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Re: [ESPResSo-users] LBM, speed of sound, stability
From: |
Ulf Schiller |
Subject: |
Re: [ESPResSo-users] LBM, speed of sound, stability |
Date: |
Thu, 18 Dec 2014 09:30:08 +0000 |
User-agent: |
Mozilla/5.0 (X11; Linux x86_64; rv:31.0) Gecko/20100101 Thunderbird/31.3.0 |
Hi Vincent,
I'm not sure whether there is a general way to identify instabilities.
Looking for oscillations in the non-hydrodynamic modes might be an
indicator. I gather it is not obvious how to distinguish numerical
instabilities from physical ones - think about onset of turbulence for
example. Pipe flows are maybe not such a good test as they are
unconditionally stable - this was Werner Heisenberg's PhD topic with
some certainty ;-)
Cheers,
Ulf
On 18/12/14 02:45, Vincent Ustach wrote:
> Ulf,
>
> Are there are criteria that would give evidence for instabilities in a
> particular system? Would it be deviations from expected results, for
> example a non-parabolic velocity profile in pressure driven tube flow?
>
> Thanks,
>
> Vincent Ustach
>
>
> On Wednesday, December 17, 2014, Ulf Schiller <address@hidden
> <mailto:address@hidden>> wrote:
>
> 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
>
>
>
>
> --
> --Vincent Ustach
> University of California, Davis
>
--
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
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- Re: [ESPResSo-users] lbboundary, roya moghaddasi, 2014/12/05
- Re: [ESPResSo-users] lbboundary, roya moghaddasi, 2014/12/05
- Re: [ESPResSo-users] lbboundary, Ulf Schiller, 2014/12/05
- [ESPResSo-users] LBM, speed of sound, stability, Ivan Cimrak, 2014/12/17
- Re: [ESPResSo-users] LBM, speed of sound, stability, Ulf Schiller, 2014/12/17
- Re: [ESPResSo-users] LBM, speed of sound, stability, Vincent Ustach, 2014/12/17
- Re: [ESPResSo-users] LBM, speed of sound, stability,
Ulf Schiller <=
- Message not available
- Re: [ESPResSo-users] LBM, speed of sound, stability, Ulf Schiller, 2014/12/18
- Re: [ESPResSo-users] LBM, speed of sound, stability, Wink, Markus, 2014/12/18
- Re: [ESPResSo-users] LBM, speed of sound, stability, Ulf Schiller, 2014/12/18