# Converting the relaxation time of a Newtonian fluid from LB simulation units to physical units

Hi everyone,
In my simulations, I need the physical relaxation time of the fluid to calculate the Deborah number. For this I have used the following approach to convert the relaxation time from LB units to physical units. (I referred to the paper by Jonas Latt on Choice of simulation units) Now, the issue I face with the final form is that the physical relaxation time comes out to be inversely proportional to the kinematic viscosity of the fluid, which is theoretically wrong. The directly proportionality between relaxation time and viscosity was shown by Debye.

Please help me understand why this is happening and whether I’m making a mistake in my approach.

This is my first time posting on a forum. I apologize if I have not explained something properly and let me know if any clarifications are required.

Thanks.

Hello,
I think you are mixing a lot of different formulas here. I think your error is to have to different defitions for tau_{lb}. One which is the “physical” tau_{lb}, aka nu_{lb}=c_s^2 \tau_{lb}, and the other being the numerical one, \nu_{lb}=c_s^2(\tau_{lb}-0.5). These two quantities do NOT have the same units obviously.

Another error is that you are using two different characteristic time scales: in physical units t_p is defined with respect to u_p and l_p, in lb units it’s with respect to \nu_lb. This makes your characteristic times inconsistent IMO.

You may also be interested in this video about LB units:

https://tube.switch.ch/videos/85642992

Hope this helps,
Orestis

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Hello Orestis,

I followed the approach explained in the video for the choice of simulation units. Still, I face the same problem of physical relaxation time being inversely proportional to the fluid’s viscosity.

I used the viscous definition of \delta_{t}, as I’m running my simulations at low Re. So, \tau_{p} comes out to be, Please let me know if I’m making a mistake.

Also, can you please suggest a comprehensive book/material covering the conversion of simulation units into physical units?

Thanks

There is the book by Kruger et al. which is very good.

I can’t say if you are making any mistake just from the result.