Sharp and diffuse interface methods for phase transition problems in liquid-vapour flows (original) (raw)
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ESAIM: Mathematical Modelling and Numerical Analysis, 2012
In the present work we investigate the numerical simulation of liquid-vapor phase change in compressible flows. Each phase is modeled as a compressible fluid equipped with its own equation of state (EOS). We suppose that inter-phase equilibrium processes in the medium operate at a short timescale compared to the other physical phenomena such as convection or thermal diffusion. This assumption provides an implicit definition of an equilibrium EOS for the two-phase medium. Within this framework, mass transfer is the result of local and instantaneous equilibria between both phases. The overall model is strictly hyperbolic. We examine properties of the equilibrium EOS and we propose a discretization strategy based on a finite-volume relaxation method. This method allows to cope with the implicit definition of the equilibrium EOS, even when the model involves complex EOS's for the pure phases. We present two-dimensional numerical simulations that shows that the model is able to reproduce mechanism such as phase disappearance and nucleation.
Journal of Fluid Mechanics, 2021
Classical continuum-based liquid vapour phase-change models typically assume continuity of temperature at phase interfaces along with a relation which describes the rate of evaporation at the interface (Hertz-Knudsen-Schrage, for example). However, for phase transitions processes at small scales, such as the evaporation of nanodroplets, the assumption that the temperature is continuous across the liquid-vapour interface leads to significant inaccuracies (McGaughey & Ward 2002; Rana et al. 2019), as may the adoption of classical constitutive relations that lead to the Navier-Stokes-Fourier equations (NSF). In this article, to capture the notable effects of rarefaction at small scales, we adopt an extended continuum-based approach utilizing the coupled constitutive relations (CCR). In CCR theory, additional terms are invoked in the constitutive relations of NSF equations originating from the arguments of irreversible thermodynamics as well as consistent with kinetic theory of gases. The modelling approach allows us to derive new fundamental solutions for the linearised CCR model and to develop a numerical framework based upon the method of fundamental solutions (MFS) and enables threedimensional multiphase micro-flow simulations to be performed at remarkably low computational cost. The new framework is benchmarked against classical results and then explored as an efficient tool for solving three-dimensional phase-change events involving droplets.
On the Dynamics of Liquid-Vapor Phase Transition
SIAM Journal on Mathematical Analysis, 2008
We consider a multidimensional model for the dynamics of liquid-vapor phase transitions. In the present context liquid and vapor are treated as different species with different volume fractions and different molecular weights. The model presented here is a prototype of a "binary fluid mixture," and is formulated by the Navier Stokes equations in Euler coordinates. This system takes now a new form due to the choice of rather complex constitutive relations that can accommodate appropriately the physical context. The setting of the problem presented in this work is motivated by physical considerations (cf. Fan [18], Fan and Slemrod [19], Slemrod [34]). The transport fluxes satisfy rather general constitutive laws, the viscosity and heat conductivity depend on the temperature, the pressure law is a nonlinear function of the temperature depending on the mass density fraction of the vapor (liquid) in the fluid as well as the molecular weights of the individual species. The existence of globally defined weak solutions of the Navier-Stokes equations for compressible fluids is established by using weak convergence methods, compactness and interpolation arguments in the spirit of [22] [32], (see also [12], [13]).
Nuclear Engineering and Design, 2001
In several numerical methods dedicated to the direct numerical simulation of two-phase flows, the concept of a continuous enlarged interfacial zone is used. In this communication, it is shown that for liquid-vapor systems, it is possible to use this concept in a thermodynamic coherent way. Indeed, if it is considered that the energy of the system depends on the density gradient, this theory being called the Van der Waals or Cahn-Hilliard or more generally the second gradient theory, then it is possible to derive the equations that characterize the fluid motion within a 3-D liquid-vapor interfacial zone. Modifying the thermodynamic behavior of the fluid, it is shown that it is possible to increase the thickness of an interface, so that it can be captured by a 'standard' mesh without changing the surface tension nor loosing the thermodynamic coherence of the model. Several examples of application show that this method can be applied to study various physical problems, including contact line phenomena.
Numerical simulation with finite volume of dynamic liquid-vapor phase transition
The present work investigates the simulation of phase transition in compressible fluids. We postulate a local and instantaneous equilibrium with respect to phasic pressures, temperatures and chemical potentials when both phases are present. This hypothesis leads to the definition of an equilibrium equation of state (EOS) for the two-phase medium. While this thermodynamical assumption is classical, there is no explicit expression of the equilibrium EOS in the general case. We propose a simple mean to approximate this EOS when both phases are governed by a "Stiffened Gas" EOS. Then we provide an implementation of this method within a Finite-Volume numerical scheme thanks to a two-step relaxation strategy.
Journal of Computational Physics, 2001
This paper presents a new method to simulate liquid-vapor flows with phase change using a phase-field-like approach. In this method, the liquid-vapor interface is described as a three-dimensional continuous medium across which physical properties have strong but continuous variations. This continuous variation is made possible by imposing that the internal energy of the fluid depends on its density gradient. This description, called the second gradient theory, is numerically attractive since a single system of partial differential equations (PDEs) is necessary to determine the flow in the entire two-phase system, the phase change, the displacement of the interfaces, and their change in topology being a part of their solution. However, to solve these PDEs using a reasonable number of grid points on a fixed grid, the interfaces need to be artificially enlarged. It is shown that this artificial enlargement can be thermodynamically consistent if the thermodynamic behavior of the fluid is modified within the binodal curve. The consequences of this thermodynamic modification are studied in detail. In particular it is shown that, within the frame of the second gradient theory, the interface thickness and the surface tension vary with the mass and heat fluxes across the interface and that these variations increase with the thickness of the interface. As a consequence, for a given accuracy, an upper bound exists for the interfacial heat and mass fluxes that can be simulated. Examples of applications in one and two dimensions show the potentialities of the method presented, in particular to deal with moving contact lines, the description of which is a part of the second gradient theory.
Direct numerical simulations of flows with phase change
Computers & Structures, 2005
We present results from direct numerical simulations of two types of phase-change processes: directional solidification of a binary alloy, and film boiling. We use a 2D finite-difference/front-tracking method which allows the evolution of the interface between the phases to be followed. The discontinuities in material properties between the phases, as well as topological changes, are easily handled. In directional solidification, the fully coupled solute and energy equations for a dilute binary alloy without fluid flow are solved. We demonstrate the evolution of a cellular interface with rejection of solute ahead of the advancing interface and in the intercellular grooves. The numerical results for the transition from a planar to a cellular interface are in excellent agreement with linear stability theory. The film boiling problem couples the phase change with fluid flow. We study the growth and dynamics of a vapor layer adjacent to an upward facing, flat, heated surface. Vaporization of the liquid at the liquid-vapor interface continually replenishes the vapor lost due to bubble departure from the interface. (Author)
Towards numerical simulation of phase-transitional flows
2016
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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.
Meshfree One-Fluid Modelling of Liquid-Vapor Phase Transitions
Cornell University - arXiv, 2022
We introduce a meshfree collocation framework to model the phase change from liquid to vapor at or above the boiling point. While typical vaporization or boiling simulations focus on the vaporization from the bulk of the fluid, here we include the possibility of vaporization from the free surface, when a moving fluid comes into contact with a superheated surface. We present a continuum, one-fluid approach in which the liquid and vapor phases are modelled with the same constitutive equations, with different material properties. The novelty here is a monolithic approach without explicit modelling of the interface between the phases, neither in a sharp nor diffuse sense. Furthermore, no interface boundary conditions or source terms are needed between the liquid and vapour phases. Instead, the phase transition is modelled only using material properties varying with temperature. Towards this end, we also present an enrichment of strong form meshfree generelized finite difference methods (GFDM) to be able to accurately capture derivatives in the presence of jumps in density, viscosity and other physical properties. The numerical results show the ability of our proposed framework to model phase changes with large jumps.