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Unified semi-analytical wall boundary conditions for inviscid, laminar and turbulent slightly compressible flows in SPARTACUS-2D combined with an improved time integration scheme on the continuity equation.
[Thesis]. Manchester, UK: The University of Manchester; 2011.
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Abstract
Smoothed Particle Hydrodynamics (SPH) is a meshless Lagrangian numerical method ideal for simulating potentially violent free-surface phenomena such as a wave breaking, or a dam break where many Eulerian methods can be difficult to apply.Dealing with wall boundary conditions is one of the most challenging parts of the SPH method and many different approaches have been developed among (i) repulsive forces such as Lennard-Jones one, which is efficient to give impermeable boundaries but leads to non-physical behaviours, (ii) fictitious (or ghost) particles which provide a better physical behaviour in the vicinity of a wall but are hard to define for complex geometries and (iii) semi-analytical approach such as Kulasegaram et al. (2004) which consists of renormalising the density field near a solid wall with respect to the missing kernel support area. The present work extends this semi-analytical methodology, where intrinsic gradient and divergence operators are employed that ensures conservation properties. The accuracy of the physical field such as the pressure next to walls is considerably improved, and the consistent manner developed to wall-correct operators allows us to perform simulations with turbulence models. This work will present three key advances:• 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 of the form 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.• 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 novel time integration scheme, allowing us any shape for the boundary.• All boundary terms issued from the continuous approximation are given by surface summations which only require information from a mesh file of the boundary. The technique developed here allows us to correct the pressure gradient and viscous terms and hence provide a physically correct wall-shear stress so that even the diffusion equation of a scalar quantity can be solved accurately using SPH such as the turbulent kinetic energy or its dissipation in a k − ε model of turbulence.The new model is demonstrated for cases including hydrostatic conditions for a channel flow, still water in a tank of complex geometry and a dam break over triangular bed profile with sharp angle where significant improved behaviour is obtained in comparison with the conventional boundary techniques. Simulation of the benchmark test case of a square object moving in an enclosed tank is shows good agreement with the reference solution and no voids are formed within the fluid domain. The performance of the model for a 2−D turbulent flow in a channel is demonstrated where the profiles of velocity, turbulent kinetic energy and its dissipation are in agreement with the theoretical ones. Finally, the performance of the model is demonstrated for flow in a fish-pass where velocity field and turbulent viscosity field are satisfactory reproduced compared to mesh-based codes.