It is known that when the LB equation accounts for a body force presence in order to obtain correct hydrodynamic equations the moments of the distribution functions have to be modified accounting for the body force. Example: rho*u=sum(f(i)c(i))+delta_t/2Force.
Because of this fact and when using the Regularization method to recompute the non-equilibrium distribution functions, the expressions traditionally presented in the literature are incomplete. In other words, the non-equilibrium distribution functions have to account for the body force too since the body force effect is felt at first order.
Furthermore, it is also known that the computation of f(neq) can be performed through two different paths: (1) The Chapman-Enskog analysis as Latt and Chopard did http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V0T-4KFV3CT-2&_user=2459750&_coverDate=09%2F09%2F2006&_rdoc=1&_fmt=high&_orig=search&_sort=d&_docanchor=&view=c&_acct=C000057394&_version=1&_urlVersion=0&_userid=2459750&md5=ffcbc994fc53ecc5da9f76f96785886c or; (2) following an Hermite expansion projection of f(neq) as Zhan et al. did http://pre.aps.org/abstract/PRE/v74/i4/e046703.
Using the Chapman-Enskog analysis I was able to successfully obtain the regularized f(neq) accounting for the body force presence . However for the Hermite expansion approach I was not able to obtain the same correct results.
Therefore, I ask if anyone has ever tried to do the same? If so some, any help or suggestion would be very much appreciated.
Thank you in advance
I’m not sure to understand your question, but maybe you can find some help in my thesis (p.43-46).
Dear Orestis Malaspinas
First of all let me thank you for your interest in my question.
I have already read your phd thesis. It is really interesting, in particular the 2nd Chapter (the broader one).
What I meant in my previous question is:
The LBE is a result of discretizing the continuous Boltzmann equation in time, orientation and velocity spaces.
This is done by first, discretizing the continuous Boltzmann equation into its discrete velocity counterpart (DVB). This is traditionally done using the Hermite expansion methodology.
Second the DVB is discretized in space and time, using for example the trapezium method. This last step requires however that we make some change of variables in the distribution function in order to keep the LB scheme explicit.
Now, from what I observed the regularization approach is traditionally applied to the distribution of the DVB, i.e. before the space and time are discretized, or in other words, to the primitive distribution function and not the one effectively solved in the LBE.
This has not importance if a body force is not included since the same result is obtained in the end. However, when a body force is present the results will be different.
I have already been able to deduce the correct form of f(neq) through Chapman-Enskog.
However I would like to obtain the same result following the methodology of Zhan et al. (as I previously said) but using the distribution function after the change of coordinates and not the primitive one.
Since you surely have a lot more knowledge than me I would ask you if you agree with what I wrote.
If have not been clear in some point or if you would like to hear more about this I can send you a more detailed file with some formulas explaining what I have tried to put here on words…
Thank you for any help.
I’m still not sure I understand your question. You would like to get f^neq by applying the Hermite quadrature to the fbar_i instead of the f_i? Because if it is what you want to do, then you will probably need to do the Taylor-Expansion of the advection term and then you will just end doing the Chapman-Enskog expansion as in the paper by Latt and Chopard (with higher order terms).
On the other hand if your goal is to compute fbar^neq_i you just use the formula given in the literature for f_i, feq_i and the force term g_i and with the formula of change of variables
fbar_i = f_i+1/(2tau)(f_i-feq_i-tau*g_i)
you should be able to compute fbar^neq_i=fbar_i-feq_i no?
I hope this helps otherwise send me you more detailed file.
Dear Orestis Malaspinas
I have tried today to deduce again the f_bar (neq) regularized expression following the Hermite expansion approach but ended up obtaining the same erroneous result.
I have no idea the reason why I keep obtaining a different expression from the one predicted by the Chapman-Enskog analysis, but something is surely wrong as the two approaches should yield the same thing!
I have sent you an email (to your lbmethod.org email account) summarizing what I have done, what I do obtain and what I would like to obtain.
Thank you in advance for your help.