This work presents a method for simulating axisymmetric and planar free-surface flows dominated by surface tension forces. The surface tension effects are incorporated into the free surface boundary conditions through the computation of the capillary pressure. The required curvature is evaluated by fitting a least squares curve to the free surface using the tracking markers in the cell and in its close neighbors. To avoid short wavelength perturbations on the free surface, a mass-conserving local four-point stencil filter is employed. This filter is an extension of the trapezoidal subgrid undulations removal (TSUR) method. The TSUR technique consists of modifying the positions of two "internal" markers of the four-point stencil in such a way that the surface length and the curvature are minimized, while still preserving the volume. The computation of the curvature is modified at cells adjacent to solid boundaries in order to apply contact angle boundary conditions. To identify neighboring cells efficiently, an implementation is effected through a dual representation of the cell data: in addition to a matrix representation, a list structure is also employed which permits the representation of specific groups of cells and associated data. This technique was implemented in the GENSMAC code, and it has been proved to be robust. The code is shown to produce accurate results when compared with exact solutions of selected fluid dynamical problems involving surface tension. Additionally, it is demonstrated that the method is applicable to complex free-surface flows, such as containers filling, splashing drops, and bursting bubbles.

At the late stage of transitional boundary layers, the nonlinear evolution of the ring-like vortices and spike structures and their effects on the surrounding flow were studied by means of direct numerical simulation with high order accuracy. A spatial transition of the flat-plate boundary layers in the compressible flow was conducted. Detailed numerical results with high resolution clearly represented the typical vortex structures, such as ring-like vortices and so on, and induced ejection and sweep events. It was verified that the formation of spike structures in transitional boundary layers had close relationship with ring-like vortices. Especially, compared to the newly observed positive spike structure in the experiments, the same structure was found in the present numerical simulations, and the mechanism was also studied and analyzed. boundary layer transition, ring-like vortices, spike structure, direct numerical simulation PACS: 47.27.Cn, 47.27.ekThe main stages of the boundary layer transition include receptivity, linear instability, weakly nonlinear instability, formation of complex vortex structures (late stage of the transition) and breakdown to turbulences. Experiments have proved that at the late-stage of the transition similar characteristic vortex structures can be found in the turbulence boundary layers. After more than a century of studies, the early stages of the transition process, such as linear theory, weakly nonlinear and secondary instability theory, are well understood. But the late-stage of the transition is still unclear and takes further investigation, such as the formation of ring-like vortices, and appearance of the spikes which are very intensive local streamwise velocity fluctuations.The formation of ring-like vortices is one of the important issues in the transition process. Kachnaov [1] suggested the existence of the chain of ring-like vortices based on hot-film measurement. Lee et al. [2] showed their clear pictures of real ring-like vortices. The appearance and development of high-frequency spike structures is an important stage of the boundary layer transition, yet the mechanism of production and the physical nature of spike structure remains unclear. Recent years have seen more progress on the investigation of the formation of ring-like vortices and spike structure. Bake et al. [3] studied the structures of turbulence development in periodic Klebanoff boundary layer transition, and found that ring-like vortices would induce highfrequency streamwise velocity fluctuations called spikes. Borodulin et al. [4] studied the late-stage transition mechanism in the boundary layers using numerical and experimental methods, and analyzed relative vortices and spike structure. They proposed that the formation of ring-like vortices was due to self-induction. Meyer et al. [5] by investigating the flow randomization process in a transitional boundary layer found that Λ-vortices and ring-like vortices

AgradecimentosAos meus pais, Mara e Luiz, pela vida, pelo amor e carinho, e pelo apoio e compreensão.Ao meu orientador, prof. Antonio Castelo Filho, pela dedicação, pela paciência e pelo incentivo durante o decorrer deste trabalho.Aos professores do grupo de Análise Numérica, que contribuíram com valiosas sugestões e ensinamentos. Em especial ao prof. Norberto Mangiavacchi, ao prof. Murilo Francisco Tomé, e ao prof. José Alberto Cuminato.Aos amigos Ricardo, Fabrício, Luciane, Marcelo, Juliana Maria, Helton e Valdemir, pela amizade, pelas risadas e, principalmente, pela ajuda e compreensão.Aos funcionários do ICMC, em especialà Beth,à Laura, ao Paulinho eà Tatiana, pela atenção, disponibilidade e paciência.A FAPESP, pelo apoio financeiro no desenvolvimento deste trabalho. Finalmente, agradeço a todos que direta ou indiretamente contribuíram para a realização deste trabalho. ResumoEste trabalho apresenta um ambiente de simulação de escoamentos com simetria radial e superfícies livres, baseado no sistema Freeflow. O sistemá e formado por três módulos: um modelador de moldes, um simulador e um visualizador de escoamentos.O simulador implementa o método GENSMAC para a solução das equações de Navier-Stokes em coordenadas cilíndricas, utilizando diferenças finitas em uma malha diferenciada. São introduzidos os efeitos da tensão superficial e dô angulo de contato nas simulações com simetria radial.Alguns resultados de simulações utilizando este sistema e uma validação do código são apresentados, comparando simulações com soluções analíticas e experimentais, e estudando a convergência do método. AbstractThis work presents an environment for the simulation of axisymmetric free surface flows, based on the Freeflow system. The system contains three modules: a geometric model modeller, a simulator and a visualizator.The simulator implements the GENSMAC method for the solution of the Navier-Stokes equations in cylindrical coordinates, using finite differences in a staggered grid. The effects of surface tension and contact angle are introduced in the axisymmetric simulations.Some results from simulations using this system and a validation of the code are presented, comparing the simulations with analytical and experimental solutions, and studying the convergence of the method.

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