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A Smooth Particle Hydrodynamics Discretization for the Modelling of Free Surface Flows and Rigid Body Dynamics

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A Smooth Particle Hydrodynamics Discretization for the Modelling of Free Surface Flows and Rigid Body Dynamics
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  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´omez-Gesteira 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 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. 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 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 1 CEHIDRO, Instituto Superior T´ecnico, UL, Lisbon, Portugal, e-mail: ri-cardo.canelas@ist.utl.pt, ruimferreira@ist.utl.pt 2 Environmental 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 by 5 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 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. 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 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 rep-resents an effort to improve and validate the solid-fluid 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 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; G´omez-Gesteira 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 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 (Dom´ınguez et al., 2013a). A Multi-GPU code was developed to 50 further phase out the increased memory consumption by running large-scale,high-resolution simulations. Dom´ınguez 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 from 55 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 providing a 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 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 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  ) 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 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  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 compress-ible Navier-Stokes 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 & Cola-grossi, 2009), designed to stabilize the density field from high-frequency oscilla-tions, 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

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Jul 28, 2017

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