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# Density in Shan Chen Multiphase LBM

 Density in Shan Chen Multiphase LBM September 06, 2017 06:59AM Registered: 2 years ago Posts: 18
I have been trying my hand at multi phase simulations using Shan Chen LBM. For G=-120 and rho0=200, I get the rho_liquid to be 529 and rho_vap to be around 80, as expected. I also read that converting LBM to real density is done by:
rho_phy=(rho_lbm/rho_cr_lbm)*rho_cr_phy
that is, using reduced variables. rho_cr_lbm is the critical density in LB units which in this case is 200. and rho_cr_phy is 322.

this leads to a liquid density of 850 which is somewhat near the real value of 1000 but the vapor density becomes around 130 which is nowhere near the actual value.

What am I doing wrong here?

I appreciate any help provided.
Githin
 Re: Density in Shan Chen Multiphase LBM September 06, 2017 07:38AM Registered: 4 years ago Posts: 56
Dear Githin,

I am assuming that you are trying to simulate a water-air-like system, where the density ratio is of the order ~1000 (some more information on what you are trying to do would be helpful), i.e. the vapor density you are obtaining is too large. To have a smaller vapor density (and larger liquid density), you need to further decrease G. That way, you might be closer to the actual values.
However, the potential function you are using (I assume: psi = psi0 * exp(-rho0/rho)) does not support that large density ratios. You would need to use a different function for psi, which will result in a different (more stable) equation of state, e.g. as shown in

Yuan & Schaefer: Equations of state in a lattice Boltzmann model. Phys. Fluids 18, 042101 (2006).

Hope this helps, otherwise some more information about your problem might be in order.

Regards,

kk
 Re: Density in Shan Chen Multiphase LBM September 14, 2017 01:24PM Registered: 2 years ago Posts: 18
Dear kk,

Yes I am trying to simulate a water-air or water-water vapor system and yes I did use the psi function you mentioned.
I did try to implement the CS EOS as it is used in lbm.
p=rho*R*T*(1+rho+rho^2-rho^3)/(1-rho)^3-rho^2
and psi=sqrt(2*(p-c_s^2*rho)/c0g)
here i omitted b/4 and as they are taken as 4 and 1 respectively.
I was unable to get phase separation using CS EOS or any other EOS in practice. I am not sure where I am going wrong. If you are able to get phase separation, what values of T did you use? and what are the density values you obtained for liquid and vapor? mine always seemed to distribute equally in the end regardless of the liquid and vapor density i use.

Also, is the method of converting density (using reduced variables) right? I would just like to make sure. any calculation i conduct on the loss of liquid water from the system can be somewhat correct since liquid density from lbm is around 800, which is similar to 1000, right?

Githin
 Re: Density in Shan Chen Multiphase LBM September 14, 2017 01:47PM Registered: 4 years ago Posts: 56
Dear Githin,

1) to the Carnahan-Starling Equation: For the Values you are using (a=1, b=4 and I am assuming R=1), the critical values are

rho_crit = 0.130 443 884 192
T_crit = 0.094 328 703 133

So, if you simulate a static droplet with e.g.

T = 0.084899758 (T_red ~ 0.9)
rho_Liquid = 0.248056279
rho_Vapor = 0.045435027

you should obtain a stable droplet with densities somewhere close to these intial values. You are going to have to set G either to -1 or 1 to prevent the radicand in the calculation of psi from becoming negative. If that is not working, something is probably wrong with your implementation.

2) Reduced Variables: Using reduced variables is the right way to go. Since you are interested in a water-air-system, you might want to take a look at

Stiles, C. & Xue, Y.: High density ratio lattice Boltzmann method simulations of multicomponent multiphase transport of H2O in air. Comput. Fluids. 131, 81-90 (2016).

Regards,

Knut
 Re: Density in Shan Chen Multiphase LBM September 15, 2017 06:11AM Registered: 2 years ago Posts: 18
Dear kk,

Thank you for the values. I will try them out and reply with the results.

Regards,
Githin
 Re: Density in Shan Chen Multiphase LBM September 18, 2017 11:38AM Registered: 2 years ago Posts: 18
I applied your values and the phase separation is now occurring. But CS is not going below T_red=0.68 and I need to simulate as close to room temperature as possible. Also, it is showing a higher interface thickness than Shan Chen. So I am thinking of using PR EOS which should be able to go to T_red=0.58. What are the density values for this EOS? And is there any way of calculating these so that I dont have to trouble you each time?
I tried T_red=0.8, rho_liq=0.2363 and rho_vapor=0.008 with no success. (accentric factor = 0.344).
Also, what is critical temperature in Shan Chen model and how do i change the temperature there? I did a few simulations using that and want to find out what the temperature was. By default RT=1 right (in the Shan Chen EOS)?

Any help is appreciated.
 Re: Density in Shan Chen Multiphase LBM September 18, 2017 01:22PM Registered: 4 years ago Posts: 56
Dear Githin,

the values I provided are for the temperature I provided. When you want to change the temperature, you should also adjust the intial values (lower temperature -> lower vapor density -> higher liquid density). If you are closer to the actual solution, the simulation is probably going to be more stable even at lower temperatures (Yuan & Schaefer 2006 managed to go down to 0.53T_crit, see their publication). The density values can be obtained by solving the Maxwell Contruction for the given EOS (see e.g. the book from Sukop & Thorne 2006). I do not have the values at hand for the PR EOS.
If the density ratio is very large, e.g. on the order of ~1000, there is not going to be a big difference in the flow characteristics to using a density ratio of e.g. ~1100. So maybe you do not have to change the EOS... depends on what you want to do.
In the basic shan chen model, the temperature is regulated via G, where G ~ 1/T. For psi = 4 exp (-200/rho), the critical value of G is ~-92.4. (You should definetely take a look at the book from Sukop & Thorne ;-) ).

Regards,

Knut

Edited 1 time(s). Last edit at 09/19/2017 06:09AM by kk.
 Re: Density in Shan Chen Multiphase LBM September 20, 2017 09:32AM Registered: 2 years ago Posts: 18
Dear kk,
I have read the paper by yuan and schaeffer. I do get a t_red of 0.5 on SC EOS. But the problem is that at those values, the rho_red will be near 5.5 which means the water density is about 1771 which is obviously false. At the default value of G (-120) where T_red=3/4, the density obtained is very accurate but not at lower temperatures. And the density ratios here are just above 20, nowhere near the actual values. Still, i need the liquid to be accurate rather than the vapor, so thats fine.
Thanks for the other information about temperature and the density values. It was very helpful.
I need a proper EOS which will give accurate density values for temperatures as low as 0.5*Tc. Any ideas?

Regards,
Githin
 Re: Density in Shan Chen Multiphase LBM September 25, 2017 05:44AM Registered: 4 years ago Posts: 56
Dear Githin,

the SC EOS is obviously not suited for such high density ratios, but the CS EOS is. You can then do a simulation where the liquid density is properly scaled to the physical system, but the vapor density is not. If the liquid density is ~800-1000 then it does not matter whether the vapor density is 1 or 5 or 10. The influence of the vapor phase on the flow characteristics is negligible, as long as its density is "very small" compared to the density of the liquid phase.

My only other idea is to obtain more accurate results at ~Tred=0.5 is that you could toy around with the parameters (a,b) in the EOS. Still, I do not think it is worth the effort.

Regards,

kk

Edited 1 time(s). Last edit at 09/25/2017 10:13AM by kk.
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