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" Hâtez-vous lentement; et, sans perdre courage, Vingt fois sur le métier remettez votre ouvrage: Polissez-le sans cesse et le repolissez; Ajoutez quelquefois, et souvent effacez." " Gently make haste, of labour not afraid; Consider twenty times of what you've said: Polish, repolish, every colour lay, And sometimes add, but oft 'ner take away." Nicolas Boileau, De l'art poétique (chant I) To my family and my teachers Summary Towards numerical simulations of phase transitional flows Numerical simulations have extensively been applied to study single-phase industrial flows. Their extension to multi-phase situations is complicated by the absence of a corresponding turbulent theory: there exists no theory describing how turbulence interacts with interfaces and therefore it is not possible to model the contribution of the small scales in such cases. To accurately describe this problem, all scales should be computed: Direct Numerical Simulation (DNS) is required. The description of the interface as a continuous transition between the two phases dates back to the end of the 19 th century and has been experimentally verified close to the critical point. The Diffuse Interface Model (DIM) resulting from this assumption consists of a set of conservation equations similar to the Navier-Stokes (NS) equations, apart from an additional stress tensor accounting for the capillary forces and an equation of state which is valid in both phases. In this way, topological and phase changes are captured in a way which is fully consistent with the underlying thermodynamics. If these partial differential equations are solved numerically by volume discretization, the grid must be very fine to cope with the thickness and stiff gradients of the interface. To limit the computation time, the size of the grid is limited: DIM should only be used in the multi-phase regions, with a fine grid close to the interface. For the single phase regions, a coarser grid can be used. In this thesis, special boundary conditions are developed to recreate on the small scale the situations arising from the macro-scale for multi-phase channel flows of a liquid with its vapor. Two situations are identified. When a vapor bubble is away from the wall, the flow at the edges of the micro-scale domain is determined by the macro-scale. These types of boundary conditions are called open boundary conditions. The second situation arising from the description of the channel flow is the nucleation of a vapor bubble on the wall and its interaction with the substrate. The surface properties of the wall (hydrophobic/hydrophilic) as well as the heat flux imposed are recreated using so-called wall boundary conditions. Designing open boundary conditions for DIM is a complex problem. Perturbations created inside the truncated numerical domain propagate since the DIM equations support traveling waves. If the boundary values were simply imposed by the macro-scale, these waves would be reflected by the edges and perturb the interior solution. Away viii from the interfacial regions, the contribution of the capillary terms in the DIM can be neglected. Therefore, in the bulk phases, the DIM equations may be approximated by the NS equations: conventional approaches from the literature for single phase flows can be used. However, when the interfacial regions come close to the boundaries, the capillary terms are dominant and two major features prevent a straightforward extension of the NS open boundary conditions: the wave-structure description is invalid and the reference state (liquid or vapor) is not defined. Two different types of open boundary conditions are found in literature for singlephase flows. They either rely on adding an artificial absorbing material at the edges which damps the amplitude of the waves leaving the computational domain or defining an operator at the boundary which prevent waves from entering the domain. The Perfectly Matched Layer (PML) method and the non-linear characteristic approach respectively belong to these two types of boundary conditions. In this thesis, they are extended to DIM in 1-D. For the PML method, when the interface regions come close to the edges, the reference state is approximated by the equilibrium profile between the two phases. In this case, the amplitude of the outgoing waves is damped and the transition between the multi-phase strategy and the conventional approach is continuous. For the characteristic approach, when the interface comes close to the boundary, the computational domain is enlarged. This buffer region ensures that the interface region is always surrounded by two bulk phases that are effectively computed. Once the multi-phase region is entirely inside the buffer region, the bulk phase is again present at the boundary of the original domain: the buffer region is then removed and the conventional approach can again be applied. This second approach is also extended to 2-D. Unlike in 1-D, the computational domain is enlarged in both directions. This should be only locally allowed to reduce the computational costs and a new data structure is proposed: the fixed-size main computational domain is complemented with separate dynamically allocated grids to handle the buffer regions. The method has successfully been tested with simple conventional open boundary conditions in the case of a uniform mean flow. The second situation arising from the large scale simulations is a bubble interacting with a substrate. No-slip velocity boundary conditions are imposed and instantaneous wall/fluid equilibrium is assumed. This last assumption allows to derive thermodynamically consistent boundary conditions which take into account the surface properties of the wall (hydrophobic/hydrophilic). The micro-contact angle is fixed and the apparent contact angle away from the substrate results from the balance between viscous and capillary forces. Unlike the sharp interface model, the DIM alleviates the velocity singularity encountered at the contact line. The boundary conditions are tested on multiple test cases: steady state of a vapor bubble on a uniform substrate for varying contact angle, bubble nucleation on a uniform substrate without initial flow for varying heat flux and contact angle, detachment of a vapor bubble from the substrate for varying contact angle and incoming flow velocity, and finally nucleation of a vapor bubble influenced by an incoming flow. Several regimes are observed depending on the balance between the capillary and the viscous forces and the fluid/wall interactions. To model a dynamic micro-contact angle, the no-slip and instantaneous equilibrium assumptions could be relaxed.
