臺灣博碩士論文加值系統

本研究最主要的目標是發展一套可用於非結構性網格中的含相變化的兩相流數值方法。基於流體體積法(VOF)，本文提出兩套相關的計算方法。第一種方法稱為通量混合介面捕捉法(FBICS)，其是透過直接求解流體體積分率的傳輸方程式來捕捉介面的運動。在計算的過程中，為了同時維持介面的鮮明度及體積分率的界限性，本研究是使用通量混合的方式來處理網格控制面上所需要的對流通量，然而本方法的缺點在於其所獲得的介面通常佔據了多個網格的寬度。另外一套方法則稱為守衡內插介面追蹤法(CISIT)，本方法是先以內差的方式重建介面後，再以先預測後修正的方式處理介面的移動。此方法計算後所得的流體體積分率分布，除了介面所在的網格外，其餘皆是0或1的均勻分布，而且介面只有佔據一個網格的寬度。不同於PLIC法，由於CISIT法的方法相當簡單，因此可輕易地在幾何外觀較為複雜的非結構性網格中使用，而且不需任何複雜的處理便可推展至三維的問題之中。在PLIC法中，其介面重建的方式並非簡單容易處理的，而且在計算面上體積通量時必須考慮許多類型的介面形狀(二維流場中有16種情形而三維則有64種)來進行，因此也造成其介面推移的計算上相當複雜。經由許多問題的測試，可知本研究的兩套數值方法於自由表面流中模擬後所得的結果與理論解或實驗數據都相當吻合並且可在非結構性的網格中所使用。
接著為了能夠模擬含相變化的兩相流問題，本研究特別將相變化所產生的質熱傳效應導入守衡內插介面追蹤法之中。經由介面處質量及能量的跳躍邊界條件可獲得通過介面的質傳量並將此質傳量以源項的方式導入連續方程式之中。然後，對於溫度場而言本研究是將介面視為一組能量內邊界的方式來處理，並且最後將以全隱性的求解方式來計算能量方程式。此外，基於假設介面處溫度及熱傳量連續性的條件來取代跳躍邊界，本方法亦可推展至模擬無相變化的雙流體熱傳問題。將本方法應用於鄰近臨界壓力的水平平板薄膜沸騰問題中，其結果顯示在不同的壁過熱溫度下將有不同的沸騰模式出現。根據不同的壁面過熱度，沸騰的模式主要可分為五種：單氣泡模式( )、單/多氣泡模式( )、單噴流模式( )、雙氣泡模式( )及雙噴流模式( )。在單氣泡模式中，模擬所得的時均化Nusselt數與半經驗式之結果相當一致。再者，透過模擬水平圓管的薄膜沸騰問題可以證明本文所提出的含相變化之兩相流數值方法可使用於複雜幾何外觀的沸騰流動中。
本文最後將含相變化的守衡內插介面追蹤法修改為可用於計算三維兩相流的流場之中。不同於二維的流場，在網格中本研究是使用多個不共面的三角形介面來進行三維介面的重建。本方法首先透過許多不含相變化的雙流體問題來驗證其在三維流場中對於介面追蹤及預測的能力。另外，也將本方法應用至三維水平平板的薄膜沸騰流動之中。結果顯示利用本方法模擬所獲得的時均化Nusselt數與Klimenko所提出的半經驗公式之結果相當吻合，尤其是在壁過熱度為10℃的情況下。最後，無相變化的熱傳模型也用於模擬油槽內的熔解錫液滴及水中的高溫辛烷噴流之問題中。

This paper is aimed at developing a numerical method for two-phase flows with phase change on unstructured grids. In this article, two schemes are presented based on VOF (volume-of-fluid) method. The first scheme is to capture the interface by solving the advection equation of the volume fraction directly, termed as FBICS. In order to maintain the sharpness and boundedness of the interface, the convective flux through each cell face is determined by means of flux blending. The weakness of this method is that the interface region will occupy several grid spaces. In the other scheme (termed as CISIT), the interface is reconstructed first using interpolation practice, following by a predicted-correction procedure to handle the movement of the interface. Except for the interface cells, the VOF distribution is uniform, either in 1 or 0, and the interface occupies only one cell in its width. Unlike the PLIC method, the CISIT can be easily extended to unstructured grids with arbitrary geometry and 3-D problems without causing any further complication because its formulation is very simple. In PLIC, the reconstruction of the interface is not straightforward and the procedure to advance the interface is complicated because a large number of interface configurations (16 configurations for 2-D flows and 64 for 3-D flows) must be considered for determining of the flux across cell faces. Tests on a number of cases reveal that results via these two schemes in this study, which can be used on the unstructured grids, give good agreement with exact solutions or experimental data of free surface flows.
In order to simulate the two-phase flow with phase change, the CISIT method is extended to include heat and mass transfer due to phase change. The mass transfer across the phase boundary is determined by taking into account the mass and energy jump conditions at the interface and added as a source term in the continuity equation. Then, the interface is treated as an internal boundary condition in the temperature flied. Finally, the energy equation is solved in an implicit way. Besides, this method is also extended to simulate the heat transfer of two-fluid flows without phase change based on the assumption of the continuity conditions of the temperature and heat flux instead of jump conditions at the interface. Application to film boiling flow on a horizontal plate at a state near the critical pressure shows that the boiling mode will be different at different superheat temperatures. According to different superheat tempera- tures, the boiling flows can be divided into five modes: single-bubble mode ( ), single/multiple bubble mode ( ), single-jet mode ( ), double-bubble mode ( ), and double-jet mode ( ). In the single-bubble mode, good agreement with semi-empirical correlations was obtained in terms of averaged Nusselt number. Furthermore, simulation of film boiling flow on a cylinder demonstrates that this method is applicable to boiling flow with complex geometry.
Finally, the CISIT method with phase change is modified to calculate three-dimensional two-phase flows. Unlike two-dimensional flow, the interface is reconstructed with several non-coplanar triangular interfaces within the grid. First, this method was tested through computations of a number of two-fluid flows without phase change to validate the capability of tracking the interface in three- dimensional flows. In addition, this method was also applied to simulated film boiling flow on a horizontal plate. It can be shown that the space and time averaged Nusselt numbers obtained from the current simulations have good agreement with the semi-empirical correlations of Klimenko, especially for .Finally, the heat transfer model without phase change was used to simulate the molten tin droplet in oil and the octane inlet in water.

摘 要 i
ABSTRACT iii
誌 謝 v
目 錄 vi
表目錄 ix
圖目錄 x
符號說明 xv
第一章 緒論 1
1.1 簡介 1
1.1.1 兩相流簡介 1
1.1.2 薄膜沸騰簡介 5
1.2 文獻回顧 11
1.2.1 兩相流之數值方法 11
1.2.1.1 移動網格法 (Moving Grid Method) 11
1.2.1.2 前端追蹤法 (Front-Tracking Method) 11
1.2.1.3 等位函數法 (Level-set method) 12
1.2.1.4 體積追蹤法 (Volume Tracking Method) 13
1.2.1.5 流體體積法 (Volume-of-Fluid Method, VOF) 14
1.2.1.5.1 介面捕捉法 (Interface Capturing Method) 15
1.2.1.5.2 介面重建法 (Interface Reconstruction Method) 17
1.2.1.6 流體體積法與等位函數法之混合 20
1.2.2 薄膜沸騰之數值模擬 21
1.2.2.1 前端追蹤法 21
1.2.2.2 等位函數法 22
1.2.2.3 流體體積法 23
1.3 研究目的 24
第二章 數學模型 34
2.1 簡介 34
2.2 流體體積(VOF)方程式 35
2.3 統御方程式 36
2.4 表面張力模型 38
2.5 相變化之熱質傳模型 39
2.5 無相變化之熱傳模型 41
2.6 邊界條件 42
第三章 通量混合介面捕捉法(FBICS) 45
3.1 簡介 45
3.2 流體體積方程式之離散 46
3.3 通量限制函數 47
3.4 通量混合介面捕捉法 48
第四章 守恆內差介面追蹤法(CISIT) 59
4.1 簡介 59
4.2 介面重建 61
4.3 流體體積分率預測步驟 61
4.4 流體體積分率修正步驟 63
4.4.1 填充過度(over-filling, ) 64
4.4.2 耗竭過度(over-depleting, ) 65
4.4.3 填充不足(under-filling, ) 65
4.4.4 耗竭不足(under-depleting, ) 66
4.5 CITSIT計算流程 66
4.6 流體體積分率平滑化 67
第五章 速度場之數值方法 73
5.1 簡介 73
5.2 動量方程式的離散 73
5.2.1 非穩態項 74
5.2.2 對流項 74
5.2.3 擴散項 75
5.2.4 源項 76
5.2.5 動量方程式之代數方程式 78
5.3 速度與壓力之耦合 79
5.3.1 預測步驟 (predictor step) 79
5.3.2 第一次修正步驟 (first corrector step) 80
5.3.3 第二次修正步驟 (second corrector step) 83
5.3.4 PISO演算法的計算流程 85
5.4 壓力出口邊界之處理 85
第六章 熱傳及質傳之數值方法 88
6.1 簡介 88
6.2 相變化模型之熱傳數值方法 89
6.2.1 介面熱通量之計算 89
6.2.2 能量內邊界之處理 89
6.3 無相變化之熱傳數值方法 90
6.4 能量方程式之離散 91
6.5 兩相流模型之計算流程 93
第七章 FBICS法之驗證與自由表面流之應用 97
7.1 簡介 97
7.2 介面於均勻速度場之傳輸 98
7.3 介面於剪切流(Shear Flow)中之拉伸 102
7.4 二維壩體潰堤 104
7.5 壩體潰堤流經阻塊 107
7.6 水力湧潮 (Hydraulic Bore) 109
第八章 CISIT法之驗證與雙流體流場之應用 141
8.1 簡介 141
8.2 介面於均勻速度場中之傳輸 142
8.3 介面於剪切流中之拉伸 146
8.4 二維壩體潰堤 148
8.5 單一上升氣泡 (Single Rising Bubble) 150
8.6 雷利-泰勒不穩定性之問題 153
第九章 CISIT法於二維薄膜沸騰分析 188
9.1 簡介 188
9.2 一維汽化問題 189
9.3 水平平板薄膜沸騰 192
9.4 水平圓管薄膜沸騰 201
9.5多模式(multi-mode)長平板薄膜沸騰 206
第十章 CISIT法之三維應用 236
10.1 簡介 236
10.2 CISIT法之三維化處理 238
10.2.1 三維浸潤面積之計算 238
10.2.2 三維介面類型及重建 239
10.2.3 三維介面處熱通量計算及能量內邊界 240
10.2.4 平行化運算 242
10.3 三維壩體潰堤 244
10.4 氣泡運動分析 246
10.4.1 單一上升氣泡 246
10.4.2 氣泡融合 251
10.5 液滴撞擊薄層液面 254
10.6 三維水平平板薄膜沸騰 256
10.7 無相變化之熱傳問題 260
第十一章 結論 294
第十二章 參考文獻 298
簡 歷 308
論文發表 309

[1]Ishii, M., Thermo-Fluid Dynamic of Two-Phase Flow, Eyrolles, Paris, 1975.
[2]Berenson, P. J., “Film-boiling heat transfer from a horizontal surface”, ASME Journal of Heat Transfer, 83, pp. 351-358, 1961.
[3]Bromley, L. A., “Heat transfer in stable film boiling”, Chemical engineering Progress, 46 , pp. 221-227, 1950.
[4]Klimenko, V. V., “Film-boiling on a horizontal plate–new correlation”, International Journal of Heat and Mass Transfer, 24, pp. 69-79, 1981.
[5]Welch, S. W. J., “Local simulation of two-phase flows including interface tracking with mass transfer”, Journal of Computational Physics, 121, pp. 142-154, 1995.
[6]Son, G. and Dhir, V. K., “Numerical simulation of saturated film boiling on a horizontal surface”, ASME Journal of Heat Transfer, 119, pp. 525-533, 1997.
[7]Juric, D. and Tryggvason, G., “Computations of boiling flows”, International Journal of Multiphase Flow, 24, pp. 387-410, 1998.
[8]Son, G. and Dhir, V. K., “Numerical simulation of film boiling near critical pressures with a level set method”, ASME Journal of Heat Transfer, 120, pp. 183-192, 1998.
[9]Welch, S. W. J. and Wilson, J., “A volume of fluid based method for fluid flows with phase change”, Journal of Computational Physics, 160, pp. 662-682, 2000.
[10]Ramaswamy, B. and Kawahara, M., “Lagrangian finite element analysis applied to viscous free surface fluid flow”, International Journal for Numerical Methods in Fluids, 7, pp. 953-984, 1987.
[11]Fukai, J., Zhao, Z., Poulikakos, D., Megaridis, C. M. and Miyatake, O., “Modeling of the deformation of a liquid droplet impinging upon a flat surface”, Physics of Fluids A: Fluid Dynamics, 5, pp. 2588-2599, 1993.
[12]Muzaferija, S. and Peric, M., “Computation of free-surface flows using the finite-volume method and moving grids”, Numerical Heat Transfer Part B: Fundamental, 32, pp. 369-384, 1997.
[13]Unverdi, S. O. and Tryggvason, G., “A front-tracking method for viscous, incompressible, multi-fluid flows”, Journal of Computational Physics, 100, pp. 25-37, 1992.
[14]Tryggvasson, G., Bunner, B., Esmaeeli, A., Juric, D., Al-Rawahi, N., Tauber, W., Han, J., Nas, S. and Jan, Y.-J., “A front-tracking method for the computations of multiphase flow”, Journal of Computational Physics, 169, pp. 708–759, 2001.
[15]Osher, S. and Sethian, J. A., “Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations”, Journal of Computational Physics, 79, pp. 12–49, 1988.
[16]Sussman, M., Smereka, P. and Osher, S., “A level set approach for computing solutions to incompressible two-phase flow”, Journal of Computational Physics, 114, pp. 146-159, 1994.
[17]Sussman, M., Fatami, E., Smereka, P. and Osher, S., “An improved level set method for incompressible two-phase flows”, Computers and Fluids, 27, pp. 663-680, 1998.
[18]Sussman, M. and E., Fatami, “An efficient, interface-preserving level set redistancing algorithm and its application to interfacial incompressible fluid flow”, SIAM Journal on Scientific Computing, 20, pp. 1165-1191, 1999.
[19]Harlow, F. H. and Welch, J. E., “Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface”, The Physics of Fluids, 8, pp. 2182-2189, 1965.
[20]Monaghan, J. J., “Simulating free surface flows with SPH”, Journal of Computational Physics, 110, pp. 399-406, 1994.
[21]Koshizuka, S., Nobe, A. and Oka, Y., “Numerical analysis of breaking waves using the moving particle semi-implicit method”, International Journal for Numerical Methods in Fluids, 26, pp. 751-769, 1998.
[22]Hirt, C. W. and Nichols, H. D., “Volume of fluid (VOF) method for the dynamics of free boundaries”, Journal of Computational Physics, 39, pp. 201-225, 1981.
[23]Rudman, M., “Volume-tracking methods for interfacial flow calculations”, International Journal for Numerical Methods in Fluids, 24, pp. 671-691, 1997.
[24]Rudman, M., “A volume-tracking method for incompressible multifluid flows with large density variations”, International Journal for Numerical Methods in Fluids, 28, pp. 357-378, 1998.
[25]Zalesak, S. T., “Fully multidimensional flux-corrected transport algorithms for fluids”, Journal of Computational Physics, 31, pp. 335-262, 1979.
[26]Ubbink, O. and Issa, R. I., “A method for capturing sharp fluid interfaces on arbitrary meshes”, Journal of Computational Physics, 153, pp. 26-50, 1999.
[27]Muzaferija, S., Peric, M., Sames, P. and Schellin, T., “A two-fluid Navier-Stokes solver to simulate water entry”, Proceeding of Twenty- Second Symposium On Naval Hydrodynamics, pp. 638-649, Washington, DC, 1998.
[28]Darwish, M. and Moukalled, F., “Convective schemes for capturing interfaces of free-surface flows on unstructured grids”, Numerical Heat Transfer Part B: Fundamental, 49, pp. 19-42, 2006.
[29]Gaskell, P. H. and Lau, A. K. C., “Curvature-compensated convective transport: SMART, a new boundedness-preserving transport algorithm”, International Journal for Numerical Methods in Fluids, 8, pp. 617-641, 1988.
[30]Noh, W. F. and Woodward, P., “SLIC (simple line interface calculation)”, Lecture Notes in Physics, 59, pp. 330-340, 1976.
[31]Youngs, D. L., “Time-dependent multi-material flow with large fluid distortion”, in K.W. Morton and M.J. Baines (Eds.), Numerical Methods for Fluid Dynamics, pp. 273–285, Academic, New York, 1982.
[32]Rider, W. J. and Kothe, D. B., “Reconstructing volume tracking”, Journal of Computational Physics, 141, pp. 112-152, 1998.
[33]Pilliod Jr., J. E. and Puckett, E. G., “Second-order accurate volume-of-fluid algorithms for tracking material interfaces”, Journal of Computational Physics, 199, pp. 465-502, 2004.
[34]Ashgriz, N. and Poo, J. Y., “FLAIR: flux line-segment model for advection and interface reconstruction”, Journal of Computational Physics, 93, pp. 449-468, 1991.
[35]Tavakoli, R., Babaei, R., Varahram, N. and Davami, P., “Numerical simulation of liquid/gas phase flow during mold filling”, Computer Methods in Applied Mechanics and Engineering, 196, pp. 697-713, 2006.
[36]Mosso, S. J., Swartz, B. K., Kothe, D. B. and Clancy, S. P., “Recent enhancement of volume tracking algorithm for irregular grids”, Los Alamos National Lab. Rept. LA-CP-96-227, Los Alamos, New Mexico, 1996.
[37]Shahbazi, K., Paraschivoiu, M. and Mostaghimi, J., “Second order accurate volume tracking based on remapping for triangular meshes”, Journal of Computational Physics, 188, pp. 100-122, 2003.
[38]Ashgriz, N., Barbat, T. and Wang, G., “A computational Lagrangian- Eulerian advection remap for free surface flows”, International Journal for Numerical Methods in Fluids, 44, pp. 1-32, 2004.
[39]Sussman, M. and Puckett, E. G., “A coupled level set and volume-of-fluid method for computing 3d and axisymmetric incompressible two-phase flows”, Journal of Computational Physics, 162, pp. 301-337, 2000.
[40]Son, G. and Hur, N., “A coupled level set and volume-of-fluid method for the buoyancy-driven motion of fluid particles”, Numerical Heat Transfer Part B: Fundamentals, 42, pp. 523-542, 2002.
[41]van der Pijl, S. P., Segal, A., Vuik, C. and Wesseling, P., “A mass-conserving level-set method for modelling of multi-phase flows”, International Journal for Numerical Methods in Fluids, 47, pp. 339-361, 2005.
[42]Yang, X., James, A. J., Lowengrub, J., Zheng, X. and Cristini, V., “An adaptive coupled level-set/volume-of-fluid interface capturing method for unstructured triangular grids”, Journal of Computational Physics, 217, pp. 364-394, 2006.
[43]Sun, D. L. and Tao, W. Q., “A coupled volume-of-fluid and level set (VOSET) method for computing incompressible two-phase flows”, International Journal of Heat and Mass Transfer, 53, pp. 645-655, 2010.
[44]Shin, S. and Juric, D., “Modeling three-dimensional multiphase flow using a level contour reconstruction method for front tracking without connectivity”, Journal of Computational Physics, 180, pp. 427-470, 2002.
[45]Esmaeeli, A. and Tryggvason, G., “Computations of film boiling. part I: numerical method”, International Journal of Heat and Mass Transfer, 47, pp. 5451-5461, 2004.
[46]Esmaeeli, A. and Tryggvason, G., “Computations of film boiling. part II: multi-mode film boiling”, International Journal of Heat and Mass Transfer, 47, pp. 5463-5476, 2004.
[47]Esmaeeli, A. and Tryggvason, G., “A front tracking method for computations of boiling in complex geometries”, International Journal of Multiphase Flow, 30, pp. 1037- 1050, 2004.
[48]Son, G., “A numerical method for bubble motion with phase change”, Numerical Heat Transfer Part B: Fundamentals, 39, pp. 509-523, 2001.
[49]Fedkiw, R. P., Aslam, T., Merriman, B. and Osher S., “A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method)”, Journal of Computational Physics, 152, pp. 457-492, 1999.
[50]Kang, M., Fedkiw, R. P. and Liu, X.-D., “A boundary condition capturing method for multiphase incompressible flow”, Journal of Scientific Computing, 15, pp. 323-360, 2000.
[51]Tanguy, S., Menard, T. and Berlemont, A., “A level set method for vaporizing two-phase flows”, Journal of Computational Physics, 221, pp. 837-853, 2007.
[52]Gibou, F., Chen, L., Nguyen, D. and Banerjee, S., “A level set based sharp interface method for the multiphase incompressible Navier-Stokes equations with phase change”, Journal of Computational Physics, 222, pp. 536-555, 2007.
[53]Maurya, R. S., Diwakar, S. V., Sundararajan, T. and Das, S. K., “Numerical investigation of evaporation in the developing region of laminar falling film flow under constant wall heat flux conditions”, Numerical Heat Transfer Part A: Applications, 58, pp. 41-64, 2010.
[54]Son G. and Dhir, V. K., “A level set method for analysis of film boiling on an immersed solid surface”, Numerical Heat Transfer Part B: Fundamentals, 52, pp. 153-177, 2007.
[55]Son, G. and Dhir, V. K., “Three-dimensional simulation of saturated film boiling on a horizontal cylinder”, International Journal of Heat and Mass Transfer, 51, pp. 1156-1167, 2008.
[56]Welch, S. W. J. and Rachidi, T., “Numerical computation of film boiling including conjugate heat transfer”, Numerical Heat Transfer Part B: Fundamentals, 42, pp. 35-53, 2002.
[57]Agarwal, D.K., Welch, S. W. J., Biswas, G. and Durst, F., “Planar simulation of bubble growth in film boiling in near-critical water using a variant of the VOF method”, ASME Journal of heat transfer, 126, pp. 329-338, 2004.
[58]Guo, D. Z., Sun, D. L., Li, Z. Y. and Tao, W. Q., “Phase change heat transfer simulation for boiling bubbles arising from a vapor film by the VOSET method”, Numerical Heat Transfer Part A: Applications, 59, pp. 857-881, 2011.
[59]Tsui, Y.-Y., Lin, S.-W., Cheng, T.-T. and Wu, T.-C., “Flux-blending schemes for interface capture in two-fluid flows”, International Journal of Heat and Mass Transfer, 52, pp. 5547-5556, 2009.
[60]Tsui, Y.-Y. and Lin, S.-W., “A VOF based conservative interpolation scheme for interface tracking (CISIT) of two-fluid flows”, Numerical Heat Transfer Part B: Fundamentals, 63, pp. 263-283, 2013.
[61]Brackbill, J. U., Kothe, D. B. and Zemach, C., “A continuum method for modeling surface tension”, Journal of Computational Physics, 100, pp. 335-354, 1992.
[62]Tsui, Y.-Y. and Wu, T.-C., “A pressure-based unstructured-grid algorithm using high-resolution schemes for all-speed flows”, Numerical Heat Transfer Part B: Fundamental, 53, pp. 75-96, 2008.
[63]Tsui, Y.-Y. and Wu, T.-C., “Use of characteristic-based flux limiters in a pressure-based unstructured-grid algorithm incorporating high-resolution schemes”, Numerical Heat Transfer Part B: Fundamental, 55, pp. 14-34, 2009
[64]Harten, A. “High resolution schemes for hyperbolic conservation laws”, Journal of Computational Physics, 49, pp. 357-393, 1983.
[65]Sweby, P. K., “High resolution schemes using flux limiters for hyperbolic conservation laws”, SIAM Journal on Numerical Analysis, 21, pp. 995-1011, 1984.
[66]Leonard, B. P., “The ULTIMATE conservative difference scheme applied to unsteady one-dimensional advection”, Computer Methods in Applied Mechanics and Engineering, 88, pp. 17-74, 1991.
[67]Leonard, B. P., “Simple high-accuracy resolution program for convective modeling of discontinuities”, International Journal for Numerical Methods in Fluids, 8, pp. 1291-1318, 1988
[68]Leonard, B. P., “Bounded higher-order upwind multi-dimensional finite- volume convection-diffusion algorithm”, in: W. J. Minkowycz, Editor, Advances in Numerical Heat Transfer, Taylor and Francis, 1997.
[69]Yeh, J.-T., “Simulation and industrial application of inkjet”, 7th National Computational Fluid Dynamics Conference, Kenting, Taiwan, 2000.
[70]Yeh, J.-T., “A VOF-FEM and coupled inkjet simulation”, Proc. of ASME Fluids Engineering Division Summer Meeting, pp. 1-5, New Orleans, USA, 2001.
[71]Peskin, C. S., “Numerical analysis of blood flow in the heart”, Journal of Computational Physics, 25, pp. 220-252, 1977.
[72]Issa, R. I., “Solution of the implicitly discretised fluid flow equations by operator-splitting”, Journal of Computational Physics, 62, pp. 40-65, 1986.
[73]Martin, J. C. and Moyce, W. J., “An experimental study of the collapse of liquid columns on a rigid horizontal plane”, Philosophical Transactions of the Royal Society of London Series A: Mathematical and Physical, 244 pp. 312-324, 1952.
[74]Koshizuka, S., Tamako, H. and Oka, Y., “A particle method for incompressible viscous flow with fluid fragmentation”, Computational Fluid Dynamics Journal, 4, pp. 29-46, 1995.
[75]Ubbink, O., “Numerical prediction of two fluid systems with sharp interfaces”, University of London, PhD thesis, 1997.
[76]Stoker, J. J., Water wave, John Wiley &; Sons, New York, 1958.
[77]Grace, J. R., “Shape and velocities of bubbles rising in infinite liquids", Transactions of the Institution of Chemical Engineers, 51, pp. 118-120, 1973.
[78]Reimann, M. and Grigull, U., “Warmeubergang bei freier konvektion und flimsieden im kritischen gebiet von wasser und kohlendioxid”, Warmeund Stof-fubertragung, 8, pp. 229-239, 1975.
[79]Tomar, G., Biswas, G., Sharma, A. and Agrawal, A., “Numerical simulation of bubble growth in film boiling using a coupled level-set and volume-of-fluid method”, Physics of Fluids, 14, pp. 112103, 2005
[80]Son, G., “A level set method for incompressible two-fluid flows with immersed solid boundaries”, Numerical Heat Transfer Part B: Fundamental, 47, pp. 473-489, 2005.
[81]Hosler, R. E. and Westwater, J. W., “Film boiling on a horizontal plate”, ARS Journal, 32, pp. 553–558, 1962.
[82]Yiantsios, S. G. and Higgins, B. G., “Rayleigh-Taylor instability in this viscous films”, Physics of Fluids A: Fluid Dynamics, 1, pp. 1484-1501, 1989.
[83]Bhaga, D. and Weber, M.E., “Bubbles in viscous liquids: shapes, wakes and velocities”, Journal of Fluid Mechanics, 105, pp. 61–85, 1981.
[84]Hua, J. and Lou, J., “Numerical simulation of bubble rising in viscous liquid”, Journal of Computational Physics, 222, pp. 769-795, 2007.
[85]Amaya-Bower, L. and Lee, T., “Single bubble rising dynamics for moderate Reynolds number using lattice Boltzmann method”, Computers &; Fluids, 39, pp. 1191-1207, 2010.
[86]Szewc, K., Pozorski, J. and Minier, J.-P., “Simulations of single bubbles rising through viscous liquids using smoothed particle hydrodynamics”, International Journal of Multiphase Flow, 50, pp. 98-105, 2012.
[87]Bonometti, T. and Magnaudet, J., “Transition from spherical cap to toroidal bubbles”, Physics of Fluids, 18, pp. 052102, 2006.
[88]Brereton G. and Korotney, D., “Coaxial and oblique coalescence of two rising bubbles”, in: I. Sahin and G. Tryggvason (Eds.), Dynamics of Bubbles and Vortices near a Free Surface, ASME, New York, 1991.
[89]Rieber, M. and Frohn, A., “A numerical study on the mechanism of splashing”, International Journal of Heat and Fluid Flow, 20, pp. 455-461, 1999.
[90]Yarin, A. L., and Weiss, D. A., “Impact of drops on solid surfaces: self-similar capillary waves, and splashing as a new type of kinematic discontinuity”, Journal of Fluid Mechanics, 283, pp. 141-173, 1995.