Transactions on Modelling and Simulation vol 23 © 1999 WIT Press, www.witpress.com, ISSN 1743-355X 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 interpola- tion for flux are used, then the velocity at the boundary can be calculated Transactions on Modelling and Simulation vol 23 © 1999 WIT Press, www.witpress.com, ISSN 1743-355X 140 Free and Moving Boundary Problems 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<t> 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) = - <t>(p)q*(s,p)dT + (p)u*(5,p)dT (7) The velocity field can 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) Transactions on Modelling and Simulation vol 23 © 1999 WIT Press, www.witpress.com, ISSN 1743-355X 142 Free and Moving Boundary Problems 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 + - Transactions on Modelling and Simulation vol 23 © 1999 WIT Press, www.witpress.com, ISSN 1743-355X Free and Moving Boundary Problems 1 43 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 — == -- arctan — =-I I- - -log log - (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 Transactions on Modelling and Simulation vol 23 © 1999 WIT Press, www.witpress.com, ISSN 1743-355X 144 Free and Moving Boundary Problems 3.2 Singular integrals When the source point collocates with node 23, the integrals become singu- lar, 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 0 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) Transactions on Modelling and Simulation vol 23 © 1999 WIT Press, www.witpress.com, ISSN 1743-355X Free and Moving Boundary Problems 145 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 pipe filled with 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 de- scribed with 40 boundary elements. The potential <f> is approximated with continuous linear distribution over the boundary, while flux d(f>/dn is con- stant over the boundary element. Boundary condition at the pipe wall is d<p/dn = 0 which means that normal velocity is zero. The time-step for simulation is Af = 0.002s. Material properties are: a — 7.25 -lO^AT/m, p — W*kg/m*, gx = 0, #„ = -10m/5*, p - p<x> = 297V/m^. In order to see the velocity field of the liquid phase, 200 internal points are used. Transactions on Modelling and Simulation vol 23 © 1999 WIT Press, www.witpress.com, ISSN 1743-355X 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. Transactions on Modelling and Simulation vol 23 © 1999 WIT Press, www.witpress.com, ISSN 1743-355X 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. Transactions on Modelling and Simulation vol 23 © 1999 WIT Press, www.witpress.com, ISSN 1743-355X 148 Free and Moving Boundary Problems The numerical procedure is the following: 1. Set initial values for <f> at bubble surface. 2. Evaluate integrals (14), (16), (17), (19).