(Koji Kawasaki) Department of Civil Engineering, Graduate School of Engineering Nagoya University 1.,.,,,,,.,,,,,,,.,,,,.,,,,., (19

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$1673 2010 77-92 77 (Koji Kawasaki) Department of Civil Engineering, Graduate School of Engineering Nagoya University 1,,,,,,,,,,,,,,,,,,,,,, (1996 1999) $\sim$ VOF (Volume OfFluid), CADMAS-SURF$

1 (Koji Kawasaki) Department of Civil Engineering, Graduate School of Engineering Nagoya University 1,,,,,,,,,,,,,,,,,,,,,, ( ) $\sim$ VOF (Volume OfFluid), CADMAS-SURF (SUper Roller Flume for Computer Aided Design of MAritime Structure), ( , ) 2 /3 $\sim$ $\sim$ DOLPHIN-2D/3D (Dynamic numerical model Of multi-phase flow with Hydrodynamic INteractions-2/3 Dimension version) 2 VOF 21 2,,, ( ),,, 21, (a), (b) 2 21

2 78,,,,,, ALE (Arbitrary Lagrangian Eulerian), BFC (Boundary-Fitted Coordinate),,,,,, (1), (2), (3) (1),,,, (2),, ( ),,,, 3,, (2), 3, (3),,,,, Nichols et al (1980), Hirt and Nichols(1981) VOF (Volume Of Fluid), VOF,,,, (1997), Youngs(1982) PLIC(Piecewise Linear lnterface Calculation),, MARS (Multi-interface Advection and Reconstruction Solver),, Yabe and Aoki(1991) CIP (Cubic Interpolated Propagation, Constrained Interpolation Profile), Sussman et al (1994) (Level Set ) Level Set, ( ) VOF $\sim$, VOF, VOF CADMAS-SURF (, 2001 ;, 2008) 22 VOF 221 2, (21), Navier-Stokes (22), (23), VOF (24),, (21), (23), (24), $\frac{\partial u}{\partial \mathfrak{r}}+\frac{\partial w}{\ }=q$ (21) $\frac{\partial u}{\partial}+u\frac{\partial u}{\partial \mathfrak{r}}+w\frac{\partial u}{\partial z}=-\frac{1}{\rho}\frac{\phi}{\ }+V( \frac{\partial^{2}u}{\partial \mathfrak{r}^{2}}+\frac{\partial^{2}u}{\ ^{2}}1$ (22)

3 $\gamma$ 79 $\frac{\delta v}{\partial}+u\frac{\partial w}{\partial\kappa}+w\frac{\partial w}{\ }=-g- \frac{1}{\rho}\frac{\phi}{\ }+ \nu(\frac{\partial^{2}w}{\partial x^{2}}+\frac{\partial^{2}w}{\ ^{2}}1+ \frac{1}{3}\nu\frac{\partial q}{\partial z}-$ (23) $\frac{\partial F}{\theta}+\frac{\partial(uF)}{\partial\kappa}+\frac{\partial(wF)}{\ }=Fq$ (24) $q=\{\begin{array}{ll}q^{*}(z,t)/\delta\kappa_{s} (x=x_{s})0 (x\neq x_{s})\end{array}$ (25), $x$, $z$ $u,$ $w$ $x$, $z$ $t$, $p$, $g$, $\rho$, $v$, $\gamma$, $0$ $q$, (25) (, 1998) $q^{r}$ $\Delta \mathfrak{r}_{s}$ $x=x_{s}$, $x=x_{s}$ $x$ ,,,, Navier-Stokes (22), (23),, (21),, SOLA (numerical SOLution Algorithm for transient fluid flow) (Hirt et al, 1975 ;, 1998),, VOF $F$,,,, 223 VOF (a)vof, VOF $F$, $F$, $\frac{df}{dt}=\frac{\partial F}{\theta}+u\frac{\partial F}{\partial\kappa}+w\frac{\partial F}{\ }=0$ (26) $F$, $F=0$, 1,, (26) $F$, (26), $F=0$ $F=1$ $u,$ $w$, $0$ 1,, (26),, $F=0$ $F=1$, (26) $F$

$80, $F$, (21) (26), (24),, (25), Nichols et al(1980) Hirt and Nichols(1981) VOF, $Fq$, (24),, $F$ (VOF ), $F=0$, $F=1$, $0<F<1$, 22, VOF, VOF $0<F<1$,, VOF 22, $\check{}$ (b)donor-acceptor,, VOF, VOF$

4 80, $F$, (21) (26), (24),, (25), Nichols et al(1980) Hirt and Nichols(1981) VOF, $Fq$, (24),, $F$ (VOF ), $F=0$, $F=1$, $0<F<1$, 22, VOF, VOF $0<F<1$,, VOF 22, $\check{}$ (b)donor-acceptor,, VOF, VOF (24) donor-acceptor donor-acceptor, VOF $F$ donor ( ) acceptor ( ) $F$,,, VOF $F$,, VOF $F$, VOF $F$ 23(a), donor, VOF $F$ VOF $F$, 23(b) donor, VOF $F$ acceptor VOF $F$,,, donor, $23(b(3))$, donor,,, donor $23(b(4))$, $\check{}$

$81 VOF $F$ (27) $F_{i,k}^{n+1}=F_{i,k}^{n}-( \frac{rx_{i+i/2,k}-rx_{i- 1/2,k}}{\Delta\kappa_{i}}+\frac{RZ_{i,k+1/2}-RZ_{i,k- 1/2}}{\Delta z_{k}}-f_{i,k}^{n}q_{i,k}^{n}\delta l)$ (27) $RX_{i,k}=$ sign$

5 81 VOF $F$ (27) $F_{i,k}^{n+1}=F_{i,k}^{n}-( \frac{rx_{i+i/2,k}-rx_{i- 1/2,k}}{\Delta\kappa_{i}}+\frac{RZ_{i,k+1/2}-RZ_{i,k- 1/2}}{\Delta z_{k}}-f_{i,k}^{n}q_{i,k}^{n}\delta l)$ (27) $RX_{i,k}=$ sign $(u_{i,k}^{n+1}) \cdot\min k_{\lambda D} u_{i,k}^{n+1}\delta t +CFX,$ $F_{D}\Delta r_{d}\}$ (28) $\ovalbox{\tt\small REJECT}$ $=$ sign $($ win, $k+l)\cdot mink_{ad} w_{i,k}^{n+1}\delta t +CFZ,$ $F_{D}\Delta z_{d}\}$ (29) $CFX= \max\iota 1-F_{\Lambda D}) u_{i,k}^{n+1}\delta l -(1-F_{D})\Delta\kappa_{D},$ $0\}$ (210) $CFZ= \max\iota 1-F_{\Lambda D}) w_{i,k}^{n+i}\delta t -(1-F_{D})$ Az $D 0\}$ (211) (28), (29) mi-n donor, (210), (211) $\max$ donor donor $D$ $\Lambda$,, $AD$ donor $RF$, acceptor donor $D$, donor, $AD$, donor, $AD$ acceptor (c) $\gamma$, 24, $x$, $z$ $0$ 1,, $\emptyset$, VOF $0$ 23 CADMAS-SURF,, CADMAS-SURF,,, (, $2007a$ ; $2007b)$, CADMAS-SURF, (1999), (2001), (2002), (2008)

$26(a),,, 26(b),,,,, $10^{-2}m^{3}/m/s$, $10^{A}m^{3}/m/s$, (, 2004) $10\cross 10^{-3}m^{3}/m/s$,,, 233 27, 28,$

6 , 1/30, $EL+534m$ 22, 22,, 1 2, VP-DONOR, VP-DONOR $=05$,,, VOF $F$,, ,, $H=45m$, $T=12Os$ 26 26(a),,, 26(b),,,,, $10^{-2}m^{3}/m/s$, $10^{A}m^{3}/m/s$, (, 2004) $10\cross 10^{-3}m^{3}/m/s$,,, , 28, $50m$,, 3

$$\text{ _{}20}^{25}$ $ $ 83 $v$ $-$ $-$ $-$ $\cup-\cdot\cdot-$ : $-\cdot-\cdot--\cdotarrow\lambda t$ (a) $-*$ 15 26$

7 $\text{ _{}20}^{25}$ $ $ 83 $v$ $-$ $-$ $-$ $\cup-\cdot\cdot-$ : $-\cdot-\cdot--\cdotarrow\lambda t$ (a) $-*$ (b) 27 50, $H=62m$, $T=1539s$,,,,,,, 3 lom 28, (a), (b), (c),, 10 $X10^{-3}m^{3}/m1s$, $EL+66m$, $EL+75m$, $EL+66m$,,,

8 84 (a) $+$ (b) (c) 28 CADMAS-SURF,, 23,,,,,,,, (2004),, lom,,, 23,, $EL+75m$, $EL+8Om$, $EL+75m$,,,,,,,,,,,,

9 : 85,,,,,,, CADMAS-SURE 3 DOLPHIN, DOLPHIN, 31, (31), (32), (33), (34), (35) $\frac{\partial\rho}{\partial t}+\nabla\cdot(\mu_{l})=0$ (31) $\frac{\partial u}{\partial t}+u\cdot\nabla u=-\frac{1}{\rho}\nabla p+f$ (32) $\frac{\phi}{\partial t}+u\cdot\nabla p=-\kappa_{s}^{2}\nabla\cdot u$ (33) $\frac{\partial\phi,}{\partial t}+u\cdot\nabla\phi_{j}=0$ (34) $\rho=f(p)$ $\phi_{1}$, $\rho$, $u$, $p$, $F$,, $C_{s}$ $t$,,, $I$ ; $\phi_{2}$, :, : ), $A^{+}h^{+\phi=1}(0\leqq\phi_{J}\leqq 1)$ (35) 32,,, (31), (32), (33), C-CUP (CIP-Combined Unified Procedure) (Yabe and Wang, 1991),,, $arrow$ (Yabe, 1997),,,,, C-CUP, SMAC(Simplifed Marker and Cell),, $\frac{\partial\rho}{\partial t}+u\cdot\nabla\rho=0$ (36)

10 $\frac{\partial u}{\partial t}+u\cdot\nabla u=0$ 86 (37) $\frac{\partial p}{\partial t}+u\cdot\nabla p=0$ (38) $\frac{\rho^{n+1}-\rho}{\delta t}=-\rho\nabla\cdot u^{n+1}$ (39) $\frac{u^{n+1}-u}{\delta t}=-\frac{1}{\rho}\nabla p^{n+1}+f$ (310) $\frac{p^{n+1}-p}{\delta t}=-\rho C_{s}^{2}\nabla\cdot u^{n+1}$ (311) $\Delta t$,, $n+1$ (n l) $+$ $*$ $\Delta$ $t$, 31, DOLPHIN, CIP (Yabe and Aoki, 1991), SMAC,,, Brackbill et al (1992) CSF (Continuum Surface Force),, LES (Large Eddy Simulation), $U_{l}$, $\emptyset/$ $\phi_{2}$,, $\phi_{7l}$, $\phi_{3}$ $U_{l}$ $U_{l}$ CIP, (35), $C_{s}$,, 33 SMAC (39) (311), $\sim$ $\tilde{u}$, SMAC,,

11 87 $\tilde{u}$,,, $\frac{\tilde{u}-u^{*}}{\delta t}=-\frac{1}{\rho^{s}}\nabla p^{s}+f$ (312) $\nabla\cdot u^{n+1}$ (310) (312) (311),, (313) $\nabla\cdot(\frac{1}{\rho^{*}}\nabla\delta p)=\frac{1}{p^{r}c_{s}^{2}\delta t^{2}}\delta p+\frac{1}{\delta t}\nabla\cdot\tilde{u}$ (313), $\delta$p pn $=$ $+$l-p $*$ (313), $C_{s}$,,,,,, (313) $\delta p$, $u^{n+1}= \tilde{u}-\frac{\delta t}{\rho^{r}}\nabla\phi$ (314) $p^{n+1}=p^{n}+\phi$ $\rho^{n+1}=\rho^{r}-\rho^{r}\nabla\cdot u^{n+1}\delta$ (315) (316) 34, Xiao et al(1997),, $\psi$ l, 1 $\emptyset$1 (317),,,, $\Omega$l (318), (319),, $du/dt$, Newton 2, $4= \sum_{l=1}^{l}a_{l}\leq 1$ (317) $\frac{dv_{l}}{dt}=\frac{1}{m_{l}}\int_{v}\frac{du}{dt}\phi_{1l}p_{sl}dv$ (318) $\frac{d\omega_{l}}{dt}=\frac{1}{i_{l}}\int_{v}r_{l}\cross\frac{du}{dt}fi_{l}\rho_{sl}dv$ (319) $R_{l}=x-x_{0l}$ (320) $U_{l}=V_{l}+\Omega_{l}R_{l}$ (321)

$\Delta$ X 88 35 DOLPH $ N$,, 5Om $\cross 5Om$, $x,$ $z$ $=\Delta z=005m$ lom, 2Om, $t$ 0$0001s$, $\rho$ w $9988kym^{3}$, $120kym^{3}$, $p_{a}$ $1013hPa$, $\sigma$ 72X $10^{-2}N/m$, $g$

12 $\Delta$ X DOLPH $ N$,, 5Om $\cross 5Om$, $x,$ $z$ $=\Delta z=005m$ lom, 2Om, $t$ 0$0001s$, $\rho$ w $9988kym^{3}$, $120kym^{3}$, $p_{a}$ $1013hPa$, $\sigma$ 72X $10^{-2}N/m$, $g$ $980665m/s^{2}$,, slip 32,,,,,,, $-21s$, $1013hPa$,, 33,, Martin and Moyce(1952),, DOLPH $ N$ 361, 34, 3,,,, ( ) 0 $75m$, 0 $2m$ 05sin(2 $\pi$t), 0 $5m$, $y$ $800kym^{3}$ $($ $025m\cross$ $0225m\cross$ 0 $5m)$

$89, $Om/s$, $y$ $1$ )$\Gamma^{-0025m}$, $x,$ $z$ 34,, r 2Om 35m $\sim$ $\Delta$ $\Delta$ -00125m, -0025m, $z$, $z=0om\sim 1Om$, $\Delta$ 1 $0m\sim 15m$ -00125m, 0$025m$, 00001 $s$, $\rho$ w$

13 89, $Om/s$, $y$ $1$ )$\Gamma^{-0025m}$, $x,$ $z$ 34,, r 2Om 35m $\sim$ $\Delta$ $\Delta$ m, -0025m, $z$, $z=0om\sim 1Om$, $\Delta$ 1 $0m\sim 15m$ m, 0$025m$, $s$, $\rho$ w $10000kym^{3}$, $120kym^{3}$, $\sigma$ $72\cross 10^{-2}N/m$, $g$ $980665m/s^{2}$, $p_{a}$ $1013hPa$, slip 1 $($ $t=06s\sim 12s)$, $-18s$ 2 $t=23s$,, $(_{\mathcal{y}}=05)$ $y$ x-z 35,, $t=23s$,, 362,, (2006) 2 3,,

90 36 37 36,,, 0 $6s$,, 37 0 $4s$ 1 $-$ Pl,, P3, 0 $6s$ 10,,

14 ,,, 0 $6s$,, 37 0 $4s$ 1 $-$ Pl,, P3, 0 $6s$ 10,, 0$04s$,,, slip,,, 3 2, DOLPHIN,, DOLPHIN, 4,,,,,,, $\check{}$,,

91 (1999) : -VOF,, 15, pp321-326 (2004) :, pp 270-298 (2004) :, pp 3-27-3-29 (1998):,, $186p$ (1996):3 Spilling,, 43, pp96-100 (1997) : 2,, 44, pp81-85 (1998) :,, 45, pp131-135 (2001) :,, 48, pp

15 91 (1999) : -VOF,, 15, pp (2004) :, pp (2004) :, pp (1998):,, $186p$ (1996):3 Spilling,, 43, pp (1997) : 2,, 44, pp81-85 (1998) :,, 45, pp (2001) :,, 48, pp (2008):Lagrange Bingham 2,, 55, pp36-40 (2007a) :,, 23, PP (2007b) :,, 54, pp (2002):3,, 49, pp56-60 (2005):2,, 52, pp (2007):3 DOLPHIN-3D,, 54, pp31-35 (2007) : DOLPHIN-2D/3 $D$,, 23, pp (2006) :,, 53, pp (1997):, (B ), 63, 609, pp (2005) :,, $144p$ (2001) : (CADMAS-SURF), No 12, $457p$ (2008):CADMAS-SURF, No30, $368p$ (2002) : [ ] (CADMAS-SURF),,

16 92 Brackbi] $]$ No705/II-59, $pp1-17$, J U, D B Kothe and C Zemach (1992) : A continuum method for modeling surface tension, $J$ ouma] of Computational Physics, Vol 100, pp Brorsen, M and J Larsen(1987) : Source generation of nonlinear gravity waves with the boundary integral equation method, Coastal Engineering, Vol11, pp Hinatsu, M (1992) : Numerical simulation of unsteady viscous nonlinear waves using moving grid system fitted on a free surface, Jour Kansai Soc Naval Architects Japan, No217, pp 1-11 Hirt, C W and B D Nichols(1981) : Volume of fluid (VOF) method for the dynamics of free boundaries, Journal of Computationa] Physics, Vol39, pp Hirt, C W, B D Nichols and N C Romero(1975) : SOLA : A numerical solution algorithm for fluid flows, Los Alamos Scientific Laboratory, Repoit LA-5852, $50p$ Kawasaki, K(1999) : Numerical simulation of breaking and post-breaking wave deformation process around a submerged breakwater, Coastal Engineering Journal, Vol41, Nos3&4, pp Kawasaki, K (2005): Numerical Model of 2-D Multiphase Flow with Solid-Liquid-Gas Interaction, Intemational Journal of Offshore and Polar Engineering, Vol15, No3, pp Kawasaki, K and K Iwata(1996) : Numerical analysis of wave breaking due to submerged structure, Proceedings of 6th Intemational Offshore and Polar Engineering Conference, VolIII, pp Kawasaki, K and K Iwata(1998) : Numerical analysis of wave breaking due to submerged breakwater in three-dimensiona] wave field, Proceedings of the 27th Intemational Conference on Coastal Engineering, Vol 1, pp Martin, J C and W J Moyce (1952) : An experimental study of the collapse of liquid columns on a rigid horizontal plane, Philos Trans Roy Soc London Ser $A$, Vol244, pp , 1952 Nichols, B D, C W Hirt and R S Hotchkiss(1980) : SOLA-VOF-A solution algorithm for transient fluid with multiple free boundaries, Report LA-8355, Los Alamos Scientific Laboratory, University of Califomia, $119p$ Sussman, M, P Smereka and S Osher(1994) : A level set approach for computing solutions to incompressible twrphase flow, Journal of Computational Physics, Vol114, pp Xiao, F, T Yabe, T Ito and M Tajima(1997): An algorithm for simulating solid objects suspended in stratified flow, Computer Physics Communications, Vol102, pp Yabe, T(1997) : Unified solver CIP for solid, liquid and gas, Computational Fluid Dynamics Review, pp 1-16 Yabe, T and T Aoki (1991): Universal solver for hyperbolic equations by cubic-polynominal interpolation I one-dimensional solver, Computer Physics Communications, Vol66, pp Yabe, T and P-Y Wang(1991): Unified numerical procedure for compressible and incompressible fluid, Journal of The Physical Society of Japan, Vol60, No7, pp Youngs, D L(1982): Numerical methods for fluid dynamics, Academic Press, pp

Web Two-phase Flow Analyses Using Interface Volume Tracking Tomoaki Kunugi Kyoto University 1) 2) 3)

Web Two-phase Flow Analyses Using Interface Volume Tracking Tomoaki Kunugi Kyoto University 1) 2) 3) Web 11 3 2003 8 Two-phase Flow Analyses Using Interface Volume Tracking Tomoaki Kunugi Kyoto University E-mail: kunugi@nucleng.kyoto-u.ac.jp 1) 2) 3) Lagrangian 4) MAC(Marker and Cell) 5) (VOF:Volume of