The diffuse interface model for phase-transitional flows Citation for published version (APA): Gelissen, E. J. (2020). The diffuse interface model for phase-transitional flows. Technische Universiteit Eindhoven. Document status and date: Published: 02/07/2020 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. 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This work is part of the research program Open Technologieprogramma with project num- ber 13781, which is (partly) financed by the Netherlands Organization for Scientific Re- search (NWO) Domain Applied and Engineering Sciences (TTW, previously Technology Foundation STW). This work was sponsored by NWO Exact and Natural Sciences for the use of supercomputer facilities. The Diffuse Interface Model for Phase-Transitional Flows PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de rector magnificus prof.dr.ir. F.P.T. Baaijens, voor een commissie aangewezen door het College voor Promoties, in het openbaar te verdedigen op dinsdag 21 april 2020 om 16:00 door Erwin Johannes Gelissen geboren te Brunssum Dit proefschrift is goedgekeurd door de promotoren en de samenstelling van de promotiecommissie is als volgt: voorzitter: prof.dr. L.P.H. de Goey 1e promotor: prof.dr. J.G.M. Kuerten 2e promotor: prof.dr. C.W.M. van der Geld leden: prof.dr. A.A. Darhuber prof.dr.ir. C. Vuik (Technische Universiteit Delft) prof.dr. J.H. Snoeijer (Universiteit Twente) prof.dr.ir. J.A.M. Kuipers prof.dr.ir. E.H. van Brummelen Het onderzoek dat in dit proefschrift wordt beschreven is uitgevoerd in overeen- stemming met de TU/e Gedragscode Wetenschapsbeoefening. The Diffuse Interface Model for Phase-Transitional Flows A surface of separation between different thermodynamic phases is called an interface. There are two fundamentally different approaches to describe an interface in models and numerical simulation methods of multiphase flows. Interfaces can be considered as surfaces of zero thickness, i.e. a sharp interface, or they can be considered to have a non-zero thickness, i.e. a diffuse interface. Postulating the interface as sharp means that the physical properties of the fluid, such as mass density, are allowed to change discontinuously across the interface. Jump conditions are then required to describe the relationships between thermodynamic and hydrodynamic quantities on both sides of the interface. Postulating the interface to be diffuse allows for interfacial forces to be modeled as continuum forces by distributing these forces over thin but numerically resolvable layers. The important advantage of this approach is that Diffuse Interface Models (DIM) are well suited to handle topological changes of the interface, such as breakup and coalescence phenomena. Surface tension effects are incorporated by including a capillary stress tensor in the governing equations. This stress tensor ac- counts for capillary stresses at the interface and depends not only on the mass density but also on its spatial derivatives. However, one criterion needs to be met in order for simulations with DIM to be successful: the thickness of the interface can be at most one order of magnitude smaller than the length scale of the domain in order for the simulations to be computationally affordable. Investigation of droplet collisions with DIM shows the existence of various collision regimes including permanent coalescence, toroidal droplet breakup, reflexive separa- tion and stretching separation. The results also highlight the importance of modeling liquid-vapor flows under non-isothermal conditions, since variations in the tempera- ture field during the collision process are too large to be neglected and also influence the collision dynamics. Comparison of the DIM with a different method often used for multiphase flows, the Local Front Reconstruction Method (LFRM), shows an over- all good correspondence between the results of both simulation models, with both models predicting the same collision outcomes. However, differences are observed in the interface evolution and energy transfer/dissipation process during the collision, a significant portion of which can be attributed to the chosen configuration of the initial velocity field, which is not divergence-free. This enlarges an important differ- ence between the two simulation models: the used implementation of LFRM assumes incompressible flow, while DIM treats the fluid as compressible. The introduction of a divergence-free vortical initial velocity field improved the agreement between the results of both simulation models. The dynamics of droplet impingement on a heated wall can also be studied by DIM coupled with the thermal diffusion inside a solid through a set of boundary conditions that are imposed at the solid-fluid interface. These boundary conditions include the wetting properties of the solid and the roughness of the solid surface. An increased cooling rate on hydrophilic surfaces was observed due to a longer liquid-solid time of contact. The cooling rate also increases with surface roughness as compared to a i smooth surface, provided the liquid is still able to properly wet the solid. The Landau-Ginzburg theory can be used to construct a direct relationship between the interface thickness, the surface tension coefficient and the equations of state (pres- sure and internal energy) of the fluid. This direct relation makes it possible to artifi- cially enlarge the thickness of the interface, which relaxes the constraint that the real interfacial thickness should be one order of magnitude less than the domain length and enables simulations with realistic fluid properties. However, the thermodynamic behavior of the fluid also has to be modified, which affects macroscopic behavior as well. The effect of this modification is tested by performing a simulation of a two- dimensional head-on droplet collision of two methane droplets. The simulation result indicates that even though the macroscopic behavior of the fluid is affected, the effect on the overall simulation outcome is small. However, the discretization method that is currently employed proved to be unable to handle low viscosity fluids on the grid that was used, and the viscosity of the fluid had to be increased in order to ensure stability of the simulation. ii Contents The Diffuse Interface Model for Phase-Transitional Flowsi 1. Introduction1 2. Theory5 2.1. The Lennard-Jones Potential........................5 2.2. The Equation of State...........................6 2.3. The Vapor-Liquid Equilibrium....................... 10 2.4. The Van der Waals Theory of Capillarity................. 14 2.5. The Capillary Stress Tensor........................ 17 2.6. The Navier-Stokes-Korteweg Equations.................. 18 2.7. The Laplace Pressure............................ 21 3. Simulations of Binary Droplet Collisions 23 3.1. Introduction................................. 23 3.2. Numerical Method............................. 26 3.3. Initialization and Boundary Conditions.................. 29 3.4. Simulation Results............................. 31 3.5. Conclusions................................. 36 4. Comparison of the Diffuse Interface Model with the Local Front Recon- struction Method 39 4.1. Introduction................................. 39 4.2. Numerical Methods............................. 43 4.3. Simulation Setup.............................. 46 4.4. Simulation Results............................