SIMULATION OF THE TWO PHASE FLOW OF DROPLET IMPINGEMENT ON LIQUID FILM BY THE LATTICE BOLTZMANN METHOD*

GUO Jia-hong, WANG Xiao-yong

Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China, E-mail: jhguo@staff.shu.edu.cn

SIMULATION OF THE TWO PHASE FLOW OF DROPLET IMPINGEMENT ON LIQUID FILM BY THE LATTICE BOLTZMANN METHOD*

GUO Jia-hong, WANG Xiao-yong

Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China, E-mail: jhguo@staff.shu.edu.cn

A Lattice Boltzmann Method (LBM) with two-distribution functions is employed for simulating the two-phase flow induced by a liquid droplet impinging onto the film of the same liquid on solid surface. The model is suitable for solution of twophase flow problem at high density and viscosity ratios of liquid to vapor and phase transition between liquid and its vapor. The roles of the vapor flow, the density ratio of liquid to vapor and the surface tension of the droplet in the splashing formation are discussed. It is concluded that the vapour flow induced by the droplet fall and splash in the whole impinging process may affect remarkably the splash behaviour. For the case of large density ratio of liquid to vapor a crown may engender after the droplet collides with the film. However, for the case of small density ratio of liquid to vapor a “bell” like splash may be observed.

droplet impact, film, two-phase flow, Lattice Boltzmann Method (LBM), numerical simulation

Introduction

Droplet impinging upon a liquid film on the solid surface is of both academic and practical importance to many processes such as the corrosion of turbines blades, spray injection in internal combustion engines, ink-jet printing, the spray and droplet impingement cooling. In these processes, evaporation of the liquid is inevitable, and for this reason, there usually exists vapour around the liquid. Therefore, it is essential to understand the underlying mechanisms of two phase flow governing the flow phenomenon as a droplet collide onto the film.

Some experimental studies by Rioboo et al.[1], Cossali et al.[2], Wal Randy et al.[3]and Castanet et al.[4]have been performed on a droplet impacting upon a film on a solid surface. A number of numerical studies have also been reported for the simulation of the impact of a single droplet on a liquid film. Yarin and Weiss[5]analyzed the axisymmetric impingement of a single drop onto liquid film. Surface tension and gravity were taken into account, whereas viscosity and compressibility were neglected. Rieber and Frohn[6]investigated the instability of the free rim, which led to the formation of cusps, fingers and secondary droplets. The study of Roisman and Tropea[7]and Josserand and Zaleski[8]concentrated on the early stages of droplet impact on a thin liquid layer. They concluded that splashing occurred for large Weber and Reynolds numbers and involved various dynamics. In many cases, a thin liquid sheet jet appeared almost immediately after the impact. In the study of Nikolopoulos et al.[9]numerical simulations were conducted to provide a physical insight into the flow regimes resulted from impingement of droplet on film. Although experimental and numerical results have been reported on a single droplet impinging on the film, until now, few efforts have been made to elucidate the two phase flow mechanism for a droplet colliding onto the. Li et al.[10]simulated a liquid droplet impacting on a solid surface based on the Smoothed Particle Hydrodynamics (SPH).

Multiphase flow phenomena are ubiquitous in nature and in many industrial areas. Numerical simulation for multi phase flow is one of the most important and challenging problems in computational fluid dynamics. As a mesoscopic model, the LatticeBoltzmann Method (LBM) is also adept in numerical prediction of multi phase flow with phase transition[11], since the macroscopic physical parameters such as density and velocity can be determined by the averaged values of the relative values related to the particles. He et al.[12]proposed a Lattice Boltzmann Equation (LBE) formulation for non-ideal gases based on the continuous Discrete Boltzmann Equation (DBE) using a single-relaxation-time approximation. Until now, most of previous LBE simulations have been carried out for multiphase fluid with the density ratio smaller than 10 due to instability at high density ratio. Therefore, the applicability of these models has been limited to some idealized situations.

There have been increasing efforts to improve the stability of two-phase LBE models at high density ratio. Although some sharp interface models such as the single phase model proposed by Xing et al.[13]could solve the free surface flow problem, they were not fit for two phase flow with phase transition. Lee and Lin[14]proposed a stable discretization of the LBE for non-ideal gases for simulation of incompressible two-phase flows with high density and viscosity ratios which is also suitable for phase transition problem.

In this article, a LBM with two-distribution functions based on[12]is employed for simulating the twophase flow induced by a liquid droplet impinging onto the film of the same liquid on solid surface. The mechanism for the droplets spread and crown and splash formation during the droplets impact on the films is elucidated. The roles of the vapor flow and the density ratio of liquid to vapor in the splashing formation are discussed.

1. Numerical method

The DBE for isothermal non-ideal gases with discretized microscopic velocity eacan be obtained by discretizing the microscopic velocity field ξion unit lattice[12]given as where fais the particle distribution function in the α direction of a lattice model, eaithe i-component of theα-direction microscopic velocity, and ρ the density normalized to the critical density. The equilibrium distribution function is given as follows

where the weighting factor tais given according to the m dimension n velocity (DmQn) models for the sake of self-containedness.

The intermolecular force Ftcan be expressed as[12]

where κ is a constant related to the magnitude of the surface tension. The thermodynamic pressure P can be obtained from the non-ideal gas equation of state such as the Van Der Waals equation of state. For the D2Q9 lattice model, the Van Der Waals equation of state can be normalized by the critical densitycρ, the critical temperature Tc, and the reference speed of soundas follows

The surface tension in the intermolecular forcing terms can be changed in either a stress form or a potential form by using the vector identity. The stress form is the best form if momentum conservation is important, while the potential form is especially good for flows that go to equilibrium states[15]. The stress form is derived as follows

in which the modified pressure is defined as

After some algebraic manipulation, the potential form derived from Eq.(3) isThe pressure in the potential form may be written in a more useful form for the control of interface thickness and surface tension at equilibrium.

The DBE for mass and momentum Eq.(1) is transformed into the DBE for hydrodynamic pressure and momentum which may improve stability. In order

Taking the total derivative of the new variable gaobtains

The DBE for mass and momentum has been transformed into the DBE for hydrodynamic pressure and momentum, therefore, another set of distribution function for density is needed. Equation (3) with the potential form of the surface tension force is adopted for the above purpose. The discrete Boltzmann equation for density is written as[14]

In order to solve the DBEs derived above, Eqs.(10) and (12) are discretized along characteristics over time stepδt. Thus, the LBE forgais obtained as

The above LBEs are solved in three steps: Prestreaming collision step, Streaming step and Poststreaming collision step. The density, the velocity, and the hydrodynamic pressure are calculated as below from the distribution functionandwhich are obtained after the streaming step.

2. Result and discussion

Numerical simulations were conducted to investigate the two phase flow resulted from the impingement of a droplet on theliquid film. The characteristics of droplet impact on a liquid film are dependent on the diameterD, the impact velocity V, the density of the liquid ρland the vapour ρv, the viscosity of the liquid μland the vapour μv, the surface tension σ and the film thickness h. The Weber number and Reynolds number are respectively defined as We=ρlDV2/σ and Re=ρlDV/μl. Dimensionless film thickness is defined as H=h/D.

Fig.1Schematic representation of the simulation setup for droplet impacting on a thin liquid film

A schematic of the simulation domain is shown in Fig.1. Periodic boundary conditions are imposed at the left and right boundaries. Topand bottom boundaries are wall, for which no slip condition is utilized. Initially, the velocity of the vapour and the liquid in the film is equal to zero. The grid number are 2 048 and 640 in the horizontal and vertical directions respectively.

Fig .2 Flow patterns resulted by impact of a droplet on filmat ρl/ρv=1000

Figure 2 shows the flow patterns obtained by computation for the case of a droplet impacting on the film at We=1000, Re=500 and H=0.2. The density and viscosity ratios of the liquid to the vapour are fixed at ρsat/ρsat=1000 and μ/μ=25 res pectively.Dimension less time variableT is defined as T=tV/D and is zero at the first contact between the droplet and the film. As is shown in Fig.2at the beginning of the droplet impacting on the film a pressure peak isobserved in the simulation at the impact neck, where jet may appear in the vapour phase, giving birth to the crown splashing. Figure 3 shows the velocity field after the droplet impacts on the film. Vapour flow is induced by the droplet fall and splash in the whole impinging process, which leads to a vortex on the medial side of the splashing lamella in the vapour phase. From the numerical results, we can find that splash starts from jet appearing at the neck of the impact due to the large pressure gradient normal to the free surface around the bottom of the droplet at the instant when the droplet begin to collide and combine with the film. And then, radial flow arises in the film near the solid surface, which leads to the splash during the droplet impingement. A crown forms after the droplet impacts on the film. Then the crown expands in the radial direction, while on its top a rim of increasing diameter is formed. This rim may form first cusps, then fingers and finally secondary droplets due to the Rayleigh-Plateau instability. For the case of ρl/ ρv=1000, because the vapour is much sparse, it could have almost no influence on the droplet splash after it collides with the film.

Fig .3 Velocity field resulted by impact of a droplet on film at T=2.5 as ρl/ρv=1000

Fig.4 Time evolution of the diameter at the bottom of the crown as a function of time at ρl/ρv=1000

As a basis for flow analysis, we determine the radius rl,bof the crown at the bottom as a function oftime.In Fig.4, the numerically predicted dimensions of the crown are compared with the analytic expression of Yarin and Weiss[5].The agreement between the two sets of predictions is fairly good,though the mumerically predicted results are a bit larger than that of the analytic expression of Yarin and Weiss[5].

Fig.5 Flow patterns resulted by impact of a droplet on filmat ρl/ρv=10

Figure 5 shows the flow patterns obtainedby computation for the case of a droplet impacting on the film at the same Weber and Reynolds numbers and the film thickness as the above case. The density and viscosity ratios of the liquid to the vapour are fixed at ρsat/ρsat=10 and μ/μ=10. As is shown in Fig.5, after the droplet impacts on the film splash forms. The splashing lamella on its bottom expands obviously, while on its top the expansion is halted, therefore, the lamella bends inwards and a “bell” like splash is observed.

Fig.6 The velocity field resulted by impact of a droplet on film atT=2.5as ρl/ρv=10

Figure 6 shows the velocity field after thedroplet impacts on the film. Because the density difference between the liquid and the vapour is small, the drag force of the vapour flow on the liquid is considerably large. From the numerical results, we can find that the vapour flow holds back the expansion of the splashing lamella on its top obviously, however, the effect of vapour flow on the expansion of the splashing lamellacause for this phenomenon was not given. Figure 7 shows the time evolution of the diameter at the bottom of the splashing lamella at different density ratio of liquid to vapour (der). It is revealed that as the density ratio of liquid to vapour is as large as 100 and 1 000, the splashing lamella on its bottom expands at almost the same speed. As the density ratio of liquid to vapour is small at (e.g., 10) the splashing lamella on its bottom expands a little slowly.

Fig.7 Time evolution of the diameter at the bottom of the crown as a function of time

It is apparent that the vapour flow induced by the dropletfall and splash in the whole impinging process may affect remarkably the splash behaviour. When the density ratio of liquid to vapour is small, “bell” like splash may be observed. The reason isdue to the drag forceof the vapour flow on the liquid which is induced by the droplet fall and splash in the whole impinging process.

3. Conclusions

The LBM with two-distribution functions, employed for simulating the two-phase flow induced by a liquid droplet impinging onto the film of the same liquid on solid surface, is suitable for solution of two-phase flow problem at high density and viscosity ratios of liquid to vapor and phase transition between liquid and its vapour.

Through simulations, the following conclusions are reached. A vortex loop may be induced by the droplet fall and splash in the whole impinging process on the medial side of the splashing lamella in the vapour phase. The vapour flow has weak influence on the expansion of the splashing lamella on its bottom, while on its top the expansion is halted obviously by the vapour flow when the ratio of liquid to vapor is small. Thus, the vapour flow may play an important role in flow pattern and splashing shape resulted from the impingement of a single liquid droplet on to the film. For the case of high ratio of liquid to vapor a crown may engender after the droplet collides with thefilm. However, for the case of small ratio of liquid to vapor a “bell” like splash may be observed. The Weber number, the Reynolds number, the ratio of the liquid to the vapor and the thickness of the film play different roles in determining the process of droplet spread, the dimensions of the splashing lamella and the shape of the splash at each time as a droplet impacts on the film.

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January 18, 2012, Revised March 20, 2012)

* Project supported by the National Natural Science Foundation of China (Grant Nos. 10872123, 11032007), the Ministry of Education in China (Grant No. IRT0844), the Opening Fund of State Key Laboratory of Nonlinear Mechanics and Shanghai Program for Innovative Research Team in Universities.

Biography: GUO Jia-hong (1966-), Male, Ph. D., Associate Professor