Two-phase electrohydrodynamic simulations using a volume-of-fluid approach

A numerical methodology to simulate two-phase electrohydrodynamic flows under the volume-of-fluid paradigm is proposed. The electric force in such systems acts only at the interface and is zero elsewhere in the two fluids. Continuum surface force representations are derived for the electric field force in a system of dielectric–dielectric and conducting–conducting fluids. On the basis of analytical calculations for simple flow problems we propose a weighted harmonic mean interpolation scheme to smoothen the electric properties in the diffused transition region (interface). It is shown that a wrong choice of interpolation scheme (weighted arithmetic mean) may lead to a transition region thickness dependent electric field in the bulk. We simulate a set of problems with exact or approximate analytical solutions to validate the numerical model proposed. A coupled level set and volume-of-fluid (CLSVOF) algorithm has been used for simulations presented here.     

Anti-diffusion method for interface steepening in two-phase incompressible flow

In this paper, we present a method for obtaining sharp interfaces in two-phase incompressible flows by an anti-diffusion correction, that is applicable in a straight-forward fashion for the improvement of two-phase flow solution schemes typically employed in practical applications. The underlying discretization is based on the volume-of-fluid (VOF) interface-capturing method on unstructured meshes. The key idea is to steepen the interface, independently of the underlying volume-fraction transport equation, by solving a diffusion equation with reverse time, i.e. an anti-diffusion equation, after each advection time step of the volume fraction. As the solution of the anti-diffusion equation requires regularization, a limiter based on the directional derivative is developed for calculating the gradient of the volume fraction. This limiter ensures the boundedness of the volume fraction. In order to control the amount of anti-diffusion introduced by the correction algorithm we propose a suitable stopping criterion for interface steepening. The formulation of the limiter and the algorithm for solving the anti-diffusion equation are applicable to 3-dimensional unstructured meshes. Validation computations are performed for passive advection of an interface, for 2-dimensional and 3-dimensional rising-bubbles, and for a rising drop in a periodically constricted channel. The results demonstrate that sharp interfaces can be recovered reliably. They show that the accuracy is similar to or even better than that of level-set methods using comparable discretizations for the flow and the level-set evolution. Also, we observe a good agreement with experimental results for the rising drop where proper interface evolution requires accurate mass conservation.     

Finite element analysis of flow and heat transfer with moving free surface using fixed grid system

A numerical study was performed on flow and heat transfer involving moving free surfaces that occurs in mold filling processes such as casting and injection molding. In these problems, the calculation domain changes continuously and the numerical treatment of the moving interface tends to cause artificial diffusion. Among the solution algorithms based on the Eulerian method, the volume-of-fluid (VOF) method was used because the method is simple and efficient in handling the complex flow patterns inside the cavity. To solve the transport equation of free surface without artificial smearing of the interface the baby-cell method was employed in the geometric reconstruction of the free surface. Furthermore, a predictor–corrector method was adopted in the time integration of volume-of-fluid (VOF) transport equation to increase the accuracy. The proposed scheme was verified through several benchmark problems. In order to show the capability of the proposed method, several three-dimensional mold filling processes were solved. The current algorithm was applied to the floating body problem. Three-dimensional floating body problems were tested.     

Numerical analysis of flow-induced gas entrainment in roll coating

Gas entrainment by plane liquid jets which plunge into a liquid pool is analyzed by numerical simulations. The numerical model is based on the equations of incompressible newtonian fluids flow. The two-phase flow problem is described with the volume-of-fluid method. The dynamic behaviour of the interface is characterized by two similarity parameters, the capillary number Ca = uη/σ and the property number Γ = σ(ρ/η⁴g)^1/3 where u is the velocity, η the dynamic viscosity, σ the interface tension, ρ the density and g the gravitational constant. Numerical simulations are performed with the open source CFD code OpenFOAM. In the simulations the stability of the gas–liquid meniscus is tested for different sets of Ca and Γ. Critical values of Ca which indicate the beginning gas entrainment are deduced from the inspection of the simulation results. The findings of the numerical investigations agree well with corresponding experimental results.     

Simulation of axisymmetric jets with a finite element Navier–Stokes solver and a multilevel VOF approach

A multilevel VOF approach has been coupled to an accurate finite element Navier–Stokes solver in axisymmetric geometry for the simulation of incompressible liquid jets with high density ratios. The representation of the color function over a fine grid has been introduced to reduce the discontinuity of the interface at the cell boundary. In the refined grid the automatic breakup and coalescence occur at a spatial scale much smaller than the coarse grid spacing. To reduce memory requirements, we have implemented on the fine grid a compact storage scheme which memorizes the color function data only in the mixed cells. The capillary force is computed by using the Laplace–Beltrami operator and a volumetric approach for the two principal curvatures. Several simulations of axisymmetric jets have been performed to show the accuracy and robustness of the proposed scheme.     

Volume-of-fluid algorithm with different modified dynamic material ordering methods and their comparisons

Volume-of-fluid (VOF) interface reconstruction methods are used to define material interfaces to separate different materials in a mixed cell. These material interfaces are then used to evaluate transport flux at each cell edges in multi-material hydrodynamic calculations. Most of the VOF interface reconstruction methods and volume transport schemes rely on an accurate material order unique to each computational cell. Similarly, to achieve overshoot-free volume fractions, a non-intersecting interface reconstruction procedure has to be performed with the help of a ‘material-order list’ determined prior to interface reconstruction. It is, however, the least explored area of VOF technique especially for ‘onion-skin’ or ‘layered’ model. Also, important technical details how to prevent intersection among different material interfaces are missing in many literature. Here, we present an efficient VOF interface tracking algorithm along with modified ‘material order’ methods and different interface reconstruction methods. The relative accuracy of different methods are evaluated for sample problems. Finally, a convergence study with respect to mesh-size is performed.     

Inertia dominated flow and heat transfer in liquid drop spreading on a hot substrate

The present work deals with computational modeling of the fluid flow and heat transfer taking place in the process of impact of a cold liquid drop (T_d = 20-25 °C) onto a dry heated substrate characterized by different thermophysical properties. The computational model, based on the volume-of-fluid method for the free-surface capturing, is validated by simulating the configurations accounting for the conjugate heat transfer. The simulations were performed in a range of impact Reynolds numbers (Re = 2000-4500), Weber numbers (We = 27-110) and substrate temperatures (T_s = 100-120 °C). The considered temperature range of the drop-surface, i.e. liquid-solid system does not account for the phase change, that is boiling and evaporation. The model performances are assessed by contrasting the results to the reference database originating from the experimental and complementary numerical investigations by Pasandideh-Fard et al. [Pasandideh-Fard, M., Aziz, S., Chandra, S., Mostaghimi, J., 2001. Cooling effectiveness of a water drop impinging on a hot surface. International Journal of Heat and Fluid Flow, 22, 201-210] and Healy et al. [Healy, W., Hartley, J., Abdel-Khalik, S., 2001. On the validity of the adiabatic spreading assumption in droplet impact cooling. International Journal of Heat and Mass Transfer, 44, 3869-3881]. In addition, the thermal field obtained is analyzed along with the corresponding asymptotic analytical solution proposed by Roisman [Roisman, I.V., 2010. Fast forced liquid film spreading on a substrate: flow, heat transfer and phase transition. Journal of Fluid Mechanics, 656, 189-204]. Contrary to some previous numerical studies, the present computational model accounts for the air flow surrounding the liquid drop. This model feature enables a small air bubble to be resolved in the region of the impact point. The reported results agree reasonably well with experimental and theoretical findings with respect to the drop spreading pattern and associated heat flux and temperature distribution.     

The effects of confinement and inertia on the production of droplets

Recent experiments of Sibillo et al., Phys. Rev. Lett. 97:054502, (2006) investigate the effect of walls on flow-induced drop deformation for Stokes flow. The drop and the fluid in which it is suspended have the same viscosities. The capillary numbers vary from 0.4 to 0.46. They find that complex start-up transients are observed with overshoots and undershoots in drop deformation. Drop breakup is inhibited by lowering the gap. The ratio of initial drop radius to wall separation is fixed at 0.34. We show that inertia can enhance elongation to break the drop by examining Reynolds numbers in the range of 1 to 10. The volumes of the daughter drops can be larger than in the unbounded case, and even result in the production of monodisperse droplets.      

Numerical simulation of a drop undergoing large amplitude oscillatory shear

Direct numerical simulations are conducted for a Newtonian drop in a Newtonian matrix subjected to large amplitude oscillatory shear flows. In the experimental study of Guido et al. (in Rheol Acta 43:575–583, 2004), the drop shape is found to oscillate at higher harmonics of the forcing frequency when the capillary number is increased. Their phenomenological model requires a much smaller capillary number for predicting the harmonic nature of the experimental data. In this paper, computational results on the evolution of drop length and inclination angle are obtained at the same fluid and flow properties as the experiments, and are shown to reasonably reproduce the experimental data. In particular, the computed velocity fields around the drop are shown to elucidate the over-rotation, which is a mechanism for the experimentally observed harmonics.     

Development and implementation of VOF-PROST for 3D viscoelastic liquid–liquid simulations

We implement a volume-of-fluid algorithm with a parabolic re-construction of the interface for the calculation of the surface tension force (VOF-PROST). This achieves higher accuracy for drop deformation simulations in comparison with existing VOF methods based on a piecewise linear interface re-construction. The algorithm is formulated for the Giesekus constitutive law. The evolution of a drop suspended in a second liquid and undergoing simple shear is simulated. Numerical results are first checked against two cases in the literature: the small deformation theory for second-order liquids, and an Oldroyd-B extensional flow simulation. We then address the experimental data of Guido et al. (2003) for a Newtonian drop in a viscoelastic matrix liquid. The data deviate from existing theories as the capillary number increases, and reasons for this are explored here with the Oldroyd-B and Giesekus models.