Geophysical flows and the effects of a strong surface tension

In the present note we review some recent results for a class of singular perturbation problems for a Navier-Stokes-Korteweg system with Coriolis force. More precisely, we study the asymptotic behaviour of solutions when taking incompressible and fast rotation limits simultaneously, in a constant capillarity regime.Our main purpose here is to explain in detail the description of the phenomena we want to capture, and the mathematical derivation of the system of equations. Hence, a huge part of this work is devoted to physical considerations and mathematical modeling

Numerical simulation methods for phase-transitional flow

The object of the present dissertation is a numerical study of multiphase flow of one fluid component. In particular, the research described in this thesis focuses on the development of numerical methods that are based on a diffuse-interface model (DIM). With this approach, the modeling problem posed by the presence of moving boundaries in the flow domain, namely the interfaces between different phases, can be solved in a way that preserves the characteristic physical features related to the interfaces, such as surface tension and phase transitions. The first, largest part of the dissertation describes how to apply the DIM formulation that has been adopted, commonly identified as Korteweg formulation, in numerical simulations, without altering the physical parameters of the fluid. The issues of stability and accuracy of the solution, which can be severely compromised by the elliptical and dispersive nature of the set of governing equations, are extensively discussed. Therefore, before discretizing the governing equations a transformation of variables is performed, which removes the most important dispersive terms and greatly increases the stability of the numerical method. The latter is tested on several benchmark two-phase flow problems and for various grid refinements, when a Van der Waals equation of state is used and the temperature is in the vicinity of the critical value. To study the behavior of the flow when the temperature and the velocity fields are coupled, not only isothermal but also non-isothermal simulations are performed and analyzed. This includes a phasetransitional flow where the initial temperature field is such that latent heat plays a major role. Next, the feasibility of a combination of the DIM formulation with Large Eddy Simulation (LES) for turbulent multiphase flow, which is typical in several industrial applications, is explored and tested on one of the isothermal flow simulations. First the various subgrid terms resulting from filtering the governing equations are assessed in an a priori analysis, and different models for the most important subgrid terms are evaluated. Subsequently, a real LES is performed with the best subgrid model based on this analysis and its results are compared with filtered results from a direct numerical simulation. The research carried out for DIM and DIM-LES simulations is intended as the first step towards the development of models for interface mass and heat transfer that can be applied in commercial flow solvers for turbulent phase-transitional flow on industrial problems. Therefore, this research represents an ideal bridge towards the last part of the dissertation, in which a CFD (Computational Fluid Dynamics) model is developed and tested for an industrial application of turbulent phase-transitional flow: the direct-contact condensation of superheated steam injected in water. This model is implemented in the commercial CFD software package ANSYS Fluent. The purpose of this work is twofold. On the one hand, a condensation model for the mass transfer rate at the steam–water interface, based on kinetic gas theory, is tested by comparison of the results with experiments conducted at the Department of Mechanical Engineering of TU/e within the scope of the same research project. By testing the phase change model, useful information can be obtained on the grid requirements and the turbulence model. On the other hand, comparison with experiments, also conducted at TU/e, can be made for the case of steam injected in a fully developed turbulent cross-flow of water in a square duct. To this purpose, results are shown for a three-dimensional simulation performed for the assigned geometry of the experimental setup and for one set of operating conditions used in the experiments. All simulations performed with Fluent are based on a Volume-of-Fluid (VOF) multiphase formulation and on the Reynolds-averaged Navier-Stokes (RANS) equations approach for turbulent flow. Both are typically adopted in the industrial two-phase flow CFD

Hyperbolic Techniques in Modelling, Analysis and Numerics

Several research areas are flourishing on the roots of the breakthroughs in conservation laws that took place in the last two decades. The meeting played a key role in providing contacts among the different branches that are currently developing. All the invitees shared the same common background that consists of the analytical and numerical techniques for nonlinear hyperbolic balance laws. However, their fields of applications and their levels of abstraction are very diverse. The workshop was the unique opportunity to share ideas about analytical issues like the fine-structure of singular solutions or the validity of entropy solution concepts. It turned out that generalized hyperbolic techniques are able to handle the challenges posed by new applications. The design of efficient structure preserving methods turned out to be the major line of development in numerical analysis

Decomposition driven interface evolution for layers of binary mixtures: I. Model derivation and stratified base states

A dynamical model is proposed to describe the coupled decomposition and profile evolution of a free surface film of a binary mixture. An example is a thin film of a polymer blend on a solid substrate undergoing simultaneous phase separation and dewetting. The model is based on model-H describing the coupled transport of the mass of one component (convective Cahn-Hilliard equation) and momentum (Navier-Stokes-Korteweg equations) supplemented by appropriate boundary conditions at the solid substrate and the free surface. General transport equations are derived using phenomenological non-equilibrium thermodynamics for a general non-isothermal setting taking into account Soret and Dufour effects and interfacial viscosity for the internal diffuse interface between the two components. Focusing on an isothermal setting the resulting model is compared to literature results and its base states corresponding to homogeneous or vertically stratified flat layers are analysed.Comment: Submitted to Physics of Fluid

Mixing and Phase Separation of Fluid Mixtures

During the three years of the PhD project we extended the di®use interface (DI) method and apply it to engineering related problems, particularly re- lated to mixing and demixing of two °uids. To do that, ¯rst the DI model itself was validated, showing that, in agreement with its predictions, a single drop immersed in a continuum phase moves whenever its composition and that of the continuum phase are not at mutual equilibrium [D. Molin, R. Mauri, and V. Tricoli, "Experimental Evidence of the Motion of a Single Out-of-Equilibrium Drop," Langmuir 23, 7459-7461 (2007)]. Then, we de- veloped a computer code and validated it, comparing its results on phase separation and mixing with those obtained previously. At this point, the DI model was extended to include heat transport e®ects in regular mixtures In fact, in the DI approach, convection and di®usion are coupled via a nonequi- librium, reversible body force that is associated with the Kortweg stresses. This, in turn, induces a material °ux, which enhances both heat and mass transfer. Accordingly, the equation of energy conservation was developed in detail, showing that the in°uence of temperature is two-folded: on one hand, it determine phase transition directly, as the system is brought from the single-phase to the two-phase region of its phase diagram. On the other hand, temperature can also change surface tension, that is the excess free en- ergy stored within the interface at equilibrium. These e®ects were described using the temperature dependence of the Margules parameter. In addition, the heat of mixing was also taken into account, being equal to the excess free energy. [D. Molin and R. Mauri, "Di®use Interface Model of Multiphase Fluids," Int. J. Heat Mass Tranf., submitted]. The new model was applied to study the phase separation of a binary mixture due to the temperature quench of its two con¯ning walls. The results of our simulations showed that, as heat is drawn from the bulk to the walls, the mixture phase tends to phase separate ¯rst in vicinity of the walls, and then, deeper and deeper within the bulk. During this process, convection may arise, due to the above mentioned non equilibrium reversible body force, thus enhancing heat transport and, in particular increasing the heat °ux at the walls [D. Molin, and R. Mauri, "Enhanced Heat Transport during Phase Separation of Liquid Binary Mix- tures," Phys. Fluids 19, 074102-1-10 (2007)]. The model has been extended then and applied to the case where the two phases have di®erent heat con- 3 ductivities. We saw that heat transport depends on two parameters, the Lewis number and the heat conductivity ratio. In particular, varying these parameters can a®ect the orientation of the domains that form during phase separation. Domain orientation has been parameterized using an isotropy coe±cient », varying from -1 to 1, with » = 0 when the morphology is isotropic, » = +1 when it is composed of straight lines along the transversal (i.e. perpendicular to the walls) direction, and » = ¡1 when it is composed of straight lines along the longitudinal (i.e. parallel to the walls) direction [D. Molin, and R. Mauri, "Spinodal Decomposition of Binary Mixtures with Composition-Dependent Heat Conductivities," Int. J. Engng. Sci., in press (2007)]. In order to further extend the model, we removed the constraint of a constant viscosity, and simulated a well known problem of drops in shear °ows. There we found that, predictably, below a certain threshold value of the capillary number, the drop will ¯rst stretch and then snap back. At lager capillary numbers, though, we predict that the drop will stretch and then, eventually, break in two or more satellite drops. On the other hand, applying traditional °uid mechanics (i.e. with in¯nitesimal interface thick- ness) such stretching would continue inde¯nitely [D. Molin and R. Mauri, " Drop Coalescence and Breakup under Shear using the Di®use Interface Model," in preparation]. Finally, during a period of three months at the Eindhoven University, we extended the DI model to a three component °uid mixture, using a di®erent form of the free energy, as derived by Lowengrub and Coworkers.. With this extension, we simulated two simple problems: ¯rst, the coalescence/repulsion of two-component drops immersed in a third component continuum phase; second, the e®ect of adding a third component to a separated two phase system. Both simulations seem to capture physical behaviors that were observed experimentally [D. Molin, R. Mauri and P. Anderson, " Phase Separation and Mixing of Three Component Mixtures," in preparation]