from lbmpy.session import *
from lbmpy.chapman_enskog import ChapmanEnskogAnalysis
sp.init_printing()

Demo: Automatic Chapman Enskog Analysis

First, we create a SRT lattice Boltzmann method. It is defined as the set of moments, together with one relaxation rate per moment.

lb_config = LBMConfig(method=Method.TRT, stencil=Stencil.D3Q19, compressible=False, zero_centered=False)
method = create_lb_method(lbm_config=lb_config)
method

Moment-Based Method			Stencil: D3Q19	Zero-Centered Storage: ✗	Force Model: None

Continuous Hydrodynamic Maxwellian Equilibrium			$f (\rho, \left( u_{0}, \ u_{1}, \ u_{2}\right), \left( v_{0}, \ v_{1}, \ v_{2}\right)) = \frac{3 \sqrt{6} \delta_{\rho} e^{- \frac{3 v_{0}^{2}}{2} - \frac{3 v_{1}^{2}}{2} - \frac{3 v_{2}^{2}}{2}}}{4 \pi^{\frac{3}{2}}} + \frac{3 \sqrt{6} e^{- \frac{3 \left(- u_{0} + v_{0}\right)^{2}}{2} - \frac{3 \left(- u_{1} + v_{1}\right)^{2}}{2} - \frac{3 \left(- u_{2} + v_{2}\right)^{2}}{2}}}{4 \pi^{\frac{3}{2}}}$
Compressible: ✗	Deviation Only: ✗	Order: 2

Relaxation Info
Moment	Eq. Value	Relaxation Rate
$1$	$\rho$	$\omega$
$x$	$u_{0}$	$\omega$
$y$	$u_{1}$	$\omega$
$z$	$u_{2}$	$\omega$
$x^{2}$	$\frac{\rho}{3} + u_{0}^{2}$	$\omega$
$y^{2}$	$\frac{\rho}{3} + u_{1}^{2}$	$\omega$
$z^{2}$	$\frac{\rho}{3} + u_{2}^{2}$	$\omega$
$x y$	$u_{0} u_{1}$	$\omega$
$x z$	$u_{0} u_{2}$	$\omega$
$y z$	$u_{1} u_{2}$	$\omega$
$x^{2} y$	$\frac{u_{1}}{3}$	$\omega$
$x^{2} z$	$\frac{u_{2}}{3}$	$\omega$
$x y^{2}$	$\frac{u_{0}}{3}$	$\omega$
$x z^{2}$	$\frac{u_{0}}{3}$	$\omega$
$y^{2} z$	$\frac{u_{2}}{3}$	$\omega$
$y z^{2}$	$\frac{u_{1}}{3}$	$\omega$
$x^{2} y^{2}$	$\frac{\rho}{9} + \frac{u_{0}^{2}}{3} + \frac{u_{1}^{2}}{3}$	$\omega$
$x^{2} z^{2}$	$\frac{\rho}{9} + \frac{u_{0}^{2}}{3} + \frac{u_{2}^{2}}{3}$	$\omega$
$y^{2} z^{2}$	$\frac{\rho}{9} + \frac{u_{1}^{2}}{3} + \frac{u_{2}^{2}}{3}$	$\omega$

Next, the Chapman Enskog analysis object is created. This may take a while…

analysis = ChapmanEnskogAnalysis(method)

This object now information about the method, e.g. the relation of relaxation rate to viscosities, if the method approximates the compressible or incompressible continuity equation …

analysis.compressible

False

analysis.pressure_equation

$\displaystyle \left[ \frac{\rho}{3}\right]$

analysis.get_kinematic_viscosity()

$\displaystyle - \frac{\omega - 2}{6 \omega}$

analysis.get_bulk_viscosity()

$\displaystyle - \frac{1}{9} + \frac{2}{9 \omega}$

But also details of the analysis are available:

sp.Matrix(analysis.get_macroscopic_equations())

$\displaystyle \left[\begin{matrix}{\partial_{t} \rho} + {\partial_{0} u_{0}} + {\partial_{1} u_{1}} + {\partial_{2} u_{2}}\\- \frac{\epsilon \omega {\partial_{0} {\Pi_{00}^{(1)}}}}{2} - \frac{\epsilon \omega {\partial_{1} {\Pi_{01}^{(1)}}}}{2} - \frac{\epsilon \omega {\partial_{2} {\Pi_{02}^{(1)}}}}{2} + \epsilon {\partial_{0} {\Pi_{00}^{(1)}}} + \epsilon {\partial_{1} {\Pi_{01}^{(1)}}} + \epsilon {\partial_{2} {\Pi_{02}^{(1)}}} + \frac{{\partial_{0} \rho}}{3} + {\partial_{t} u_{0}} + {\partial_{0} (u_{0}^{2}) } + {\partial_{1} (u_{0} u_{1}) } + {\partial_{2} (u_{0} u_{2}) }\\- \frac{\epsilon \omega {\partial_{0} {\Pi_{01}^{(1)}}}}{2} - \frac{\epsilon \omega {\partial_{1} {\Pi_{11}^{(1)}}}}{2} - \frac{\epsilon \omega {\partial_{2} {\Pi_{12}^{(1)}}}}{2} + \epsilon {\partial_{0} {\Pi_{01}^{(1)}}} + \epsilon {\partial_{1} {\Pi_{11}^{(1)}}} + \epsilon {\partial_{2} {\Pi_{12}^{(1)}}} + \frac{{\partial_{1} \rho}}{3} + {\partial_{t} u_{1}} + {\partial_{1} (u_{1}^{2}) } + {\partial_{0} (u_{0} u_{1}) } + {\partial_{2} (u_{1} u_{2}) }\\- \frac{\epsilon \omega {\partial_{0} {\Pi_{02}^{(1)}}}}{2} - \frac{\epsilon \omega {\partial_{1} {\Pi_{12}^{(1)}}}}{2} - \frac{\epsilon \omega {\partial_{2} {\Pi_{22}^{(1)}}}}{2} + \epsilon {\partial_{0} {\Pi_{02}^{(1)}}} + \epsilon {\partial_{1} {\Pi_{12}^{(1)}}} + \epsilon {\partial_{2} {\Pi_{22}^{(1)}}} + \frac{{\partial_{2} \rho}}{3} + {\partial_{t} u_{2}} + {\partial_{2} (u_{2}^{2}) } + {\partial_{0} (u_{0} u_{2}) } + {\partial_{1} (u_{1} u_{2}) }\end{matrix}\right]$