Refined Boundary Conditions for the Smoothed Particle Hydrodynamics

Researcher: Martin Ferrand

Supervisor(s): Prof. D. Laurence, Dr. B. Rogers
Sponsor: EDF
Start Date: 01/10/09 End Date: 31/09/10
Keywords: Boundary conditions in SPH

Overall Research Aim

State of the art The SPH (Smoothed Particle Hydrodynamics) method is nowadays widely used for modelling complex, rapid and turbulent free-surface flows, including fluid-structure interactions, turbulence, multiphase, etc. Under the ERCOFTAC.org association a large group of international experts focused on SPH methods was developed, mostly under the impulse of Damien Violeau EDF R&D and Ben Rogers at UniMAN/MACE who developed the “Spartacus” and “SPHysics” SPH codes respectively. See http://wiki.manchester.ac.uk/spheric/ Spartacus is applied to the simulation of schematic or real, 2-D or 3-D flows, such as the behaviour of an oil spill near an oil containment boom or the flow over a dam spillway. The SPH method has many advantages, among which the simplicity and a broad range of potential applications. However, some drawbacks subsist, partly due to its Lagrangian character, in particular in the treatment of boundary conditions. The goal of the proposed work is to develop some numerical solutions to partly overcome the issues listed above. Another issue is the treatment of turbulent boundary layers with SPH, since there is no equivalent to elongated near wall cells as used in Eulerian codes (with dx/dy aspect ratios up to 10 or 50), and non-local nature of SPH discretisation.

General Objectives of the Project The work proposed here consists of developing numerical tools allowing a better treatment of boundary conditions and numerical stability in SPH. One will focus on the following questions:

  • Consistent wall boundary condition: Bonet and Rodriguez-Paz, among other authors, suggest an novel approach to model solid walls, based on variational Mechanics. It allows the modelling of impermeable solid walls without the use of dummy particles, artificial repulsive forces or other expensive and difficult techniques, and moreover satisfies the conservativity of momentum. This method will be tested, validated and compared to others.
  • Thin turbulent boundary layers at high Reynolds numbers are difficult to capture with SPH (homogeneous isotropic discretisation) and must be modelled by “wall functions”. The partly “non-local” nature of SPH discretisation (kernels overlap the solid boundaries in a large range of manners) will be challenging (as opposed to the special “first cell at the wall” treatment for Eulerian codes). Morever “wall functions” mostly assume steady-state / equilibrium situation (neglecting time dependent or convective terms) which is not suitable in the Lagrangian SPH framework. Recent developments by T. Craft et al. for Eulerian codes at UniMAN/MACE could prove very useful for near-wall turbulent boundary layer in the SPH framework.
  • Inlet-outlet boundary conditions: recent publications investigate the possibility to model in-flow and out-flow conditions, to allow the modelling of a prescribed flowrate at one side of the computational domain, for example. Research will be carried out in order to propose a consistent scheme for this type of problems.

Research Progress

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Dealing with wall boundary conditions is one of the most challenging parts of the SPH method and many different approaches have been developed by authors including Kulasegaram et al. (2004), Oger et al. (2007), Di Monaco et al. (2009), Kajtar and Monaghan (2009), Marongiu et al. (2009) and De Leffe et al. (2009). Accurate boundary conditions are of great interest in fields such as studying turbulence close to the wall which is the overall aim of this current research effort.

The present work is based on Kulasegaram et al. (2004) which consists of renormalising the density field near a solid wall with respect to the missing kernel support area. This methodology, combined with the Lagrangian formalism, defines intrinsic gradient and divergence operators which are variationally consistent and ensure conservation properties.

  • The time integration scheme used for the continuity equation requires particular attention, and as already mentioned by Vila (1999), we prove there is no point in using a dependence in time of the particles' density if no kernel gradient corrections are added. Thus, by using a near-boundary kernel-corrected version of the time integration scheme proposed by Vila, we are able to simulate long-time simulations ideally suited for turbulent flow in a channel in the context of accurate boundary conditions.
  • As mentioned by De Leffe et al. (2009), the method of Kulasegaram et al. (2004) defines an inaccurate gradient operator which provides non consistent behaviour, we have developed corrections of differential operators analogous to Di Monaco et al. (2009) and De Leffe et al. (2009) for slightly compressible viscous Newtonian fluids, but all boundary terms issued from the continuous approximation are given by surface summations which only use information from a mesh file of the boundary.
  • In order to compute the kernel correction, Feldman and Bonet (2007) use an analytical value which is computationally expensive whereas Kulasegaram et al. (2004) and De Leffe et al. (2009) use polynomial approximation which can be difficult to define for complex geometries. We propose here to compute the renormalisation term of the kernel support near a solid with a time integration scheme, allowing us any shape for the boundary.

Below we plot examples of 2D periodic channel flow which show in (a) and (b) volume loss with conventional continuity equation. In (c) we show that the improved time integration scheme conserves volume while in (d) we also use the new computation of corrected kernel.

fig1_chanels_wiki.PNG



Last Modification: r10 - 2010-03-03 - 11:12:36 - MartinFerrand


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