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

[Ulf Schiller]

Hi Ivan,

yes, of course the grid Reynolds number is not the only limitation. One
also needs a small Mach number, i.e. u/c_s = \sqrt(3)*u*h/a, and this is
kind of the CFL for LB  - the flow must not be faster than the lattice
velocities c_i. But this would lead to a different source of errors,
namely compressibility artefacts. The two conditions show that LB,
unlike other methods, can not be made more stable by just reducing the
time step h. The reason is, and this is what the Chapman-Enskog
expansion is all about, that LB operates in the diffusive scaling limit
a=\epsilon*L and h=\epsilon^2*T, i.e., taking the Mach number to zero at
*fixed* Reynolds number. It is also the reason why LB can never reliably
approximate anything beyond hydrodynamics. Many people overlook that, if
they try to reduce the Mach number by using a smaller time step, they do
in fact increase the grid Reynolds number.

Best wishes,
Ulf

On 18/12/14 08:51, Ivan Cimrak wrote:
> Dear Ulf,
> 
> Thank you for your explanation. I do not understand all the details, but
> at least now I have a glue.
> 
> The reason why I asked was that I was a bit concerned about the
> resulting fraction a^2/(3*\nu*h). Normally, with e.g. a CFL condition
> for finite difference methods, you have a fraction with space
> discretization step in numerator and the time step in denominator, e.g.
> dt/dx in 1D case.
> 
> Here, we have space discretizaion in numeator and time discretization in
> denominator.
> 
> 
> Thanks,
> Ivan
> 
> 
> 
> Dňa 17.12.2014 19:10 Ulf Schiller  wrote / napísal(a):
>> 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

Phone: +44 (0)20 7679 5300