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orestis.malaspin
lattice-boltzmann
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9ca5f761
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9ca5f761
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1 year ago
by
orestis.malaspin
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lattice-boltzmann.com/content/1-dimensionless-lbm.md
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22 additions, 4 deletions
lattice-boltzmann.com/content/1-dimensionless-lbm.md
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22 additions
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4 deletions
lattice-boltzmann.com/content/1-dimensionless-lbm.md
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22
−
4
View file @
9ca5f761
...
@@ -97,6 +97,11 @@ $$
...
@@ -97,6 +97,11 @@ $$
\l
eft[M
\c
dot L^{-2D}
\c
dot T^D
\r
ight]
\l
eft[M
\c
dot L^{-2D}
\c
dot T^D
\r
ight]
$$
$$
By defining the characteristic velocity of our particles by $
\x
i_0$ (due to thermal agitation)
By defining the characteristic velocity of our particles by $
\x
i_0$ (due to thermal agitation)
$$
\b
egin{equation}
\x
i_0^2=
\f
rac{k_BT}{m},
\e
nd{equation}
$$
the non-dimensional quantities of interest become
the non-dimensional quantities of interest become
$$
$$
\b
egin{equation}
\b
egin{equation}
...
@@ -120,11 +125,24 @@ $$
...
@@ -120,11 +125,24 @@ $$
\m
athrm{Kn}=
\f
rac{
\t
au
\x
i_0}{L},
\m
athrm{Kn}=
\f
rac{
\t
au
\x
i_0}{L},
\e
nd{equation}
\e
nd{equation}
$$
$$
is the Knudsen number.
is the Knudsen number. As for the Navier-Stokes equations we see that the non-dimensional BGK equation is parametrized
with an unique transport coefficient which is the Knudsen number in this case. In the next episodes of this series
As for the Navier-Stokes equations we see that the non-dimensional BGK equation is parametrized
with an unique transport coefficient which is the Knudsen number. In the next episodes of this series
we will make the link between the BGK and the Navier--Stokes equations and show how we discretize
we will make the link between the BGK and the Navier--Stokes equations and show how we discretize
the BGK equation in order to simulate weakly compressible fluid flows.
the BGK equation in order to simulate weakly compressible fluid flows.
Finally we want to express the non-dimensional Maxwell-Boltzmann distribution as a function of only non-dimensional quantities and it reads
$$
\b
egin{equation}
{f^{eq}}^
\a
st(
\b
m{x}^
\a
st,
\b
m{
\x
i}^
\a
st, t^
\a
st) =
\f
rac{
\r
ho^
\a
st(
\b
m{x}^
\a
st,
\b
m{
\x
i}^
\a
st, t^
\a
st)}{(2
\p
i
\t
heta^
\a
st(
\b
m{x}^
\a
st,
\b
m{
\x
i}^
\a
st, t^
\a
st))^{D/2}}
\e
xp
\l
eft(-
\f
rac{
\l
eft(
\b
m{
\x
i}^
\a
st-
\b
m{u}^
\a
st(
\b
m{x}^
\a
st,
\b
m{
\x
i}^
\a
st, t^
\a
st)
\r
ight)^2}{2
\t
heta^
\a
st(
\b
m{x}^
\a
st,
\b
m{
\x
i}^
\a
st, t^
\a
st)}
\r
ight),
\e
nd{equation}
$$
where $
\t
heta^
\a
st$ is given by
$$
\b
egin{equation}
\t
heta^
\a
st =
\f
rac{k_B T}{m
\c
dot
\x
i_0^2}
\e
nd{equation}
$$
In the next episode we will discuss in more details, the link between the description of a fluid in the Boltzmann framework
and in the Navier--Stokes framework.
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