(483j) Three-Dimensional Numerical Simulation of Spherical Drops in Liquid-Liquid and in Liquid-Gas Systems with Emphasis on the Influence of the State of Internal Motion on Drag

The drag exerted on spherical liquid drops is investigated by numerical simulations, in which both the inner and the ambient phase are solved in a full three-dimensional domain. Drops in both a liquid (liquid-liquid systems) and a gaseous ambient phase (liquid-gas systems) are studied. The drops are fixed in space and a constant, uniform inflow velocity of the ambient phase is prescribed. It is shown that the assumption of axisymmetry, used in many studies, is often not valid.

The time-accurate simulations were performed with the DLR THETA code, a Finite-Volume solver for incompressible flows. For the velocity in the momentum fluxes a central fourth order scheme was used, whereas the pressure and the other transport equations were solved with a second order central scheme. The time was discretized using a Crank-Nicolson scheme. The DLR THETA code is used for many years and is well validated (e.g. Ivanova et al. (2009), Probst et al. (2015)). For the present investigation, a boundary condition for the surface of the droplet was implemented, such that the shear stress and the velocity are continuous across the interface and no mass flux normal to the interface is possible.

The drag exerted on spherical drops in a uniform stream is governed by three parameters: The Reynolds number of the outer fluid, the viscosity ratio λ of inner to outer fluid and the density ratio κ of inner to outer fluid. For the basic validation of the present droplet simulations, an excellent agreement with other numerical studies over a wide range of parameters is found. However, for some cases, major discrepancies can be observed. For example, Feng & Michaelides (2001) state that the effect of the density ratio (κ) on drag is minimal, whereas in the present study κ has a significant influence in some cases. These differences are attributed to the fact that non-axisymmetric flow fields emerge inside the drops, which cannot be captured by their simulations as they assume an axisymmetric solution.

Simulations of cases, studied experimentally by Thorsen et al. (1968), were performed. They observed a â??great and sudden increase at a well-defined value of the drop diameterâ? for certain substances in liquid-liquid systems. Simulation results show, that this change in drag coefficient is due to different states of the internal circulation. Besides the axisymmetric Hillâ??s vortex solution, steady plane symmetric solutions and unsteady solutions are found. With the present, full three-dimensional simulations, drag coefficients from simulation are in good agreement with most of their experimental results.

Present results show that different internal flow modes are also visible in liquid-gas systems. However, in all cases simulated, only a minor influence of these different internal motions on the drag is found.

To conclude, the assumption of axisymmetric flow fields inside droplets is not justified in many cases. For liquid-liquid systems, a significant influence of the inner flow field on drag was found, whereas the influence in liquid-gas systems is negligible. Nonetheless, the state of the internal motion will influence the mass/heat transfer inside droplets, a topic which should be researched in the future.

References
Feng, Z.-G. & Michaelides, E. E. (2001), â??Drag coefficients of viscous spheres at intermediate and high reynolds numbersâ??, Journal of Fluids Engineering 123(4), 841â??849.
Ivanova, E., Di Domenico, M., Noll, B. & Aigner, M. (2009), â??Unsteady Simulations of Flow Field and Scalar Mixing in Transverse Jetsâ??, ASME paper GT2009-59147 .
Probst, A., Johannes, L., ReuÃ?, S., Knopp, T. & Kessler, R. (2015), â??Scale-resolving simulations with a low-dissipation low-dispersion second-order scheme for unstructured finite-volume flow solversâ??, AIAA paper 2015-0816 .
Thorsen, G., Stordalen, R. & Terjesen, S. (1968), â??On the terminal velocity of circulating and oscillating liquid dropsâ??, Chemical Engineering Science 23(5), 413 â?? 426.