Computation of the rise or fall of a drop through a suspending liquid with no mass transfer: Axisymmetric, transitional and complex regimes
We investigate the buoyant motion of a liquid drop through a suspending liquid for three different liquid-liquid pairs and different drop sizes, covering a range of parameter space including axisymmetric, transitional and complex regimes, in cases for which there is no mass transfer between the drop and suspending liquid. We also investigate the accuracy of different numerical techniques for solving immiscible two-phase problems. In a number of free-surface flows, computations using an interface-fitted finite-element approach can lead to spurious velocities. Here, we use an Arbitrary Lagrangian-Eulerian (ALE) moving-mesh approach to investigate this using four test problems for which analytic solutions are available. An interface-fitted technique based on a stabilized Galerkin finite-element method is applied to the four test problems. We investigate Laplace’s equilibrium for a two-dimensional circular interface, an oscillating inviscid two-dimensional circular interface, damped capillary waves. We also consider relaxation to Laplace’s equilibrium for a spherical interface, neglecting mesh and interface motion. Two different approaches to account for interfacial tension are demonstrated for the spherical interface: one using differential geometry arguments and the other using a higher-order least-squares fit to the interface. We show that these approaches have approximately the same convergence rate and accuracy when used with second-order interpolation of velocity and pressure, with the least-squares fitting approach being significantly more accurate when linear interpolation is used. Application of artificial interfacial shear stress, including interfacial viscosity used to mitigate spurious velocities, is demonstrated and the oscillating inviscid circular interface case is shown to effectively dampen spurious velocities without adversely affecting the solution if the amount of damping is judiciously chosen. The first investigation of a buoyant drop is for the n-butanol and water pair, which form two distinct liquid phases over a wide range of composition and temperature. The rise of a butanol-rich drop (the “organic” phase) through a water-rich suspending fluid (the “aqueous” phase) has become a standard test case for assessing the fidelity of approaches to the computational approximation of two-phase flows, in part because the viscosity ratio is typical of many liquid/liquid systems of interest. Here, we identify the transition between steady, axisymmetric flow in which drop ascent is strictly vertical, and unsteady, three-dimensional motion, in which the drop trajectory deviates from the vertical. We also fill in a significant gap in understanding three-dimensionality in this test case. We use a three-dimensional ALE moving-mesh technique incorporating an interface-fitted grid to study the motion over a range of drop sizes at a single temperature. Computations are also performed with a volume-of-fluid (VOF) approach for three of the larger drop sizes. Computed transient rise behavior is favorably compared to previous experimental and computational results. The second investigation of a buoyant drop is a three-dimensional computation using the VOF approach of nitrobenzene drops freely descending in water. Results are reported for drop sizes with diameters of 2 - 4 mm (based on the volumes of spherical drops with equivalent diameters) covering several flow regimes, from steady axisymmetric laminar flows without separation for the smaller drops, to flows with highly complex, transient and three-dimensional characteristics for the larger drops. Comparison to experimental results shows good agreement. We also report on a new phenomenon, where following an initial transient, a 4 mm drop descends vertically and rotates about a vertical axis coinciding with the drop’s center-of-mass, leaving a helical wake. Transient development of the helical wake and its long-time structure are analysed in detail, and mechanisms leading to the rotational motion are discussed. The flow associated with the wake is shown to have a special point reflectional symmetry with respect to the drop’s center-of-mass. In addition, two long-time solutions were found for a 3.8 mm drop. Depending on initial conditions, we compute an approximately axisymmetric solution with periodic vortex shedding and shape oscillations, or another solution, similar to that for the 4 mm drop, in which the drop rotates, producing a helical wake. To the best of our knowledge, there has been no previous report of a rotating drop descending nearly vertically with a helical wake. Lastly, we report on a liquid CO2 drop freely rising in seawater, under conditions relevant to ocean carbon sequestration, computed by three-dimensional numerical simulations using the VOF approach. This system has the peculiar property that the flow internal to the drop becomes complex, while the external flow remains nominally laminar and smooth due to the lower kinematic viscosity of the drop liquid compared to the seawater. The transition of the flow internal to 5 and 6 mm CO2 drops, from a relatively simple laminar flow to a highly complex one with a large range of length scales, is demonstrated. Further investigation of the transition for the 6 mm drop reveals that the main mechanism responsible is an azimuthal wave instability, similar to that observed for free vortex rings. To the best of our knowledge, there has been no previous report of an azimuthal instability internal to a buoyant liquid drop.
Arbitrary Lagrangian-Eulerian method, Multiphase flow, Two-phase flow, Moving mesh, Interface-fitted mesh, Adaptive mesh refinement, Volume-of-fluid Method, Terminal velocity, Wake instability, Helical wake, Drop rotation, Liquid-liquid interfaces, Drop shape oscillations, Carbon dioxide drops, Subsea carbon sequestration, Surface tension, Butanol-water drops, Nitrobenzene drops, Multiple long-time solutions, Drop trajectories
University of Illinois - Urbana-Champaign
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University of Illinois at Urbana-Champaign