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TWiki> CfdTm Web>CfdTmResearch>ResearchSummary0007 (2010-03-03, MartinFerrand)

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

** 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.

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.

I | Attachment | Action | Size | Date | Who | Comment |
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PNG | fig1_chanels_wiki.PNG | manage | 97.5 K | 2010-02-26 - 11:58 | MartinFerrand |

Topic revision: r10 - 2010-03-03 - 11:12:36 - MartinFerrand

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