# Lattice dimensions for Turbulence Simulations

I have been looking at the PDF posted at lbmethod.org about choosing lattice dimensions and I have questions regarding the same process for turbulent flows. In every example, there is the consideration of a reference velocity and length scale with regard to the mean flow and domain dimensions, respectively. The issue I have is that in turbulent flows, the characteristic length scale is generally an integral length scale, such as an energy containing eddy length, and the RMS velocity. With these values and the kinematic viscosity, I can calculate Re, but I don’t exactly know how it affects my unit conversion. Let me explain two unit conversions I came up with based on the document and other forum threads:

1. Define tau = 1.5 (for example)
Determine lattice viscosity: nu_lb = 3tau - 0.5 (Assuming (c_s)^2 = 1/3)
Determine time step: nu_lb = dt/(dx)^2
nu_d --> dt = nu_lb/nu_d*(dx)^2 (where d signifies a dimensional value)
Scale physical velocity field: u_lb = dt/dx*u_d (The reason for this is that I have a preformulated homogeneous isotropic turbulent velocity field)

The other way I figured it could be done is:
2. Define lattice velocity for incompressibility: u_lb = 0.01
Determine time step: u_lb = dt/dx*u_d --> dt = (u_lb/u_d)dx (In this case, u_d would be the rms velocity value I discussed)
Determine lattice viscosity: nu_lb = dt/(dx)^2
nu_d
Determine relaxation: tau = (1/3)*nu_lb + 0.5

Both assume a given dx based on a domain length and cell count, and result in a non-dimensional velocity field as well as time step to convert between physical and lattice velocities. The problem is that neither of these processes are affected by Reynold’s number which bothers me, so I can’t convince myself that either is correct. Am I missing a step or a value in the equations?

Thank you!

Hi,
In dimensionless units, the Reynolds number is just the inverse of the viscosity (because the reference length and reference velocity are equal to one). Thus, in both cases you describe, you have Re = 1 / u_d, and all your calculations are affected by the Reynolds number.

Thank you for your help, but a couple of things are still unclear from what you have said.

In the PDF on choosing dimensions, Eq 12 essentially non-dimensionalizes the velocity u_d based on the scaling del_t / del_x. I assume those scaling factors must have dimensions, but in the example at the bottom of the same page, they are listed as simply being numbers with no dimensionality. Should they have units?

Secondly, you say that Re = 1 / u_d, but the example in the PDF directly calculates the value with u_0,p = 2 cm/min. In my case, where I have a complete field of velocities to be scaled, does that mean that I must rescale twice: once to force u_0,p = 1 and then convert all u_d to lattice space? Or if I calculate Re with a value of u_0,p not equal to one can I go directly to lattice space from an array of u_d that does not have u_0,p = 1? In this discussion, I am considering u_0,p to be equal to the RMS value of the velocity field.

I appreciate any help you can give me.

Hi,

You can define the discrete parameters dx and dt in different ways. They can for example be defined to convert from physical units to lattice units (u_p = dx/dt u_lb), or they can be defined to convert from a dimensionless system to lattice units (u_d = dx/dt u_lb). Both choices are fine as long as you use them consistently. In the first case, dx has length units (“meters”) and dt has time units (“seconds”), because u_p has physical units. In the second case, dx and dt have no units, because u_d is dimensionless. Of course, if you like to have units so you can write down a dimensional analysis, you can continue saying that u_d has dimensions of length per time, dx dimensions of length, and dt dimensions of time. This is essentially just a matter of choosing the vocabulary.

The Reynolds number is Re = U*L/nu, no matter which system of units you chose. In the dimensionless one, U=L=1, so nu = 1/Re.

If you like to refer to a dimensionless system, then you have to rescale twice. If you don’t like the dimensionless system and convert directly from physics to LB, then you rescale once.

Thank you! That clarifies a lot.