A phase-field method for interface-tracking simulation of two-phase flows

For interface-tracking simulation of two-phase flows, we propose a new computational method, NS-PFM, combining Navier-Stokes (NS) equations with phase-field model (PFM). Based on the free energy theory, PFM describes an interface as a volumetric zone across which physical properties vary continuously. Surface tension is defined as an excessive free energy per unit area induced by density gradient. Consequently, PFM simplifies the interface-tracking procedure by use of a standard technique. The proposed NS-PFM was applied to several problems of incompressible, isothermal two-phase flow with the same density ratio as that of an air-water system. In this method, the Cahn-Hilliard (CH) equation was used for predicting interface configuration. It was confirmed through numerical simulations that (1) the flux driven by chemical potential gradient in the CH equation plays an important role in interfacial advection and reconstruction, (2) the NS-PFM gives good predictions for pressure increase inside a bubble caused by the surface tension, (3) coalescence of liquid film and single drop falling through a stagnant gas was well simulated, and (4) collapse of liquid column under gravity was predicted in good agreement with other available data. Then, another version of NS-PFM was proposed and applied to a direct simulation of bubble nucleation of a non-ideal fluid in the vicinity of the critical point, which demonstrated the capability of NS-PFM to capture liquid-vapor interface motion in boiling and condensation. INTRODUCTION For last two decades, much attention has been paid to the phase-field model (PFM) as one of the useful tools to well understand complex phenomena involving self organization of mesoscopic structures in multi-component fluids, such as twophase flows (bubbly flow [1], drop breakup [2], RayleighTaylor instability [3], phase change [4,5], etc.), solidification of binary alloys [6] and polymer membrane formation [7]. Based on the so-called Cahn-Hilliard theory [8], PFM describes an interface as a volumetric transition zone with a finite width between pure components (phases), across which all the physical properties vary steeply but continuously. A freeenergy functional, which has a double-well potential of fluid density and depends on square of local density gradient, allows the coexistence of two phases with diffusive interface without imposing topological constraints on interface as phase boundary. In the free-energy theory, surface tension is defined as an excessive energy per unit area induced by the density gradient inside diffusive interface, enabling calculation of the continuous body force without using interfacial curvature and normal vector. As a result, the PFM-based method for twophase flows does not necessarily require conventional algorithms for the advection and reconstruction of interface and continuum surface force model [9-11], unlike front-tracking, level-set and volume-of-fluid (VOF) methods [12-14]. This feature simplifies interface-tracking calculation on a fixed spatial grid. The PFM method therefore has attractive advantages over the other methods, easy implementations of multi-dimensional interface advection and associated heat and mass transfer across the interface [1-5,15]. PFM methods are categorized into two types according to basic equations; a direct numerical method using Navier-Stokes (NS) equations (referred to as NS-PFM hereafter) [3,4], and a lattice Boltzmann method (LBM) [1,2,5] which uses mesoscopic kinetic equations for the velocity distribution of a number density of fictitious fluid particles [16, 17]. Both types 1 Copyright © 2005 by ASME have been applied only to two-phase flows with a small density difference because of numerical instability. To overcome the difficulty, Inamuro et al. [1] recently proposed a two-phase LBM which is applicable to incompressible two-phase flows with the density ratio up to 1,000. The purposes of this study are to examine PFM for interface-capturing and tracking simulation, and to develop a PFM-based computational method for two-phase flows. First, we propose a NS-PFM by extending the above-mentioned LBM [1] for simulating immiscible, incompressible, isothermal two-phase flows with a high density ratio. Comparing with the LBM, the proposed method would not only save computational cost, but also have flexibility in selection of space and time discretizations. In order to verify the NS-PFM, several interface-tracking simulations of two-phase flows are carried out for fluids with the same density ratio as that of an air-water system. Second, we took heat transfer into account in the NSPFM for direct numerical simulation of phase change of compressible non-ideal fluid in the vicinity of the critical point. NOMENCLATURE A long-range interaction of van-der-Waals fluid particles a width of liquid column B short-range interaction of van-der-Waals fluid particles g gravitational acceleration vector g magnitude of gravity (=|g|) H height of liquid column or position of drop in z direction I second-rank isotropic tensor k thermal conductivity n aspect ratio of liquid column P pressure tensor including surface-tension force P’ pressure including excess free energy of interface T parameter of free energy function (and/or temperature) t time u flow velocity vector x,y,z position in Cartesian coordinate system Greek Letters ∆t time increment Γ mobility of index function φ in Cahn-Hilliard equation φ index function to indicate interface profile η chemical potential κS surface tension parameter κφ interface thickness parameter μ viscosity of fluid ρ mass density of fluid σ surface tension Subscript G gas phase L liquid phase Superscript * dimensionless time BASIS OF PHASE-FIELD METHOD (NS-PFM) In this section, we briefly state basic equations of the proposed numerical phase-field methods (NS-PFM) for interface-tracking simulations of two-phase flow, in which Navier-Stokes (NS) equations are combined with the phasefield model (PFM). Isothermal two-phase fluids with a high density ratio The proposed NS-PFM for immiscible, incompressible, isothermal gas-liquid flows solves a set of a continuity equation, momentum conservation equations, and a CahnHilliard (CH) equation describing time evolution of diffusive interface profile [1, 3, 8], 0 t ρ ρ ∂ + ⋅∇ = ∂ u , (1) ( ) ( { ) } 1 T t μ ρ ∂ ⎡ ⎤ + ⋅∇ = −∇⋅ + ∇⋅ ∇ + ∇ ⎣ ⎦ ∂ u u u P u u , (2) ( ) [ ] t φ φ φ η ∂ + ∇⋅ = ∇⋅ Γ ∇ ∂ u . (3) In the CH equation (3), the scalar variable φ is an index to describe interface profile [1], which is continuously distributed in the whole flow field. In this study, the chemical potential η is derived from a van-der-Waals free energy [1, 5] as follows, 2 ln 2 1 1 T T A B B φ φ η φ κ φ φ ⎛ ⎞ φ = − + − ∇ ⎜ ⎟ − − ⎝ ⎠ , (4) where κφ is a parameter to control interfacial thickness at T < 8A/(27B). For simplicity, the mobility Γ was set to be constant. The density ρ is defined as a continuous function of φ [1], ( ) / 2 sin 2 2 L G L G L G L G φ φ φ ρ ρ ρ ρ ρ π φ φ ⎛ ⎞ + − + − = + ⎜ ⎟ ⎜ ⎟ − ⎝ ⎠ , (5) where φG and φL are arbitrary thresholds for the index φ to distinguish the gas and liquid phases. The tensor P is expressed as, ( ) 2 ' S S P κ ρ κ ρ ρ = − ∇ + ∇ ⊗∇ P I , (6) where κS denotes the strength of surface tension σ, and P’ is the effective pressure including the free-energy increase κS |∇ρ |. The parameter κS is set to be constant in the whole flow field, and determined from the definition of σ on a flat interface [8],