Chorin’s method, Incremental Pressure Correction Scheme (IPCS) and Crank-Nicolson’s method (CN) are three numerical methods that were investigated in this study. These methods were here used for solving the incompressible Navier-Stokes equations, which describe the motion of an incompressible fluid, in three different benchmark problems. The methods were stabilized using residual based artificial viscosity, which was introduced to avoid instability. The methods were compared in terms of accuracy and computational time. Furthermore, a theoretical study of adaptivity was made, based on an a posteriori error estimate and an adjoint problem. The implementation of the adaptivity is left for future studies. In this study we consider the following three well-known benchmark problems: laminar 2D flow around a cylinder, Taylor-Green vortex and lid-driven cavity problem. The difference of the computational time for the three methods were in general relatively small and differed depending on which problem that was investigated. Furthermore the accuracy of the methods also differed in the benchmark problems, but in general Crank-Nicolson’s method gave less accurate results. Moreover the stabilization technique worked well when the kinematic viscosity of the fluid was relatively low, since it managed to stabilize the numerical methods. In general the solution was affected in a negative way when the problem could be solved without stabilization for higher viscosities.

Este é o primeiro trabalho que apresenta simulações tridimensionais completamente adaptativas de um modelo de campo de fase para um fluido incompressível com densidade de massa constante e viscosidade variável, conhecido como Modelo H. Solucionando numericamente as equações desse modelo em malhas refinadas localmente com a técnica AMR, simulamos computacionalmente escoamentos bifásicos tridimensionais. Os modelos de campo de fase oferecem uma aproximação física sistemática para investigar fenômenos que envolvem sistemas multifásicos complexos, tais como fluidos com camadas de mistura, a separação de fases sob forças de cisalhamento e a evolução de micro-estruturas durante processos de solidificação. Como as interfaces são substituídas por delgadas regiões de transição (interfaces difusivas), as simulações de campo de fase requerem muita resolução nessas regiões para capturar corretamente a física do problema em estudo. Porém essa não é uma tarefa fácil de ser executada numericamente. As equações que caracterizam o modelo de campo de fase contêm derivadas de ordem elevada e intrincados termos não lineares, o que exige uma estratégia numérica eficiente capaz de fornecer precisão tanto no tempo quanto no espaço, especialmente em três dimensões. Para obter a resolução exigida no tempo, usamos uma discretização semi-implícita de segunda ordem para solucionar as equações acopladas de Cahn-Hilliard e Navier-Stokes (Modelo H). Para resolver adequadamente as escalas físicas relevantes no espaço, utilizamos malhas refinadas localmente que se adaptam dinamicamente para recobrir as regiões de interesse do escoamento, como por exemplo, as vizinhanças das interfaces do fluido. Demonstramos a eficiência e a robustez de nossa metodologia com simulações que incluem a separação dos componentes de uma mistura bifásica, a deformação de gotas sob cisalhamento e as instabilidades de Kelvin-Helmholtz. / This is the first work that introduces 3D fully adaptive simulations for a phase field model of an incompressible fluid with matched densities and variable viscosity, known as Model H. Solving numerically the equations of this model in meshes locally refined with AMR technique, we simulate computationally tridimensional two-phase flows. Phase field models offer a systematic physical approach to investigate complex multiphase systems phenomena such as fluid mixing layers, phase separation under shear and microstructure evolution during solidification processes. As interfaces are replaced by thin transition regions (diffuse interfaces), phase field simulations need great resolution in these regions to capture correctly the physics of the studied problem. However, this is not an easy task to do numerically. Phase field model equations have high order derivatives and intricate nonlinear terms, which require an efficient numerical strategy that can achieve accuracy both in time and in space, especially in three dimensions. To obtain the required resolution in time, we employ a semi-implicit second order discretization scheme to solve the coupled Cahn-Hilliard/Navier-Stokes equations (Model H). To resolve adequatly the relevant physical scales in space, we use locally refined meshes which adapt dynamically to cover special flow regions, e.g., the vicinity of the fluid interfaces. We demonstrate the efficiency and robustness of our methodology with simulations that include spinodal decomposition, the deformation of drops under shear and Kelvin-Helmholtz instabilities.