Determination of Velocity at the Gas-Liquid Interface by BEM Using Potential Flow Laboratory for Fluid Dynamics and Thermodynami

Determination of velocity at the gas-liquid interface by BEM using potential flow

Z Rek Laboratory for Fluid Dynamics and Thermodynamics, Faculty of Mechanical Engineering, University of Ljubljana,

Askerceva 6, SI-1000 Ljubljana, Slovenia EMail: zlatko. rek@fs. uni-lj. si

Abstract

The scope of the paper is determination of velocity at the gas-liquid interface by Boundary Element Method using potential flow. Velocity on the bubble boundary has to be known in order to update the interface position. When evaluating integrals with their source point at the boundary caution must be exercised due to the hyper-singularity of the kernel. They are evaluated by moving the source point from boundary into domain. After that the integrals are evaluated analytically, and finally the source point is moved back to the boundary using a limiting process. As an example the simulation of a rising air bubble in a watter-filled pipe is given.

1 Introduction

The Boundary Element Method* (BEM) has proved to be a very efficient method for solving potential flow problems. It can also be used in bubble dynamics and other moving boundary problems^ which appear in a number of areas of engineering. Using the standard BEM approach, where potential at the gas-liquid interface boundary is governed by Bernoulli's equation, the normal velocity is obtained from the solution of the system of linear equations. To update the potential, the velocity at the gas-liquid interface has to be known. Until now, the tangential component of the velocity has been computed by numerical derivation of potential along the boundary^. It is well known that this approach introduces additional numerical error in the iterative process. If linear boundary elements with constant interpolation for flux are used, then the velocity at the boundary can be calculated

explicitly.

2 Basic equations

With the assumption that an incompressible and inviscid liquid is used, bubble motion is governed by Laplace's equation

^=0 (1) where is liquid velocity potential. At the gas-liquid interface Bernoulli's equation is used

d 1 Poo — P o

Here POO is reference liquid pressure, p the pressure inside bubble, p the liquid density, a the surface tension, K, the surface curvature, QJ the gravity acceleration and Xj position vector. The interface position is obtained from

%y + %,Af, n = 0, 1, 2, . . . (3)

Eq. (1) is transformed into a boundary integral equation using the standard BEM approach

(4)

It is solved in a discrete form

(5) by dividing boundary F into N^ boundary elements. In our case, elements with linear interpolation for potential and constant interpolation for flux are used, to avoid problems with undefined unit normal at the corners. For the required boundary conditions: potential at FI and flux at F2

0 = 0 on FI (6)

%a-^ = W-3 - orn Fa on on eq. (5) is solved for flux at FI and potential at F2. The potential value in the domain points is calculated from eq. (4) explicitly

4>(s) = - (p)q*(s,p)dT + (p)u*(5,p)dT (7)

The velocityfiel dca n be recovered from

(8)

Transactions on Modelling and Simulation vol 23 © Fre1999e WIT and Press, Movin www.witpress.com,g Boundary Problem ISSN 1743-355Xs 14 1

or in a discretised form

by deriving eq. (7). The 2D fundamental solution and its derivatives which appear in eqs. (4) and (8) are */ x 1 , ro u (s,p) = —log— (10)

***(•.!>) J_jg , * ^ ^ '^' A^ ^j o^ ~2 v•*••*•;

v 9U*(5,») j. c,*. /xx, '•"' = -«?f = -2;-y-" (12)

(13) <,c.rt - ^i = i^-v.

Here TO is an arbitrary constant, 5 = (sxi,^) source point, p= (pxnPxz) £ F integration point, r^ = \/(sz, -px,-)(^- "Pi,-) distance between source and integration point, dsp = (5%., - PXJ)^XJ distance between source point and tangent line to the boundary at integration point.

3 Integration

Due to the linear geometry and elliptic fundamental solution, analytical integration is used instead of numerical integration^. Matrices [H], [G], [Hx.] and [Gxj] are composed from nodal contributions of integrals g, {h}, gxj and {/&%,} over the boundary element (see fig. 1)

g = J u*(s,p)dr =^t (logro - log y/a -h brj -f erf*} dr] (14)

= I r]»q*(s, (16) Jr,

(17) r. %- _i a + 677 + C7?2 ^ '

(19)

where ii , 12, 23 are nodes of the discretised model, a^, = (xj^ -f Xj J/2, % - S2, a = 5% - a%ss - o%, 6 = -2^. - a%6%, c = 6 j3 - Xji)(xjs - Xji) element length, and D — —b* + 4ac.

Figure 1: Source point and boundary element.

3.1 Regular integrals

When the source point 5 does not collocate with the nodes of the discretised model, then integrals (14), (16), (17) and (19) are regular. With regard to the position of the source point, two cases are possible:

3.1.1 Point 5 on the tangent line through p

. TO \/a . \fa — \fc g = 2 + log — -0- + ^T= log V V (20) a-c Vc \/o + \/c

fc° = ft*= 0 (21)

v/0- (22) a - c]

(23) V~c(a- a - c

(12o + -

c(a-c) ^ *

3c(a - c)3

; -QS,.)] n^ . A/a-

3.1.2 Point s not on the tangent line through p

^ -2c 64-2c , , g = 2 + — — I arctan — •= -- arctan — •=- J 4- (25) \/JD

2 logro - log V(o + c)2 - 62 + jog 4c a -f o + c

LO 2 / 6-2c h = -- ?= arctan — •== — arctan — ==- (26) VS \ ^ '

1 b ( 6-2c 6 + 2c\ \ 11 aa-6 + c — I arctan — ==- -arcta n — =-I I-- -log lo g - (27) V j2c^^a ^ '

^ ^ _ ^^an ^h: I 4- (28)

a-6 2c ^ a + b + c

, o _ ; + b(s^. - <%%,.) - bbg.. - 2c(s%, - a^)

.j - a,,) + 266,jd,p - On*,} ( b-2c I arc tan V 7)4-2r\ arctan

i ga s H- 2ac(ga.. - Qa - bc(sy. - a% 4- _ * " cD[(a -f c)2 - 62] *' dsp - Dbn.,.+] ( 6 - 2c 1 arctan — %= — ( ou ) \ 6+ 2c\ rix. , a - 6-f c arctan

3.2 Singular integrals

When the source point collocates with node 23, the integrals become singular, so evaluation must be performed carefully, especially with integrals (17) and (19). The procedure is the following: point s is moved into the domain distance e in the negative direction of the unit normal (see fig. 2). In this case it stands as s^ = a^. - en^., and the constants become: a — e^, b = 0 and c — ig/4. Because the integrals are now regular, they are evaluated and the limit is taken for e —> 0.

—en

Figure 2: Source point is moved from boundary into domain.

3.2.1 Weakly-singular integrals

3.2.1.1 s =

9= T(l- -flogro) (31) Z7T

3.2.1.2 s -

(32)

3.2.1.3 5 = 23

9= (1-logLC + logro) (33)

3.2.2 Strongly-singular integrals

^ = ^ = 0 (34)

TJ 1 L/C

|2 7T ^ 2e

(e) = |n=j (36) c—fu ^ 3.2.3 Hyper-singular integrals

+ Lg) arctan ff -

1 1 1 /i^. = limh^.(e) —-—eijHxi (40)

where are: en = 622 = 0, e%2 = 1 and 621 = -1.

4 Example: Free-rising bubble

To illustrate the described method, a free-rising bubble in a pipefille dwit h water is simulated. The diameter and height of pipe are D = 10cm and H - 50cm. Bubble diameter is d = 1cm and its initial position is h = 2cm from the bottom end. The pipe is discretised with 10 horizontal boundary

elements and with 50 vertical boundary elements, while the bubble is described with 40 boundary elements. The potential is approximated with continuous linear distribution over the boundary, while flux d(f>/dn is constant over the boundary element. Boundary condition at the pipe wall is

= 297V/m^. In order to see the velocity field of the liquid phase, 200 internal points are used.

146 Free and Moving Boundary Problems

t = 0.05s

t = 0.045

t = 0.03s

t = 0.02s

t - 0.01s

Figure 3: Bubble shape and boundary velocity.

Free and Moving Boundary Problems 147

_ \ / /^-"\ /

t = 0.05s

- / \ _

t = 0.04s

t = 0.03s

t = 0.02s

t = 0.01s

Figure 4: Velocity field in the liquid phase.

148 Free and Moving Boundary Problems

The numerical procedure is the following:

1. Set initial values for at bubble surface.

2. Evaluate integrals (14), (16), (17), (19).

3. Solve system of equations for 0 and d(j>/dn (5).

4. Explicitly compute velocity at boundary and in domain (9).

5. Use Bernoulli's equation to update potential at bubble surface (2).

6. Move bubble surface (3).

7. Go to step 2.

The results for thefirs tfe w time-steps for bubble shape and boundary node velocities are shown infig .3 . One can see the bubble deformation from the initial circle. Fig. 4 shows the liquid phase velocity distribution around the rising bubble.

5 Conclusion

The paper shows that velocity at the gas-liquid interface can be calculated explicitly. This is true only if constant interpolation for flux is used, while velocity potential can be interpolated with continuous functions. With the proposed method, the full velocity vector is obtained from the Boundary Integral Equation and no numerical derivation of potential is needed to find tangential velocity.

References

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4. Blake, J.R., Boulton-Stone, J.M. & Tong, R.P., Boundary integral methods for rising, bursting and collapsing bubbles, BE Applications

in Fluid Mech., Advances in Fluid Mech., ed. H. Power, Vol. 4, Comp. Mech. PubL, Southampton, pp. 31-71, 1995.

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