# poiseuille flow with inamuro method

Hello everyone,

I used the Inamuro boundary condition for poiseuille flow with periodic boundary condition and a constant body force to drive the flow in the channel.

Inamuro said that with his method the velocity profile remains the same for different values of Tau (relaxation time) and the no slip condition satisfies.

Now, according to my understanding, by increasing Tau we actually DECREASE the Reynolds number in the channel, because by increasing Tau the value of viscosity increases and due to the velocity experision : u(y)=u0*[1-(y/L)^2] for a channel with 2L width and a bodyforce F to run the flow, we have u0=2FL^2/(Rho*Viscosity). so by increasing the viscosity we actually decrease the value of u0 and therefore the Reynolds number. Is it true?

In the other hand, when we use the bounce back, by increasing the value of Tau, the slip velocity at the walls increases and that means by decreasing the reynolds number the slip velocity at the wall increases??!! Is it true?

Hi,
when you increase tau you increase the viscosity. Then if you keep the velocity, and the size of your domain constant, this means that you indeed decrease your Re number. Note that you have a certain number of free parameters in your simulations. Usually one parameter is the spatial resolution (or a caracteristic length) then you have the temporal resolution (which is equivalent to fix some reference velocity or viscosity) and finally you have the Reynolds number or the the viscosity. Note that you can impose the different parameters in different ways. You should have a look at this link for more details. Therefore saying that increasing tau means increasing the slip velocity for the bounceback BC is not really relevant because it really depends on what happen to the other parameters.

If you can give us more details it would be easier to give you a meaningful answer.

Dear Orestis,

First of all, thank you for your help,

My code has two input values,

1- a constant body Force to run the flow ( which is fixed to 0.0002 in my simulations)
2- Tau

then according to equation u(y)=u0*[1-(y/L)^2] for a channel with 2L width and a bodyforce F to run the flow, we have u0=2FL^2/(Rho*Viscosity). ( I get these equations from the paper “On boundary conditions in lattice boltzmann method”)

and then we can calculate other parameters such as Re. for example in my code when I increase the value of Tau, the maximum velocity in my profile decreses and it is in agreement with increasing Re.

I also get similar results in investigating Couette flow. for example the velocity profile with Tau=1 in the method mentioned in Succi’s book for Couette flow is similar to the velocity profile with Tau=20 with in inamuro method. I also have sent my results to your Email,

by the way I think the Re should increase to get the results of Inamuro, I also get the same results but by increasing Tau I think Re decreses, How could it possible?

1- a constant body Force to run the flow ( which
is fixed to 0.0002 in my simulations)
2- Tau

I guess that you also have the spatial resolution (or L the caracteristic length)…

I also get similar results in investigating
Couette flow. for example the velocity profile
with Tau=1 in the method mentioned in Succi’s book
for Couette flow is similar to the velocity
profile with Tau=20 with in inamuro method. I also
have sent my results to your Email,

You cannot compare simulations only based on the relaxation time. You have to keep the Reynolds number constant in order to be able to do comparisons.

What I suggest you to do is to fix some Re (Re=1 for example). You also give some characteristic length and some characteristic velocity (not bigger than 0.02 usually). This enables you to compute the viscosity (and therefore tau). Then you can compare the quality of your boundary conditions. When you compare two different reynolds numbers then you do not expect to have the same results, because the equations you are solving are not equivalent.

Hi and Thank you so much,

But I Knew it, I have two completely similar problems ( velocities and characteristic length is the same) with two different boundary conditions.

the only thing I do is to change the Tau ( and all the other parameters will be the same ).

So, can i ask you another question? it can really help to my understanding.

I hope that you read the paper of Inamuro " A non-slip boundary condition for lattice boltzmann simulations"
The question is that in what condition Figure4 in this paper will be correct?

When can we have U/Umax=0.8 at the solid wall (it means that slip velocity increases) ? Does it happen by increasing Re ( Re=V*D/nou , in this case V should increase faster because according to this paper nou increases up to 7) ? or by keeping Re the same and increasing Tau( V and Tau will increase at the same rate)?

What does Inamuro mean for increasing Tau? Does he mean that increasing Tau cause Re to increase or increasing Tau will increase the fluid viscosity at the same Reynolds number?

Thanks a lot

I think that you have to test it yourself, because the authors do not state clearly what is kept constant between the simulations…

Hi,

I am sorry but I have another simple question,

Which value of Tau in LBM corresponds to the viscosity of water? Since we always set Rho equal to 1, I doubt a little and I want to be sure.

Thanks a lot

Hello,

water at 20°C has a viscosity of eta_p = 10e-3 Pa s, i. e. nu_p = 10e-6 m^2 / s (kinematic). With the relation

``````
nu_p = dx^2 / dt * (tau - 1 / 2) / 3

``````

where nu_p is the physical kinematic viscosity, you can calculate the value of tau, once dx and dt are set. The other way around you can set a value for tau and then find dx and dt. This depends on your preferences.
But if you lattice is already fixed and your time step as well, then the above relation directly returns your tau.

Timm