Fluid-solid interaction
This research was undertaken by Dr. Julian Mindel and Dr. Xavier Garcia during their PhD studies at Imperial College under the supervision of Dr. J-P Latham, and Prof. Chris Pain, with assistance from AMCG’s, Mr. Dimitrios Pavlidis, Dr. Gerard Gorman, Dr. Matt Piggott and Dr. Jiansheng Xiang.
In applications with VGeST, two-way coupling between solids and fluids is realized using a dual mesh approach. The mathematical formulation has some similarities to the immersed boundary method and is presented in cited references. One mesh is used across the whole solution domain on which the fluids equations are solved and the second mesh contains a finite element representation of the solid structures. Adaptive meshing resolves down onto the complex geometry of the solids at the level of detail necessary, hence addressing one of the main challenges – the accuracy of the flow field near the solid surfaces and the capture of boundary layer effects. The forces and volume fraction from the DEM or FEMDEM structure model are mapped onto the fluids mesh using FEM mapping and updated hydraulic forces are returned to the explicit transient dynamic FEMDEM modelling of the solids.
To illustrate this, the figure shows how a domain with two solids spheres is meshed. Adaptivity refines the mesh near regions of high gradients in solid concentration to capture the shape of the interfaces. As the approach involves solving the Navier-Stokes equations for two fluids, the drag forces on the solid boundaries do not require further parameterization. Instead, the forces arise naturally from the solution of the governing fluid equations. This approach virtually eliminates meshing efforts because a very coarse arbitrarily chosen initial mesh can be used and adaptivity completes the complex fluids mesh unoccupied by solids. This has proven to be extremely powerful for one-way coupled flow problems e.g. when applied to flow within complex void geometries inside the pore space of sand packs. See Applications.
Large particles (i.e. relative to the grid size) are exceptionally well resolved with the help of Fluidity’s adaptive mesh optimisation. However, representing the shape and size of an arbitrary number of smaller particles through a complex fully resolved computational grid would be extremely difficult, as this would require high amounts of resolution and the mesh to evolve with the motion of these particles. It seems a logical cost-effective solution to approximate and parameterise effective forces acting on particles which, in comparison to other dimensions in the problem to be solved (i.e. minimum element size), are much smaller in size. Furthermore, if the larger particles were to somehow break or fragment, the method should be able to transition from the non-parameterised model to the parameterised model in a smooth straightforward manner. The potential to handle this range of sizes is embedded in the coupling methods adopted in VGeST. The flexibility of this approach is important as it may be used to adopt the parameterisation of particle flow, as opposed to resolving each individual particle, and for modelling both stationary and moving particles independent of size. Adaptive mesh optimisation within the fluid domain allows for greater resolution in areas of interest such as those that define the geometry of the larger particles, as well as those of dynamic importance, such as fluid vortices and boundary layers.
This means that in the application to storm damage of coastal structures and breakwaters, for example, the adaptive/moving mesh will follow the armour units should they become unstable and move through the domain, putting necessary fine scale resolution around the armour units (anisotropically in boundary layers) and conforming well to the boundaries of these unusual-shaped rock or concrete blocks. Compared to alternative approaches for multi-fluid structure interactions with free surfaces and large solid movements, there are many advantages in employing an adaptive approach in combination with solid FEMDEM models.
This two phase adaptive mesh approach to coupling has produced excellent drag and terminal velocity results in two-way coupled problems. These together with coupled DEM/fluids simulating collisions of steel spheres in silicone oil reverting to terminal velocities can be seen below and are reported in Xiang, Mindel, et al. (2007).
Pressure drops close to Ergun’s predictions for full Navier-Stokes flows through stationary packed beds of grains with Reynolds Number, Re from 1 to 100, lend further support to the two phase coupling approach adopted. See Garcia et al. (2010) and flow through grain packs.
References
Xiang, J., Mindel, J.E., Latham, J.-P., Pain C.C., Piggott M.D., Guises, R., Garcia, X. Munjiza A. 2007. Proceedings of the Fifth International Conference on Coastal Structures, Venice, Italy, July 2-4, 2007. Published by World Scientific Publishing Co. Pte. Ltd. 2009, 1441-1452.
Garcia, X., Pavlidis, D., Gorman, G.J., Gomes, J.L.M.D.A., Piggott, M.D., Mindel, J.E., Aristodemou, E., Pain, C., Latham, J.-P. ApSimon, H.M. 2010. A two-phase adaptive finite element method for solid-fluid coupling in complex geometries, International Journal for Numerical Methods in Fluids. (In Press).