Contribution to development of meshless methods for free and moving boundary problem

The purpose of this dissertation is the development of a novel, diffuse approximate meshless method in connection with the phase field formulation for solution of free and moving boundary problems. Free and moving boundary problems arise in a wide variety of scientific and technological applications. The common feature of such problems is the topological evolution of the interfaces between the phases, which leads to the formation of various flow patterns and regimes that strongly depend onthe physical properties of the phases. Evolution of deformed interfaces remain challenging from the physical and the computational points of view due to the strong effectsof surface tension and the discontinuities across the interfaces. We deal with the numerical modelling and simulation of two-phase flowusing diffuse interface method, namely, phase field methodin the present dissertation. Both phases are considered to be Newtonian, incompressible and immiscible. The problem is formulated with coupled Navier-Stokes and Cahn-Hilliard equations. Navier-Stokes equationsgovern the flow of the two fluids and Cahn-Hilliard equationis used for representation of the surface tension and to describe the evolution of the interface. The diffuse approximate method is structured with second order polynomial shape functions, Gaussian weights, local domain support and upwinding.The pressure-velocity coupling is performed by an incremental pressure correction scheme.The governing equations are solved in two-dimensions (2D) and in axisymmetry by using explicit time discretization. The performance of the method is tested on the well known 2D Rayleigh-Taylor instability problem usingthree different physical models: (I) a model with large density variation and surface tension, (II) a model with Boussinesq formulation for small density variation, and (III) a model with phase field dependent density for small density variation without the surface tension.The assessment of the method is carried out based on the sensitivity analysis by using different values of the shape parameter and the number of nodes in the local subdomain for the model(I). Furthermore, the simulations are performed on three different node arrangements 64x256, 128x512 and 192x786 to demonstrate the node density convergence. The effect of the Atwood numberAt = 0.01, 0.1, 0.3 and 0.5 on the height of the bubbles and the spikes is carried out for the models without surface tension effect. The fine-node arrangement results are compared with the previously published results obtained by staggered Marker-And-Cell method and finite volume method results,calculated by using open source finite volume method based computational fluid dynamics code Gerris.The meshless results demonstrate that the dynamics of Rayleigh-Taylor instability can be efficiently evaluated by a combination of phase-field method and meshless solution procedure. By comparing the results of all aforementioned models with the referenceresults from literature, it is found that the surface tension significantly changes the shape of the moving boundary between the fluids and that the largest curvature appears on both left and right tail of the mushrooms. Furthermore, the sensitivitystudy shows that when using nine nodes in local subdomain for shape parameterc= 2.5 and 5, the results are almost the same, howeverfor c= 10,simulations show that the heavy front moves faster than the lighter one and a significant bend of both left and right tails appears. When using different number of nodes in local subdomain (e.g. eleven and thirteen) with c= 10, the results are similar demonstrating that the simulations are not sensitive tothe number of nodes in local subdomain and are in close agreement with the finite volume results. The meshless results are also verified by reproducing the moving boundary dynamics, consistent with the previously published results: for an initially symmetric perturbation of the interface, the symmetry of the heavy and light fronts of the Boussinesq model is preserved for a long time. However, for the variable density model, the related symmetry is lost despite the fact that the flow starts symmetrically.The method is afterwards applied also in axisymmetry for aproblem of two co-flowing Newtonian, incompressible and immiscible fluids with different material properties that yield dripping or jetting of the core fluid. An assessment of the novel method is carried out based on the nodedensity convergence in terms of calculated dimensionless jet length. The meshless results are compared withthe finite volume resultsprovided with an open source numerical toolbox OpenFOAM. A sensitivity study of the various process parameterssuch as capillary number and viscosity ratiois also studied and verified against the previously published results.It is found that there is no significant difference in the calculated jet length for discretisations: 30x400, 45x600 and 60x800 and the results with node arrangement 30x400 are reasonably accurate. The meshless results demonstrate excellent agreement with the finite volume results in terms of drop size and temporal behaviorof interphase boundary. The meshless results are also in good agreement with the finite difference results in terms of dimensionless limiting length and volume of the drop. The combination of diffuse approximate method is also suitable for tackling the axisymmetric forced-flow moving boundary two-phase flow problems.This represented work deals with a pioneering attempt in solution of 2D Rayleigh-Taylor instability and the axisymmetric forced-flow moving boundary two-phase flow problems by a meshless solution of the phase field formulation.The results showthat the combination of diffuse approximate method and phase field method is capable to handle free surface flows with large topological changes and provide a valuable numerical tool for solving immiscible convective hydrodynamics.Furthermore, extensive experiments are carried out using the combination of diffuse approximate method and phase field method for the numerical simulation of gas focused micro-jets formed bygas dynamic virtual micro-nozzles. After such experiments, it is found that the current version of the methodis not yet suitablefor gas focused micro jets due to very large density difference of liquid (water) and gas (helium) and many very sharp edges of the nozzle. In the perspective, there is a need to further explore aspecial treatment of sharp edges and pressure velocity coupling scheme to handle very large density variations