Numerical simulation of a water droplet splash: Comparison between PLIC and HRIC schemes for the VoF transport equation
Introduction
The study of multiphase flows is of great importance since there is a wide range of phenomena described by the interaction between fluids. Among them there are natural processes such as the rain drop, which can cause soil erosion and is also a medium to transport bacteria and spores. The impact of rain drops also influences in the air-sea gas exchange as well as in the damping of wave motion as explained by [1].
There are also industrial processes involving multiphase flows such as coating, painting, fuel injection and irrigation. To effectively achieve the desired results in each of these processes, the instruments must be correctly characterized and applied. For example, the jet spray used in painting and coating process, has to be accurately applied onto the surface of the subject in study as represented in the works of [2] and [3]. In the case of fuel injection in internal combustion engines, the spray has to be correctly characterized to achieve improvements in the combustion process reducing fuel consumption and engine emission as pointed out in the works of [4], [5], [6]. According to [7], the correct usage of irrigation instruments leads to a decrease of water consumption as well as avoid soil erosion. Noticing that sprinkler irrigation is responsible for half of annual consumption of water, the optimization of this process is extremely important.
As mentioned by [8], the understanding of the physics involving the multiphase flow phenomena is a fundamental step in order to improve these processes. For that purpose, numerical methods have been increasingly used in complex engineering problems, providing results in scenarios where experiments may not be feasible. The Computational Fluid Dynamics (CFD) approach has proven a powerful, valuable tool to reach the state-of-art in engineering, reducing costs and development time.
One of the major difficulties in simulating these phenomena is to accurately predict the interaction between the fluids. Different methodologies can be used to track the interface between two fluids, such as the front-tracking, level set and VOF (volume of fluid) methods.
According to [9] the level set approach is based on the transport of a function, which is important only at the interface between fluids. This method proved to be accurate on capturing complex topological changes. Nevertheless, this method is not mass conservative. On the other hand, the VOF method solves a scalar convective equation through the computational domain, being locally and globally mass conservative. A disadvantage of this method is the computational time required to compute the advection term of the scalar function.
Four different test cases were used to validate the interface tracking in the work of [10]. The test cases included the translation of a solid body, rotation of a solid body, a single vortex and a complex deformation field. The main findings are that the level set presented good agreement on the cases that the body being tracked does not deform, otherwise there is no conservation of mass. In the work of [11], structured adaptive mesh refinement (SAMR) was used along with the Linear Weighted, Essentially Non-Oscillatory (LWENO) scheme to minimize the errors caused by spatial discretization on the level set method. The main objective of his work was to minimize these errors to improve the mass conservation. The main finding is that the combination of these techniques on the simulations presented very low mass conservation errors. The authors also highlighted that this code is easy implemented in parallel coding in comparison to the hybridization of VOF and level set methods.
In the work of [9], the VOF method and a coupling of level Set and VOF methodologies were used to capture the shape of a rising bubble. This work was carried out to compare these models. The test case consisted of a bubble rising in a two dimensional channel, and accounted for breaking up, coalescence and deformation of the bubble. These are fundamental phenomena to examine the accuracy of the models. The main findings were that the coupled level set and VOF method required a higher computational time. However, the VOF method required a finer mesh to capture the interface between the bubble and the liquid accurately.
The scalar value attributed to the phase of each flow does not represent the location nor the shape of the interface between fluids in the VOF methodology. To account for these missing details a numerical scheme to reconstruct the interface must be used. According to [8] algebraic discretization schemes for the advection term of the VOF equation such as HRIC and CICSAM, are computationally cheaper when compared to geometrically schemes such as the PLIC scheme. On the other hand, the geometrical schemes yields better results.
A comparison of the effect of four different interpolation schemes for density interpolation were tested in the work of [8] for the CICSAM method. The numerical simulations presented in his work consisted of a droplet falling onto a deep pool. The analyses of the results consisted on the comparison the numerical results and the physical experimentation of [1]. The author compared the crater and the ascending jet height formed by the impact of the impinging droplet on the pool. The results presented a better agreement for the volume fraction weighted scheme over the central difference, first-order upwind and second order upwind when compared with experimental data.
In addition to the physical experiments, [1] also carried out numerical simulations for the impingement of a droplet on a pool. For the simulations carried out in his work the PLIC scheme was used for the advection term of the VOF method. Considering the huge difference of density of the gas and the liquid phases, the gas phase density was neglected. The main findings are that the absence of the gas phase did not affect the representation of the physics involved in some cases. The numerical simulation was capable of predicting with good agreement the crater formation. However, the bubble formation due to the droplet impact is not represented correctly once that the gas phase is assumed as vacuum, i.e. the trapped bubble disappears after the crater closure.
As the VOF method is mass conservative it becomes an attractive option for the simulations. In this work, the latter is employed to simulate the impact of a water droplet in a water pool. Besides being important in many natural and industrial processes, such problem was reliably experimentally documented, becoming an interesting case to validate the numerical procedures as illustrated in the works of [8] and [1].
The main object of this paper is to compare two different schemes for the advection term of the VoF transport equation. The schemes are PLIC, which is geometric and therefore exact, and HRIC, which is an algebraic scheme. The relevance of the paper is to point out the advantages and drawbacks of each of the schemes used for the advection term of the VOF formulation. The tested case is a droplet impacting in a deep water pool. For that purpose, the CFD software Convergent Science Inc.’s CONVERGE ™ CFD was used. This software uses an adaptive mesh refinement technique (AMR) to better capture the interface between fluids.
Section snippets
Physical model
The physical model consists in a droplet impacting onto a water pool as depicted in Fig. 1. Fig. 1(a) represents the complete model and Fig. 1(b) represents the sketch of a quarter of the model. A spherical droplet of radius () 2.9 mm is released from a certain height, which results in different impact velocities. This impact creates a disturbance in the initially quiescent water pool. This disturbance in turn creates a crater that can be experimentally and numerically measured.
The crater is
Mathematical model
The Volume of Fluid (VOF) method is used to simulate multi-phase flow of gases and liquids. This technique is used to track the interface between the fluids, in which the immiscible fluids share momentum and energy. To track the interface the volume fraction () of each cell is calculated throughout the computational domain. The value of the volume fraction can be found in three different situations: 0 representing only liquid in the cell, 1 representing only gas, or the value can be in between
Model description
The tested cases are presented on Table 2 with dimensions in accordance with Fig. 1, where is the drop diameter (2.9 mm). Case I has an impact velocity of 1.55 m/s and case II of 2.5 m/s.
Some simplifications were adopted to reduce computational cost and mesh size in simulations. The drop is released near the free surface of the water pool (0.2 mm) with its impact velocity. Fortunately, even the gap between the droplet and the pool surface being only 7% of the droplet size, it was enough to
Results and discussions
This section presents the results for the simulations and the comparison with the experimental data obtained by [1]. Case I and case II are tested for both schemes HRIC and PLIC. To track the surface between the air and the water a sampling line was traced in the middle of the pool as illustrated in Fig. 10.
Fig. 11 illustrates the instant of for both the HRIC and PLIC. Note that there is numerical diffusion for the HRIC scheme, as indicated by the interface smearing in Fig. 11. This
Final remarks
The main findings can be summarized as follows:
The PLIC scheme maintains a sharper interface than the HRIC scheme.
The PLIC scheme is more accurate and computationally efficient than the HRIC.
The PLIC scheme was more efficient than the HRIC in simulating both cases presented in this paper.
The numerical models still need to be improved so they can correctly reproduce the experiments regarding the crater formation.
Although more efficient to simulate the case herein studied, the PLIC scheme
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
We would like to express gratitude to Convergent Science Inc. for their assistance and for the use of the CONVERGE™ CFD software. Acknowledgement is given to Universidade Federal de Uberlândia for providing the computational resources used for the CFD simulation presented and for Programa de Pos-graduação em Engenharia Mecânica – FEMEC/UFU for fostering research activities. This study was financed in part by the Coordenação de Aperfeioamento de Pessoal de Nivel Superior - Brasil (CAPES) -
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