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A Numerical Method for Two-Phase Flow Based on a Phase-Field Model∗ Naoki TAKADA∗∗ and Akio TOMIYAMA∗∗∗ For interface-tracking simulation of incompressible two-phase fluids with high density ratios, a new numerical method was proposed by combining Navier-Stokes equations with a phase-field model based on a van der Waals-Cahn-Hilliard free-energy theory. The method was applied to several benchmark problems. Major findings are as follows: (1) The volume flux derived from a local chemical potential gradient in the Cahn-Hilliard equation leads to accurate volume conservation, autonomic reconstruction of gas-liquid interface, and reduction of numerical diffusion and oscillation. (2) The proposed method gave good predictions of pressure increase inside a bubble caused by the surface tension force. (3) A single liquid drop falling in stagnant gas and merging into a stagnant liquid film was successfully simulated.
Key Words: Computational Fluid Dynamics, Multi-Phase Flow, Numerical Analysis, Interface Tracking, Cahn-Hilliard Free Energy Theory, Phase-Field Model, Lattice Boltzmann Method
1.
Introduction
The objectives of this study are (1) to understand the capability of a phase-field model (PFM)(1) – (3) in the advection and reconstruction of a two-phase fluid interface and (2) to propose a numerical method for two-phase flow with a high density ratio. The PFM has been used in mesoscale simulations of the solidification of a binary alloy(4) , polymer membrane formation in a highly functional material design platform project(5) , etc. It is one of several useful tools for studying the organization process of microscopic structures through phase transformation(3) . In the PFM, an interface is described as a finite volumetric zone across which the physical properties (mass density, concentration, viscosity, etc.) vary steeply and continuously. The shape of the interface is determined so as to minimize the free energy of the system(6), (7) , and therefore, no phase boundary condition for interface is required for the interface in PFM-based numerical simulations. In contrast with conventional interface-tracking ∗
∗∗
∗∗∗
Received 10th February, 2006 (No. 04-0186). Japanese Original: Trans. Jpn. Soc. Mech. Eng., Vol.71, No.701, B (2005), pp.117–124 (Received 2nd March, 2004) National Institute of Advanced Industrial Science and Technology (AIST), 1–2–1 Namiki, Tsukuba-shi, Ibaraki 305–8564, Japan. E-mail:
[email protected] Graduate School of Science and Technology, Kobe University, 1–1 Rokkodai-cho, Nada-ku, Kobe-shi, Hyogo 657–8501, Japan
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methods(8), (9) , such as volume-of-fluid and front-tracking methods(10) , surface tension in the PFM is given as surface free energy per unit area caused by a local density gradient. The PFM reconstructs the interface autonomously by taking a chemical potential gradient into account. Therefore, the effect of surface tension force on flow fields can be treated without complicated topological calculation of the interfacial profile(10) . In addition, a PFM-based method does not necessarily require conventional algorithms such as Donor-Accepter(10) , FLAIR(11) , MARS(12) , and CIP(13) for interface advection and reconstruction. As a result, the PFM-based method can easily reproduce interfacial displacement caused by phase changes or dissolution(2) . It also has the potential to simulate complex two-phase flows more efficiently than other methods, because the computational cost of PFM simulation does not depend on the interfacial area concentration or degree of interfacial deformation, but on the spatial and temporal resolutions. Thus, the PFM can be regarded as a suitable method for tracking multiple bubbles or drops with repeated coalescence and breakups. It is also expected that the PFM excels in computing two-phase flow through porous media with microscopic inner structures, since the wettability of a solid surface can be represented by a contact angle in simple boundary conditions(1) . Traditional PFM-based methods for two-phase flows have solved the Navier-Stokes equations only under conditions of low density ratios, such as an immiscible twoJSME International Journal
637 phase flow at about 1:0.9(1) and a liquid-vapor flow around the critical point(2) . Swift et al. proposed a gas-liquid twophase fluid model(14) by introducing the concept of the PFM into a lattice Boltzmann method (LBM)(15), (16) using mesoscopic kinetic equations for the velocity distribution of the number density of fictitious fluid particles. An advanced LBM, developed from the model of Inamuro et al.(17) , can stably simulate two-phase flow at a highdensity ratio by introducing an index function to represent the interfacial profile(18) . In this study, we propose a numerical method for two-phase flows combining NavierStokes equations with the advantages of the LBM and traditional PFM methods. This method is applicable to twophase flow problems at high-density ratios. In addition, the selections of time and space discretization schemes and solution algorithms are more flexible in comparison with those in the LBM. 2. Two-Phase Interface Model Based on Statistical Thermodynamics 2. 1 Free-energy theory In this section, we explain the interface modeling of two-phase fluids at a high-density ratio based on a freeenergy theory. We will consider a two-phase system, numbered as 1, which consists of an incompressible isothermal fluid with an inhomogeneous density field, ρ. A liquid phase L and a gas phase G are pure fluid regions with constant densities ρG and ρL respectively. A gas-liquid interface is described as a finite zone across which ρ varies steeply. An equilibrium state of the system, in which two phases coexist stably, corresponds to a state with a minimum value of the Helmholtz free-energy functional: κ1 ψ1 (ρ) + |∇ρ|2 dV, (1) Ψ1 = 2 V where the first term in the integral is the bulk free energy and the second is the surface free energy with constant κ1 . Equation (1) gives the following pressure tensor Pαβ (3) including surface tension effects ∂ρ ∂ρ Pαβ = Pδαβ + κ1 , (2) ∂xα ∂xβ κ1 δΨ1 − Ψ1 = p − κ1 ρ∇2 ρ − |∇ρ|2 , (3) P=ρ δρ 2 ∂ψ1 (4) p=ρ − ψ1 , ∂ρ where the subscripts α and β denote the indices in a Cartesian coordinate system. Instead of the equation of state, Eq. (4), the two-phase LBM proposed by Inamuro et al.(17), (18) introduces another free-energy functional Ψ2 of a virtual system 2 in addition to Ψ1 of the actual fluid system 1, in order to control surface tension and interfacial thickness and to handle twophase flow problems at a high-density ratio ρL /ρG , κ2 Ψ2 = ψ2 (φ) + |∇φ|2 dV, (5) 2 V JSME International Journal
where φ is an index function describing the interfacial profile and the constant κ2 is related to the interfacial thickness. In this study, we use the following van der Waals model for ψ2 : φ ψ2 (φ) = φ T ln − aφ , (6) 1 − bφ which simplifies the behavior of a single-component twophase fluid system around the critical point. The constants a and b represent the strengths of long-range attraction and short-range repulsion between fictitious particles in system 2. In the van der Waals model, an inhomogeneous distribution of φ is formed through phase separation when the temperature parameter T is set to be lower than the critical value TC = 8a/(27b). In this work, the resulting regions in the fluid system having maximum and minimum values φmax and φmin are regarded as the liquid and gas phases, respectively. The two-phase coexisting equilibrium state is achieved by the current of φ induced by a local gradient of the chemical potential, η: η=
δΨ2 ∂ψ2 = − κ2 ∇2 φ. δφ ∂φ
(7)
The density ρ is defined as a function of φ and is distributed continuously in the whole region, including the interfacial zone: φ − (φL + φG )/2 ρL + ρG ρL − ρG + sin ρ= π 2 2 φL − φG for φG < φ < φL , (8) ρ = ρL
for
φ ≥ φL ,
(9)
ρ = ρG
for
φ ≤ φG ,
(10)
where φL and φG are the threshold values to distinguish the liquid and gas phases. PFM-based numerical methods, including the latest series of two-phase lattice Boltzmann methods, reproduce two-phase interface motions based on the governing equations that utilize Eqs. (2) – (7) as closure relations. Therefore, these methods can be regarded as multi-scale and multi-physics computational fluid dynamics methods combining fluid dynamics with non-equilibrium statistical thermodynamics. 2. 2 Pre-evaluation of surface tension The surface tension σ in the PFM is equivalent to the free-energy increase induced by the density gradient. The following definition of σ for a flat interface is derived from Eq. (2)(3) , +∞ 2 ∂ρ dx, (11) σ ≡ κ1 ∂x −∞ where x is a coordinate perpendicular to the interface. In this study, the value of σ is determined as follows(20) . ( 1 ) φmax and φmin are calculated using a given set of parameters, a, b, and T , in Eq. (6) according to the Maxwell construction rule in system 2. Series B, Vol. 49, No. 3, 2006
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Fig. 1 Profile of φ for a flat interface
Fig. 2 Profile of ρ for a flat interface
( 2 ) φ(x) is computed under the equilibrium condition at a given value of κ2 : η(φ) = constant (= η(φmax ) = η(φmin ))
(12)
( 3 ) The density distribution ρ(x) is computed by substituting φ(x) into Eq. (8). ( 4 ) The density gradient is calculated using a fourthorder central difference scheme. ( 5 ) Equation (11) is integrated numerically using Simpson’s rule. ( 6 ) The value of κ1 is determined as σ/(integral value of Eq. (11)). Figures 1 and 2 show theoretical solutions of the index function φ(x) and density profile ρ(x) at ρL /ρG = 801.7 across a flat interface. The fluid interface corresponds to the continuous transition zone in the density field, the thickness of which increases with the parameter κ2 . In this study, a = b = 1 and T = 0.293 are used in Eq. (6)(18) , and gas-liquid interfaces in a two-dimensional flow field are represented by contour lines with the medium value between φmin and φmax . 3. A PFM-Based Numerical Method for Two-Phase Flows with High-Density Ratios PFM-based numerical methods have mainly been applied to two-phase flows with low-density ratios(1), (2) . In this section, we propose a PFM method applicable to twophase flow with a high-density ratio, such as an air-water Series B, Vol. 49, No. 3, 2006
system, based on the traditional PFM methods and the two-phase LBM of Inamuro(18) . 3. 1 Basic equations The proposed method for incompressible isothermal two-phase flows solves a set of continuity equation, Navier-Stokes equation including Eq. (2), and the CahnHilliard equation with the advection of φ by velocity u, ∂ρ ∂ρ + uβ = 0, (13) ∂t ∂xβ ∂Pαβ ∂uβ ∂uα ∂uα ∂uα ∂ ρ + uβ =− µ + + ∂t ∂xβ ∂xβ ∂xβ ∂xα ∂xβ (14) +(ρ − ρk )gα , (k = L or G) ∂φ ∂φuβ + = S. (15) ∂t ∂xβ The pressure tensor Pαβ in Eq. (14) is evaluated by substituting the effective pressure P (18), (21) , which is the combination of P in Eq. (3) and the interfacial free energy κ1 |∇ρ|2 , into P on the right-hand side of Eq. (2). The last term on the right-hand side of Eq. (14) expresses the buoyant force due to gravity gα , and the viscosity µ varies continuously with the density ρ between µL and µG (18) : µL − µG µ = µG + (ρ − ρG ). (16) ρL − ρG The right-hand side term S of the Cahn-Hilliard equation (15) denotes the variation of φ induced by the gradient of η. Equation (15) at a local equilibrium state, S = 0, has the same form as the advection equations for a volume of fluid function and a level-set function in other conventional interface-tracking methods. 3. 2 Numerical techniques and procedure Conventional discretization schemes and solution algorithms are applicable to Eqs. (14) and (15). In this study, scalar and vector variables are located in a staggered arrangement. The advection term in Eq. (14) is calculated with a third-order upwind difference scheme(22) , while fourth-order and second-order central difference schemes are applied to the gradient of the effective pressure P and the viscous term in Eq. (15), respectively. The velocity u and P are calculated using the projection algorithm and the successive over-relaxation scheme. In time marching, u and φ are updated by applying the second-order RungeKutta scheme to Eqs. (14) and (15), respectively, as follows: ( 1 ) The velocity un+1/2 at the intermediate time t + ∆t/2 is calculated explicitly using un , ρn , and P n , etc., in Eq. (14) at the time t = n∆t (where n is the number of the time step). ( 2 ) The index function φn+1/2 at t+∆t/2 is calculated explicitly using φn and un at t = n∆t in Eq. (15) using the improved Euler scheme. ( 3 ) The velocity un+1 at the next time t + ∆t is calculated using un+1/2 and φn+1/2 in Eq. (14) using the revised Euler scheme. JSME International Journal
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(a) Fig. 4 Fig. 3 Negative diffusivity of index function φ
( 4 ) The index function φn+1 at t +∆t is calculated using φn+1/2 and un+1/2 in Eq. (15) according to the improved Euler scheme. 3. 3 Cahn-Hilliard equation In this study, the right-hand-side term S of the CahnHilliard equation (15) is defined according to the twophase LBM proposed by Inamuro et al.(18) as follows: ∂φ ∂φ ∂ ∂ S= Γ Uδαβ + κ2 , (17) ∂xα ∂xβ ∂xα ∂xβ κ2 U = φη − ψ2 − |∇φ|2 , (18) 2 where the positive constant Γ is the parameter for the mobility of φ. As a result, Eq. (15) takes the following form. ∂φ + ∇ · (φu + ΓJ) = 0 (19) ∂t J = −φ∇η = −ζ∇φ + κ2 φ∇(∇2 φ) (20) ζ = T (1 − bφ)−2 − 2aφ
(21)
As shown in Fig. 1, the interfacial thickness decreases with the value of parameter κ2 (20) . This is because the diffusion factor ζ on φ in Eq. (21) takes negative values in the central region of the interface, 0.293 5 < φ < 0.374 8 (see Fig. 3). On the other hand, the term with κ2 increases the thickness by its positive diffusivity in that zone. Consequently, in the PFM, the interface retains its thickness owing to the above-mentioned mechanism of the index function φ. 3. 4 Advection and reconstruction of the interface Numerical simulations of unsteady advection of φ in two dimensions were carried out to verify the interfacetracking capability of the formulated Cahn-Hilliard equation (19). A single circular-shaped interface of diameter d = 32 was placed in a space (x,y) that was discretized uniformly with square cells of unit width ∆x = ∆y = 1. The first benchmark was a linear translation of the interface in a uniform flow at constant velocity, u = v, in the x and y directions. The computational domain was surrounded with periodic boundaries in both the x and the y directions. In the first case, the Courant number C = u∆t/∆x was 4×10−3 (u = 0.1 and ∆t = 0.04), and κ2 = 0.04 and Γ = 0 and 1. The JSME International Journal
(b)
(c)
Two-dimensional linear translation of circular-shaped interface. (a) Initial condition and the profile after translation for (b) Γ = 1 and (c) Γ = 0
conservation of φ in the whole domain was ensured by setting a finite difference of the advection term according to a finite volume scheme given by ∂F x,i −F x,i+3/2 + 27(F x,i+1/2 − F x,i−1/2 ) + F x,i−3/2 , ∂x 24∆x (22) where the subscript i denotes the cell number in the x direction, and F x,i+1/2 on a cell surface of the ith cell represents the summation of x-direction volume fluxes of φ due to the flow velocity u and the current J given by Eq. (20). The scalar variable φ on the cell surface i+1/2 was calculated by interpolating the values at four neighboring cells as follows: 1 (23) φi+1/2 = [9(φi+1 + φi ) − φ x,i+2 − φ x,i−1 ]. 16 As for the calculation of the third-order differential term on the right-hand side of Eq. (20), the following fourthorder central difference scheme was applied to the firstorder differential term on ∇2 φ: ∂∇2 φ −∇2 φi+2 + 27(∇2 φi+1 − ∇2 φi ) + ∇2 φi−1 , ∂x i+1/2 24∆x (24) where ∇ φ was computed using a fourth-order central difference scheme. Figure 4 (a) shows the initial condition, and Fig. 4 (b) and (c) shows numerical predictions of φ(x,y) at the dimensionless time t∗ = 6.25 for Γ = 1 and Γ = 0. Initially, φ = φmin was set inside the circular zone in Fig. 4 (a). The current J in Eq. (20) was evaluated using κ2 = 0.04. Comparing (b) with (c), it is clearly confirmed that J plays an important role in reducing numerical oscillation and diffusion of φ and in retaining interfacial thickness. Figure 5 depicts the development of the variation ratio (N − N0 )/N0 (where N is the number of the cells and N0 is its initial value) inside the circular-shaped fluid. Figure 5 also shows that the initial circular area was also conserved within a 5% error. As shown in Fig. 6, even at higher Courant numbers C = 1.25 × 10−2 and 2.5 × 10−2 , the initial circular shape and thickness were well kept until t∗ = 156.25 for κ2 = 0.1 and Γ = 12. In addition, the 2
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Fig. 5
(a) Schematic of domain
(b) C = 2.5 × 10−2
(c) C = 5 × 10−2
(d) C = 10−1
Volume conservation in two-dimensional linear translation simulation (C = 4 × 10−3 )
Fig. 8 Interface profile drawn as a contour line of the index function φ in two-dimensional rotation simulation
(a) C = 1.25 × 10−2 Fig. 6
(b) C = 2.5 × 10−2
Interface profile drawn as a contour line of the index function φ in two-dimensional linear translation simulation
Fig. 9 Volume conservation in two-dimensional rotation in a circular vortex
Fig. 7
Volume conservation in two-dimensional linear translation simulation
circular-shaped area was conserved within ±1% error at C = 5 × 10−2 and 10−1 (see Fig. 7). In the latter cases of higher C, the initial values of φ inside and outside the circular-shaped area were set to be 0.263 and 0.403 respectively: they were smaller than those at C = 4.0 × 10−3 . This operation made it possible to balance the chemical potential η more exactly over the whole flow field, thus preventing the generation of local diffusive current of φ dependent on interfacial curvature (Gibbs-Thomson effect(1) ). Series B, Vol. 49, No. 3, 2006
The second benchmark of interfacial advection was a rotation of a circle in a non-uniform flow velocity field with a constant angular velocity under the condition ∆t = 0.025, κ2 = 0.1, and Γ = 6. As shown in Figs. 8 and 9, the interface retained its initial shape and thickness adequately. The circular-shaped area was conserved within ±1% error at various Courant numbers; C = 2.5 × 10−2 , 5 × 10−2 and 10−1 . As a result, the Cahn-Hilliard equation (19) allowed accurate advection and reconstruction of the interface while conserving each phase volume in twophase flows. In addition, it possessed numerical stability. In addition to C, the mobility factor Γ and the interfacial thickness parameter κ2 affect numerical stability. As expected from Fig. 4, numerical stability and interfacialshape conservation can be improved by the mobility Γ. In JSME International Journal
641 our previous works(20), (23), (24) , we determined that (1) Γ can take a larger value with a decreasing time step, ∆t, (2) numerical instability takes place when Γ is too large, and (3) the maximum value of Γ for suitable solutions depends not only on the parameters a, b, T , and κ2 , which determine interfacial thickness profile, but also on the state of the flow field (velocity and phase distribution). Then, we determined the largest Γ possible for two-phase flow simulations at high-density ratios by iteratively conducting the above-mentioned benchmark tests on interfacial advection. Numerical stability is also improved by increasing the interfacial thickness proportional to κ20.5 (1) . When increasing the thickness, more spatial meshes are needed on a uniform regular grid for resolving the whole flow system with a given dimension. When κ2 decreases at a given cell width, the surface tension cannot be evaluated accurately because the numerical discretization error in the gradient of ρ increases as the interfacial thickness decreases. In this PFM method, we therefore selected the value of κ2 that gave the interface the minimum thickness necessary to ensure steep and continuous variations of φ and ρ inside the interface on a given grid size.
(a) Pressure
(b) Velocity
Fig. 10 Neutrally-buoyant bubble in stagnant liquid
4. Numerical Simulation of Gas-Liquid Two-Phase Flows The applicability of the proposed PFM-based method to two-phase flows at high-density ratios was examined through numerical simulations of single bubbles and single drops in two dimensions. The simulated system comprised two fluids with density ratio ρL /ρG = 801.7 and viscosity ratio µL /µG = 73.76. It is equivalent to an air-water system at room temperature and atmospheric pressure. The computational domain was discretized uniformly using square cells, ∆x, ∆y = 1. The values of κ2 , φG and φL were 0.1, 0.275, and 0.380, respectively. Thus, the thickness of the computed gas-liquid interface ranged from four to five cells (Fig. 2). 4. 1 Pressure and velocity in a gas-liquid two-phase fluid A numerical simulation of a neutrally buoyant bubble of diameter d = 20∆x in a stagnant liquid was carried out to investigate the characteristics of pressure and velocity fields predicted by the proposed method. As shown in Fig. 10 (a), the effective pressure P inside the bubble took a high and uniform value due to the surface tension effect. On the other hand, the predicted velocity field (Fig. 10 (b)) was induced mainly by the discretization error of density gradient in the pressure tensor term. It had also been observed in the results obtained with LBM(15) . However, the magnitude of such an unphysical velocity was too small to affect the whole flow field. It is supposed that the parasitic current can be reduced by improving the spatial resolution inside the interfacial zone and a finite difference scheme for the density gradient. JSME International Journal
Fig. 11 Pressure increase inside bubble neutrally-buoyant in stagnant liquid at σ = 4.31 × 10−4 , ρL /ρG = 801.7, and µL /µG = 73.76
4. 2 Surface tension To verify the accuracy of the pressure increase ∆P caused by surface tension σ, ∆P for a two-dimensional neutrally buoyant bubble was calculated under the conditions of ρL = 1.0×10−3 , κ1 = 1.71×103 and σ = 4.31×10−4 . The open circles in Fig. 11 are the predicted pressure increases. They agreed well with Laplace’s law, ∆P = σ/R (the solid line in the figure). It was therefore confirmed that the surface tension effect is well realized in the proposed method. 4. 3 Free-fall single drop A numerical simulation of a two-dimensional drop of diameter dD falling through a stagnant gas was carried out. As shown in Fig. 12, the computational domain of width 6dD and height 4dD , including a circular-shaped drop of dD = 20∆x, was surrounded by stationary non-slip walls. Initially, the bottom wall was covered with a flat liquid film of thickness 4∆y, and the drop was placed at height y = 3.25dD . The effect of gravity gy = −2.0 × 10−2 was taken into account only in the liquid phase region. The Morton and E¨otv¨os numbers of this system, M = 1.52 × 10−12 and Eo = 13.2, were equivalent to those of a 10 mm drop in an air-water system under normal gravity. The parameters were set as follows: ρL = 1.0, µL = 1.67 × 10−3 , σ = 6.06 × 10−2 , ∆t = 1.25 × 10−2 , Γ = 12.0, and κ1 = 0.240. Series B, Vol. 49, No. 3, 2006
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Fig. 12 Schematic of two-dimensional computational domain
(a) t = 225.0
(d) t = 250.0
(b) t = 231.25
(e)
(c) t = 237.5
t = 275.0
(f) t = 325.0
Fig. 13 Time series of snapshot of velocity field and interface of two-dimensional drop and liquid film under gravity for M = 1.52 × 10−12 and Eo = 13.2
The predicted velocity field and interfacial profile are shown in Fig. 13. The drop contacted the surface of the liquid film at t = 231.25 (Fig. 13 (b)). Then, it coalesced into the film at t = 237.5 (Fig. 13 (c)). Gas between the drop and the film was rapidly ejected just before contact, as shown in Fig. 13 (a) and (b). Then, two single waves propagated outward in opposite horizontal directions, while a pair of vortices generated by the gas jet was rising up in the gas phase (Fig. 13 (c) – (f)). Figure 14 shows predicted pressure distributions. The maximum pressure spot appeared between the drop and the liquid film just before contact at t = 225.0, and moved onto the surface of the bottom wall after coalescence at t = 231.25. Series B, Vol. 49, No. 3, 2006
Fig. 14 Snapshots of pressure field of two-dimensional drop and liquid film in gas under gravity
Fig. 15 Time series of liquid phase volume in two-dimensional drop simulation
The pressure decreased rapidly in the whole flow field for t = 237.5 – 250.0. At t = 231.25, the difference between the maximum and minimum pressures was 0.206, which corresponded to 5.05 × 102 Pa, i.e., 0.8% of ambient pressure in the air-water system. Then, the conservation of fluid volume was evaluated using the ratio of the number of cells N, for which the value of φ is less than (φmin +φmax )/2, to its initial value N0 . Figure 15 shows the time evolution of N/N0 . The number N was approximately constant while the drop was falling through the gas. It varied with errors between −2% and +1% in a short period (225.0 < t < 231.25) just before coalescence. After coalescence, N decreased at a constant rate until t = 325, which resulted in about −6% maximum variation. It should be noted that the function φ was conserved in the whole flow field by the finite volume discretization of the Cahn-Hilliard equation (19). The variation of φ was due to the deformation of the diffusive interface with given threshold values on φ. The local volume flux of φ due to the Gibbs-Thomson effect(1) is also one of the factors that cause variations in N. However, the variation of N was small in comparison with those of conventional interfacetracking methods. The above results demonstrate that the JSME International Journal
643 proposed numerical method can be used to compute twophase flows. Undesirable variations in fluid volume in the PFM-based method can be reduced by setting lower values for the interfacial thickness and mobility parameters κ2 and Γ. The simulation was conducted on a personal computer (Intel Pentium M processor with 1 GHz frequency and Microsoft Windows XP operating system), and took about 40 minutes to compute 26 000 time-step iterations until t = 325. 5.
(3)
(4)
(5)
Conclusions
In this study, we investigated the interface-tracking characteristics of the Cahn-Hilliard equation in a phasefield model (PFM) based on the free-energy theory. The proposed method for two-phase flow with high density ratios combined Navier-Stokes equations with the advantages of traditional PFM methods and a lattice Boltzmann method (LBM). Through verification of the numerical simulations of two-dimensional incompressible, isothermal two-phase flow, the following results were obtained. ( 1 ) The Cahn-Hilliard equation allowed accurate advection and reconstruction of interface through conservation of volumes of each phase due to instantaneous local balance of chemical potential in a non-equilibrium state. ( 2 ) The equation of two-phase fluid motion, including the pressure tensor derived from a free-energy functional, yielded the surface tension defined as an interface free energy per unit area. ( 3 ) The false velocity component caused by the discretization error of density gradient was negligibly small and did not affect the flow field calculation in the same way as LBM. ( 4 ) A single liquid drop falling through a stagnant gas and merging into a stagnant liquid film was successfully reproduced. ( 5 ) The proposed method is useful for two-phase flows at density ratios about 800. It is also suggested that the proposed method possesses a potential in simulating two-phase flows at high density ratios. The PFM is particularly useful for interface tracking and can be combined with other conventional computational fluid dynamics techniques. It also can be extended to other coordinate systems, and accommodate different levels of numerical accuracy and stability. Moreover, other free-energy functions of double-well form(14), (16) can be adopted in the Cahn-Hilliard equation, instead of the van der Waals model used in this study.
(6)
References
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