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A. MALIDI, S. DUFOUR AND D. N’DRI2Rising Drops ProblemreferencedropsTotal Area of the Drops100 2 4TimeFigure 18. Rising and coalescence of two bubbles: mass conservation.from each other. The non-dimensional groups are given by Re = √ N = 30 and Eo = We = 100.Finally, the density ratio is given by = 2 = 1 = 2 and the viscosity ratio is = 2 = 1 =2.Figure 17 illustrates the deformation and coalescence of the two rising bubbles at varioustime steps. The shape of the drops are similar to what we can observe in Reference [28], thedierence coming from the fact that these results come from 3-D simulations. We observethat coalescence occurs after 29 time steps (˜t =3:43). Figure 18 illustrates the variation of thetotal mass of the bubbles (non-dimensional area) for the duration of the simulation, which isvery small. It is unfortunately dicult to nd quantitative results in the literature in order toperform an accurate validation for this problem.7. CONCLUSIONA time integration strategy is proposed for modelling free surface ows in the context ofEulerian interface capturing. An adaptive second-order accurate backward dierentiation formulais used to discretize the transient term of the Navier–Stokes equations, and the implicitmidpoint rule is used for the transport equation of the marker variable. The adaptive schemeallows the automatic determination of a time-step size that follows the physics of the problemunder study. It is shown that this mixed strategy gives accurate results for steady-state freesurface ows, and that it reduces mass loss for transient multiuid ows. Such a study isnovel to our knowledge. It also seems that it is the rst time that the ABDF scheme wasused in an applied context.In order to keep the focus of this study on time integration strategies, several componentsof the numerical model necessary for performing accurate free surface ow simulations wereCopyright ? 2005 John Wiley & Sons, Ltd.Int. J. Numer. Meth. Fluids (in press)
A STUDY OF TIME INTEGRATION SCHEMES FOR MODELLING FREE SURFACE FLOWSnot discussed in this paper. Maintaining the region of transition of the pseudo-concentrationsharp but smooth is very important in order to perform the accurate modelling of surfacetension. Mesh adaptation in the vicinity of the free surface is also important for accuratelymodelling interfacial physics. The accurate numerical modelling of the capillary force is alsoa delicate matter. These questions were studied in References [5, 6, 16, 20] and we are stillmaking progress developing more accurate algorithms for addressing these questions. Futurework also includes the development of an axisymmetric model which should allow us toobtain numerical results that are closer to what is observed experimentally. We also plan tostudy applied (industrial) free surface ows involving the dynamics of drops and bubbles suchas emulsions and jet break-ups.APPENDIX A: COMPUTING THE AMOUNT OF MASS FOR EACH FLUIDVerifying mass conservation for each uid is not a straightforward process in the contextof Eulerian free surface capturing. The fact that we are working with non-structured meshesmakes it more dicult to nd an algorithm for computing the area of the regions 1 and 2 .When the interface does not go through an element, i.e. there are no points x in the elementsuch that F(x)= 1 2, the area of the triangle gives the quantity of mass of uid 1 or 2 in thatelement. The other possibility is to see the interface S go through an element, as illustrated inFigure A1. A rst approximation will be obtained by locating the two points of intersectionof the free surface with the sides of the element (the points 2 and 5 in Figure A2(a)). Thequantity of uid 1 is approximated by the area of the triangle with vertices 1; 2 and 5, andthe quantity of uid 2 by the area of the quadrangle with vertices 2; 3; 4 and 5. We canthen repeat this process by subdividing the element, as illustrated in Figure A2(b), and bycomputing the area occupied by each uid in the subtriangles, as described above. We thenrepeat this subdivision process (cf. Figure A2(c)) until convergence of the computed area ofthe regions occupied by each uid in that element, for a given tolerance. We observe that wecompute the mass for each uid to machine precision after 3 or 4 levels of subdivisions.Figure A1. Elementary free surface ow problem.Copyright ? 2005 John Wiley & Sons, Ltd.Int. J. Numer. Meth. Fluids (in press)
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