# Benchmark

Hello

I wonder if there exist a benchmark problem regarding flow of one or two fluids in a granular medium. I need these benchmark problems for verification of my MRT LBM code.

Amir

If you mean by Darcy law : V=K*i we don’t have K(permeability). we can use empirical equations to estimate K but this equations are for 3D media while my simulation is 2D and i used 2dq9. so this simulation doesn’t have any accuracy in initial steps. can you explain more about this, may be i’m wrong.

Regards,
Amir

Let me know if you are using pore-scale flow solver (Navier-Stokes) or macroscopically averaged (Darcy law) flow in granular porous media. I mean, at what scale you are solving flow field?

I used LBM to simulate granular medium and for example in permeability simulation i used average macroscopic velocity of all lattice of one section of my domain to calculate permeability of my domain. My domain simulation is 10050 Cm^2 (100005000 Lattice). If i’ve got your point.

Regards,
Amir

There is a way to calculate hydraulic conductivity (K, not permeability) in every macroscopic flow model using some LBM parameters, like we do for kinematic viscosity or diffusion coefficient.

Darcy law v=Ki, note that, K is hydraulic conductivity not permeability.

which LBM based macroscopic model you are working with?

First i used SC BGK 2DQ9 but when i saw that the results depend on relaxation time and because of excessive use of bounceback in porous media it’s appropriate to use MRT, so i changed collision operator of my code to MRT and now i want to verify my code.

Regards,
Amir

Look up following article
Qinjun Kang , Dongxiao Zhang , Shiyi Chen. Immiscible displacement in a channel: simulations of fingering in two dimensions. 2004.
There are different benchmark problems given to test SC model.
What are viscosity ratio and density ratio for binary fluids in your problem?

This is a really good article and i will check my code with it surely. but in fact what I want is a benchmark to verify my codes (MRT and BGK) in porous media this means that i want to check which of them has a better result considering excessive bounceback due to granular media.

I have another question, as I told I coded SC LBM based on sukop’s Book but when i used MRT i found that adding body force by shift velocity doesn’t have good results and i read in this forum that it is better to add body force to distribution function so i read Guo’s article but I’m confused. Please help me in this subject.

The point is that I have to simulate two-phase fluids and consequently there are interaction forces that I think are time dependent.

Thank you very mach.
Amir

You can do simple Poiseuille flow simulation and back calculate width of channel. This will give you error due to bounceback.

Pore-scale investigation of viscous coupling effects for two-phase flow in porous media. Huina Li, Chongxun Pan, and Cass T. Miller. PRE. 2005

You can add 3wt(exgr_x + eygr_y) in f’s after collision step to include body force. Hope this helps.

About the formula you wrote ,Does it mean that i must calculate interaction forces in X and Y directions (gr_x and gr_y) and then add this by weighted function to my distribution function after collision step and no other changes should be made. Is this true? How about the term added to calculate the macroscopic velocity that describe in Guo’s article?

Thank you very mach.
Amir

Yup, that’s right. I don’t know about Guo’s method.

Amir

I imply your suggestion about adding interaction force, my results are very well in Poiseuille Flow but when i want to simulate contact angle of one droplet on surface my results are not good for some of the variables of G (adhesion coefficient) with considering that when i add the interaction forces by shifting velocity instead of added interaction forces to distribution function my results are not bad. I calculate interaction force from shan-chen method that is described in sukop’s book. Your help is appreciated.

Amir

set relaxation parameter using approach given in following article:

Pore-scale investigation of viscous coupling effects for two-phase flow in porous media. Huina Li, Chongxun Pan, and Cass T. Miller. PRE. 2005

Let me describe my code completely:

First I calculate composite macroscopic velocity by formula No.95 of sukop’s book and then calculate collision step and after collision step I calculate Interparticle Forces by formula 98 and Fluid-Surface Forces by formula 66 of sukop’s book for each fluid in X and Y directions and then add sum of these forces to my distribution functions by weighted function in this way for example for direction e5 (F1(5,i,j)=Tempf_5+1.D0/12.D0*(FORCE_X_1+FORCE_Y_1)) and for e1 (F1(1,i,j)=Tempf_1+1.D0/3.D0*(FORCE_X_1)). My results are very good for Poiseuille Flow for different relaxation times but it cannot simulate contact angle of droplet on surface. As your suggestion about the set of relaxation times I set s7=s8=1/tou=1/3(1/s7-1/2) and s4=s6=8(2-s7)/(8-s7) and s1=s2=s4 but there is not any change! when i set all relaxation times to 1/tou the results are acceptable. It’s driving me crazy. Please help me.

Amir

Amir

what are G and Gads? Dr. Huang and Dr. Sukop has an article in PRE about selecting G and Gads. check if your G and Gads are good values. Gads control wetting or contact angle.

My G is about 1.8 and Gads is about -0.4 to 0.4 , Before i changed my collision operator to MRT I checked my code by article “Proposed approximation for contact angles in Shan-and-Chen-type multicomponent multiphase lattice Boltzmann models”(PHYSICAL REVIEW E 76, 066701 (2007)). and everything is mostly good. in my simulation density and viscosity and multi relaxation time of two fluids are the same. I think something about the composite macroscopic velocity of two fluids or adding forces is wrong, I guess my code doesn’t have any bug, I don’ know, what do you think?

Thank you very mach.
Amir

Check if you can do simple bubble test and verify your results against Laplace law. If this works, then your fluid-fluid interaction (surface tension) is good, later you can debug your fluid-solid (adhesion) interaction.
just a thought, try G=1 and Gads=0.1.
what kind of results are showing up? are you getting “Nan” ?

I checked my code as your suggestion by Laplace law as described below:

1. If I add interaction forces by shifting velocity and set all parameters of relaxation time matrix to 1/tau the results are very good but if I set S0=S3=S5=0 two fluids are mixed and bubble disappears and if I set s0=s5=0 and s3=1/tau I get “NAN”.

2. If I add interaction forces to distribution function for all relaxation times equal to 1/tau two fluids are again mixed with each other and bubble is disappeared and for other relaxation time parameters i get “NAN”.

But for Poiseuille Flow when I add interaction forces by shifting velocity the results are very good for all conditions of relaxation time parameters except when s3=0 this means that i can’t set s3=0 but when I add interaction forces to distribution function the results are very good even when s0=s3=s5=0.

The form of my relaxatione time matrix is [s0,s1,s2,s3,s4,s5,s6,s7,s8] and G=1.9. Viscosity of two fluids is the same and equal to 0.1666667 and RHO1=RHO2= 1.

What do you think?