Review of potential flow solutions for velocity and shape of long isolated bubbles in vertical pipes

Alexandre Boucher; Roel Belt; Alain Liné

doi:10.1515/revce-2021-0026

Published by De Gruyter December 20, 2021

Review of potential flow solutions for velocity and shape of long isolated bubbles in vertical pipes

Alexandre Boucher , Roel Belt and Alain Liné

From the journal Reviews in Chemical Engineering

https://doi.org/10.1515/revce-2021-0026

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Abstract

The motion of elongated gas bubbles in vertical pipes has been studied extensively over the past century. A number of empirical and numerical correlations have emerged out of this curiosity; amongst them, analytical solutions have been proposed. A review of the major results and resolution methods based on a potential flow theory approach is presented in this article. The governing equations of a single elongated gas bubble rising in a stagnant or moving liquid are given in the potential flow formalism. Two different resolution methods (the power series method and the total derivative method) are studied in detail. The results (velocity and shape) are investigated with respect to the surface tension effect. The use of a new multi-objective solver coupled with the total derivative method improves the research of solutions and demonstrates its validity for determining the bubble velocity. This review aims to highlight the power of analytical tools, resolution methods and their associated limitations behind often well-known and wide-spread results in the literature.

Keywords: bubble nose shape; elongated bubble; potential flow; terminal velocity; vertical flow

Corresponding author: Alexandre Boucher, Transfert-Interface-Mixing Group, Toulouse Biotechnology Institute (TBI), Université de Toulouse, CNRS, INRAE, INSA, Toulouse, France, E-mail: aboucher@insa-toulouse.fr

Funding source: TotalEnergies S.E.

Acknowledgements

The author was allowed by CALMIP to run a numerical simulation on the regional super-computer Olympe. Extensive help was received from the LEMMA group for the use of NiceFlow CFD software. Moreover, the authors are indebted to Pr. A. Ahmadi (TBI Toulouse, France) for his guidance on high-performance resolution solvers and the fruitful talks on this subject.

Author contributions: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
Research funding: This project was supported financially by TotalEnergies S.E.
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

Appendix A: Bessels functions, properties and representations

Definition

Bessel functions were discovered by Daniel Bernoulli, a Swiss mathematician, and called after Friedrich Bessel. They are the solution of the Bessel differential equation

(A.1) x 2 d 2 y d x 2 + x d y d x + ( x 2 − n 2 ) y = 0

for all integer no-equal to zero. Two kinds of function are the solution of this equation, but we will only consider here the Bessel function of the first kind. These functions have the property to be defined and continuous in x = 0 and are defined by

(A.2) J n x = x 2 n ∑ p = 0 ∞ − 1 p 2 2 p p ! n + p ! x 2 p .

Figure 14 presents the representative curves of J_i with i = 0, 1, 2, 3. From this result, we can see that

(A.3) J 0 ( 0 ) = 1 and ∀ i ≥ 1 , J i ( 0 ) = 0 .

$Figure 14: Graphical representation of Bessel function J i with i = 0, 1, 2, 3 on x ∈ [ 0 , 10 ] $x\in \left[0,10\right]$$

Figure 14:

Graphical representation of Bessel function J_i with i = 0, 1, 2, 3 on x ∈ [ 0 , 10 ]

The zero of the first Bessel function of first kind, β i are summed up in Table 1 for i = 1, …, 10.

Properties

The different order of Bessel functions are linked by the following recurrence relations:

(A.4) J n + 1 ( x ) = n J n ( x ) x − J n ′ ( x ) ,

(A.5) J n + 1 ( x ) + J n − 1 ( x ) = 2 n x J n ( x ) ,

(A.6) J n + 1 ( x ) − J n − 1 ( x ) = − 2 J n ′ ( x ) .

Specifically, we have

(A.7) J 1 ( x ) = − J 0 ′ ( x )

and

(A.8) d d x ( x n J n ( x ) ) = x n J n − 1 ( x ) .

Moreover, Bessel functions are orthogonal and we have

(A.9) ∀ i ≠ j , ∫ 0 1 x J n β i x J n β i x d x = 0 .

Taylor expansions

The Taylor expansion of the Bessel and exponential function used in Bendiksen (1985) are given in what follows:

(A.10) J 0 ( β i ξ ) = − ( β i 2 ) 2 ξ 2 + 1 ( 2 ! ) 2 ( β i 2 ) 4 ξ 4 − 1 ( 3 ! ) 2 ( β i 2 ) 6 ξ 6 + o ( ξ 8 ) …

(A.11) J 1 ( β i ξ ) = ( β i 2 ) ξ + 1 1 ! 2 ! ( β i 2 ) 3 ξ 3 + 1 1 ! 2 ! 3 ! ( β i 2 ) 5 ξ 5 + o ( ξ 7 ) …

(A.12) e − β i η = − 2 ( β i 2 ) η + 2 2 2 ! ( β i 2 ) 2 η 2 − 2 3 3 ! ( β i 2 ) 3 η 3 + 2 4 4 ! ( β i 2 ) 4 η 4 + o ( η 5 ) …

Stream function derivatives

The radial and axial velocities in the liquid all around the gas bubble can be determined by deriving the stream function with respect to η and ξ respectively:

u ( ξ , η ) = − 1 ξ ∂ ψ ∂ η = ∑ i = 1 ∞ k i β i J 1 ( β i ξ ) e − β i η ,

v ( ξ , η ) = 1 ξ ∂ ψ ∂ ξ = − v 0 m − v m ξ 2 + ∑ i = 1 ∞ k i β i J 0 ( β i ξ ) e − β i η .

It is then interesting to know the nth derivative of these functions with respect to η and ξ.

Radial velocity u

From multiple derivations with respect to ξ, if n is an odd number, then ∃ p ∈ ℕ so that n = 2p + 1, and we have

(A.13) ∂ n u ∂ ξ n | 0 = ( − 1 ) p ∑ i = 1 ∞ n k i β i n + 1 2 n ,

else

(A.14) ∂ n u ∂ ξ n | 0 = 0 .

Hence, only the axial velocity derivatives with an even order are different from zero when deriving for the radial component ξ. For instance, for n = 1:

(A.15) ∂ u ∂ ξ | 0 = ∑ i = 1 ∞ k i β i β i J 0 β i ξ − J 2 β i ξ 2 e − β i η | 0 = ∑ i = 1 ∞ k i β i 2 2 .

Multiple derivations with respect to the axial variable η gives

(A.16) ∂ n u ∂ η n = ( − 1 ) n ∑ i = 1 ∞ k i β i n + 1 J 1 ( β i ξ ) e − β i η ,

and at the stagnation point, i.e. ( η , ξ ) = ( 0 , 0 ) , we have

(A.17) ∂ n u ∂ η n | 0 = 0 .

Axial velocity v

In the case where n is an even number, i.e. ∃ q ∈ ℕ so that n = 2q, the successive derivatives of the axial velocity with respect to ξ are:

(A.18) ∂ n v ∂ ξ n | 0 = ( − 1 ) q ∑ i = 1 ∞ ( n − 1 ) k i β i n + 1 2 n − 1 ,

else

(A.19) ∂ n v ∂ ξ n | 0 = 0 .

Multiple derivations with respect to η give

(A.20) ∂ n v ∂ η n = ( − 1 ) n ∑ i = 1 ∞ k i β i n + 1 J 0 ( β i ξ ) e − β i η ,

and at the stagnation point:

(A.21) ∂ n v ∂ η n | 0 = ( − 1 ) n ∑ i = 1 ∞ k i β i n + 1 .

Appendix B: Derivation of the stream function governing equation of Lai (1964)

The equations of Lai (1964), also inspired from Long (1953), are expressed in cylindrical co-ordinates (r, θ, z). The velocity components are (u, w, v) in these co-ordinates, and the equation of motion for a steady inviscid flow are

(B.1) u ∂ u ∂ r + v ∂ u ∂ z − w 2 r = − ∂ ∂ r ( P ρ L + Ω ) ,

(B.2) u ∂ w ∂ r + v ∂ w ∂ z − u w r = 0 ,

(B.3) u ∂ v ∂ r + v ∂ v ∂ z = − ∂ ∂ z ( P ρ L + Ω ) ,

where z is an axis of symmetry, ρ L is the density of the liquid phase (constant) and Ω is the potential of the external forces (gravity).

Equation (A.2) can be expressed as follows:

(B.4) u ∂ ( r w ) ∂ r + v ∂ ( r w ) ∂ z = 0 ,

it yields that r w is a function of the Stokes stream function ψ alone, and we assume

(B.5) ( r w ) 2 = f ( ψ ) .

If we introduce the simplifying variables:

(B.6) χ 1 = 1 2 ( u 2 + w 2 + v 2 ) + P ρ L + Ω , and χ 2 = ∂ u ∂ z − ∂ v ∂ r ,

Eqs. (A.1) and (A.3) can be written as

(B.7) ∂ χ 1 ∂ r = w 2 r + 1 2 ∂ w 2 ∂ r − v χ 2 ,

and

(B.8) ∂ χ 1 ∂ z = 1 2 ∂ w 2 ∂ z + u χ 2 .

From the last two equations and the continuity equation,

(B.9) ∂ ( r u ) ∂ r + ∂ ( r w ) ∂ z = 0 ,

we have

(B.10) u ∂ χ 1 ∂ r + v ∂ χ 1 ∂ z = 0 .

Consequently, χ 1 is also a function of ψ alone, and for instance χ 1 = H ( ψ ) .

In cylindrical co-ordinates the Stokes stream function is linked to the radial and axial velocities as follows

(B.11) u = − 1 r ∂ ψ ∂ z , v = 1 r ∂ ψ ∂ r .

This together with the fact that

(B.12) w 2 2 + 1 2 ∂ w 2 ∂ r = 1 2 r 2 ∂ r 2 w 2 ∂ r = 1 2 r 2 ∂ f ψ ∂ r

leads to a new formulation of Eqs. (A.8) and (A.9) as

(B.13) ∂ H ( ψ ) ∂ r = 1 2 r 2 ∂ f ( ψ ) ∂ r − 1 r ∂ ψ ∂ r χ 2 ,

(B.14) ∂ H ( ψ ) ∂ z = 1 2 r 2 ∂ f ( ψ ) ∂ z − 1 r ∂ ψ ∂ z χ 2 .

Multiplying Eq. (A.13) by dr, Eq. (A.14) by dz, and summing them, we have

(B.15) d H = 1 2 r 2 d f − χ 2 r d ψ , or − χ 2 r = d H d ψ − 1 2 r 2 d f d ψ ,

and with

(B.16) χ 2 = − 1 r ( ∂ 2 ψ ∂ z 2 + ∂ 2 ψ ∂ r 2 − 1 r ∂ ψ ∂ r ) ,

we find back Eq. (2.8) from Lai (1964)

(B.17) ∂ 2 ψ ∂ z 2 + ∂ 2 ψ ∂ r 2 − 1 r ∂ ψ ∂ r = r 2 ∂ H ∂ ψ − 1 2 ∂ f ∂ ψ .

Appendix C: Spherical bubble shape for negligible surface tension

Neglecting surface tension in Eq. (2.39) assumed that the curvature term is also neglected. Hence, all the successive derivatives of the curvature term are equal to zero near the stagnation point, i.e.,

(C.1) K 2 | 0 = 0 , K 4 | 0 = 0 , K 6 | 0 = 0 , …

and from the definition of the curvature derivative with respect to the interface equation, we have that,

(C.2) Z 4 | 0 = 3 ( Z 2 | 0 ) 3 , Z 6 | 0 = 45 ( Z 2 | 0 ) 5 , K 2 | 0 = 1575 ( Z 2 | 0 ) 7 , …

In addition, if we express the interface equation under its Taylor expansion around the stagnation point, we have

(C.3) Z ( r ) = Z 2 | 0 2 r 2 + 3 ( Z 2 | 0 ) 3 24 r 3 + 45 ( Z 2 | 0 ) 5 720 r 6 + …

Rearranging the last expression knowing that Z 2 | 0 = − 1 / R c yields

(C.4) Z r = − R c 1 2 r R C 2 + 1 4 r R C 4 + 1 8 r R C 6 + … .

From here, we can recognise the series expansion,

(C.5) ( 1 − x ) 1 / 2 = 1 − x 2 − x 2 8 − x 4 16 − 5 x 4 128 + …

and the equation of the shape becomes

(C.6) Z ( r ) = R c [ ( 1 − ( r R c ) 2 ) − 1 ]

which becomes after some manipulations

(C.7) r 2 + Z r + R c 2 = R c 2 .

The latter equation is the equation of a circle of radius R_c going through the origin of the coordinate system. Neglecting the surface tension in the governing equations fixes the bubble nose’s shape by imposing a circle shape. Considering the inertia-dominated regime where the effects of surface tension are neglected in most cases, experimental studies showed that the nose of an elongated bubble was very close to a spherical shape, corroborating the result that has just been found analytically.

Appendix D: Shape solution in the literature

Some of the literature data on the solution of the potential problem for infinitely long confined gas bubble in the pipe are summed up in this appendix. Very few results are shared in the literature, and only the results of Dumitrescu (1943) and Nickens (1984) can be shared.

For the sake of simplicity, the results are given in order to be used directly with Eq. (2.23).

Dumitrescu (1943)

In Dumitrescu (1943), the dimensionless terminal velocity is expressed as

(D.1) v 0 m = − ∑ i = 1 n k i β i .

Consequently, Dumitrescu’s k_i will be of the opposite sign as those in Eq. (2.23).

Dumitrescu determined the solution for different dimensionless curvature radius at the nose of the bubble. He calculated the solution for R₀ = 0.7 and R₀ = 0.8 and concluded that a curvature radius of R₀ = 0.75 was the best choice to also fit the solution he calculated in the liquid film. The solution is thus not explicitly given in this paper. However, the mean value of the R₀ = 0.7, and R₀ = 0.8 solution has been given in Table 4 as it also corresponds to the terminal Froude solution for the bubble given by Dumitrescu for R₀ = 0.75.

Table 4:

Dumitrescu’s solution expressed in the formalism used in Eq. (2.23).

k ₁	k ₂	k ₃	F r ∞
0.105	0.00855	0.00337	0.496

Nickens (1984)

In Nickens (1984) the results of his resolution method are given in terms of d i δ i to take the author notation. For the relation,

(D.2) ∑ i = 1 n d n δ n = 1

which is also equivalent to Eq. (2.33) using the notation of this paper, we have that

(D.3) d i δ i = k i β i v 0 m , i . e. k i = d i δ i β i v 0 m

it yields the following solutions (see Tables 5 and 6),

Table 5:

Nickens’ solution without surface tension term in the equation Σ = 0 and for stagnant liquid v_m = 0. N is the number of unknowns in the system, i.e. the number of k_i, and N_D is the maximum order of Taylor’s expansions in the resolution.

N	N_D = 2	N_D = 4	N_D = 6	N_D = 8	N_D = 10	N_D = 12
1	0.1332	0.1404	0.0819	0.1060	0.1043	0.1054
2		−0.0184	0.0143	0.0104	0.0110	0.0100
3			0.0059	0.0016	0.0018	0.0023
4				0.0003	0.0002	0.00005
5					0.00003	0.0001
6						−0.00003
F r ∞	0.511	0.409	0.474	0.499	0.498	0.498

Table 6:

Nickens’ solution with surface tension term in the equation Σ ≠ 0 and for stagnant liquid v_m = 0. N is the number of unknowns in the system, i.e. the number of k_i, and N_D is the maximum order of Taylor’s expansions in the resolution.

N	N_D = 2	N_D = 4	N_D = 6	N_D = 8	N_D = 10	N_D = 12
1	0.1218	∅	0.0451	0.0924	0.0926	0.0965
2		∅	0.0060	0.0149	0.0150	0.0127
3			0.0037	0.0021	0.0023	0.0003
4				0.0006	0.0004	0.0002
5					0.00002	0.0001
6						−0.00001
F r ∞	0.4667	∅	0.5049	0.4879	0.4893	0.4893

Appendix E: Rankine solid developments for potential flow analysis

The conformal transformation associated with a source in a uniform flow is presented in what follows. The resulting shapes of the stream lines are known as the Rankine solid, which seems intuitive to suit the shape of a rising bubble in an infinite domain. The latter geometry is presented in Figure 12.

The source m is located at point O(0,0) and the flow is defined as U = − U ∞ e z → . The stream function associated with this conformal transformation is defined in Milne–Thomson (1996) by

(E.1) ψ = − 1 2 U ∞ r 2 sin 2 θ + m cos θ ,

where m also denote the strength of the source, and ( r , θ ) are the spherical coordinate associated with the three-dimensional Rankine body. The liquid velocity around the bubble can be expressed in spherical coordinate through:

(E.2) U L → = ‖ q r = − 1 r 2 sin θ ∂ ψ ∂ θ = U ∞ cos θ + m r 2 q θ = 1 r sin θ ∂ ψ ∂ r = − U ∞ sin θ

Let’s call the point A(a, π) the stagnation point associated with an arbitrary gas-liquid interface. At the stagnation point, q r = q θ = 0 , and from the last equation we have the following relation between a, m and U ∞ :

(E.3) m = U ∞ a 2 .

For the sake of simplicity, a cylindrical coordinate system ( e ξ → , e η → ) will be adopted for the rest of the developments. In this plane, we have

{ tan θ = − ξ η + a r 2 = ξ 2 + ( η + a ) 2

and liquid velocity becomes,

U L → = ( m ξ [ ξ 2 + ( η + a ) 2 ] 3 / 2 ) e η → − ( U ∞ − m η + a [ ξ 2 + ( η + a ) 2 ] 3 / 2 ) e η → .

The shape of the Rankine body is given by the stream line going through the stagnation point, which yields in the initial spherical coordinate system,

(E.4) r = a sin θ / 2 .

The last equation leads after few mathematical developments, to a quadratic equation where only one root is physically permitted. The final result in the cylindrical plane is

(E.5) ξ 2 = − 1 2 ( η + a ) 2 + 2 a 2 − 1 2 ( η + a ) ( η + a ) 2 + 8 a 2 .

Note that from the above equation, when η → − ∞ , the width of the Rankine body in the radial direction tend asymptotically to ξ = 2 a . The gas bubble’s width is thus limited to 2a, which seems to be a good representation of the gas–liquid interface of a Taylor bubble in the fully developed liquid film zone.

Appendix F: The interface description using curvilinear coordinates

This section goes back the work originally done by Ha-Ngoc (2002) on the gas–liquid interface description. He used a curvilinear formulation, with the apex of the bubble as the origin of this new set of coordinate (s = 0). Assuming that ξ I ( s ) and η I ( s ) are the new coordinates associated with the interface in the moving frame attached to the bubble, the Bernoulli’s equation can be written as

(F.1) ξ i − ∑ ( κ 0 − κ ) = u i 2 .

Taking the derivative with respect to s yields,

(F.2) ξ ′ ( s ) + ∑ κ ′ ( s ) = 2 u i u i ′ .

If we introduce the angle θ ( s ) between the pipe axis and the tangent to the interface, so that the interface results from

(F.3) { ξ i ( s ) = ∫ 0 s cos ( θ ( t ) ) d t η i ( s ) = ∫ 0 s sin ( θ ( t ) ) d t

we get

(F.4) cos θ s + ∑ κ ′ s = 2 u i u i ′ .

On a stream line, the velocity at the interface u i can be written under the following form:

(F.5) u i ( s ) = | 1 η i ( s ) ∂ ψ ∂ n ( s ) | .

The curvature using the new set of coordinates can be expressed as

(F.6) κ ( s ) = − η i ′ ′ ( s ) ξ i ′ ( s ) + η i ′ ( s ) ξ i ′ ′ ( s ) + ξ i ′ ( s ) η i ( s ) = − θ ′ ( s ) + cos ( θ ( s ) ) η i ( s )

which gives by derivation with respect to s:

(F.7) κ ′ ( s ) = − θ ″ ( s ) − θ ′ ( s ) η i ( s ) + cos ( θ ( s ) ) η i 2 ( s ) sin ( θ ( s ) ) .

Injecting all the above relations into Bernoulli’s equation gives

(F.8) cos ( θ ( s ) ) − ∑ [ θ ″ ( s ) + θ ′ ( s ) η i ( s ) + cos ( θ ( s ) ) η i 2 ( s ) sin ( θ ( s ) ) ] = 2 η i ( s ) ∂ ψ ∂ n ( 1 η i ( s ) ∂ ψ ∂ n ( s ) ) .

The latter equation gives a unique solution if two boundary conditions are given, one for s → + ∞ and another one at the apex of the bubble s = 0.

In the developed film the liquid thickness is constant, and the gas-liquid interface is parallel to the walls. Consequently,

(F.9) lim s → + ∞ θ ( s ) = 0

and

(F.10) lim s → + ∞ κ ( s ) = lim s → + ∞ 1 η i ( s ) = 1 R ∞ ,

where R ∞ is the radius of the bubble when the fully developed liquid film region is reached. Moreover, this yields lim s → + ∞ θ ′ ( s ) = 0 . The second condition should be expressed at the nose of the elongated bubble, and if we suppose a smooth interface (i.e. the derivatives of θ, ξ, and η exist in s = 0) and θ = π / 2 at the stagnation point, we have

(F.11) u i ( 0 ) = 0 , u i ′ ( 0 ) = 0 .

Hence, Bernoulli’s equation becomes

(F.12) cos ( θ ( 0 ) ) + ∑ κ ′ ( 0 ) = 0 .

If the dimensionless tension surface is not equal to zero, Σ ≠ 0 , then κ ′ ( 0 ) = 0 , which is equivalent to

(F.13) lim s → 0 d κ ( s ) d s = lim s → + ∞ θ ″ ( s ) = 0

i.e. the bubble nose is spherical at the point s = 0.

This formulation of the problem allows Ha-Ngoc (2002) to solve with numerical methods the system made of Poisson’s equation (Eq. (2.7)) and a functional equation on θ. The solution is not analytical but numerical and does not provide an analytical (or semi-analytical) solution for the bubble velocity and shape that we were looking for. Nevertheless, the numerical procedure has been used in Miksis et al. (1981), Vanden–Broeck (1994, 1995) and Kang and Vanden-Broeck (2000) for free surface problems.

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Received: 2021-04-11

Accepted: 2021-08-17

Published Online: 2021-12-20

Published in Print: 2023-04-25

Review of potential flow solutions for velocity and shape of long isolated bubbles in vertical pipes

Abstract

Acknowledgements

Appendix A: Bessels functions, properties and representations

Definition

Properties

Taylor expansions

Stream function derivatives

Radial velocity u

Axial velocity v

Appendix B: Derivation of the stream function governing equation of Lai (1964)

Appendix C: Spherical bubble shape for negligible surface tension

Appendix D: Shape solution in the literature

Dumitrescu (1943)

Nickens (1984)

Appendix E: Rankine solid developments for potential flow analysis

Appendix F: The interface description using curvilinear coordinates

References

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