Test-case number 29b: The velocity and shape of 2D long bubbles in inclined channels or in vertical tubes (PA, PN) Part II: in a flowing liquid Hien Ha-Ngoc, Institute of Mechanics, 264 Doi Can, Hanoi, Vietnam. E-Mail: [email protected] Jean Fabre, Institut de M´ecanique des Fluides de Toulouse, All´ee du Pr. Camille Soula, 31400 Toulouse, France. E-Mail: [email protected]

1

Practical significance and interest of the test-case

In flowing liquid, the motion of the long bubbles results from the complex influence of both buoyancy and mean motion of the liquid. Following Nicklin et al. (1962) the velocity of these long bubbles would result as the sum of two terms: V = C0 U + V∞

(1)

where V 8 is the bubble velocity in a stagnant liquid and U is the mean liquid velocity far from the bubble head. Such decomposition does not mean that V is a linear function of U since C 0 is a dimensionless number that may depend on U through the other dimensionless parameters of the problem. Because the motion of a long bubble does not depend on its length, C 0 represent therefore the influence of the liquid velocity profile imposed far from the bubble head – it’s called the distribution coefficient. The experimental results of Nicklin et al. (1962) in vertical pipe have shown that: C 0 increases from 0.9 for negative U (downward flow) to a maximum of 1.8 near U =0 and then decreases toward an asymptotic value of 1.2 for liquid Reynolds number Re L =UD/ ν greater than 8000. Therefore C 0 is not only sensitive to the velocity profile but also to the flow direction. In downward flow, bubble may become asymmetrical: such a situation is observed experimentally (Griffith & Wallis, 1961, Nicklin et al. , 1962, Martin, 1976) but it is neither explained theoretically nor reproduced by numerical simulation. An analytical investigation on C 0 for an axis-symmetrical bubble in upward flow in a vertical tube was carried out by Collins et al. (1978) using inviscid theory. Like in stagnant liquid, the viscosity effects is negligible when inertia dominates. The condition is satisfied provided Nf = D3/2 g 1/2 /ν > 300 (Fabre & Lin´e, 1992). In fact, viscosity acts essentially to develop the liquid velocity profile far ahead of the bubble, but it has a negligible influence near the bubble front. For an inviscid axis-symmetrical flow, the Stokes stream function satisfies a Poisson equation which is solved by applying the boundary conditions at the bubble surface. Collins et al. (1978) have obtained an approximate solution for the bubble velocity for both laminar and turbulent velocity profile upstream the bubble. In this approach, the condition of constant pressure was satisfied locally at the bubble nose. Bendiksen (1985) used the same approach to include the effect of surface tension. In this paper, we provide analytical and numerical solutions for the velocity and shape of 2D long bubbles (plane and axis-symmetrical) moving in a flowing liquid. The analytical solution is an extension of the Benjamin (1968) solution for the horizontal case without surface tension, whereas the numerical solutions are obtained for the inertial regime with surface tension. The Poisson equation for the stream function with a source term determined by the velocity profile given ahead of the bubble is solved by an algorithm based on the boundary element method (BEM). The bubble shape is determined by satisfying the condition of constant pressure all along the bubble surface (Ha-Ngoc, 2003). For

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Test-case number 29b by Hien Ha-Hgoc and J. Fabre

plane bubbles, in contrast to the case of a stagnant liquid where the source term in the Poisson equation vanishes, the source term always exists in flowing liquid. Moreover, in most cases, this is a non-linear function of the stream function. The numerical algorithm requires, therefore, a domain discretization for the integration of the source term and an additional iterative loop for treating its non-linearity character. The results obtained by the method are in good agreement with different theoretical, experimental and numerical results of Zukoski (1966), Collins et al. (1978), Bendiksen (1985), Mao & Dukler (1990, 1991)(Ha-Ngoc, 2003).

2

Definitions and model description

The reference calculations to be proposed are made in dimensionless units. The length scale of the problem is the channel haft-width a or the tube radius R. The velocity scale is (gD)1/2 , where D is the channel width (2a) or the tube diameter (2R). In flowing liquid, the flow may be characterized by four dimensionless parameters: • The bubble Reynolds number, Re Re =

VD ν

(2)

ReL =

UD ν

(3)

U F rL = √ gD

(4)

ρgD2 , σ

(5)

• The liquid Reynolds number, Re L

• The liquid Froude number, Fr L

• The E¨ otv¨os number, Eo

Eo =

where ν is the kinematic viscosity of the liquid, ρ, the liquid density, σ, the surface tension and g, the gravity. The dimensionless bubble velocity is presented by the bubble Froude number, Fr, defined as : V Fr = √ . gD

(6)

Like in the case of stagnant liquid, it is expected that there exists a critical bubble Reynolds number Re ∗ such as for Re>Re ∗ the bubble velocity does not depend on the viscosity. In this flow regime, the liquid can be considered as inviscid and the bubble motion is characterized by three dimensionless parameters: the liquid Reynolds number Re L , the liquid Froude number Fr L and the E¨otv¨ os number Eo. Because the flow is assumed inviscid, the liquid Reynolds number Re L is used essentially to determine the velocity profile imposed far ahead of the bubble. For small Re L , the liquid flow is laminar and has a parabolic distribution. For turbulent flow, the velocity distribution is a function of Re L and can be determined by empirical correlation. For axis-symmetrical flow in tubes,

Test-case number 29b by Hien Ha-Hgoc and J. Fabre

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in the cylindrical coordinate system the following correlation is employed (Collins et al. , 1978, Bendiksen, 1985):
ux (r) = umax 1 − γr2 − (1 − γ) r2n , (7) where u max is the maximum velocity on the tube axis, γ and n depend on the Reynolds number and can be determined by the wall friction law. According to Bendiksen (1985), using the following wall friction law: p 1/fw = 3.5 log10 (ReL ) − 2.6, (8) where f w is the friction factor, the following values of γ and n yields: 7.5 , 4.12 + 4.95(log10 (ReL ) − 0.743) log (ReL ) − 0.743 γ n = (γ − 1)[ 10 − (1 − )]−1 − 1. log10 (ReL ) + 0.31 2 γ=

(9) (10)

The parabolic distribution can be obtained from (7) by setting γ=1. For the case of plane bubbles, a turbulent velocity distribution similar to that of (7) is also used in the calculations. For the relation between the stream function and the vorticity far upstream the bubble to be uniform, the solutions have to be restricted to a specific range of Froude numbers [Fr L − , Fr L + ]. This restriction comes simply from the solution method which requires the elimination of vorticity by the stream function to obtain the equations of the problem. In the calculations, the contact angle, θ0 =π/2, is always used for case when the bubble touches the upper wall and the criterion of maximum velocity (Garabedian, 1957) is used to select the physically observable solution.

3

Motion in horizontal and inclined channel

For horizontal channel, the exact solutions for bubble velocity, V, and the liquid film thickness, d, can be obtained by an extension of the Benjamin (1968) method for the case when surface tension is negligible (Fabre & Ha-Ngoc, 2003). The solutions are given under the form of Taylor series for small liquid Froude number (Fr L