Re: [ESPResSo-users] LBM, speed of sound, stability

[Wink, Markus]

Hello everybody,

a practical question, probably stupid, but anyways.
As Ulf wrote: "you need to make sure that h*c_s^2/\nu is small to avoid 
nonlinear instabilities. h is the LB timestep, c_s is the speed of sound, and 
\nu is the kinematic viscosity"

Is the LB timestep h the one you invoke in the tcl script as tau? For example 
having a h=0.1, so you write "tau 0.1" for the lbfluid? 
Unfortunately the user's guide just tells that it is "the LB timestep", but I 
am not sure, if it is the same.

Greetings

Markus 

-----Ursprüngliche Nachricht-----
Von: address@hidden [mailto:address@hidden Im Auftrag von Ulf Schiller
Gesendet: Mittwoch, 17. Dezember 2014 19:10
An: address@hidden
Betreff: Re: [ESPResSo-users] LBM, speed of sound, stability

On 17/12/14 12:12, Ivan Cimrak wrote:
> Hi all,
> 
> In one of his emails Ulf Shiller explained that:
> "you need to make sure that h*c_s^2/\nu is small to avoid nonlinear 
> instabilities. h is the LB timestep, c_s is the speed of sound, and 
> \nu is the kinematic viscosity. In the D3Q19 model, c_s^2=1/3*a^2/h^2, 
> so
> a^2/(3*\nu*h) must be small. It may work with values O(1) but it is 
> not guaranteed."
> 
> 
> Ulf, could you please give me the reason why this is necessary? And 
> what does it mean "is small"? Are the values 0.1 - 0.99 ok?

Hi Ivan,

the standard lattice Boltzmann algorithm is typically thought to be second 
order accurate in time, however, if you look at the discretisation of the 
collision operator (usually Crank-Nicolson), the error is actually of the order 
O((h/\tau)^3) where \tau is the viscous relaxation time (or BGK relaxation 
time). The latter is related to the viscosity by \nu=c_s^2*\tau where c_s is 
the speed of sound. Hence the grid Reynolds number h/\tau=h*c_s^2/\nu needs to 
be small. Now, in LB there is a subtle cancellation of errors of the 
Crank-Nicolson discretisation and the splitting error, such that the standard 
LB algorithm approximates the slow manifold of solutions to the discrete 
velocity model even at values of \tau/h beyond unity (an intriguing side effect 
of this is that the exact solution of the collision operator does produce 
excessive decay of shear waves due to the lack of said cancellation). Another 
way to phrase it is that the LBM disconnects from kinetic theory and can work 
in the over-relaxation regime (i.e. negative eigenvalues of the collision 
operator). Some details of the derivation are given in 
http://dx.doi.org/10.1016/j.cpc.2014.06.005 and references therein (in 
particular Brownlee et al. and Paul Dellar). In practise, instabilities may 
arise at the higher moments and couple into the Navier-Stokes dynamics. I'll 
mention in passing that coupling particles to the LB fluid involves singular 
forces that may also affect stability.
If this actually occurs will depend on the characteristics of the flow under 
consideration; for laminar flow and non-stiff coupling there is probably no 
problem.

Best wishes,
Ulf

--
Dr Ulf D Schiller
Centre for Computational Science
University College London
20 Gordon Street
London WC1H 0AJ
United Kingdom