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A. MALIDI, S. DUFOUR AND D. N’DRI1.6Taylor’s Problem1.41.2Time-Step Sizes10.80.60.40.200 20 40 60 80 100 120 140 160Time StepsFigure 13. Time-step sizes for Taylor’s problem, using the ABDF scheme (Ca =0:082).Table I. Comparison of experimental and numericalresults for Taylor’s problem: =0:2, =27:3.Exper.Numer.Ca D D 0.082 0.0635 −28.0 0.1042 −28.00.164 0.0786 −34.0 0.1391 −34.00.246 0.0788 −35.0 0.1438 −35.00.329 0.0798 −35.0 0.1470 −35.0Taylor [24] dened a deformation parameter D asD = L − BL + Bwhere L and B are the longest and shortest semi-axes of the drop cross-section, in orderto measure the deformation of a drop. For various values of Ca, we computed thedeformation parameter D and the rotation angle of the drop. The results are comparedto the experimental data of Bentley and Leal [25] in Table I. It can be seen that the deformationparameter is nearly constant, this being the consequence of the rotational ow. Theconcave deformation curve, illustrated in Figure 14, is typical of this situation. Our overestimationof the deformation parameter is probably caused by the fact that we perform a2-D simulation of a 3-D problem. However, the computed angle of rotation reached by thenumerical drops is accurate.Copyright ? 2005 John Wiley & Sons, Ltd.Int. J. Numer. Meth. Fluids (in press)
A STUDY OF TIME INTEGRATION SCHEMES FOR MODELLING FREE SURFACE FLOWS0.2experimental datanumerical resultsD00 0.6Figure 14. Deformation curve for Taylor’s problem: =0:2, =27:3.Ca6.3. Numerical validation: buoyancy-driven rising bubblesThe velocity and deformation of a single rising bubble due to buoyancy will rst be studied.This test is based on the results of Sussman and Smereka [26]. Let us consider a bubbleof non-dimensional radius ã = 1, which is initially lying at rest in another uid, in ageometry of dimensions [−3; 3] × [0; 12]. This system is submitted to a constant body forceg =(0; −g), where g is the gravitational acceleration. Since the uids have a density ratioof = 1 = 2 =0:0011 (the uid of the bubble is lighter), buoyancy generates a ow whichmakes the bubble rise. The viscosity ratio for this test is given by = 1 = 2 =0:0085.For this problem, the Navier–Stokes equations with the CSF model (5) arenon-dimensionalized as1St˜@ũ@˜t +˜(ũ · ˜∇)ũ = 1 Re ˜∇ · ˜ + 1 Fr ˜ ˜g + 1 We ˜ ˜∇F (18)where, in our case, the Reynolds number is given bythe Froude number is dened asRe = au 0 2 2=9:8Fr = u2 0ag =0:76Copyright ? 2005 John Wiley & Sons, Ltd.Int. J. Numer. Meth. Fluids (in press)
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