# analysis of boundary conditions

Hi All,

I was wondering if anyone knows of some good articles related to the analysis of boundary conditions, other than for standard bounce-back schemes. For example, something for Zou and He boundaries would be nice. I know there are claims of the order of accuracy of this method, an easy expression for mass leakage, and we can numerically show some stability properties/issues. However, I have seen very little in the way of concrete mathematical analysis (although there is some very nice work for bounce-back by people like Ginsburg and Junk).

This is a problem I have become quite interested in so Id like to know of any existing literature (in terms of theory, not just numbers and examples).

Thanks a lot,
Tim

Hi Tim

He, X.Y., Zou, Q.S., Luo, L.S., and Dembo, M. Analytic solutions of simple flows and analysis of nonslip boundary conditions for the lattice Boltzmann BGK model. Journal of Statistical Physics 87, 115-136, 1997.

Hiya,

Yes, that’s a very good paper indeed. I was wondering if there are any more similar ones that are particularly interested in stability and convergence from a mathematical viewpoint. Junk also has some very nice work (JCP 2005) but it’s primarily concerned with bounce-back.

Tim

Hi dear Tim,

for extrapolation method I suggest " a coupled lattice BGK model for Boussinesq equation" and " an extrapolation method for pressure and velocity boundary condition in LBM" by Zhaoli Guo, Baochang shi.
for Regularized BC I suggest “Hydrodynamic limit of lattice Boltzmann equations” by LATT, Jonas.
“Viscous ?ow computations with the method of lattice Boltzmann equation” by Yu D,Mei R . This paper reviews different techniques of boundary conditions
for moving BC I offer “A dynamic boundary model for implementation of boundary conditions in lattice-Boltzmann method” by Jinfen and Sangmo Kang.
“Straight velocity boundaries in the lattice Boltzmann method” by Jonas Latt. in this paper 5 BC models are compared.
you said zou and he BC so “Implementation of on-site velocity boundary conditions for D3Q19 lattice Boltzmann simulations” by Martin Hecht would help you in D3Q19 Zou and He model.
also we have symmetry condition , periodic BC, Open BC, Equilibrium and Non-Equlibirion distribution function Model and so on. It would be better if u tell me exactly what kind of BC you like to work on it.

Best Regards,
Arman

Hi Arman,

Thank you for your suggestions. I’m actually working on a slightly different method to all of those. I suppose it is most like Zou and He in the sense that a) it is on-site and b) it is local. However, my current interest is mainly with analysis of methods, and I mean this in a mathematical sense.

For example, we can do an asymptotic analysis of the LBE with diffusive scaling to assess the properties of our scheme (Junk et al). We can do a similar thing when we have bounce-back boundary conditions.

Zou and He’s method, for example, is less stable than bounce-back. We can show this through high Re simulations but to understanding this and quantify this we need some numerical analysis (ie, some maths, probably, and not just examples).

I was wondering if anyone knows of any attempts to do this, or something similar, for methods other than bounce-back.

Thanks.

Hi Tim,
Great work, I checked my paper’s archive and my university library but I could not find a good one which is really close to your research. I guess “Straight velocity boundaries in the lattice Boltzmann method” by Jonas Latt is the same work as you want to do.
Honestly, your finding will help my future research. please share your finding in this forum or inform me via email.
Thanks a lot in advance and Good Luck,
Arman

Hello!
Anyone able to explain the first result in the He, X.Y., Zou, Q.S., Luo, L.S., and Dembo, M. mentioned above?
The same derivation can be found in Zou, He 1995. Here, it is precisely the last step of equation (6) that I do not understand. How can you get rid of the distribution functions f_5…f_8 ?

The He et al paper shows that the LBE can be cast as a recurrence relation which, in certain situations (e.g. Poiseullie flow) can be solved analytically (it’s worth noting the work of irena Ginzburg here, too). The f5…f8 contain f at a neighbouring node and these can be written in terms of u. The algebra gets a bit tedious, but it is manageable.

This work shows that bounce-back BGK introduces a numerical slip. This is NOT physical. Note also that this paper shows the unique LBE solution is a shifted parabola, so there can not be any Knudsen layers.

Tim

## pleb01 Wrote:

The f5…f8 contain f at a
neighbouring node and these can be written in
terms of u. The algebra gets a bit tedious, but it
is manageable.

Well, I tried, but: The last step where the f5…f8 are eliminated is unclear to me. What exactly do you have to substitute for f_5, for instance?

It’s best to look at things in pairs in the definition of momentum

f5(j-1)-f6(j-1)=rho*u(j-1)-f1(j-1)+f3(j-1)-f8(j-1)+f7(j-1)

f8(j+1)-f7(j+1)=rho*u(j+1)-f1(j+1)+f3(j+1)-f5(j+1)+f6(j+1)

You can use the given expressions for f for (f8(j-1)-f7(j-1)) and for (f5(j+1)-f6(j+1)), and once you’ve done this you have what you need, ie (f5(j-1)-f6(j-1) + (f8(j+1)-f7(j+1))

Thank you for your reply! But I think the substitution you gave does not eliminate the f_5…f_8 in Eq (6) of Zou,He-1995! Your substitution is helpful only in the first step.
What I really need to solve in the last step is how

(1-1/tau)[rhou[sub]j-1[/sub] + rhou[sub]j+1[/sub] - ( f1(j-1) - f3(j-1) - f7(j-1) +f8(j-1) ) - ( f1(j+1) - f3(j+1) + f5(j+1) - f6(j+1) )]
= 1/3
(1-1/tau)[rhou[sub]j-1[/sub] + rhou[sub]u+1[/sub] - rhou[sub]j[/sub]],

in the last line of Eq. (6).

It certainly works for the He Zou Luo and Dembo paper (which I believe is the original fully published version), so it will work with anything equivalent (with perhaps a bit of notation tinkering). Doing what I say above will leave you with f1 and f3 terms at both j+1 and j-1, which we know, and also f5, f6, f7 and f8 all at the same node (j) and in a nice combination (specifically, ff5-f6+f8-f7). This can we written in terms of momentum (ie rho*u-f1+f3)=…_).

It definitely works. I can see from your quote that there are f8(j-1) and f5(j+1) etc terms. As I said in my previous post, you can use the expressions for f to find these. You’ll then have the f8(j) etc terms that I mention above. f1 and f3 cause no problems.

Could you provide some hints on how to derive the formula for the slip velocity, Eq (20) in the He Zou Luo and Dembo paper, for the Bounce Back rule. Because now I am stuck with that… Well done!

Equation 20 is again difficult/messy because of the recurrence relation, and because the previously derived expression for U_s isn’t valid in this case. This means you have to go a step back and apply the recurrence relation to the f_i at the wall, which will involve things like f_7(j=0). f_7 etc evaluated at j=0 involves the velocity at j=1 and indeed f(j=1). Playing with this in a similar way as before will yield equation 20 (if the algebra gets to tedious you could use something like maple to help). Equation 22 is a lot simpler to find, I think.

Good luck!

Hi, pleb!
I agree, Equation 22 is no problem. Did you manage to solve the recurrences of the paper with Maple or some similiar tool? If so I would be highly interested!
On more question regarding Eq. (20): Do I have to solve for an expression like

``````
u_1 = (u_0 + u_2)/2 + G*(dx^2)/(2*nu) + [wall-influence],

``````

like Eq . 17 again or is the trick different from the derivations before?
thanks!

I manage to get rid of all die PDFs, but I end up with a different slip velocity.

You know what u1 is from its definition (ie f1-f3+…). The f1-f3 bit is no problem. Now use the bounce back condition to replace the unknowns (f5 and f6) with the knows (f7 and f8 at y=0). This gives you an expression that involves f7 and f8 at both y=0 and y=1. Using the recurrence relation of the fs at y=0 leaves you with only f7 and f8 at y=1. Now use the recurrence relation again. This will give you a u2 term, which we have an expression for, and you can proceed as before (that is, use the definition of momentum to find f7 and f8 at y=2). I’ve done this by hand a couple of times for various different things, but because the algebra can get a bit messy/tedious/boring you could use a (symbolic) package like Maple to help you.

Good luck!

## pleb01 Wrote:

You know what u1 is from its definition (ie
f1-f3+…). The f1-f3 bit is no problem. Now use
the bounce back condition to replace the unknowns
(f5 and f6) with the knows (f7 and f8 at y=0).
This gives you an expression that involves f7 and
f8 at both y=0 and y=1. Using the recurrence
relation of the fs at y=0 leaves you with only f7
and f8 at y=1.

it gives

rhou[sub]1[/sub] = c(f1sub[/sub]-f3sub[/sub]) - 1/(6tau) * rhou[sub]1[/sub] - 1/6d[sub]x[/sub]/crhoG + c1/tau * (f8sub[/sub]-f7sub[/sub]).

Now use the recurrence relation
again.

this gives

rhou[sub]1[/sub] = c(f1sub[/sub]-f3sub[/sub]) - 1/(6tau) * rhou[sub]1[/sub] - 1/6d[sub]x[/sub]/crho*G

• c1/tau * [1/(6tau) * rhou[sub]2[/sub] + 1/6d[sub]x[/sub]/crhoG + c*(tau-1)/tau * (f8sub[/sub]-f7sub[/sub])] .

This will give you a u2 term, which we have
an expression for, and you can proceed as before
(that is, use the definition of momentum to find
f7 and f8 at y=2).

I recognize that I could solve the equation for (f8-f7) at j=2 now. But what would the “definition of momentum” be useful for in this? Using the LBE-recurrence also on the f1-f3 i would get an expression in u1 and u2 for my (f8-f7).

I’ve done this by hand a couple
of times for various different things, but because
the algebra can get a bit messy/tedious/boring you
could use a (symbolic) package like Maple to help
you.

Good luck!

thanks! We have already worked out the solution of the LBE in the interior [i.e. u(j) for j being interior nodes] , so we know what u at j=2 is, for example. The definition of momentum will allow you to write f8-f7 in terms of rhou2, f1-f3, and f5-f6 (at j=2). Again, we have the LB solution for u2, so this bit is fine, and we have the recurrence relation for the difficult fs. f5 and f6 will introduce fs at neighbouring points, so in our full expression we have things like rhou1, f1-f3 and f5-f6 (at j=1), which can be manipulated. We did a very similar thing when we we looking for the analytic solution in the bulk - that is, we played around with fs and momentum and j+1 and j-1 to give us something convenient at j. Once we are left with only terms involving u or G (or f1 and f3) we are in good shape because we know all these in the bulk - G is constant here and we have already analytically solved the LBE for this flow (in the bulk).

## pleb01 Wrote:

Once we are
left with only terms involving u or G (or f1 and
f3) we are in good shape because we know all these
in the bulk - G is constant here and we have
already analytically solved the LBE for this flow
(in the bulk).

Sure - it’s always the same in principle! In the bulk you can use the symmetry of the recurrence in the y-direction to get rid of them. But doing as you described at the boundary does not geht you rid of all the fs because you will always have either f5-f6 or f8-f7 remaining.