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

A. MALIDI, S. DUFOUR AND D. N’DRIThis can be explained by the various properties of the IMR scheme [21]. This scheme issecond-order accurate, it introduces nearly no articial numerical oscillations, and it is A-stable, yet it introduces little undesirable damping compared to other A-stable schemes, whichmakes this scheme interesting for an advection equation. Constant-time-step size schemes areknown to give accurate results for a transport equation. Adaptive time integration schemesare suitable when the diusion term is non-negligible.6. OVERALL TIME INTEGRATION STRATEGY6.1. The time integration strategyBased on the results of the previous sections, the overall time integration strategy we proposefor modelling free surface ows using an Eulerian free surface capturing approach is composedof the ABDF scheme for the Navier–Stokes equations coupled with the IMR scheme for thetransport equation. Since we solve the system of partial dierential equations (2) and (3) ina coupled manner, the time steps used with the IMR scheme are the ones computed by theABDF scheme.This strategy allows us to combine the previously described properties of both schemes,and to avoid their drawbacks. The BDF scheme for the transport equation is too diusive,which would lead to the deformation of the pseudo-concentration and to mass loss. Moreover,an adaptive time integration scheme is not useful for an hyperbolic equation. In practice, weobserve that the time-step size reaches a constant value which corresponds to the chosentolerance, and it does not vary thereafter. The IMR scheme for the Navier–Stokes equationscould lead to numerical oscillations in the discretization of the velocity components.The proposed methodology is used to solve free surface ow problems of various nature inthe following subsections. First, we will model Taylor’s problem using the proposed methodology.This problem is known to reach a steady-state for specic operating conditions. TheABDF scheme should smoothly handle this problem. We will then study transient bubbleinteraction problems, which should be modelled accurately, with little mass loss, with thehelp of the IMR scheme.6.2. Numerical validation: Taylor’s problemLet us consider an initially undeformed ‘2-D circular drop’ of radius a = 1, located at thecentre of a computational domain of dimensions [−5; 5] × [−5; 5]. The mesh grid size for thissimulation is x =0:2. We are interested in the steady-state shape reached by the drop whensubmitted to a Cartesian linear ow. The far eld ow is described byu(x)= 1 ( )1+ 1 − x2 ˙ (17)−1+ −1 − )(ywhere ˙ is the shear rate of the ow and is a ow-type parameter. For example, the case = 1 corresponds to an extensional ow, while = 0 expresses a pure shear ow. The owsdescribed by Equation (17) are generated using the four-roll mill, developed by Taylor [24].The apparatus consists of four rotating cylinders, in the middle of which a drop is placed. ByCopyright ? 2005 John Wiley & Sons, Ltd.Int. J. Numer. Meth. Fluids (in press)