Falling liquid film flow is widely used in many processes. Supplementary to experimental studies, NavierStokesbased models have been employed for describing film flow phenomena. These models are often disadvantageous since they are either strongly limited in their generality or need enormous computational resources. In this investigation, a new approach is proposed for modelling flow by lattice Boltzmann methods. Therefore, the wellknown ShanChen model (Shan and Chen, 1993) has been employed to an isothermal falling liquid film. The validity of the implementation has been checked against some singlephase and twophase reference cases. Test series have been conducted for three different Reynolds numbers without external disturbances and for one Reynolds number with sinusoidally pulsating inlet velocity. The computational results show that lattice Boltzmann methods are capable to model falling liquid film flow and that the flow morphology is in qualitatively good agreement with other numerical and experimental works.
Falling liquid film flow is a critical operation in many engineering processes, such as densification of suspensions or emulsions or the separation of mixtures, liquefaction in the condenser of a power plant or heat and mass transfer in a heat pipe. Experiments and previous simulations have shown that the film becomes wavy from certain Reynolds numbers
The theoretical and numerical description of falling liquid films is still a challenge. Existing models are either limited in their generality or need enormous computational resources. These are evolution equations of the film thickness (also known as Benney equation, longwave equation), (weighted residual) integrated boundary layer equations (IBL, WIBL, WRIBL), and the full NavierStokes equations, with increasing computational demand.
Based on the twodimensional NavierStokes equations, evolution equations of the film thickness have been designed, by introducing a stream function and assuming long disturbance wavelengths and Reynolds number of the order of one. First research has been performed by Yih, 1955 [
Shkadov, 1967 [
Miyara, 1999 [
Lattice Boltzmann methods are a relatively new approach for simulating flow phenomena. Based on molecular gas dynamics, first models were presented in the late 1980s [
The aim of this study is to provide an approach for falling liquid film flow modelling by employing lattice Boltzmann methods. For this purpose, we assume isothermal twophase flow that can be described by the model of Shan and Chen, 1993 [
For solving the hydrodynamics of a falling liquid film, the ShanChen model [
The lattics Boltzmann equation of the particle density distribution function of a component
Velocity directions
The vector
The interphase force leads to the effects of phase separation and surface tension. From (
The fluidwall forces can be calculated by
The computation of the gravitational force can be performed with
By using a ChapmanEnskog expansion (see, e.g., [
A falling liquid film is investigated, flowing down an inclined wall with an angle
Geometry of the computational domain (black box); A: wall, B: liquid inlet velocity, C and D: walls, E: gradientfree outlet;
The applied discretisation scheme (D2Q9) can be used only for uniform lattices, wherein the spacing is limited by two dependencies as follows.
Necessary resolution of film thickness (Kuzmin and Derksen, 2011 [
For neglecting compressibility effects,
Grid independence test for
The boundary conditions are shown in Figure
The outlet (E) was assumed to be free of gradients, which is realised by setting all
For initialising the domain, the film was supplied with a constant thickness
The previously described model has been implemented in a selfwritten serial C++program, which has been validated with single and twophase benchmark tests. Computations have been conducted for Poiseuille flow and flow around a cylinder which showed an excellent performance of the code for singlephase flows. The twophase benchmarks were the transition from a liquid square to a circle and the following determination of the surface tension as well as the investigation of the contact angle formation at the threephase points [
The squaretocircle computations have been performed by defining squares of different sizes being transformed to circles, whose radii have been measured. The pressure difference between inside and outside of the drop has been also determined. In Figure
Evaluation of the pressure difference depending on curvature of the drop.
The characteristic velocity
By employing the present implementation, the following test cases have been investigated for a vertical wall (
nondisturbed flow at
sinusoidally disturbed flow at
The parameters for the computations are shown in Table
Parameters for computation.
Re = 5  Re = 8  Re = 20  Re = 20.1  


0.115  0.135  0.183  0.184 

43.50  59.51  109.5  109.9 

0.578  0.676  0.917  0.919 

115.5  135.2  183.3  183.8 

2.655  2.272  1.674  1.672 

0  0  0  0.05 

0  0  0  45 

2.452  2.452  4.660  4.415 
Ka/10^{−3 }  28.58  3.544  0.050  0.049 
 

90.0  

1.004809 × 10^{−6}  

−120.0  

−187.16  

528.8  

86.4  

14.33 
The hydrodynamics of falling liquid films without artificial disturbance at the inlet have been studied for three different Reynolds numbers representing specific changes in the flow morphology according to Gross et al., 2009. [
In the following sections, the wave shapes depending on time and space are shown as well as the velocity fields and the characteristic wave velocities.
Figures
Evolution of the interface with
The wave evolution for
In Figure
Evolution of the interface with
Evolution of the interface with
The wave evolution for
The figures shown here demonstrate that it is possible to model the hydrodynamics of falling liquid films by lattice Boltzmann methods, but it should be mentioned that the direct comparison of these results with experimental data is difficult, due to the limitations of the model; the fluid densities are coupled with the surface tension by the cohesion parameter
For better insight, two sections of the flow field are shown in Figures
Section of the velocity field for
Section of the velocity field for
Figure
In Figure
Both the wave velocity of a solitary wave and the velocity of a wave producing region can be evaluated by measuring wave positions at two consecutive points in time. In Table
Wave velocity of solitary waves (solitary w.) and beginning wave production (b. w. production).
Re 



Solitary w.  b. w. production  
5  —  32.59 
8  126.86  42.74 
20  387.50 

It can be seen in Table
Apart from falling liquid films with a natural transition to wavy interface, computations have been performed with a sinusoidally disturbed film, and the results are compared to those obtained by Gao et al., 2003 [
In Figure
Comparison of the present results for
In the present investigation, falling liquid films on a vertical plane have been modelled with the ShanChen model, which is an isothermal multiphase and multicomponent lattice Boltzmann model based on interaction potentials. The implementation of the model has been validated against various single and twophase flow problems and proved its validity.
The simulation of falling liquid films has been conducted for cases with Reynolds numbers of
To conclude, it can be stated that the lattice Boltzmann methods are capable to simulate falling liquid films but the constraints given by the ShanChen model limit their generality. The investigation of liquid film flow with heat transfer, larger Reynolds numbers, and other geometrical parameters will be conducted in the future.
The authors greatly acknowledge the valuable comments and suggestions given by the unknown reviewers.