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The force-coupling method (FCM) developed by Maxey and Patel (2001) was modified and applied to trace the trajectories of spherical bubbles with solid-like and slip surfaces. Careful comparison was made to the experimental results of Takemura et al. (2000, 2002a, 2002b). First, the result obtained by use of the original version of the FCM was compared to the experimental results. It was found that the original FCM was not feasible for tracing spherical bubble trajectories. Then, a correction was made to the FCM calculation of the bubble velocity by renormalization in terms of the bubble Reynolds number, which could very well trace the trajectory of the bubble with a solid-like, no-slip surface, but not that of a bubble with a slip surface. Finally, a substantial correction was made to the monopole term of the FCM, which could trace the trajectory of a bubble with a solid-like or slip surface very well even for the Reynolds number up to 20.

The study of the multiphase flow of water including air bubbles has been attracting interest of many researchers, not only because of a lot of mechanical and chemical engineering applications, but also out of physical interest. When the diameter of an air bubble is around several tens of micrometers, it is called a microbubble, and a nanobubble if the diameter is around hundreds of nanometers. Experimental studies of microbubbles are now well developed in engineering applications, such as drag reduction of pipe flow or boundary layer flow [

Theoretical and numerical studies of microbubble and nanobubble flows are on the other hand very retarded because numerical calculation of such flows requires a significant computational load. An appropriate numerical method has not been found in spite of many methods proposed so far. Such methods include the volume of fluid method (VOF) [

The force-coupling method (FCM) originally proposed by Maxey and Patel [

Note that there exist a few experimental studies on the bubble motion in the quiescent fluid. Takemura et al. [

In the present paper, we proposed corrections to the original FCM in order to trace the trajectories of spherical bubbles with no-slip and slip surfaces. The bubble with a no-slip surface or a slip surface will be called the no-slip bubble or the slip bubble, respectively, henceforth in this paper. Then, we compared the results of the numerical simulation with the experimental results of Takemura et al. [

In the original version of the FCM, the incompressible Navier-Stokes equations with a force term representing the interaction with bubble,

∂ u i ∂ t = − u j ∂ u i ∂ x j − 1 ρ f ∂ p ∂ x i + ν ∂ 2 u i ∂ x j 2 + g i + f b , i , ( i = 1 , 2 , 3 ) (1)

and the equation of continuity,

∂ u i ∂ x i = 0 , (2)

are used as the basic equations, where the Einstein’s summation convention is used. u i is the velocity vector of the fluid, t the time, ρ f the density of the fluid, p the pressure, ν the kinematic viscosity, g i the gravitational acceleration vector. f b , i represents the body force acting on the fluid from N bubbles given by

f b , i = ∑ n = 1 N { F i ( n ) Δ M ( x i − Y i ( n ) , σ M ) + G i j ( n ) ∂ ∂ x j Δ D ( x i − Y i ( n ) , σ D ) } , (3)

where i = 1 , 2 , 3 indicate the x , y , z -direction components, respectively. f b , i consists of two parts, the force monopole term (FMT, the first term on the right-hand side of Equation (3)) and the force dipole term (FDT, the second term on the right-hand side of Equation (3)). Here, x i is the spatial position vector in the fluid, and Y i ( n ) is the position vector of the center of the n-th bubble. Δ M and Δ D are the force ranges of FMT and FDT, respectively, in terms of the Gaussian distribution function given by

Δ M ( x i , σ M ) = 1 [ 2 π ( σ M ) 2 ] 3 / 2 exp [ − x i 2 2 ( σ M ) 2 ] , (4)

Δ D ( x i , σ D ) = 1 [ 2 π ( σ D ) 2 ] 3 / 2 exp [ − x i 2 2 ( σ D ) 2 ] , (5)

according to [

FMT represents the reaction to the drag force acting on the bubble due to the influence of its translational motion. F i ( n ) is the force acting on the fluid from the n-th bubble, which in the original FCM [

F i ( n ) = 4 3 π R 3 ( ρ f − ρ b ) ( g i − d U i ( n ) d t ) , (6)

where ρ b is the density of the bubble. The velocity of the n-th bubble, U i ( n ) , is given by the following equation in terms of the Gaussian distribution function.

U i ( n ) = ∭ u i ( x i , t ) Δ M ( x i − Y i ( n ) , σ M ) d x 1 d x 2 d x 3 , (7)

which means that the bubble moves with the fluid velocity averaged nearly over the bubble region.

FDT represents the influence of the velocity gradient around the bubble, where G i j ( n ) is the sum of two parts, an antisymmetric part, T i j ( n ) , and a symmetric part, S i j ( n ) . T i j ( n ) represents the torque on the fluid from the bubbles and is given by the equation,

T i j ( n ) = 1 2 ϵ i j k T k ( n ) ( e x t ) (8)

where ϵ i j k is the Eddington’s alternating sign tensor, and T k ( n ) ( e x t ) is an external torque on the n-th bubble. The symmetric part, S i j ( n ) , represents a stresslet acting on the fluid and is given by the following equations for a no-slip bubble,

S i j ( n ) = 20 3 π ν ρ f R 3 E i j ( n ) , (9a)

and for a slip bubble,

S i j ( n ) = 8 3 π ν ρ f R 3 E i j ( n ) , (9b)

respectively. E i j ( n ) is the rate of strain, which determined to satisfy the constraint,

E i j ( n ) = 1 2 ∫ ( ∂ u i ∂ x j + ∂ u j ∂ x i ) Δ D ( x i − Y i ( n ) , σ D ) d x 1 d x 2 d x 3 = 0 , (10)

for each bubble using an iteration method [

The calculation procedure of the original FCM is summarized as follows:

1) Give the initial conditions.

2) Get F i ( n ) from the initial conditions (Equation (6)).

3) Obtain the FMT from Δ M ( x i , σ M ) , and F i ( n ) (Equations (4), and (6)).

4) Get G i j ( n ) from Δ D ( x i , σ D ) , T i j ( n ) , S i j ( n ) , and E i j ( n ) (Equations (5), (8), (9), and (10)).

5) Obtain the FDT from Δ D ( x i , σ D ) , and G i j ( n ) (Equations (5), (8), and (9)).

6) Acquire f b , i from the FMT and the FDT (Equation (3)).

7) Solve the Navier-Stokes equations including f b , i and obtain the velocity field u i ( x i , t ) (Equations (1), and (2)).

8) Obtain the bubble velocity U i ( n ) from u i ( x i , t ) and Δ M ( x i , σ M ) (Equations (1), (2), (4), and (7)).

9) Solve the next-time step bubble position from U i ( n ) and Δ t .

10) Return to Step 2.

In the present paper, N was put to be 1, because we studied only a single bubble case, and the affix (n) will be deleted henceforth.

FCM was originally developed for the fluid including solid particles. Therefore, the calculation results using the original version of FCM were first compared with the results by Takemura et al. [

First, the equation of U i in FCM (Equation (7)) was changed for a no-slip bubble considering the drag coefficient of a sphere over a wide range of the Reynolds number. It is known that the drag coefficient of a spherical solid particle, D n , is very accurately approximated by the following equation in the range of the bubble Reynolds number, R e ∞ , from 0.01 to 20 [

D n = 6 π ν ρ f R U ∞ ( 1 + 0.1315 R e ∞ 0.82 − 0.05 log 10 R e ∞ ) , (11)

where R e ∞ = 2 R U ∞ / ν , and U ∞ is the terminal velocity of the particle in the quiescent fluid. Then, the equation for determining the velocity of a no-slip bubble is changed to

U i = ∭ [ u i ( x i , t ) Δ M ( x i − Y i , σ M ) ⋅ 1 1 + 0.1315 R e 0.82 − 0.05 log 10 R e ] d x 1 d x 2 d x 3 , (12)

where R e = 2 R ( u ˜ i − U i ) 2 / ν , and u ˜ i is the fluid velocity though which the bubble travels. The system of Equations (1), (2), (3), (6), and (12) instead of Equation (7) will be called the renormalized FCM (RFCM) henceforth in this paper.

Next, the reaction of a no-slip bubble to the fluid was considered in terms of the drag force instead of the acceleration term, d U i / d t , for FMT in Equation (6). The following equation is introduced,

F i = − 4 3 π R 3 ( ρ f − ρ b ) g i + 6 π ν ρ f R ( U i − u ˜ i ) ( 1 + 0.1315 R e ∞ 0.82 − 0.05 log 10 R e ∞ ) , (13)

taking into consideration of the drag force formula (11). The terminal velocity, U ∞ , in the infinite quiescent fluid is given by

U ∞ = 2 9 g i R 2 ν ⋅ 1 1 + 0.1315 R e ∞ 0.82 − 0.05 log 10 R e ∞ . (14)

Therefore, R e ∞ is obtained by solving the following equation.

R e ∞ = 4 9 g i R 3 ν 2 ⋅ 1 1 + 0.1315 R e ∞ 0.82 − 0.05 log 10 R e ∞ . (15)

Note that R e ∞ is uniquely determined if R is specified.

The drag coefficient of a spherical slip bubble, D s , is known to be very accurately approximated by the following equation in the Reynolds number range from 0 to 100 [

D s = 4 π ν ρ f R U ∞ { 1 + [ 8 R e ∞ + 0.5 ( 1 + 3.315 R e ∞ 0.5 ) ] − 1 } . (16)

Then, the reaction from a slip bubble to the fluid was considered in terms of the drag force instead of the acceleration term d U i / d t for FMT in Equation (6) as

F i = − 4 3 π R 3 ( ρ f − ρ b ) g i + 4 π ν ρ f R ( U i − u ˜ i ) { 1 + [ 8 R e ∞ + 0.5 ( 1 + 3.315 R e ∞ 0.5 ) ] − 1 } . (17)

Using the drag force formula (16), U ∞ is given by

U ∞ = 1 3 g i R 2 ν ⋅ 1 1 + [ 8 R e ∞ + 0.5 ( 1 + 3.315 R e ∞ 0.5 ) ] − 1 , (18)

where R e ∞ is obtained by solving the following equation.

R e ∞ = 2 3 g i R 3 ν 2 ⋅ 1 1 + [ 8 R e ∞ + 0.5 ( 1 + 3.315 R e ∞ 0.5 ) ] − 1 . (19)

Now, R e ∞ is uniquely determined if R is specified. The system of Equations (1), (2), (3), (7), and (13) instead of Equation (6) will be called the modified FCM-n (MFCM-n) for a no-slip bubble henceforth in this paper, while the system of Equations (1), (2), (3), (7), and (17) instead of Equation (6) will be called the modified FCM-s (MFCM-s) for a slip bubble.

The calculation domain of the present study was a cube, the length of each side of which was 0.06 m, as shown in

The test section used in the experiment by Takemura et al. [

numerical calculations, because the measurement in the experiment was conducted in a relatively small region of x where the rising bubble reached a nearly asymptotic state.

^{2} and the kinematic viscosity, ν , and the bubble radius, R, were taken to be equal to be the experimental conditions. L shows the distance from the bubble center to the nearest vertical wall.

Because of the periodic boundary condition in the x-direction of the present study, a pressure gradient opposite to the gravitational acceleration was needed to keep the flow in the calculation domain quiescent as a whole. A bubble moving in the calculation region was accelerated from the initial stationary state by the buoyancy force. The initial position of the bubble center was put at the point ( 0 , 0 , 0.03 − L 0 ) , where L 0 , being the initial value of L, was put to be 2R.

The Navier-Stokes equations were solved by the simplified marker and cell method (SMAC). The fourth-order central difference scheme was used for spatial differentiation, the Adams-Bashforth and the Crank-Nicolson method were used for the advection and viscous terms, respectively, in the time advancement. In order to solve the pressure equation, the bi-conjugate gradient stabilization method (Bi-CGSTAB) was used.

In order to study the applicability of the proposed formulas, RFCM, MFCM-n, and MFCM-s, comparison with the experimental data for the trajectory of a rising bubble by Takemura et al. [

Here, an outline of the experimental study by Takemura et al. [

obtained by the present simulations and compared with the experimental results.

Takemura et al. introduced the wall distance Reynolds number, R e L , defined in terms of L as

R e L = U ∞ L ν , (20)

and analyzed the force acting upon bubbles moving parallel to the wall under the assumption of the Oseen approximation when a wall exists in the infinite fluid. They modified the formula for the lift force obtained by Vasseur and Cox [

I D − n ⋅ U 3 U 1 = 3 U 3 4 U 1 π R e L 2 ∫ 0 ∞ ∫ 0 2π ξ 2 i cos φ [ ( e − 2 ξ ξ − e − 2 χ χ ) + 2 ( e − ξ − e − χ ) 2 χ − ξ ] d ξ d φ , (21)

I D − s ⋅ U 3 U 1 = U 3 2 U 1 π R e L 2 ∫ 0 ∞ ∫ 0 2π ξ 2 i cos φ [ ( e − 2 ξ ξ − e − 2 χ χ ) + 2 ( e − ξ − e − χ ) 2 χ − ξ ] d ξ d φ , (22)

where U 1 is the bubble velocity in the vertical direction (x-direction) and U 3 is that in the horizontal direction (z-direction), χ is defined by the following equations.

χ = R e L η , η 2 = k 1 2 + k 2 2 + i k 1 , k 1 = ξ cos φ R e L , k 2 = ξ sin φ R e L ,

where i denotes an imaginary unit. Takemura et al. [

3 π ν ρ f R U 1 R e ∞ I L − n = 6 π ν ρ f R U 3 { 1 + 0.1315 R e ∞ 0.82 − 0.05 log 10 R e ∞ } + 3 π ν ρ f R U 3 R e ∞ I D − n , (23)

2 π ν ρ f R U 1 R e ∞ I L − s = 4 π ν ρ f R U 3 { 1 + [ 8 R e ∞ + 0.5 ( 1 + 3.315 R e ∞ 0.5 ) ] − 1 } + 2 π ν ρ f R U 3 R e ∞ I D − s , (24)

where I L − n and I L − s are the inverse Fourier transform of the horizontal (z-direction) components of the lift force acting on a no-slip bubble and a slip bubble, respectively. Finally, I L − n and I L − s are given by

I L − n = 2 R e ∞ ⋅ U 3 U 1 ( 1 + 0.1315 R e ∞ 0.82 − 0.05 log 10 R e ∞ + R L R e L I D − n ) , (25)

and

I L − s = 2 R e ∞ ⋅ U 3 U 1 { 1 + [ 8 R e ∞ + 1 2 ( 1 + 3.315 R e ∞ 0.5 ) ] − 1 + R L R e L I D − s } . (26)

Because the data for I L − n or I L − s were given by Takemura et al. [

(25) based on the calculation results using RFCM in the present study. This figure also shows that the results by RFCM are in good agreement with the experimental data for R e ∞ = 0.9 , 2.5, and 5.

In order to study the applicability of MFCM-n to the no-slip bubble motion, the calculation results by MFCM-n were compared with those by RFCM and the experimental data of Takemura et al. [

MFCM-n is applicable for R e ∞ = 17.7 , and probably for R e ∞ ≤ 17.7 as RFCM.

Although RFCM was applied to predicting the trajectory of a no-slip bubble, it proved to be inapplicable to a slip bubble. Even if the Reynolds number was about 1 or less, the difference was about by 50%. Because the boundary conditions for no-slip and slip bubbles are different, the drag force is larger for a

no-slip bubble than that for a slip bubble and the terminal velocity of a slip bubble is at least 1.5 times larger than that for a no-slip bubble. Therefore, the lift force for a slip bubble is larger than that for a no-slip bubble if the Reynolds number is same (Equations (15), (19), (25), and (26)). The MFCM-s was applied to trace a slip bubble trajectory.

Three new methods for simulating multiphase flow consisting of water and air bubbles were proposed based on FCM in order to trace the trajectories of a spherical bubble with a slip or no-slip surface rising near the wall in the quiescent fluid. The calculation results were compared with the experimental data by Takemura et al. [

The first method is called RFCM, where the induced bubble velocity was renormalized according to the formula of the drag force for a no-slip bubble according to [

The second method is called MFCM-n, where FMT was modified according to the drag force equation given in [

The third method is called MFCM-s, where FMT was modified according to the drag force equation given in [

In the present paper, it is found that the three newly proposed methods can correctly trace the trajectory of a bubble near a vertical wall for relatively large Reynolds numbers, regardless of whether the bubble has a slip or no-slip surface. In future study, we investigate the effects of the value of the Gaussian distribution’s width, σ , in FMT and the modification of G i j ( n ) in FDT and develop simulation methods to deal with the fluid which includes many bubbles.

S. Yanase would like to give his cordial thanks to the Ministry of Education, Culture, Sports, Science and Technology for the financial support through the Grant-in-Aids for Scientific Research No. 15K05814 and 15H0419919.

Guan, C., Yanase, S., Matsuura, K., Kouchi, T. and Nagata, Y. (2017) Application of the Modified Force-Coupling Method of Tracing the Trajectories of Spherical Bubbles with Solid-Like and Slip Surfaces. Open Journal of Fluid Dynamics, 7, 657-672. https://doi.org/10.4236/ojfd.2017.74043