A study of time integration schemes for the numerical modelling of ...

More documents

Recommendations

Info

$Guide to using LaTeX on a Linux machine - University of Leeds$

$MATH 2650 Calculus of Variations Examples 1 The practice ...$

A. MALIDI, S. DUFOUR AND D. N’DRI(a) t = 0.0 (b) t = 3.6 (c) t = 4.8 (d) t = 6.0Figure 15. Buoyancy-driven rising of a single bubble.and the Weber number is given byWe = a2 g 2=7:6where the reference velocity u 0 is the experimentally observed [27] steady-state rise velocityof an air bubble, which is u 0 =21:5cm=s. The Strouhal number St is again set to 1. The otherquantities were previously described.Figure 15 illustrates the deformation of the rising bubble at various time steps. The simulationwas performed for the complete bubble. Our methodology performs well in maintainingthe symmetry of the bubble, and its vertical path. Since this simulation is performed usinga Cartesian frame of reference, the computed shapes of the bubble dier slightly from theones illustrated in Reference [26]. The computed rising speed of the bubble is underestimatedwhen compared to the non-dimensional rising speed which should be equal to 1, this beingcaused again by our 2-D approximation of an axisymmetric phenomena. Finally, usingour methodology, we measure a numerical mass loss of the order of 8% over 80 time steps(t =0;:::;6:5).We will now study the rise and interaction of two bubbles due to buoyancy. It is a commonpractice in the literature to describe bubbles dynamics problems using the Eotvos number,also known as the Bond numberEo = 2gL 2 0and the Archimedes number, also known as the Galileo numberN = 2 2 gL3 0 2 2Copyright ? 2005 John Wiley & Sons, Ltd.Int. J. Numer. Meth. Fluids (in press)
A STUDY OF TIME INTEGRATION SCHEMES FOR MODELLING FREE SURFACE FLOWSFigure 16. Bubbles rising problem.(a) t = 0.00 (b) t = 2.39 (c) t = 2.99 (c) t = 3.43Figure 17. Buoyancy-driven rising and coalescence of two bubbles.The Morton number is also seen in the literature:M = g4 2 2 = Eo33 N 2However, it is possible to still work with the non-dimensionalized equations (18) by deningthe reference velocity as u 0 = √ gL 0 , which gives us a relation between the non-dimensionalgroups (Re; Fr; We) and (N; Eo). In our case, Re = √ N, Fr = 1 and We = Eo.Our test problem, described in Reference [28], is illustrated in Figure 16. It consists of twovertically aligned bubbles with same diameter d =2:6 mm, located at a distance of h =0:4mmCopyright ? 2005 John Wiley & Sons, Ltd.Int. J. Numer. Meth. Fluids (in press)
Page 1 and 2: INTERNATIONAL JOURNAL FOR NUMERICAL
Page 3: A STUDY OF TIME INTEGRATION SCHEMES
Page 6 and 7: A. MALIDI, S. DUFOUR AND D. N’DRI
Page 8: A. MALIDI, S. DUFOUR AND D. N’DRI

A study of time integration schemes for the numerical modelling of ...

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?