A filling process in an open tank will be simulated by the composite of modified MUSCL and modified BDS scheme in this section. The tank is a square and vertical plate with a gate at the bottom. Initially, the tank is filled with air (Fig. 46). The water enters the tank through the gate and fills the tank as the time pass by.
The computational domain of the tank in our simulation is a square space with side length of 0.152m. The height of the gate is 0.038m. The flow is considered to be two-dimensional with the assumption of a laminar and incompressible flow under the body and surface tension force. The properties of the fluids are shown as:
3 3
The inlet velocity of the gate is given by some approximation from [46]. The volume of the water inside the tank can be evaluated from the photographs by using the curve fitting method. The volume inside the tank can be written as a function depended on time, shown as:
2
which implies:
( ) in( )
where A can be considered as the height of the gate in two-dimension, Q is the volume inside the tank, t is the calculational time, and U is the inlet velocity depended on t. in
The boundary conditions of walls are treated as non-slip condition and the top of the tank is a pressure fixed outlet. The meshes used in the calculation involve the uniform and
quadrilateral mesh with 28 28× , 40 40× and 80 80× grids and the triangular mesh with 902, 2024 and 3584 cells (Fig. 47, Fig. 48 and Fig. 49).
Fig 5.50 to 5.55 shows the volume fraction distribution with time evolution. The counter is 0.05 to 0.95 with 10 levels. The results show that the flow with the particular velocity enters the tank as a wall jet along the bottom. When the front of the flow touches the right wall, the high pressure gradient will be introduced into the vicinity of stagnation point. This effect will make the flow jump along the vertical wall. To continue, the flow falls back and forms a gravity wave in the x-direction. The front of flow goes to the left wall along the x-axis and jumps again when the front touches the vertical wall. The tank will be filled by degrees.
The volume fraction distribution in the 28 28× and 40 40× mesh produces more numerical diffusions on the interface. The interface in the 80 80× mesh is sharper than them. It shows that the finer mesh can present well results for the filling process of two-fluid flow. The results also show that our code can do well on the triangular mesh. The position of the leading edge at the bottom on the quadrilateral and triangular mesh is plotted in Fig 56.
Then, the figure 5.57 shows the water volume inside the tank. It presents the data in quadrilateral and triangular meshes from our simulation and the data from the inlet condition which is mentioned in (5.6). This figure can demonstrate the accuracy of our numerical method which is applied in the filling process in an open tank and show that our numerical result is very close to the data in [46].
Chapter6 Conclusion and Future Work
A method for capturing the interface of two-fluid flow has been present. The interface movement is solved by an indicator function of the volume fraction. The composite scheme is used to solve the advection equation of the volume fraction. The aim of this study is to develop a composite scheme with switching function in volume-of-fluid method for two-fluid flow. The scheme switches smoothly between the modified MUSCL and modified bounded downwind scheme. Its results of test cases have compared with the results of other schemes.
It can not only maintain the sharpness at the interface but also enhances the accuracy, even on the triangular mesh. In the future work, there are two missions in order to improve our numerical method. First, the volume-of-fluid method is usually used in three-dimension system because many actual applications, such as casting process, are applied in three-dimensions. Second, there are a lot of problems about the heat transfer in the two-fluid flow. How to remain the sharpness at the interface is a quite significant problem.
Chapter7 Reference
[1] Azcueta, R., Hadzic, I., Muzaferija, S. and Peric, M., “Computation of flows with free surface”, MARNET-CFD First Workshop – Barcelona, 1999.
[2] Pericleous, K. A., Chan, K. S. and Cross, M., “Free surface flow and heat transfer in cavities: the SEA algorithm”, Numerical Heat Transfer, Part B, v27, pp.487-507, 1995.
[3] Yang, Z., Peng, X. F. and Ye, P., “Numerical and experiment investigation of two phase flow during boiling in coiled tube”, Int. J. of Heat and Mass transfer, v51, pp.1003-1016, 2008.
[4] Wang, Y., Basu, S. and Wang, C. Y., “Modeling two-phase flow in PEM fuel cell channels”, J. of Power Source, v179, pp.603-617, 2008.
[5] Muzaferija, S. and Peric, M., “Nonlinear water waves interaction, computational mechanics publications”, in: Mahrenholtz O. and Markiewicz M. (Eds.), Computation of free surface flows using interface tracking and interface capturing methods. Chap.2, 1998.
[6] Floryan, J. M. and Rasmussen, H., “Numerical methods for viscous flows with moving boundaries”, Appl. Mech. Rev., v42, pp.323-341, 1989.
[7] Hayes, R. E., “Numerical Simulation of Mold Filling in Reaction Injection Molding”, Polym. Eng. Sci., v31, pp.842-848, 1991.
[8] Muzaferija, S. and Peric, M., “Computation of free-surface flows using the finite-volume method and moving grids”, Numerical Heat Transfer, Part B, v32, pp. 369-384, 1997.
[9] Tryggvasson, G., “A front tracking method for the computations of multiphase flow”, J.
Comput. Phys., v169, pp.708–759, 2001.
[10] Osher, S. and Sethian, J. A., “Fronts propagating with curvature-dependent speed:
algorithms based on Hamilton Jacobi formulations”, J. Comput. Phys., v79, pp.12–49, 1988.
[11] Li, X.L., “Study of three-dimensional Rayleigh-Taylor instability compressible fluids through level set method and parallel computation”, Phys. Fluids A, v5, pp.1904-1913, 1993.
[12] Harlow, F. H. and Welch, J. E., “Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface”, Phys. Fluids, v8, No.12, pp. 2182-2189, 1965.
[13] Patel, N. R. and Briggs, D. G., “Mac Scheme In Bound Ary-Fitted Curvilinear Coordinates”, Numerical Heat Transfer, v6, n4, , pp.383-394, Oct-Dec, 1983.
[14] Chen, S., Juhnson, D. B. and Radd, P. E., “Velocity boundary conditions for the simulation of free surface fluid flow”, J. Comput. Phys., v116, n2, pp. 262-276, 1995.
[15] Noh, W. F. and Woodward, P. R., “SLIC (Simple Line Interface Method)”, Lecture Note in Physics, v59, pp. 330-340, 1976.
[16] Chorin, A. P., “Flame advection and propagation algorithms”, J. Comput. Phys., v35, pp.1-11, 1980.
[17] Youngs, D. L. “Time-dependent multi-material flow with large fluid Distortion”, in:
K.W. Morton, M.J. Bianes (Eds.), Numerical Methods for Fluid Dynamics, Academic Press,
New York, pp. 273–285, 1982.
[18] Ramshaw, J. D. and Trapp, J.A., “A numerical technique for low-speed homogeneous two-phase flow with sharp interfaces”, J. Comput. Phts., v21, pp.438-453, 1976.
[19] Hirt, C. W. and Nichols, B. D., “Volume of fluid (VOF) method for the dynamics of free boundaries”, J. Comput. Phys., v39, pp. 201-225, 1981.
[20] Shyy, W., Thakur, S. and Wright, J., “Second-order upwind and central difference schemes for recirculating flow computation”, AIAA Journal, v30, pp. 923-932, 1992.
[21] Leonard, B. P., “A stable and accurate convective modeling procedure based on quadratic interpolation”, Computer Methods in Applied Mechanics and Engineering, v.19, pp.56-98, 1979.
[22] Zalesak, S. T. , “Fully multi-dimensional flux corrected transport algorithms for fluid flow”, J. Comput. Phys., v31, pp.335-362, 1979.
[23] Harten, A. “High resolution schemes for hyperbolic conservation laws”, J. Comput. Phys., v49, pp. 357-393, 1983.
[24] Leonard, B. P., “Simple high-accuracy resolution program for convective modeling of discontinuities”, In. J. Numer. Methods Fluids, v8, pp.1291-1318, 1988
[25] Van Leer, B., “Towards the ultimate conservative difference scheme. . Monotonicity Ⅱ and conservation combined in a second-order scheme”, J. Comput. Phys., v14, pp.361-370 1974.
[26] Sweby, P. K., “High resolution schemes using flux limiters for hyperbolic conservation laws”, SIAM J. Numer. Analysis., v21, No.5, pp. 995-1011, 1984.
[27] Gaskell, P. H. and Lau, A. K. C., ”Curvature-compensated convective transport: SMART, a new boundedness-preserving transport algorithm.” Int. J. Numer. Meth. Fluids, v8, pp.
617-641, 1988.
[28] Jasak, H., Weller, H. G. and Gosman, A. D., “High Resolution NVD Differencing Scheme for Arbitrary Unstructured Meshes”, Int. J. for Numer. Meth. Fluids, v31, pp.431-449,
1999.
[29] Roe, P. L., “Large scale computations in fluid mechanics, Part2”, In Lectures in Applied Mathematics, Vol. 22, pp. 163-193, 1985.
[30] Darwish, M. S., “A new high-resolution scheme based on the normalized variable formulation”, Numer. Heat Trans, Part B, v24, pp. 353-371, 1993.
[31] Van Leer, B., “Towards the Ultimate Conservative Difference Scheme V. A Second-Order Sequel to Godunov's Method”, J. Comput. Phys., v135, No.2, pp. 229-248, 1997.
[32] Ubbink, O. and Issa, R. I., “A method for capturing sharp fluid interfaces on arbitrary meshes,” J. Comput. Phys., v153, pp.26-50, 1999.
[33] Muzaferija, S., Peric, M., Sames, P. and Schellin, T., “A Two-Fluid Navier-Stokes Solver to Simulate Water Entry”, Twenty-Second Symp. On Naval Hydrodynamics, Washington, D.C., pp.638-649, 1998.
[34] Darwish, M. and Moukalled, F., “Convective schemes for capturing interfaces of free-surface flows on unstructured grids.”, Numer. Heat Trans., Part B, v49, pp.19-42, 2006.
[35] Ubbink, O., “Numerical prediction of two fluid systems with sharp interfaces,” thesis submitted from Department of Mechanical Engineering Imperial College of Science, Technology & Medicine, January 1997.
[36] 鄭東庭，「利用流體體積法之雙流體計算」，國立交通大學機械所，碩士論文，2007。
[37] Brackbill, J.U. et al., “A continuum method for modeling surface tension”, J. Comput.
Phys., v100, PP.335-354, 1967.
[38] Batchelor, G. K., ”An introduction to fluid dynamics,” Cambridge: Cambridge University Press, 1967.
[39] Adamson, A. W., “Physical chemistry of surfaces,” New York: John Wiley & Sons, 1976.
[40] Orlanski, "A simple boundary condition for unbounded hyperbolic flows", J.
Computational Physics, v21,pp.251-269,1976.
[41] Issa, R. I., “Solution of the implicitly discretised fluid flow equations by operator-splitting,” J. Comput. Phys, v62, pp. 40-65, 1985.
[42] 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.
[43] Rudman, M., “Volume-tracking methods for interfacial flow calculations”, Int. J. Numer.
Methods Fluids, v24, pp.671-691, 1997.
[44] Martin, J. C. and Moyce, W. J., “An experimental study of the collapse of liquid columns on a rigid horizontal plane,” Philos. Trans. Roy. Soc. London, vA244, pp.312-324, 1952.
[45] Koshizuka, S., Tamako, H. and Oka, Y., “A particle method for incompressible viscous flow with fluid fragmentation,” Comput. Fluid Dynamics JOURNAL, v4(1), pp.29-46, 1995.
[46] Hwang, W.-S. and Stoehr R.A., “Modeling of fluid flow”, Metal Handbook, 9th Ed., ASM International, Metals Park, OH, v15, PP.867-876.
NVD form Flux limiter function
SUPERBE
Table 4.1 The NVD equation and flux limiter function of linear and non-linear difference
Scheme Hollow circle Hollow square
UDS 1.2274 1.2568 CUS 0.4223 0.5156 DDS 0.3245 0.2074 BDS 0.2362 0.3399 CDS 0.6027 0.6933 MUSCL 0.4943 0.5802 SUPERBEE 0.3123 0.4271
M-MUSCL 0.4623 0.5520 M-BDS 0.1283 0.1544 Table 5.1 Errors of different schemes in Co=0.25 (quadrilateral mesh)
Scheme Hollow circle Hollow square
UDS 1.2281 1.2566 CUS 0.5392 0.5971 DDS 0.7116 0.6572 BDS 0.2258 0.3523 CDS 0.7011 0.7599 MUSCL 0.5190 0.5935 SUPERBEE 0.3171 0.4476
M-MUSCL 0.4957 0.5639 M-BDS 0.2818 0.2408 Table 5.2 Errors of different schemes in Co=0.75 (quadrilateral mesh)
Scheme Hollow circle Hollow square
UDS 1.0846 1.0418 CUS 0.3469 0.3402 DDS 0.2587 0.2415 BDS 0.1797 0.1806 CDS 0.4305 0.4169 MUSCL 0.3460 0.3388 SUPERBEE 0.2282 0.2246
M-MUSCL 0.3419 0.3349 M-BDS 0.1745 0.1635 Table 5.3 Errors of different schemes in Co=0.25 (triangular mesh)
Scheme Hollow circle Hollow square
UDS 1.0846 1.0417 CUS 0.3809 0.3732 DDS 0.5324 0.5618 BDS 0.1787 0.1828 CDS 0.4653 0.4472 MUSCL 0.3638 0.3565 SUPERBEE 0.2378 0.2342
M-MUSCL 0.3596 0.3528 M-BDS 0.1819 0.1684 Table 5.4 Errors of different schemes in Co=0.75 (triangular mesh)
Scheme Hollow circle Hollow square
CICSAM 0.1267 0.1280
HRIC 0.3259 0.3714
SUPERBEE+MUSCL 0.3812 0.4429
M-BDS+M-MUSCL 0.1173 0.1666
Table 5.5Errors of different composite schemes in Co=0.25 (quadrilateral mesh)
Scheme Hollow circle Hollow square
CICSAM 1.0000 1.0041
HRIC 0.2276 0.2438
SUPERBEE+MUSCL 0.4031 0.4612
M-BDS+M-MUSCL 0.1889 0.2521
Table 5.6Errors of different composite schemes in Co=0.75 (quadrilateral mesh)
Scheme Hollow circle Hollow square
CICSAM 0.1556 0.1514
HRIC 0.1813 0.1869
SUPERBEE+MUSCL 0.2697 0.2660
M-BDS+M-MUSCL 0.1652 0.1641
Table 5.7Errors of different composite schemes in Co=0.25 (triangular mesh)
Scheme Hollow circle Hollow square
CICSAM 0.2090 0.2055
HRIC 0.2111 0.1793
SUPERBEE+MUSCL 0.2841 0.2824
M-BDS+M-MUSCL 0.1920 0.1734
Table 5.8Errors of different composite schemes in Co=0.75 (triangular mesh)
(a)
(b)
Figure 1.1 The method of two-fluid flow (a) Lagrangian (b) Eulerian scheme
Figure 1.2 Front tracking method
Figure 1.3 Maker and cell method
(a) original distribution (b) Young’ VOF
(c) SLIC with x-sweep (d) SLIC with y-sweep
(e) Chorin with x-sweep (f) Chorin SLIC with x-sweep
Figure 1.4 Line techniques
Figure 1.5 Donor and accepter cell configuration
Figure 2.1General form of the conservation law
Figure 2.2 VOF method on the Eulerian grids
Figure 2.3 Continuity of the velocity and discontinuity of the momentum
Figure 2.4 Fluid arrangements and the sign of the curvature
Figure 3.1 Illustration of the primary cell P and the neighbor cell nb with a considering face
Figure 4.1 The relationship of a control volume and its neighbor cells
Figure 4.2 The CBC constraint in NVD
Figure 4.3 The CBC constraint in NVD
Figure 4.4 The TVD condition in TVD diagram
Figure4.5 Linear schemes in Normalized Variable Diagram
Figure4.6 Linear schemes in TVD Diagram
Figure 4.7 SMART
Figure 4.8 MUSCL
Figure 4.9 SUPERBEE
Figure 4.10 STOIC
Figure 4.11 OSHER
Figure 4.12 BDS
Figure 4.13 Van Leer
Figure 4.14 CHARM
Figure 4.15 Modified BDS
Figure 4.16 Modified MUSCL
(a) (b)
Figure 4.17 The NVD of (a) HYPER-C and (b) UQ (Co= 0.5)
Figure 4.18 The switching function of CICSAM scheme
Figure 4.19 The switching function of HRIC scheme
Figure 4.20 The switching function of the composite of M-MUSCL and M-BDS
Figure 5.1 (a) The constant velocity field and initial position of hollow circle
Figure 5.1 (b) The constant velocity field and initial position of hollow square (2,1)
Vuv=
(2,1) Vuv=
Figure 5.2 Triangular computational mesh in uniform density flow (22478 cells)
Figure 5.3 (a) The exact distribution of the hollow circle
Figure 5.3 (b) The exact distribution of the hollow square
Co=0.25
Upwind Difference Scheme Cubic Upwind Scheme Downwind Difference Scheme
Bounded Downwind Scheme Central Difference Scheme MUSCL
SUPERBEE Modified MUSCL Modified BDS
Co=0.75
Upwind Difference Scheme Cubic Upwind Scheme Downwind Difference Scheme
Bounded Downwind Scheme Central Difference Scheme MUSCL
SUPERBEE Modified MUSCL Modified BDS
Figure 5.4 The final shape of hollow circle from different schemes in quadrilateral mesh
Co=0.25
Upwind Difference Scheme Cubic Upwind Scheme Downwind Difference Scheme
Bounded Downwind Scheme Central Difference Scheme MUSCL
SUPERBEE Modified MUSCL Modified BDS
Co=0.75
Upwind Difference Scheme Cubic Upwind Scheme Downwind Difference Scheme
Bounded Downwind Scheme Central Difference Scheme MUSCL
SUPERBEE Modified MUSCL Modified BDS
Figure 5.5 The final shape of hollow square from different schemes in quadrilateral mesh
Co=0.25
Upwind Difference Scheme Cubic Upwind Scheme Downwind Difference Scheme
Bounded Downwind Scheme Central Difference Scheme MUSCL
SUPERBEE Modified MUSCL Modified BDS
Co=0.75
Upwind Difference Scheme Cubic Upwind Scheme Downwind Difference Scheme
Bounded Downwind Scheme Central Difference Scheme MUSCL
SUPERBEE Modified MUSCL Modified BDS
Figure 5.6 The final shape of hollow circle from different schemes in triangular mesh
Co=0.25
Upwind Difference Scheme Cubic Upwind Scheme Downwind Difference Scheme
Bounded Downwind Scheme Central Difference Scheme MUSCL
SUPERBEE Modified MUSCL Modified BDS
Co=0.75
Upwind Difference Scheme Cubic Upwind Scheme Downwind Difference Scheme
Bounded Downwind Scheme Central Difference Scheme MUSCL
SUPERBEE Modified MUSCL Modified BDS
Figure 5.7 The final shape of hollow square from different schemes in triangular mesh
Co=0.25 Co=0.75
CICSAM
HRIC
SUPERBEE+MUSCL
Modified BDS+Modified MUCSL
Figure 5.8 The final shape of hollow circle from different composite schemes in quadrilateral mesh
Co=0.25 Co=0.75
CICSAM
HRIC
SUPERBEE+MUSCL
Modified BDS+Modified MUCSL
Figure 5.9 The final shape of hollow square from different composite scheme in quadrilateral mesh
Co=0.25 Co=0.75
CICSAM
HRIC
SUPERBEE+MUSCL
Modified BDS+Modified MUCSL
Figure 5.10 The final shape of hollow circle from different composite scheme in triangular mesh
Co=0.25 Co=0.75
CICSAM
HRIC
SUPERBEE+MUSCL
Modified BDS+Modified MUCSL
Figure 5.11 The final shape of hollow square from different composite scheme in triangular mesh
Figure 5.12 The volume fraction distribution in a shear flow field
Figure 5.13 Triangular mesh in uniform density flow (22494 cells)
Forward 2 seconds Backward 2 seconds
CICSAM
HRIC
SUPERBEE+MUSCL
Modified BDS+Modified MUCSL
Figure 5.14 The volume fraction distribution in shear flow with Co=0.25 (quadrilateral mesh)
Forward 4 seconds Backward 4 seconds
CICSAM
HRIC
SUPERBEE+MUSCL
Modified BDS+Modified MUCSL
Figure 5.15 The volume fraction distribution in shear flow with Co=0.25 (quadrilateral mesh)
Forward 8 seconds Backward 8 seconds
CICSAM
HRIC
SUPERBEE+MUSCL
Modified BDS+Modified MUCSL
Figure 5.16 The volume fraction distribution in shear flow with Co=0.25 (quadrilateral mesh)
Forward 16 seconds Backward 16 seconds
CICSAM
HRIC
SUPERBEE+MUSCL
Modified BDS+Modified MUCSL
Figure 5.17 The volume fraction distribution in shear flow with Co=0.25 (quadrilateral mesh)
Forward 2 seconds Backward 2 seconds
CICSAM
HRIC
SUPERBEE+MUSCL
Modified BDS+Modified MUCSL
Figure 5.18 The volume fraction distribution in shear flow with Co=0.75 (quadrilateral mesh)
Forward 4 seconds Backward 4 seconds
CICSAM
HRIC
SUPERBEE+MUSCL
Modified BDS+Modified MUCSL
Figure 5.19 The volume fraction distribution in shear flow with Co=0.75 (quadrilateral mesh)
Forward 8 seconds Backward 8 seconds
CICSAM
HRIC
SUPERBEE+MUSCL
Modified BDS+Modified MUCSL
Figure 5.20 The volume fraction distribution in shear flow with Co=0.75 (quadrilateral mesh)
Forward 16 seconds Backward 16 seconds
CICSAM
HRIC
SUPERBEE+MUSCL
Modified BDS+Modified MUCSL
Figure 5.21 The volume fraction distribution in shear flow with Co=0.75 (quadrilateral mesh)
Forward 2 seconds Backward 2 seconds
CICSAM
HRIC
SUPERBEE+MUSCL
Modified BDS+Modified MUCSL
Figure 5.22 The volume fraction distribution in shear flow with Co=0.25 (triangular mesh)
Forward 4 seconds Backward 4 seconds
CICSAM
HRIC
SUPERBEE+MUSCL
Modified BDS+Modified MUCSL
Figure 5.23 The volume fraction distribution in shear flow with Co=0.25 (triangular mesh)
Forward 8 seconds Backward 8 seconds
CICSAM
HRIC
SUPERBEE+MUSCL
Modified BDS+Modified MUCSL
Figure 5.24 The volume fraction distribution in shear flow with Co=0.25 (triangular mesh)
Forward 16 seconds Backward 16 seconds
CICSAM
HRIC
SUPERBEE+MUSCL
Modified BDS+Modified MUCSL
Figure 5.25 The volume fraction distribution in shear flow with Co=0.25 (triangular mesh)
Forward 2 seconds Backward 2 seconds
CICSAM
HRIC
SUPERBEE+MUSCL
Modified BDS+Modified MUCSL
Figure 5.26 The volume fraction distribution in shear flow with Co=0.75 (triangular mesh)
Forward 4 seconds Backward 4 seconds
CICSAM
HRIC
SUPERBEE+MUSCL
Modified BDS+Modified MUCSL
Figure 5.27 The volume fraction distribution in shear flow with Co=0.75 (triangular mesh)
Forward 8 seconds Backward 8 seconds
CICSAM
HRIC
SUPERBEE+MUSCL
Modified BDS+Modified MUCSL
Figure 5.28 The volume fraction distribution in shear flow with Co=0.75 (triangular mesh)
Forward 16 seconds Backward 16 seconds
CICSAM
HRIC
SUPERBEE+MUSCL
Modified BDS+Modified MUCSL
Figure 5.29 The volume fraction distribution in shear flow with Co=0.75 (triangular mesh)
Figure 5.30 Comparison of errors in the shear flow with Co=0.25 (quadrilateral mesh)
Figure 5.31 Comparison of errors in the shear flow with Co=0.75 (quadrilateral mesh)
Figure 5.32 Comparison of errors in the shear flow with Co=0.25 (triangular mesh)
Figure 5.33 Comparison of errors in the shear flow with Co=0.75 (triangular mesh)
Figure 5.34 Schematic of the broken dam
Figure 5.35 Experimental results of a collapsing water column by Koshizuka
Figure 5.36 Schematic representation of the non-uniform and quadrilateral mesh with 56 36× grids in broken dam
Figure 5.37 Schematic representation of the triangular mesh with 4506 cells in broken dam
Figure 5.38 Schematic representation of the triangular mesh with 12354 cells in broken dam
Figure 5.39 Numerical results of the broken dam on the uniform and quadrilateral mesh with 48 28× grids
Figure 5.40 Numerical results of the broken dam on the uniform and quadrilateral mesh with 120 70× grids
Figure 5.41 Numerical results of the broken dam on the non-uniform and quadrilateral mesh with 56 36× grids
Figure 5.42 Numerical results of the broken dam on the triangular mesh with 4506 cells
Figure 5.43 Numerical results of the broken dam on the triangular mesh with 12354 cells
Figure 5.44 The position of the leading edge in broken dam
Figure 5.45 The height of the collapsing water in broken dam z
b
Figure 5.46 Schematic representation of the filling process in an open tank
Figure 5.47 Schematic representation of the triangular mesh with 902 cells in filling process
Figure 5.48 Schematic representation of the triangular mesh with 2024 cells in filling process
Figure 5.49 Schematic representation of the triangular mesh with 3584 cells in filling process
Figure 5.50 The volume fraction distribution and velocity field of the filling process on the uniform and quadrilateral mesh with 28 28×
Figure 5.51 The volume fraction distribution and velocity field of the filling process on the uniform and quadrilateral mesh with 40 40× grids
Figure 5.52 The volume fraction distribution and velocity field of the filling process on the uniform and quadrilateral mesh with 80 80× grids
Figure 5.53 The volume fraction distribution and velocity field of the filling process on the triangular mesh with 902 cells
Figure 5.54 The volume fraction distribution and velocity field of the filling process on the triangular mesh with 2024 cells
Figure 5.55 The volume fraction distribution and velocity field of the filling process on the triangular mesh with 3584 cells
Figure 5.56 The position of leading of the filling process in the open tank
Figure 5.57 The water volume inside the tank