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91:?"=+Immersed Free Surface Method for Simulation of Mesoscale Gas-liquid-solid Multiphase Flows ŎĢó¹, û¹, ûrŲðrùű 7-3-1, E-mail: masahiro@chemsys.t.u-tokyo.ac.jpÄč, û¹, ûrŲðrùű 7-3-1, E-mail: osamu_koike@sogo.t.u-tokyo.ac.jpÅģÆº, û¹, ûrŲðrùű 7-3-1, E-mail: yukiyama@chemsys.t.u-tokyo.ac.jp
Masahiro Fujita, The University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-8656
Osamu Koike, The University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-8656
Yukio Yamaguchi, The University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-8656
An immersed free surface model for a simulation of mesoscale gas-liquid-solid multiphase flows is presented. Using
the model, a contact angle boundary condition on particles at a free surface is implicitly satisfied, so that one cancalculate efficiently menisci and capillary forces among the many particles. A three-way coupling among particle
motion, fluid motion and free surface motion is accomplished using the immersed free surface model in conjunction
with a level set method and an immersed boundary method. An accuracy of the present method is validated through thecomparisons with analytical solutions of the capillary forces exerted on spherical particles at a free surface.
Sij 6.#Ƌ4"0Đ|l#Ûĺ¼#P_9cŮ
6ĝ3
 3ƌCSF [J`#|İR=K` FsƋIBM#ÒR=K` FpƋŴƆ6¢5ƋLLNS ñĮÏ$Eq. (2)Ő43ƌ

A@ 
**!Ąňßĺ¼ŉ Ûĺ¼îđ#²üg´ÊnĚQb
DC"0}Œ43ƌ#³¢ƋÛĺ¼ƋĐ|ƋĊđĤƄ#Ů6ŖÛĺ¼Ľ#āūÕãůĮ6Ŗý3-"$ƋZEC?d`Ċđ«ĒĨĐAX]adA^c ÝŔ !3ƌ$
Đ|#Ŭŀ|Ŧ{ ãĴ3hź"Ŧ1µm vh#C?d`6ZEC?d` ¾Ņ3 Ƌ#Ûĺ¼$Đ|¼#ęì"0P_9cŮ63."ƋÛĺ¼" ¼Ÿ/ƂƀĊ6ņâ!4%!1!ƌ1"ŴŔ!#$ƋĊđĤƄ
"½­3Ûĺ¼ ŐƄÑ6Ƌĺ¼Ÿ#ĉĸĨq}ġ"0Ů3 3ƌZEC?d`$UcLï 9:dN
dï."Ä-ƋÛĺ¼#Ů$m"ĉĸ Ļß"íų43ƌ
ōŇ1$4*"ƋŊģŐƄ6ç!ZEC?d`Đ|«|pĨĐ"Â3AX]adA^c[J`6ŷĥ(1)ƌ
$ immersed boundary method (IBM)(2)6ġ«|ŐƄ#Ĕ2!
µĤúx6żĦ"j3 "02Ƌ¸ï#«|ĺ¼#ĜĦ!
AX]adA^c6ŏƌeñƋĊđ«ĒĨĐ#³¢ƋĊđĤƄ#«|ĺ¼"$¬ Ļß ŐƄ#u"ƋŐƄÑł fĨéŗł" ƌ ¬/Ļß £ĂƋŐƄÑ#µĤúx6żĦ"2æ-"$ƋfĨéŗł"ł3ŐƄÑ6|İ"¶ë3Çº ÝŔ3ƌùĩıêĀ3ŊģŐƄ°-ť+[J`$Ƌ«|ĺ¼Ű"wàĊđĤƄ6āã3 "0ƋfĨéŗł" ŐƄÑ
6wàĊđĤƄ" ŐƄÑ"¶ëƋ1" continuum surface
force (CSF)[J`(3)6ġ|İ"¶ë3ñĎ3ƌ#ñĎ"0ĺ¼Ů ĤƄŮ ñ¤"Ŭã4ƋĐ|Ů6¢5fŬãAX]adA^c ň"!3ƌvh$ùĩıġ143íųñĮÏ ïŘķĎ6řŧØƋs¨êĀ3ŘķĎ ĊđĤƄ#Ûĺ¼" ĉĸ6ćĪ"Řķ 3
6īƌ

!

!v
= "!v # $v " $p + $ # µ ($v + $vT )!t+! g + $ # S + Fs + Fp

(2)

Fp = !!(vp " v) #t(3)Ƌ

!$ĺ¼Ű 1Ƌ4v· 0 !3«ĨŹïƋvp $ĺ¼Ű#ÿ¼Ė#ŪÍR=K`2ƋEq. (4)j143ƌ

vp = V + ! " rp(4) V $ĺ¼#kŭŪÍR=K`Ƌ" $ĺ¼#ŕŪÍR=K`Ƌ
rp $ĺ¼lÜ 1#ĨÂR=K`3ƌSij #Ë®$ 0 2Ƌî$ Eq. (5)j143ƌ

(Sij (r1 ,t1 )Skl (r2 ,t2 ) = 2kBT µ !ik! jl + !il! jk)!! (r1 " r2 )! (t1 " t2 )(5)

 kB $U`HWc¾ïƋ! $=bMG;d#J`FƋ! ( )$J`FŹï3ƌ* level set Ď(5)"ÙƋĺ¼Ű6¥-
Řķƈ±" 3ĊđĤƄ#ĭĐñĮÏ$ Eq. (6)Ő43ƌ

!!
+ v " #! = 0!t(6)eñƋĺ¼#kŭ 0&¨ţŮ$ Eq. (7)Ő43ƌ

mB@%>(4$&;6+*ĊđpĨĐ6ÁÍÊ#3ƃ¬ńßĐ| ņƋšŵ
½  Eq. (1)#Ŭŀ#Ï6ġ3ƌ
(1)!"v = 0ZEC?d`Đ|#ŮñĮÏ ƃ¬ń Landau-Lifshitz
Navier-Stokes (LLNS)ñĮÏ(4)6èġ3ƌLLNS ñĮÏ$ NS ñĮ
Ï# Ť"Đ|¼#ęì"03ĪĜĦìÞIcE` S =dV= Fc + Fh + Fv + Fedt(7a)!
d!
= Tc + Thdt(7b)IFc Tc $44éŗR=K` éŗK`=R=K`Ƌ
Fh Th $44Đ|R=K` Đ|K`=R=K`ƋFv
Fe $44 van der Waals  ƂƀĊ3ƌƋ
1Copyright © 2012 by JSFM

5 26

Fh = ! # Fp ! "( !p ! ! ) dV}(8a)

Th = ! # rp " Fp dV

((8b)

V{V)

(a) t = 0 µ s

"p $ĺ¼ÁÍƋV $ĺ¼ƈ±6īƌ* Fc Tc $ discreteelement method (DEM)(6)ġ143ĻÓß ƋFv Fe $Derjaguin-Landau-Verwey-Overbeek (DLVO)ğŜ(7)"² 2 |ŸVIcA\` j143ƌ
¡õŸCIGQ#ïŘķĎ6vh"īƌ* Řķƈ±
Eq. (6)6ŖaR`DGKŹï6ĭĐ3ƌ
ą"ĺ¼Ű#aR`DGKŹï6ćwàĊđĤƄ6āã3(8)ƌ
wàĊđĤ
Ƅ$ĺ¼ŐƄéŗŕúx6ēƋĺ¼Ű"}143öƄ4%Ƌyá#ÕěŌƌ#ØƋ Řķƈ±aR`DGK
Źï6øƋ÷Ø"đĨ|İ ½30aR`DGK
Źï6ć3ƌvg#aR`DGKŹï#ĭĐ"$ƋĲŸÈ6 5 ąļÍ weighted essentially non-oscillatory (WENO)C<dY(9) ƋõŸİ6 3 ąļÍ total variation diminishing (TVD)
Runge-Kutta Ď(10)6ġŖ ƌ#0"ĺ¼ŐƄ#éŗŕúx6ēaR`DGKŹï#Ê Ú141ƋŦ{Heaviside Źï6ġĺ¼Ű6¥, ƈ±đĨ|İĜ6ŘķƋđĨ|İĜ 1Đ|#ÁÍ Ļß~ï#Ê6Č-3ƌ! Ƌ!#Ê6Č-3Ž".aR`DGKŹï Ŧ{ Heaviside Źï6ġ3ƌ#Ø simplified marker and cell (SMAC)Ď(11)6ġ Eq.
(1) Eq. (2)6ŖŪÍ³ ¬³6Ú3."ƋEq. (7)6Euler explicit ĎŖĺ¼ŪÍ6Č-3ƌvg#åƇ"0ƋwàĊđĤƄ" ŐƄÑ Eq. (3) Eq. (8)6ũ Eq. (7)"Ã43-Ƌĺ¼" ĉĸ Ŋė"Řķ43ƌ! ƋŘķƈ±$ħñ| ƋeŤ#Ŷ ĺ×# 1/10 #Ĵñ|ŘķD`ƈ±63ƌ#ĊđĤƄ"Ëŏ!ñ¤"$¦øµĤúx6śƌ(b) t = 1.5 µ s(c) t = 2.5 µ sC@  7)
*ùĩıêĀ3ŊģŐƄ°-ť+[J`"0ƋĊđĤƄ ĺ¼ŐƄ#éŗŕ"ÞŮ3 6īƌFig. 1 "ī
#$ƋÁÍĈ 100 #ĊđĤƄ6đĨ 1ĊĨ"ĺ× d = 1µmƋ
éŗŕ# = 30#ĞÕĺ¼ Šũ3AX]adA^c3ƌ$ĺ¼#ŮñĮÏ$Ŗ Ƌĺ¼" 1m/s #g¤ e¾ŪÍ6j3ƌªl#ƁŨô#Ƅ$$ = 0 #ĶƄ2ƋĊ
đĤƄ6ŐƌFig. 1(b)"ī0"ƋĊđĤƄ 1ĺ¼ ĳ 
3 Ƌĺ¼ŐƄ#éŗŕúx6ē0"ĤƄ$¶Õ3ƌ#Ø Fig. 1(c)"ī0"Ƌĺ¼#ĭ"4ĤƄ$Ð Î%
43 Ƌ"$ Fig. 1(d)"ī0"ƋÐ Î%4ĤƄ ő3ƌĤƄ őØ$ Fig. 1(e)"ī0"ƋĊđĤƄ$.
#ËƄÕě"ä3ƌ#0"ĺ¼ŐƄ#éŗŕúx6ũƋ
ĺ¼Ů ĊđĤƄŮ"Öƅ6)
Ī -14ƌeñƋĊđĤƄŮ$ĺ¼" ĉĸ6ũĺ¼Ů"Öƅ6)ƌ#-ą"Ƌĺ¼ ĤƄ6ũů3Ž"ĤƄ"¯ħ!ñ¤"ĺ¼" Ńĉĸ6Řķ3ƌFig. 2 "ī0"ƋƂĆĞÕĺ¼#ċË¹"fĨéŗł ()eŋ3ƋĂ!éŗŕ#"Â3g¤ Ńĉĸ6ī>_O Fig. 3 3ƌ#Ńĉĸ#ŖýŖ$ Eq. (9)Ő43ƌ

Fv = !" d cos #&,#  </2 C04-6(d) t = 2.64 µ s(e) t = 3.5 µ s(9)%$ŐƄÑ3ƌ>_O"ī0"Ƌéŗŕ#ÌĹ
©"5Ƌĺ¼" Ńĉĸ#Řķ$ŖýŖ '7Ō eŋ3ƌFig. 1 Free surface deformation caused by a penetrating sphere.
2

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5 26Fv&,#  </2 C04-6

LFlFig. 2 Free surface deformation caused by a protruded sphere.

Fig. 4 Free surface deformation caused by two protruded spheres.

Fig. 3 Vertical capillary force exerted on a protruded sphere.

Fig. 5 Lateral capillary force exerted between two protruded spheres.

ĺ¼ŐƄ#éŗŕúx"0ĺ¼*52#ĊđĤƄ ¶Õ3-Ƌœï#ĺ¼ ½­3³¢"$¶ÕĊđĤƄ Ŵ!
2¢Ƌ¡ĺ¼*52#ĊđĤƄÕě ƃÂĬ !3ƌ#ƃÂ
Ĭß"0£įĺ¼#³¢"$ăĉĸÐ  ƋĤƄgĺ
¼#ŊÉž¢ĝŞ ĥĠ3ƌFig. 4 "ī0"ƋƂĆĞÕ
ĺ¼#ċË¹"éŗŕ# = 60˚#fĨéŗł eŋ3Ƌĺ¼lÜŸŢſ L "Â3ăĉĸÐ#¹ 6ī>_OFig. 5 3ƌ d/2 << L #ƋăĉĸÐ#Ŧ{ŖýŖ$ Eq. (10)j143(12)ƌ

(2) Kajishima, T., Takiguchi, S., Hamasaki, H. and Miyake, Y., :“Turbulence Structure of Particle-Laden Flow in a Vertical Plane
Channel Due toVortex Shedding”, JSME Int. J. Ser. B, 44 (2001),
pp. 526-535.
(3) Brackbill, J. U., Kothe, D. B. and Zemach, C., "A ContinuumMethod for Modeling Surface Tension," Journal of ComputationalPhysics, 100, (1992), pp. 335-354.
(4) Landau, L. D. and Lifshitz, E. M., "Fluid Mechanics," Pergamon
Press, London, (1959).
(5) Chang Y. C., Hou, T. Y., Merriman, B. and Osher, S., "A Level Set
Formulation of Eulerian Interface Capturing Methods for
Incompressible Fluid Flows," Journal of Computational Physics,
124 (1996), pp. 449-464.
(6) Tsuji, Y., Tanaka, T. and Ishida, T., "Lagrangian numericalsimulation of plug flow of cohesionless particles in a horizontal
pipe," Powder Technology, 71 (1992), pp. 239-250.
(7) Israelachvili, J. N., "Intermolecular and Surface Forces," AcademicPress, London, (1992).
(8) Fujita, M., Koike, O. and Yamaguchi, Y., "Computation ofCapillary Interactions among Many Particles at Free Surface,"
Physical Review Letters, (submitted).
(9) Jiang, G. -S. and Peng, D., "Weighted ENO Schemes for Hamilton
Jacobi Equations," SIAM Journal on Scientific Computing, 21(2000), pp. 2126-2143.
(10) Shu, C. -W. and Osher, S., "Efficient Implementation of EssentiallyNon-oscillatory Shock-Capturing Schemes," Journal ofComputational Physics, 77 (1988), pp. 439-471.
(11) Amsden, A. A. and Harlow, F. H., "The SMAC Method: A
Numerical Technique for Calculating Incompressible Fluid Flows,"
Los Alamos Scientific Laboratory Report LA-4370 (1970).
(12) Kralchevsky, P. A. and Nagayama, K., "Particles at Fluid Interfaces
and Membranes," Elsevier, Amsterdam, (2001).

Fl = 0.5!" d 2 cos2 # L(10)

>_O"ī0"ƋL Ä³¢6ŻƋĺ¼" ăĉĸ#Řķ$ŖýŖ 0 eŋ3ƌL Ä³¢#ieŋ$Ŧ{ŖýŖ ćĪ! !3- ņ143ƌD@ ĵī0"ƋŊģŐƄ°-ť+[J`6ġŘķĎ"02ƋĊđĤƄ#ĺ¼" ĉĸ ćĪ"Řķ 3
ĪŚ4ƌ ĉĸ6¥,ĺ¼#ŮñĮÏ6Ŗ "0Ƌĺ¼ŮƋĐ|Ů 0&ĤƄŮ#fŬãAX]adA^c ćĪ"¿ĝ 3 ņ143ƌ$#AX]adA^cĿþ6ī! ƋŝĕÔò"ĤƄg#Ûĺ¼#ĉĸ"03ŊÉž¢AX]adA^c6ľt3o¾3ƌ
! Ƌùĩı$ƉşƊŁ¢ĩı»z Ů§3ZEAX]adA^cƍ@cEdA8Y 2011 #ĩıŷĥ#eŰ ŏ54
3ƌ@cEdA8Y#ďÀ"$TdYSdBƉhttp://nanotech.t.u-tokyo.ac.jp/Ɗ6Ę4ƌ
8'0
(1) Fujita, M. and Yamaguchi, Y. "Simulation Model of Concentrated
Colloidal Nanoparticulate Flows," Physical Review E, 77 (2008),026706.

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