# Are the Reynolds numbers in physical and LB unit Differ?

Dear LB Researchers
hi,
I have a question about Reynolds number in physical and LB unit bases. consider the fluid flow through a 2-d channel with the width of W and the inlet velocity U all in the physical world. in physical unit Reynolds=UD/viscosity
Timm defined Reynolds number in LB unit as Reynolds=(U_lb
Ny)/viscosity_lb where, Ny is the number of lattice nodes in y-direction . I confused. if we compare to Reynolds number we can see that they are not same but we expect that dimensionless number Reynolds remain unchanged in physical and numerical world. I will be appreciated if someone give me her/him evaluable comments

Hello,

if you want to simulate the same flow as in a physical system, the Reynolds numbers (physical and numerical) must be the same. This condition can be used to find the correct simulation parameters. For example: If you know the physical Re, and you have (in lattice units) the characteristic length and velocity, you can directly compute the numerical viscosity.
You have to be careful how you define your Reynolds number: If you have a cylinder in a channel, you might take the cylinder diameter or the channel diameter for its definition. When you compare the Reynolds numbers, you have to make sure that the definitions are equivalent, i.e., you should either take the channel diameter or the cylinder diameter in both cases.
The cleanest way to start, I believe, is first to think of accurate definitions of all quantities. Then you can start computing derived quantities like the Reynolds number.

Timm

hi
Timm,
Thank you replied my question very soon.
Consider we are simulating steady state incompressible 2-d channel flow using d2q9 LB where, the channel physical dimensions is 2m in length and 1m in width. Assume the physical Reynolds is a known parameter Reynolds=10 with inlet velocity Uin=0.01m/s and viscosity =0.001 m^2/sec ( Reynolds=10.01/0.001=10) and we persist to keep above mentioned physics in our LB simulation. For our simulation assume we generated a calculational domain with 201 101 nodes (lengthwidth) then Dx=Dy=0.01 . if we choose relaxation time to be 0.8 as you prefer in your valuable paper (shear stress…) with known inlet velocity Uin,lb=0.01 then Reynolds number in LB units will be (0.01100)/0.1=10 . If the mesh size is not set primarily and I determine it using the similarity of Reynolds numbers in physical and LB, I can reach to mesh 201*101. Please tell me if the above analysis is correct (specially about the characteristic length I have used in LB based Reynolds)
Thank you Timm

Yes, this is correct. I first write down the definitions of the dimensionless quantities (e.g., Reynolds number) in two version: for physical units and for lattice units. Then I choose the lattice values of some quantities (e.g., viscosity, time step, lattice constant, or velocity). The last missing quantity I compute from the similarity of the dimensionless quantities in the physical and the lattice system, similar as you did it in your example. If I am not content with the result, I choose different values for the initial quantities. it is always difficult to get a good compromise of accuracy, computing time etc.

Thank you so much Timm .