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LBM equation in the relaxation approximation
Equation

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In the relaxation approximation, it is assumed that the distribution f_i(\vec{x},t) tends to relax at a time \tau to an equilibrium distribution f_i^{eq}(\vec{x},t) according to equation\\n\\n

$\displaystyle\frac{df_i}{dt}=-\displaystyle\frac{f_i-f_i^{eq}}{\tau}$



which has in the discrete approximation the equation

$f_i(\vec{x}+c\vec{e_i}\delta t,t+\delta t)=f_i(\vec{x},t)+\displaystyle\frac{1}{\tau}(f_i^{eq}(\vec{x},t)-f_i(\vec{x},t))\delta t$

where the term of the differences in the distribution functions represents the collisions.

ID:(8489, 0)


Velocidades Microscopicas
Description

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Las velocidades microscópicas según las dimensiones son el numero de dimensiones (D1, D2, D3) y el número de puntos (Q3 - para D1, Q9 - para D2, Q15 o Q19 - para D3):

ModeloVelocidadesIndex
1DQ3(0)i=0
'(s)i=1, 2
2DQ9(0,0)i=0
'(s,0), (0,s)i=1,...,4
'(s,s)i=5,...,8
3DQ15(0,0,0)i=0
'(s,0,0), (0,s,0), (0,0,s)i=1,...,6
'(s,s,s)i=7,...,14
3DQ19(0,0,0)i=0
'(s,0,0), (0,s,0), (0,0,s)i=1,...,6
'(s,s,0), (s,0,s), (0,s,s)i=7,...,18

donde los $s$ puede asumir los valores -1 y 1.

ID:(8491, 0)


Distribution in Balance (Gas of Particles)
Equation

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The equilibrium distribution can be approximated by a distribution of Maxwell Boltzmann

$f_i^{eq}=\displaystyle\frac{m}{2\pi kT}e^{-m|c\vec{e}_i-\vec{u}|^2/2kT}$

Where m is the mass of the particle, T the system temperature and k the Boltzmann constant.

ID:(8490, 0)