# grid size and uniform flow

Hi Dear All,
I need some help related to two basic things:
In case of circular cylinder we know the following:
Re=ud/v (u inflow velocity, d diameter and v kinematic viscosity)
v=ud/Re (from here we can calculate the kinematic viscosity v)
t=3v+.5 ( t is the single relaxation time, v is the kinematic viscosity)

1. I want to know if i want to kept the single relaxation time constant and to study the effect of inflow velocity for fixed reynolds number like 100. From above equations if i change the inflow velocity this means kinematic viscosity are changed and the reynolds also changed not remaining fixed.
So how i can set these parameters that the single relaxation time must be fixed and also the reynolds number and i can only change the inflow velocity u and to study the influence of u and keeping the Mach number less than 0.1.

2. The other basic thing i want to know if i want to study the grid sensitivity of LBM. Like 126x96, 177x122 and 348x224.
I know if the grid size change the time step also need to change. I want to know if the grid size change how the time step can be changed. Because in LBM we know that the lattice constant and time step need to be unity. Most of the time i used the time step equal to 1. But if i want to use like 0.05, 0.02 so how i can set the grid size for this.

Thanks in advance these are basic questions and a lot of my friends know about these things. So if some can explain these with some examples.
Best Regards
Khan

Dear Khan,

have a look at this recent thread.

Timm

Hi Dear Timm,
I read that thread it’s a nice explanation about how to increase the lattice resolutions. If you don’t mind i need some help more. I study your recent paper and you explain the effect of single relaxation time in detail. For example in your paper you study the influence of different tau for Mach number 0.1, 0.01, 0.2 and 0.4. Can you give here a little explanation that how you can get different tau for the same Mach number. I know that if we change the Mach number we can get different tau from the following equations:

1. Re=Ud/V (u inflow velocity, d diameter and v the kinematic viscosity)
2. v=Ud/Re (from here we can get the kinematic viscosity)
3. tau=3*v+1/2
from above three equations it shows to me that if i use the one value for inflow velocity U i can get the value of tau from equation (3). But if i want to get another tau value for the same inflow velocity U so what i need to do.
Can you explain this via an example that how to get different tau values for fixed Mach number.
Thanks again and i really appreciate your kind suggestions and help from time to time.
Best Regards
Khan

There is an equation in our article (Eq. 34) which tells you everything you need. It contains the four parameters Ma, Re, tau and N (number of lattice nodes along a given axis, i.e. the inverse resolution). If Re and Ma are fixed, the equation states that one has to change N in order to increase or decrease tau. Basically, this equation is all you need for the computation of the simulation parameters of simple flows.

Hi Dear Timm,
Thanks a lot. Dear Timm one more thing after reading your paper. There is a statement " The accuracy of the boundary conditions also depends on tau. In case of bounceback rule it introduce a slip velocity at the walls, which depends on the relaxation time tau."
I want to know dear friend if i can study the flows around circular cylinder for stationary walls at the top and bottom. How i can observed this that there is slip velocity can from any one below:

1. From drag coefficient
2. From lift coefficient
3. From vorticity contour lines
4. From streamlines
Or some other criteria for observing this slip velocity in case of bounceback.
Thanks in advance and Best Regards
Khans

Dear Khan,
I have never computed the slip velocity in a system simulated by the LBM, thus I do not know how you can do it efficiently. I think you have to consult some articles.
Does anybody else know how to do this?
Timm

Hi Dear Timm,
Thanks for reply again. Related to your (equation(34)) i need some help but first i want to explain something that you can understand my question what i want to ask.
For example for Re=100 i simulate flow around a circular cylinder.
First i choose the inflow velocity (Ui=0.0438, d is the diameter of the cylinder is equal to 20)

1. we calculate tau like this: tau=Uid/v (where v is the kinematic viscosity and we can find from this equation Re=Uid/v so v=Uid/Re) thus we get the tau value is 0.518.After knowing tau we just say (omega=1/tau) so omega value is 1.9305019
2. Then we need to be sure that Re is fixed it;s 100. From first step we know the equation for Re and from that equation we find kinematic viscosity, v. so we can use this equation (Re=Uid3.0/(1.0/omega-0.5) and we get that Re=100.
Same like your equation (34) in your paper I know that Re=Uid/v and Ma=Ui/Cs
so from these two equations i know Ma/Re=(Ui/Cs)(v/Uid)=v/Csd then we know about v=CsCs(tau-1/2)hh/dt (where h is the lattice speed constant and dt is the time step) if i kept h=dt=1 so i have Ma/Re=(CsCs(tau-1/2))/Csd which is equal to
Ma/Re=(1/sqrt(3)(tau-1/2))/d where the speed of sound of fluid is 1/sqrt(3). Now we have four dimensionless parameters Ma, Re, tau and d(diameter of the cylinder). I want to find tau from here for Ui=0.0438 means Ma=0.025 and Re=100. so the equation for tau is
(tau=sqrt(3)d(Ma/Re)+1/2) so this mean tau value is 0.50866. Now i want like step 2 there we know that Re=100.
so i want to find now Re from this equation so our equation will be like this (Re=2
sqrt(3)dMa/(2*tau-1)) so i get the Re=200 not Re=100.
My dear friend Timm i want to know if i kept the Re and Ma fixed and studying different tau so from your equation we need to increase Nz or to decrease tau.
Can you explain little bit more if you have inflow velocity Ui=0.0438 this mean you have Ma=0.025 , diameter of the cylinder is (d=20) and Re=100. So how i can study different tau for fixed Re and Ma that the Re will be same in all cases.
If you explain in one example it will help more my friend.
One thing more my friend in your equation (34) Nz along one axis it’s about the streamwise direction or the vertical direction of the computational domain.
Thanks again i hope you will help through an exmple.
Best Regards
Khan

As I told you: In order to keep Re fixed, you can change Ma, tau and N according to Eq. (34) in our paper. If additionally Ma should be held constant, your only choice varying tau is to change N, namely N \propto (tau - 0.5).
For example, let us assume that you have tau = 1.0 and N = 80. Now you want to see what happens taking tau = 0.6 at fixed Ma and Re. Then you have to compute the ratio (1.0 - 0.5) / (0.6 - 0.5) which is 5. Your new numerical grid size then is 80 / 5 = 16. Actually, this is the way we have created figs. 2 and 3 in our paper.
Is it clearer now?
Timm

Hi Dear Timm,
Thanks for answer related to my question. I still little bit confused. If you can help one last time more.
We know that Re=Ud/v (U inflow velocity, d circular cylinder diameter, v is the kinematic viscosity), Ma=U/Cs and v=CsCs(tau-1/2). Now from above relations i know that Ma/Re=Cs*(tau-1/2)/d.
Now i want to to calculate different tau and fixed Re=100, Ma=0.025 (this means U=0.04325) and d=20 with computational domain Xmax=1001, Ymax=201.
Now i want to calculate tau=1.0, 0.9, 0.7 and 0.6.
So in such cases Ma and Re is fixed so what about the computational domain for different tau and what about the cylinder diameter d, which one will be changed in these.
Thanks and i hope you will answer for such different tau values.
Khan

This is exactly what I told you in my last post. Please re-read it carefully. Changing tau from tau[sub]old[/sub] to tau[sub]new[/sub] leads to a change in lattice length scale by N[sub]new[/sub] = N[sub]old[/sub] * (tau[sub]new[/sub] - 0.5) / (tau[sub]old[/sub] - 0.5). This is valid for both the total grid size, but also for the cylinder diameter.

Hi Dear Friend Timm,
I need just some confirmations. N means to change the Xmax or Ymax or need to change both.
Second if i used oldtau=1.0 and newtau=0.8 then Nnew=Nold*(0.8-0.5)/(1.0-0.5)
then i want to check newtau=0.7 so it will be like this Nnew=Nold*(0.7-0.5)/(0.8-0.5) so here i used tauold 0.8 not 1.0 and the Nold i will used that i get for the combination of oldtau=1.0 and newtau=0.8.
Is this process is correct.
Best Regards and Thanks a lot.
Khan

I mean you just have to solve equation 34. There is nothing more to it. So your procedure is correct.
You then have to change all length scales in your system, including x-, y- and z-axes as well as all diameters, radii etc.