Simulations of Droplets falling on a solid surface Using Phase-Field Method

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1 APCOM & ISCM th December, 013, Singapore Simuations of Dropets faing on a soid surface Using Phase-Fied Method T. Sakakiabara¹, *T.Takaki 1, and M.Kurata 1 Graduate Schoo of Science and Technoogy, Kyoto Institute of Technoogy, Matsugasaki, Sakyo, Kyoto, , Japan. Nucear Science and Engineering Directorate, Japan Atomic Energy Agency, Shirakata, Tokai-mura, Ibaraki, , Japan. *Corresponding author: takaki@ kit.ac.jp Abstract Due to the accident inoing the Fukushima-Daiichi nucear power pants, it becomes necessary to construct a numerica scheme to precisey eauate the process of metdown, incuding phase transformation among soid, iquid and gas phases. In this study, we constructed a mode for gasiquid two-phase fow with a high density ratio. We used the phase-fied method to express a dropet of moten nucear fue fowing down a wa. By performing a dam break simuation using the deeoped mode, we confirmed the mode s aidity. We aso performed a numerica simuation of a dropet faing down a soid surface with wettabiity. The wettabiity was modeed by setting the boundary condition of the phase-fied ariabe. As a resut, we confirmed that the deeoped mode can express the typica characteristics of a faing dropet on a wa. Keywords: Two-phase fow, Phase-fied method, Naier-Stokes equation, Contact ange, Dropet Introduction The deeopment of a simuation mode abe to accuratey eauate the metdown process was made urgent by the accident at the Fukushima-Daiichi nucear power pants. This process incudes phase changes, such as meting and soidification, as we as the faing down of moten materia. In order to simuate gas-iquid two-phase fow incuding phase change, we needed to choose an interface tracking method. The oume of fuid (VOF) method (Tomiyama, Sou, Minamigawa and Sakagushi, 1991; Minato, Ishida and Takamori, 000; Tan, Aoki, Inoue and Yoshitani, 011) and the ee-set method (Osson and Kreiss, 005; Tan, Aoki, Inoue and Yoshitani, 011) are widey used as interface tracking methods for gas-iquid two-phase fow. Since the VOF method uses a sharp interface, appying it to treat compicated morphoogies is difficut. The ee-set method requires re-initiaization of the adection equation, eading to arge cacuation costs. Therefore, in this study we used the phase-fied method as an interface tracking method. The phase-fied method has mutipe adantages; it can automaticay construct an interface of compex shape and express interface migration simpy by soing a time eoution equation. The biggest reason for appying the phase-fied method in this study is that it enabed the expression of phase changes among gas, iquid and soid states using the muti-phase-fied method. The purpose of this study was to construct the gas-iquid two-phase fow mode which can express a faing dropet on a wa using the singe phase-fied method. Numerica mode and cacuation technique Goerning equations of the two-phase fow mode In this study, we constructed a mode to simuate gas-iquid two-phase fow with a high density ratio in order to express dropets faing on a soid surface. The phase-fied method was used as an interface tracking method and was couped with the Naier-Stokes equation for incompressibe fows. In phase-fied method, we used the Cahn-Hiiard equation with an adection term of conseration form, where the phase-fied ariabe φ was regarded as a consered quantity to keep a constant iquid oume. The phase-fied ariabe φ was defined as 0 in the gas phase and 1 in the iquid phase, and it continuousy and sharpy changed from 0 to 1 in the interface region. The Naier-Stokes equation incudes surface tension force and graity force terms, considered body 1

2 force in preious work (Anderson, McFadden, and Wheeer, 1998; Inamuro, Ogata, Tajima and Konishi, 004; Takada, Matsumoto, Matsumoto and Ichikawa, 008; Borcia, Borcia and Bestehorn, 006; Borcia and Bestehorn, 007). The goerning equations of the gas-iquid two-phase fow mode are φ + ( φu ) = M φ η t 1 (1) η = a φ + W φ( 1 φ) φ u = 0 () u ρ + u u = p + µ u + φa φ + ( ρ ρ )g, (3) t where u denotes the eocity ector; M φ, phase-fied mobiity; η, chemica potentia; a, the gradient coefficient; W, the energy barrier; ρ, density; p,pressure; µ, iscosity; and g, the graitationa acceeration ector. The gradient coefficient a and energy barrier W are reated to physica properties by the foowing equations. a = 3δγ b (4) W 6γb = δ (5) Here, δ denotes interface width; γ, the interface energy between gas and iquid; and b, the coefficient to be reated to the interface ( b. ). We assumed that the density ρ and iscosity µ continuousy changed in the interface region, with a change in the phase-fied ariabe, according to foowing equations. ρ = ρ φ + ρ ( 1 φ) (6) µ = µ φ + µ ( 1 φ) (7) The subscripts and represent the gas and iquid phases, respectiey. Numerica scheme The Cahn-Hiiard equation was soed by a perfecty expicit method. The Lapacian discretization of the phase-fied ariabe and chemica potentia was eauated with a fourth-order centra difference scheme. The adection term was eauated with a third-order upwind scheme. Time integration was eauated with a first-order forward difference scheme. The soution of the fow fied was obtained using the SMAC method. The Poisson equation for pressure was discretized by a second-order scheme and the sparse matrix was soed using the SOR method. The diffusion term was eauated with a second-order centra difference scheme. The adection term and the time integration were cacuated with the same scheme as the Cahn-Hiiard equation. Dam break probem The dam break probem is we-known for aidity erification of gas-iquid two-phase fow cacuation code. Therefore, in this study we confirmed the reiabiity of our cacuation code using a two-dimensiona cacuation of this probem. The computationa mode of a water-air system is shown in Figure 1. The initia water coumn width a and height n 3 a were a = m and n a = m (aspect ratio n = ); the same aues were used in a preious experiment by Martine and Moyce (Martine and Moyce, 195). The physica properties were as foows ρ = 1000 kg/m, µ = Pa s, ρ = 1.6 kg/m, µ = Pa s, γ = J/m, δ = 4 x and

3 4 5 M = m /( J s). The attice number was set to The attice sizes, x and y, and the φ 3 6 time increment t were x = y = m and t = s. In a boundaries, we set the foowing conditions: n φ = 0, n η = 0, n u = 0 and n p = 0, where n denotes the norma ector. Equation (8) represents a static contact ange of 90 degrees and Equation (9) represents the fact that fuids cannot pass through the wa boundaries. Figure 1. Computationa mode for the dam break probem. Time ariations of the front coordinate z and height ξ when the water coumn is broken by the graity force are shown in Figure. Here, z, ξ and respectie times are nondimensionaized by =, H ξ /( n a ) Z = z / a, T z nt g / a = and T h t g / a As shown in Figure (a), the front position moed more rapidy in the simuation than in the experiment. This is because the experiment was not perfecty two-dimensiona een though it was performed in a thin region of the thickness dimension. In addition, the initia rectanguar water coumn and its sudden breaking were difficut to express in the experiment. In Figure (b), the time ariation of the height corresponds faithfuy. Athough there are some discrepancies between the numerica and experimenta resuts, we see reasonabe agreement between them and can thus confirm the reiabiity of our cacuation code. =. (a) front (b) height Figure. Time ariations of front and atitudina contact ines in the dam break probem. 3

4 Dropet faing on a soid surface with wettabiity Boundary conditions for wettabiity In order to gie wettabiity to the boundary, we introduced an idea which impements the geometry shown in Figure 3. This mode is different from that used in preious work (Briant, Papatzacos and Teomans, 00); it directy gies wettabiity to the boundary. The contact ange θ formed by the interface energy between soid and iquid and between iquid and gas (back ectors in Figure 3) is geometricay identica to the ange formed by the norma ector n of the wa and the outward interfacia norma ector φ (red ectors in Figure 3). Then, by cacuating the inner product of the two red ectors, the boundary condition giing wettabiity to the eft boundary is gien by the foowing equation. φ = ± x cos θ φ 1 cos θ y if cosθ < 0 then sign otherwise, sign is is + (16) Figure 3. Boundary condition of wettabiity. Numerica conditions and resuts We conducted the numerica simuation of a dropet faing down a soid surface with wettabiity. The eftmost part of Figure 4 shows the computationa domain and initia condition. The eocity of the initia semicircuar dropet was set to zero. The boundary condition of the phase-fied ariabe on the eft wa was set to the Neumann condition considered wettabiity gien by Equation (16).We set the contact ange to θ = 70. The other boundary conditions were identica to those used for the simuation of the dam break probem in the preious section. The other parameters were aso 7 5 unchanged from the preious simuation, with the foowing exceptions: M = m /( J s), 5 9 x = y = m and t = s. φ 4

5 Figure 4 shows the morphoogica changes of a dropet on a soid surface with wettabiity. The upper part of the dropet becomes thin and the ower part expands as time progress. The bottom and side of the dropet become fat. In the simuation, we assumed phase-fied mobiity. If we used a arger aue, the dropet shape woud tend to be round due to the infuence of curature. To perform quantitatie simuation, we needed to accuratey identify the aue of phase-fied mobiity. Time ariations of adancing and receding contact anges are shown in Figure 5. In the eary stage of faing, the contribution from the boundary condition is arger than that from the graity force. Therefore, the adancing and receding contact anges approached the static contact ange θ = 70. With time, because the contribution from the graity force became arger, the adancing contact ange increased and the receding contact ange decreased to minimize the system energy. Figure 4. Computationa domain and morphoogica changes of dropet on a wa surface with wettabiity. Figure 5. Time ariations of adancing and receding contact anges. 5

6 Concusions We constructed a mode for gas-iquid two-phase fow with a high density ratio by using the phasefied method. Using the two-dimensiona dam break probem, we confirmed that our cacuation code can reasonaby simuate gas-iquid two-phase fow with a high density ratio. Next, we conducted a simuation of a dropet faing down a soid surface with wettabiity, where the wettabiity was modeed by setting the boundary condition of the phase fied ariabe. It was obsered that adancing and receding contact anges are changed by the contributions of the boundary condition and the externa force. References Hirt, C. W. and Nichos, B. W. (1981), Voume of Fuid (VOF) Method for the Dynamics of Free Boundaries. Journa of Computationa Physics 39, pp Tomiyama, A., Sou, A., Minamigawa, H. and Sakaguchi, T. (1991), Numerica Anaysis of a Singe Bubbe with VOF Method. Transactions of the Japan Society of Mechanica Engineers, Series (B) 57, pp Takada, N., Misawa, M., Tomiyama, A. and Hosokawa, S. (001), Simuation of Bubbe Motion under Graity by Lattice Botzmann Method. Journa of Nucear Science and Technoogy 38, pp Tan, N., Aoki, T., Inoue, K. and Yoshitani, K. (011), Numerica Simuation of Two-Phase Fow Drien by Rotating Object. Transactions of the Japan Society of Mechanica Engineers, Series (B) 77, pp Osson, E. and Kreiss, G. (005), A conseratie ee set method for two phase fow. Journa of Computationa Physics 10, pp Anderson, D.M., McFadden, G.B., Wheeer, A.A. (1998), DIFFUSE-INTERFACE METHODS IN FLUID MECHANICS. Annua Reiew of Fuid Mechanics 30, pp Inamuro, T., Ogata, T., Tajima, S. and Konishi, N. (004), A attice Botzmann method for incompressibe two-phase fows with arge density differences. Journa of Computationa Physics 198, pp Takada, N., Matsumoto, J., Matsumoto and N., Ichikawa, N. (008), Appication of a Phase-Fied Method to the Numerica Anaysis of Motions of a Two-phase Fuid with High Density Ratio on a Soid Surface. Journa of Computationa Science and Technoogy, pp Borcia, R., Borcia, I.D. and Bestehorn, M. (006), Drops on an arbitrariy wetting substrate: A phase fied description. Physica reiew E 78, Borcia, R. and Bestehorn, M. (007), Phase-fied simuations for drops and bubbes. Physica reiew E 75, Martine, J.C. and Moyce, W.J. (195), An experimenta study of the coapse of iquid coumns on a rigid horizonta pane. Phiosophica Transactions of the Roya Society of London Series A 44, pp Briant, A.J., Papatzacos, P. and Yeomans, J.M. (00), Lattice Botzmann simuations of contact ine motion in a iquid-gas system. Phiosophica Transactions of the Roya Society of London Series A 360, pp