, namely the continuityand NavierStokes equations, and Newton’s equations for rigid body dynamics.The numerical solution, based on Smoothed Particle Hydrodynamics (SPH),resolves solidﬂuid 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. A
δ
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 experiments are carried out in order to compare the results with analitical andknown numerical solutions.
Keywords:
Smooth Particle Hydrodynamics, Rigid Body Dynamics,Multiphase, Freesurface, Meshless methods
1
CEHIDRO, Instituto Superior T´ecnico, UL, Lisbon, Portugal, email: ricardo.canelas@ist.utl.pt, ruimferreira@ist.utl.pt
2
Environmental Physics Laboratory (EPHYSLAB), Universidade de Vigo, Ourense,Spain, email: 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 by
5
considering the solid material perfectly rigid. A model that is capable of providing meaningful solutions for all of these areas and is scalable, from the computational 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 latter
15
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 collection
of Moving Particle Simulation (MPS) ﬂuid particles, rigidiﬁed by default. Thishas become the standard approach due to its simplicity and elegance. Monaghan 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 a
30
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 represents an eﬀort to improve and validate the solidﬂuid descriptions. It uses the
35
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 to
40
the continuity equation, controlling the density ﬁeld ﬂuctuations and contributing for the mitigation of known solidﬂuid interface deﬁciencies (Colagrossi &Landrini, 2003). The implementation has been already validated for interaction between ﬂuid and ﬁxed structures (Crespo et al., 2011; G´omezGesteira
et al., 2012). The DualSPHysics code enables simulation of millions of particles
45
2
at a reasonable computation time by using GPU cards (Graphics ProcessingUnits) as the execution devices. This allows to somewhat alleviate the previously expressed concerns about requirements of scale and resolution, since thecomputations are made up to two orders of magnitude faster than on normalCPU systems (Dom´ınguez et al., 2013a). A MultiGPU code was developed to
50
further phase out the increased memory consumption by running largescale,highresolution simulations. Dom´ınguez et al. (2013b) showed that very higheﬃciency was achieved for hundreds of GPUs using the MultiGPU implementation of DualSPHysics.A set of numerical experiments and analytical solutions are recovered from
55
the literature in order to provide a benchmark for the results achieved withthe revised DualSPHysics implementation. Previous eﬀorts with the methodology focus mainly on practical applications (Rogers et al., 2010), not providing
a more systematic study of the fundamental properties of these systems. Important features that the model should respect include free stream consistency,
60
simple dynamics of a buoyant body with various densities and the correct recovery 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 FORMULATION
65
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 easily labelled as ”particles”, and their individual angular velocity is disregarded,hence ”smooth”. The method relies heavily on integral interpolant theory (Monaghan, 2005), that can be resumed to the exactness of
A
(
r
) =
Ω
A
(
r
)
δ
(
r
−
r
)
d
r
,
(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 approximation 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
h
then
W
(
r
−
r
,h
) = 0 if

r
−
r
 ≥
h
, withlim
h
→
0
W
=
δ
and
Ω
W
(
r
,h
)
d
r
= 1, where
h
is the smoothing length anddeﬁnes the size of the kernel support (Liu, 2003). This leads to
A
(
r
) =
Ω
A
(
r
)
W
(
r
−
r
,h
)
d
r
,
(2)3
known as the integral interpolant. An approximation to discrete Lagrangianpoints can be made, by a proper discretization of the integral
A
i
≈
j
A
j
W
(
r
ij
,h
)
V
j
,
(3)called the summation interpolant, extended to all particles
j
,

r
ij

=

r
i
−
r
j
≤
h
, where
V
j
is the volume of particle
j
and
A
i
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
V
j
W
(
r
ij
,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 gradientes
are 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 compressible NavierStokes system
d
v
dt
=
−
∇
pρ
+
µ
∇
2
v
+
g
(5)
dρdt
=
−
ρ
∇
v
,
(6)where
v
is the velocity ﬁeld,
p
is the pressure,
ρ
is the density and
µ
and
g
are 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
dρ
i
dt
=
−
ρ
i
j
(
v
i
−
v
j
)
∇
W
(
r
ij
,h
) + Φ
i
,
(7)where
m
i
is the mass of particle
i
and Φ
i
is a diﬀusive term (Molteni & Colagrossi, 2009), designed to stabilize the density ﬁeld from highfrequency oscillations, written asΦ
i
= 2
δhc
0
j
(
ρ
j
−
ρ
i
)
r
ij
·
∇
W
(
r
ij
,h
)

r
ij

2
+
η
2
m
j
ρ
j
,
(8)4