Bifurcation(s) and physical/lattice boltzmann units (BIG SUMMARY and questions)

Hello everybody!

I want to present my problem with LBM units conversion.

I have studied/read:

- T. Krüger …; Shear stress in lattice Boltzmann simulations; Phys. Rev. E 79, 046704 (2009)
- J. Gotz; Numerical Simulation of Bloodflow in Aneurysms using the Lattice Boltzmann Method; PhD Thesis
- J. Latt; Choice of units in lattice Boltzmann simulations
- i hope all posts on this forum connected with this subject

I try to simulate the blood flow in at least 3 vessels (1 vessel is a input vessel next we have bifurcation (first vessels is divided) and we have 2 vessels as outputs; obviously this is the simplest configuration because these output vessel can also be divided. (simple visulization :):), L - length)

<-------------L----------->

… output

…

…

input …

…

…

…

…

… output

I code this problem using D3Q19 LBM BGK with constant pressure BC and it looks that it works good but now I want to introduce real data/units.

What I have:

- the pressure at the input (Pin_f) (begin of the first vessel) in [Pa] and output pressure (Pout) (two ends of the second vessels) (Pin_f, Pout_f)
- the blood flow in [ml/min or m3/s] (Q)
- blood viscosity in [cm2/s] (nu_f)
- blood density in [g/cm3] (rh_f)

Now I want to present my view/solution and I mark places where I have the little or strong doubt (based on information which I collected on this forum and proposed resources):

Legend/key:

- nu = viscosity
- rh = density
- _f = value with/in physical units
- _lb = value with/in lb units
- dt = delta time
- dx = delta x
- u = velocity

####1.STEP

dx_f = L / N, where N - number of lattice boltzmann nodes

dx_lb = 1

####2.STEP####

u_f = Q / PI*r^2

where:

Q - blood flow

r - radius of first vessel

####STEP 3#### (SMALL DOUBT)

I fix/set u_lb, for example:

- 1, one lattice node in one lattice time unit
- 0.1, one lattice node in ten lattice time units

and we can calcaulte:

dt_lb = dx_lb / u_lb

####STEP 4####

dt_f = u_lb * dx_f / u_f

####STEP 5####

nu_lb = nu_f * dt_f / dx_f^2

####STEP 6#### (SMALL DOUBT)

tau is calcaulted from: nu_lb = (u_lb^2 / 3) * (tau-1/2) * dt_lb

where

cs^2 = u_lb^2 / 3

when dt_lb = 1, then we have simpler equation which is used very often

nevertheless

We receive the same tau for u_lb=1 and u_lb=0.1

because changes in u_lb introduce changes in nu_lb and dt_lb

which cancel out, and therefore the tau is the same

Do I think good ???

####STEP 7####(here my uncertainty is strong) (BIG DOUBT)

p_lb = p_f * dt_f / dx_f^2

Is this equation for pressure good/right ???

where rh_lb = p_lb / cs^2

I used incompressible model from:

Lattice Bolztmann Model for the Incompressible Navier-Stokes Equation; He and Luo, 1997

with constant pressure BC at inlet and outputs of vessels

so I need also p_lb0 or obviously rh_lb0 ???

How can I calculate/establish it ???

Maybe rh_lb0 = 1 ???

####STEP 8####(here my uncertainty is strong) (BIG DOUBT)

DENSITY

I have the physical density rh_f = 1,005 g/cm3 and I don’t know where I should use it ???

What does this physical density change in LBM calculation ???

I have one theory but I am not sure if it is correct.

Maybe this physical density has an influence on physical viscosity ???

Becasue I have found that nu is a kinematic viscosity (nu_f) and we have also dynamic viscosity (dnu):

nu_f=dnu_f / rh_f

and there is this influence of physical density.

So finally, I shouldn’t use rh_f in LBM calculactions ???

Thank very much for any help.

And I hope that this summary which I wrote will be usable for others.

Regards