Research areas and projects
Multi-component fluid flow
Introduction | lattice Boltzmann method | Binary flow | Result 1 | Result 2 | Result 3 | Applications | References
Binary flow
We use a Rothman-Keller type interface. Even though not the most used model to handle binary fluid, it has been found to be very handy for implementing multicomponents. Its main advantage over other interfaces models resides in that it does not need any embedded physics such as interface tracking and that it remains a purely local model. This crucial properties really eased the implementation.
The Gunstensen interface
Recalling the standard densities of probability f's defined above in (I), the Gunstensen model uses R's and B's corresponding now to a density of red (R) and blue (B) fluid. On a pure node (red or blue), the fluid undergoes a normal LBGK collision. On an interfacial node (where blue and red fluid coexist), the new R's and B's collide similarly but through their sum - colourless f's (f=R+B) are collided. This says the LBGK collision model is well established, 'the collision of the red and blue fluid at the interface will follow the same correct hydrodynamics'.
After having collided the f's (sum of red and blue), a segregation process is applied to separate the two species. A re-colouring process maximises the colour flux across the interface, minimising the diffusion of the red fluid into the blue and vice versa (practically, it achieves it with a great efficiency). This step is very straight forward and the core is contained in a look up table allocating the order of re-colouring. This interfacial model taken alone does not have any surface tension embebed and would be of very low interrest and utility.
The Guntensen surface tension method
Guntensen desined a way to impress surface tension to his model. As mentioned previously, it has to be forced in the algorithm as an additional step. It is added to the collided colourless densities before the re-colouring process (see previous section). The Guntensen approach is a very pragmatic one: the surface tension of a drop induces a diminution in the droplet's interfacial area which induces a pressure step accross the interface (c.f. the Laplace law). From this statement, it is reasonable to assume that contracting articially the interfacial area of a droplet not subjected to any surface tension would indeed produce the required interfacial pressure step and therefore surface tension. Guntensen's approach consist therefore in depleting the links parallel to the interface and feeding to the links perpendicular to it through the following pertubation step
where A is the surface tension parameter,
is the angle with the ith link, 
is the angle of the colourfield (assumed to be perpendicular to the interface), and cc is a modulator ensuring constant forcing throughout the interfacial domain.
The Lishchuk surface tension method
Lishchuk approach to impose surface tension to the Guntensen's interface is the reverse engineer from Guntensen's original method. It consists in forcing the inner pressure of the droplet to build up with respect to its surrounding though a net bodyforce acting on the interface, from the surrounding into the drop. This pressure step causes the interfacial length to decrease slightly and produces the wanted surface tension as well. The body force imposed on the interface is as follows
where F is the forcing term, s is the surface tension parameter, div(n) is the local curvature, rN is the interfacial function defined as follow: rN=(rB-rR)/(rB+rR).
It is important to note that the Lishchuck method provides much better drop's isotropy and reduced micro-current activity. This enhanced behaviour is probably due to its 2nd order gradients quantities, that makes it less local than the Gunstensen method.
The algorithm can be summarised as such
Improvement of the RK model
The major artefact of this model is the so-called microcurrents, which seems to be a common artefact for any interface using the LBM. These microcurrents drive a small however unwanted flow in the simulation which leads to anisotropy of a drop at rest. It should be noted that these microcurrents are easily overcame by the presence of a flow. We have implemented the RK model by tuning the perturbation to the densities with respect to the lattice. We observed a noticeable diminution of the microcurrents and a significant improvement in the isotropy [see mmdupin for more details].
As Thompson's [Thompson] reported it in his thesis, the surface tension applied by in the RK model is responsible for the microcurrents. He notice some dependence between the applied perturbation constant and the intensity of the unwanted flow. This has been confirmed by the presence of microcurrents within a monochromatic flow where a surface tension is applied along an imaginary circle
The anisotropy of the microcurrents and the explicit shape of the lattice (here D2Q9) within the microcurrents
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suggest that a non-negligeable unbalance of the surface tension between the long (i=1,3,5,7) and short links (i=2,4,6,8). The idea is therefore to modulate one with respect to the other in order to get them balanced and reduce the microcurrents. We define
where short/long refer to the lattice's links. We computed a series of drop supposed to be at rest and calculated the activity of the micro-current (defined as being the lattice-sum of the velocities) and the isotropy of the drop at steady state (defined as being the standard deviation of the distance of the interface to the best fitted circle):
From this graph
- the microcurrents activity can be highly reduced taking
~2.1 - the isotropy of the drop depends highly on the microcurrents activity
We then calculated the interfacial tension for given parameters 
and 
and found two expressions for the surface tension (to a first order value in 
)
where 
The ratio of these two surface tensions gives = 2.1213!
This calculation confirmed the results we obtained and validated the improvement. Further measurements using a Fourier fit to fit the interface shown that this model ensures a steady state isotropy of less than a 10th of a node.
It should be noted however that Lishutz et al have implemented a way to put surface tension by having greatly reduced microcurrents. This is by adding a bodyforce perpendicular to the interface and inversely proportional to the radius of curvature (see [sergey] for more details). The main drawback of this method is that it requires non local calculations (i.e. gradients to find the radius of curvature) which annihilates the local property of the RK model.
Difference of viscosity
Having a difference of viscosity between fluids is a necessary step towards real engineering flow. This section describes a test we have implemented to validate the hydrodynamics of the viscosity difference. The following configuration is considered
The velocity profile of the 'water' is given analytically by
Where the variables have their usual meaning. The profile of the air is very very similar to this equation. The programme was 1 lattice unit long, 200 lattice unit wide (100 water + 100 air), water was simulated by a viscous fluid, air is simulated by a non viscous fluid. We obtained the following flow profile for the water
The correlation coefficient of these two curves is 0.99998 . The air profile accords its analytical solution with the same accuracy. The same degree of accuracy was found for various values of 
, g and 
. We have investigated the stability of the system and the accuracy to the model within a wide range of viscosities. The model has been found to be very stable up to a viscosity ratio of about 600. The stability decreases with the forcing of the system (body force g and surface tension) but also when one of the viscosity reaches the boundaries (
=0 and =2).
The wall-liquid wetting dictates how a liquid sticks to a given surface. It is necessary to control it in any simulation dealing with walls. It is even more important when two or more fluids are in contact with a wall, giving rise to a three-component system (liquid A, liquid B, wall). We handle wall-wetting by applying a perturbation very similar to the one used and improved for the fluids' interface. It has been inspired by the classical analysis of the wetting that considers the different surface tensions as forces acting on the same volumelet of fluid, and consequently, having the same 'nature'.
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It is therefore reasonable to treat the wall-wetting in the same way that we treat the interfacial surface tension. The perturbation is consequently modified as such
We obtained the following shapes for different contact angles
Bigger systems we used to quantify the isotropy of the wetting drop. The shapes obtained were round to a 10th of a node. The following video shows a rectangular drop wetting the surface. Note that the simulation does not reach steady state.
Capillarity has not been fully studied yet. However, rising fluid through a capillary has been observed using this model
Dynamic contact angle
Very little is known or understood about this controversial phenomena. It is typically observed for moving droplets and two angles can be defined
This can also be observed when a drop is condensing or evaporating. Two angles can be measures, an advancing (A) and a receding (R). The (static) contact angle 
lies somewhere between them.
A drop on the surface of a window will drop if the following condition is met
Where the parameters have their usual meanings. The parameters of the following simulation are such that the drop is clamped to the surface because its does not met the latter equation.
