cfd simulation of microscopic two-phase...

VI International Conference on Computational Methods for Coupled Problems in Science and Engineering COUPLED PROBLEMS 2015

B. Schrefler, E. Oñate and M. Papadrakakis (Eds)

CFD SIMULATION OF MICROSCOPIC TWO-PHASE FLUID MOTION ON SOLID BODY WITH EDGES AND HETEROGENEOUSLY-

WETTED SURFACE USING PHASE-FIELD MODEL

NAOKI TAKADA*, JUNICHI MATSUMOTO*,

SOHEI MATSUMOTO* AND KAZUMA KURIHARA*

* National Institute of Advanced Industrial Science and Technology (AIST) 1-2-1 Namiki, Tsukuba, Ibaraki 305-8564, Japan e-mail: [email protected], www.aist.go.jp

Key words: Computational Fluid Dynamics, Multiphase Flow, Wettability, Diffuse-interface Model, Conservative Level-set Method, Lattice Boltzmann Method.

Abstract. Our objective in this study is to examine the applicability of a computational fluid dynamics (CFD) method to simulation of motions of microscopic two-phase fluid on solid surfaces with edges for evaluating fluidic devices and for predicting underground fluid flows through porous media. The method adopts the phase-field model (PFM) for fluid-fluid interfacial dynamics and the lattice-Boltzmann method (LBM) as numerical scheme for solving a set of two-phase fluid-dynamics equations. Based on the free-energy theory, the PFM reproduces an interface as a finite volumetric zone between phases without imposing topological constraints on interface as phase boundary. Wettability of solid surface to fluid is taken into account through minimizing the free energy of the fluid plus the surface energy per unit area. The LBM assumes that a macroscopic fluid consists of fictitious mesoscopic particles repeating collisions with each other and rectilinear translations with an isotropic discrete velocity set. The simply-iterative particle-kinematic operation in a semi-Lagrange form is useful for high-performance computing of complex fluid flow systems. The CFD method therefore has an attractive advantage over the others, efficient simulation of motions of multiple fluid particles on partially-wetted and textured solid surfaces. The major findings in preliminary immiscible liquid-liquid simulations are as follows: (1) the method simulated well departure of single droplet from flat surface due to buoyancy in agreement with the semi-empirical prediction; (2) the droplet had larger static contact angle on hydrophobic grooved surface than that on flat surface in a similar way with available data; (3) the droplet moved more easily under external forcing in the tangential direction to the grooves than that under forcing in the normal direction in qualitative agreement with experimental observations.

1 INTRODUCTION

Microscopic gas-liquid and liquid-liquid flows with a fluid-fluid interface on a heterogeneously or homogeneously wettable solid surface are widely encountered in various science and engineering fields. It is often difficult to experimentally observe such flows and measure the velocity and the pressure simultaneously in three dimensions or to analyse

Naoki Takada, Junichi Matsumoto, Sohei Matsumoto and Kazuma Kurihara.

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theoretically them by the classical continuum dynamics approach based on a sharp-interface model. Computational fluid dynamics (CFD) simulations facilitate the understanding and prediction of the two-phase flows for flexible and accurate control of fluid-particle motion and position; as its result, micro-fluidic devices and micro-electro-mechanical-systems (MEMS) device fabrication processes can be optimally designed [1-3].

In last two decades, the phase-field model (PFM) has been attracting much attention from many researchers as one of the mesoscopic models which efficiently simulate behaviours of multi-phase multi-component systems [4]. Based on the free-energy theory [5, 6], the PFM describes an interface as a finite volumetric zone between different phases, across which physical properties vary steeply but continuously. Two-phase coexistence is allowed by a free-energy functional which has a double-well potential of an order parameter (i.e. mass density, molar concentration) and its squared local gradient term, without imposing topological constraints on interface as phase boundary. The contact angle, depending on the energy balance among three types of interfaces, is obtained from a potential of the solid surface through a simple boundary condition of the gradient of the order parameter on the surface [7, 8]. As a result, the PFM-CFD methods do not necessarily require conventional elaborating algorithms for advection and reconstruction of interfaces [4, 7-18]. The method therefore has an advantage over others, efficient simulation of motions of multiple fluid-fluid interfaces attached on solid bodies with edges and partially-wetted textured surfaces. We have recently evaluated two types of diffuse-interface advection equation based on the PFMs [9, 16] to the simulation of two-phase fluid motion on solid surface [17, 18].

Our objective in this study is to examine the applicability of the PFM-CFD method that we have proposed [17, 18] to simulation of microscopic two-phase fluid motion on solid surfaces for evaluating fluidic devices and for predicting underground fluid flows. This paper is organized as follows. In the next section, the basic equations in the PFM-CFD method are outlined for immiscible isothermal liquid-liquid two-phase flow. The third section describes the numerical solution scheme based on the lattice-Boltzmann method (LBM) [8, 9, 19]. In the fourth section, we present preliminary simulation results by use of the method for three-dimensional single droplet attached on flat solid surface in stagnant liquid under gravity, and 3D droplet on textured solid surface. The conclusions are described in the last section.

2 BASIC EQUATIONS IN PHASE-FIELD MODEL (PFM)

The PFM-based CFD methods for simulating incompressible isothermal viscous two-phase fluid flows generally adopt the following set of mass and momentum conservation equations with an interface advection equation [4, 9, 16-18]:

0 u (1)

1Ip

t

uu u τ F (2)

( ) 0Dt

u (3)

where u denotes the velocity; t, the time; , the density; p, the pressure; , the viscous stress


3

tensor; FI, the interfacial-tension force; , the order parameter which indicates each phase and interfacial regions with its continuous values; and D(), the diffusivity. In the well-known Cahn-Hilliard (CH) form [4-10, 12-15], the diffusion term of Eq. (3) includes the following coefficient:

( )( ) CHD M

(4)

20( )

(5)

where MCH denotes the mobility; , the chemical potential; 0, the double-well potential on ; and , the capillary coefficient which is related with interfacial width and tension.

In the method we have proposed [17, 18], the conservation-modified Allen-Cahn (AC) form [11, 16] is adopted to Eq. (3). The diffusion coefficient is defined as follows:

1( )( ) 1ACD M

(6)

1( ) 1 tanh

2 2

(7)

where MAC denotes the mobility; and , the signed distance in the direction normal to the interface from the central position at = 0. The profile of across an interface in an equilibrium state is theoretically expressed by use of Eq. (7). In the fluid system, two phases correspond to the regions of = 0 and 1 respectively, between which the interface is regarded as the finite volumetric region of 0 < < 1 with width 40.5.

In each of the CH and AC forms, the diffusion term of Eq. (3) disappears in an equilibrium state. The AC equation, Eq. (3) with Eq. (6), is fully equivalent to a conservative level-set equation [20] in one-step form by which both of interfacial re-initialization and advection calculation are done at the same moment [16]. The same idea for removing the interfacial-curvature dependency on diffusion of from the original AC equation [11, 16] is applicable to the diffusion coefficient of the CH equation, Eq. (4) [17].

In the same way as the other PFM-CFD methods [7, 8, 10, 15], a wetting boundary condition on solid surface is taken into account in the present method through the following equation [17, 18],

S S n (8)

where nS is the unit vector normal to the solid boundary and the parameter S is called as wetting potential. The static contact angle W of a fluid phase on the surface depends on the local value of S. The angle can be flexibly set to be heterogeneous on textured solid surfaces by changing the value of S at arbitrary positions on the fluid-solid boundaries.

In addition to Eq. (8), the following constraints are imposed on the stationary solid surface as a non-slip and no-flux boundary [15, 17, 18].

0S p n (9)


4

0S n (10)

u 0 (11)

3 NUMERICAL METHOD FOR TWO-PHASE FLOW

For solving the above-mentioned set of Eqs. (1), (2) and (3) with Eqs. (6) to (11), we have proposed a numerical scheme [17, 18] based on the lattice Boltzmann method (LBM) [8-10, 19]. The LBM assumes that a macroscopic fluid consists of fictitious mesoscopic particles repeating collisions with each other and linear translations with a set of isotropic discrete velocities. The main variables in the LBM are distribution functions of number density of the particles grouped with their velocities. Time evolutions of the distribution functions, fa and ga, at position x and at time t are expressed by

1, ,eqa aI

a a a aa f

f ff f t f t

t

Fe x x

e (12)

1, ,eqa

a a a ag

gg g t g t

t

e x x (13)

where ea is the particle velocity vector; the subscript a, index of the velocity set; f and g , the relaxation times for fa and ga respectively in the BGK approximation collision operator; and the superscript eq denotes a local equilibrium state.

In this study, Equations (12) and (13) are discretized respectively into the semi-Lagrangian forms with second-order accuracies in both space and time as follows [8-10, 19]:

, ' , , ,eqa a a a a a a

f

tf t f t t f t t f t t

x x e x e x e

(14)

2, , ' 3 a I

a a af t t f t w tc

e Fx x (15)

, , , ,eqa a a a a

g

tg t t t g t g t g t

x e x x x (16)

where t is the constant increase in time, x+eat denotes the position neighboring x in the direction of the vector ea, and wa is the weight parameter. In three-dimensional Cartesian coordinate system (x, y, z), the space is divided uniformly into unit cubic cells with sides

x y z c t (17)

where c is the lattice constant. The particles are set to be rest or to move according to three-dimensional 15-velocity model [8-10, 19]. The scalar and vector variables of two-phase fluid, , u, p and , are co-located at each of the spatial cell centers, and defined with fa and ga by

eqa aa a

f f (18)

eqa a a aa a

f f u e e (19)


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2 / 3p c (20)eq

a aa ag g (21)

The kinematic viscosity and the constant part of the diffusion coefficient Eq. (6), D0 = MAC , are defined by the following equations, respectively [8-10, 17-19]:

2

3 2f

c t

(22)

20 2g

tD c

(23)

where is the parameter to control the number densities of the moving and rest particles in a stationary equilibrium state of ga

eq. The boundary conditions for fa and ga of the moving particles are taken into account as

follows. On non-slip and non-flux solid walls, bounce-back condition is applied to the particles moving with ea from fluid side to solid side, which go back to the fluid side in the opposite directions of -ea. On inflow and outflow boundaries, it is assumed that the particles coming into the computational domain from the outside would be in a local equilibrium state for pressure and fluid velocity given at the outside.

One of main features of the LBM scheme for solving Eqs. (1)-(3) is the simple particle-kinematic operation of iterative local collisions and linear translations in the discrete conservation form on an isotropic spatial lattice. It is useful for high-performance computing on multi-phase fluid dynamics.

4 NUMERICAL RESULTS

4.1 Departure of droplet from solid surface due to buoyancy

For evaluating the PFM-CFD method [17, 18], single droplet in a stagnant continuous liquid phase under gravity was simulated. The computational domain was divided into 646464 cubic cells with sides x = y = z in the coordinate system (x, y, z), which was surrounded with stationary non-slip flat solid wall boundaries. In the initial condition, the droplet took a hemispherical shape with a radius of 16x on the bottom wall with homogeneously-wetted smooth surface at the static contact angle W = 60 or 90 to the continuous phase. The buoyant force acting on the droplet was taken into account inside the dispersed-phase region where 0.5 through volumetric forcing at each cell in an upward direction perpendicular to the bottom wall. The two-phase fluid was assumed as an oil-water system under the condition of interfacial tension = 51 mN/m, density difference between the phases of 108.2kg/m3 and equal viscosities of 1.0 mPas at room temperature and at pressure 1atm. The droplet volume VD was set within the range of 65.0l to 1.767ml in the actual system.

As shown in Fig. 1, the droplet departs from the wall in cases of (A) at W = 90 and (C) at W = 60 for lager VD than that in other cases of (B), (D) and (E) at each W. It is also confirmed that the departure takes place more easily at smaller W. The numerical results on


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departure and non-departure agreed with the semi-empirical prediction (see Fig.2).

Figure 1: 3D simulation of oil droplet with volume VD on horizontal flat solid surface with W in stagnant water (continuous phase) under gravity g.

Figure 2: Diagram of departure of droplet from horizontal flat solid surface under gravity g in terms of static contact angle W and volume VD.

(b)W = 60

(a) Static contact angle W = 90(A) VD = 1.767ml (departure) (B) VD = 0.382ml (non-departure)

(D) VD = 0.113ml (non-departure)

(E) VD = 0.065ml (non-departure)

(C) VD = 0.524ml (departure)

xzO

g

0 20 40 60 80 100 120 14010-4

10-3

10-2

10-1

100

101

102

103

104

Continuous-phase static contact angle W [deg.]

Dro

plet

volu

me

VD

=

d D3 /

6[u

l]

(A)

(B)(C)

(D)(E)

51 mN/m

Fritz’s eq. :

Numerical results

0.0209D Wdg

0 140W

: departure

: non-departure


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4.2 Droplet on textured solid surface

The numerical method was applied to micro-fluidic problem of a droplet attached on a textured solid surface in a stagnant liquid. In the Cartesian coordinate system (x, y, z), the computational domain was uniformly divided into 929296 cubic cells and surrounded with solid wall boundaries on the both sides in the x-axis and z-axis directions and with periodic boundaries in the y-axis direction. Initially a spherical-shaped droplet with diameter dD = 48x was placed at the center of the bottom wall, which was patterned to have rectilinear grooves of width wF = 6x and their intervals of wS = wF along the x-axis. The static contact angle W of the droplet on the flat solid surface was set at 130, 113 or 101 uniformly. After time t = 0, external force G had been applied to the droplet. The two-phase fluid was assumed as an immiscible liquid-liquid system with equal density and viscosity for dimensionless parameters of Oh = /(dD)1/2 = 1.1410-2 and Eo = |G|dD

2/= 1.7610-2, which were set under the condition of = 998.2kg/m3, = 1mPas, = 32mN/m and dD = 240m.

Figure 3 shows the steady-state shapes of the droplet on the grooved surface under downward G in the z direction. At each case of W, the apparent contact angle on the y-z plane became larger than that on the x-z plane in the tangential direction to the grooves. In other cases, as shown in Fig. 4, the droplet moves when G has been applied in the x-axis direction, whereas the droplet remained at the initial position on the surface when applying G in the y-axis direction. The numerical simulation results in terms of the apparent contact angles and the motion of the droplet agreed qualitatively well with experimental data.

(a) W = 130 (b) W = 113 (c) W = 101

Figure 3: Steady-state shape of droplet on grooved solid surface under downward external force G for static contact angle W and Oh = /( dD)1/2 = 1.1410-2 and Eo = |G|dD

2/ = 1.7610-2

G

0t

G

0t

yzO

xzO


8

(a) Force components Gx 0 and Gy = Gz = 0

(b) Force components Gx = 0, Gy 0 and Gz = 0

Figure 4: Snapshot of droplet on grooved solid surface under external force G = (Gx, Gy, 0) for W =113, Oh = 1.1410-2 and Eo = 1.7610-2

5 CONCLUSIONS

In this study, we examined the applicability of the computational fluid dynamics (CFD) method based on the phase-field model (PFM) [17, 18] to the numerical simulations of motion of microscopic two-phase fluid on heterogeneously-wetted and textured solid surfaces. The PFM-CFD method, that adopted the lattice Boltzmann method (LBM) [8-10, 19] for solving the Navier-Stokes equations with the conservation-modified Allen-Cahn equation [16, 20], was applied to preliminary simulations of an immiscible liquid-liquid two-phase fluid on flat and textured solid surfaces.

The following major findings were obtained: - The PFM-CFD method simulated well departure of droplet from flat solid surface

due to buoyancy in agreement with semi-empirical prediction. - The droplet on a rectilinearly-grooved hydrophobic solid surface had larger static

contact angle than that on a flat surface. The droplet on the textured surface moved more easily in the tangential direction to the grooves than in the transverse direction to them. These numerical results agreed qualitatively with experimental data.

Gx

0t

0t

Gy= 0

xzO

yzO

0t

0t

Gx= 0

Gy

xzO

yzO


9

The above-mentioned results prove that the PFM-CFD method can be useful for numerically analyzing the microscopic two-phase fluid motions on solid surfaces for optimal design and evaluation of micro-fluidic devices and micro-fabrication processes, and also for understanding and prediction of underground water flowing with heat and mass transfer.

ACKNOWLEDGEMENTS

This research was granted by the Japan Society for the Promotion of Science (JSPS) through the “Funding Program for World-Leading Innovative R&D on Science and Technology (FIRST Program),” initiated by the Council for Science and Technology Policy (CSTP), and also supported by JSPS KAKENHI Grant Number 26420128.

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