<div>Incompressible single-phase and two-phase flows are widely encountered in and underlie many engineering applications. In this thesis, we aim to develop efficient methods and algorithms for numerical simulations of these classes of problems. Specically, we present two schemes: (1) a modied consistent splitting scheme for incompressible single-phase flows with open/out flow boundaries; (2) a three-dimensional hybrid spectral element-Fourier spectral method for wall-bounded two-phase flows.</div><div><br></div><div><div>In the first part of this thesis, we present a modied consistent splitting type scheme together with a family of energy stable outflow boundary conditions for incompressible single-phase outflow simulations. The key distinction of this scheme lies</div><div>in the algorithmic reformulation of the viscous term, which enables the simulation of outflow problems on severely-truncated domains at moderate to high Reynolds numbers. In contrast, the standard consistent splitting scheme is observed to exhibit a numerical instability even at relatively low Reynolds numbers, and this numerical instability is in addition to the backflow instability commonly known to be associated with strong vortices or backflows at the outflow boundary. Extensive numerical experiments are presented for a range of Reynolds numbers to demonstrate the effectiveness and accuracy of the proposed algorithm for this class of flows.</div></div><div><br></div><div><div>In the second part of this thesis, we present a numerical algorithm within the phase-field framework for simulating three-dimensional (3D) incompressible two-phase flows in flow domains with one homogeneous direction. In this numerical method, we represent the flow variables using Fourier spectral expansions along the homogeneous direction and C0 spectral element expansions in the other directions. This is followed by using fast Fourier transforms so that the solution to the 3D problem is obtained by solving a set of decoupled equations about the Fourier modes for each flow variable. The computations for solving these decoupled equations are performed in parallel to effciently simulate the 3D two-phase</div><div>ows. Extensive numerical experiments are presented to demonstrate the performance and the capabilities of the scheme in simulating this class of flows.</div></div>

Le modèle numérique que nous avons développé au cours de cette thèse présente deux caractéristiques principales : un modèle dilatable pour l'eau et la prise en compte de domaines ouverts. Les difficultés associées au premier aspect concernent l'adaptation de la loi d'état de l’eau au modèle dilatable sous l’approximation à faibles nombres de Mach, tandis que celles associées au second sont relatives à la mise en œuvre de conditions aux limites numériques de sortie compatibles avec l'algorithme de projection utilisé. Les résultats de simulations d'écoulement de convection mixte en canal horizontal chauffé par le bas ont été confrontés à celles utilisant l'approximation de Boussinesq et aux expériences. / The 3D numerical model which we developed in this thesis presents two main features: a Low-Mach-Number approximation for water along with an open boundary condition formulation. Indeed, the difficulties related to the former point stand in a computationally efficient adaptation of the water equation of state in the framework of Low Mach number approximation, whereas the difficulties related to the latter concern the introduction of Open Boundary Conditions in the projection algorithm used. We have computed a mixed convection flow in a horizontal channel uniformly heated from below and compared the results obtained with both the Boussinesq approximation and experimental results.