well well …
I really like this question … and I’d really really love the hear the answer of Mr. Latt.
I always have the same kind of problems anytime that I try to think about LB method in term of its micro-mesoscopic nature. Then I wont investigate the first part of your message … actually your question. Sorry
On the other hand I personally agree with you numerical experiment and I want to share you guys what I think. Of course I would be pleased to be taught and then flogged by my Mr. Latt and Orestis.
. Fixing the Reynolds (characteristic length (L) * characteristic velocity (U) / viscosity (nu) ) number, you fix hydrodynamics;
. You also fix your geometry (and then the characteristic length L );
… I would write then Re/L =U/nu eq 1.
then I write something in lattice units: L-----> Nx (number of lattice nodes is chosen constant)
U------>Nx/T (where T is the number of iterations)
nu------> cs2*(tau-0.5) (where cs2 is the speed of sound
and tau the relaxation time)
I rewrite then eq.1 in lattice units
Re/Nx=Nx/Tcs2(tau-0.5) -------------------> Re/Nx^2= 1/Tcs2(tau-0.5)
then If i choose bigger and bigger tau, in order to keep constant the ratio Re/Nx^2, I have to decrease the value of T (the number of iteration get smaller ------> and then we “speed up” the convergence … at least this is what I think … But I use to say many bullshit … then Mr. Latt, please help)
-------------- BUT --------------- (and here you can stop because I will star talking about things that I really don’t know about … but which puzzle me a lot )
Can we make tau bigger and bigger and bigger and then increase and increase the speed of converge of our simulation? Well, I would like answer : “Of course not… and I think that one of reasons is the Knudsen number (the ratio between the mean free path of our “gas”-particles divided by the characteristic Length of our system) which has to be kept small (see Jonas Latt thesis (pag. 18 ))”. Now, in some way that is not complete clear to me, the Knudsen number is proportional to the relaxation time and then… I cannot make tau bigger and bigger and bigger.
Now, if we think that bigger tau is bigger the viscosity-----> … probably we should state that with LB method we cannot model extremely high viscous fluid because we would should have high Knudsen number.
And more? from the kinetic theory point of view (or at least something that I think to be kinetic theory ), if I have an high Kundsen number I’m looking at dilute gases (or microfluids… see Jonas thesis pag 18.) . Wow…
Is then any “link”… or parallelism… between dilute gases and high viscous fluid?
My answer to the first question is: “We can model high viscous fluid thank to the fact that the real viscosity of the fluid we model (viscosity in real units) does not depend only on tau but also on the chosen deltaX and deltaT of the simulation. Playing with deltaX and deltaT we can make our fluid more and more viscous keeping tau small.”
But we all know that to increase tau is the usual way to model a more viscous fluid then I want to give my answer also to the second question … and please let me know what you think.
I see the parallelism in this term. Both dilute gases and high viscous fluid need time to let an “instability” travel in the system and return to the “equilibrium”…
Voila … I’m ready to be punished…