Continuous distribution functions ?
Grid refinement algorithm
Validations
Grid refinement in LBM based on continuous
distribution functions
Denis Ricot1
1 Renault
Simon Marié1,2
Pierre Sagaut2
- Research, Material and Advanced Engineering Department
2 d’Alembert Institute, Université Paris VI
5th ICMMES - 16-20 june 2008
D. Ricot, S. Marié, P. Sagaut
le-logo
Grid refinement in LBM based on continuous distribution fun
Continuous distribution functions ?
Grid refinement algorithm
Validations
Grid refinement problem
• For simulating complex engineering flows, mesh refinement technique
is necessary
• LB distribution functions g α are not conserved through a refinement
interface
[Filippova & H anel,
1998]
• Spatial and time interpolation methods can not be used directly on the
distribution functions to evaluate the unknown data
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D. Ricot, S. Marié, P. Sagaut
Grid refinement in LBM based on continuous distribution fun
Continuous distribution functions ?
Grid refinement algorithm
Validations
Grid refinement schemes
• Vertex centered approach
with ([Yu et al., 2002][Dupuis & Chopard, 2003]) our without rescaling ([Lin & Lai,
2000])
rescaling and interpolations applied on the post-collision
1998])
distribution functions ([Yu et al., 2002][Filippova & H anel,
spatial interpolation performed before time interpolation [Yu et al.,
2002][Dupuis & Chopard, 2003]
• Volumetric approach (cell centered) [Rohde, 2004][Chen et al., 2005 (PowerFLOW)]
Explode (c → f ) and coalesce (f → c) distribution functions :
mass conservative scheme
No rescaling of distribution functions
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D. Ricot, S. Marié, P. Sagaut
Grid refinement in LBM based on continuous distribution fun
Continuous distribution functions ?
Grid refinement algorithm
Validations
Presentation outline
Continuous distribution functions ?
Grid refinement problem
DVBE and LBE
Numerical verification
Grid refinement algorithm
Rescaling procedure
Overlapping nodes
Algorithm description
Validations
Grid parameters
Pulse propagation in uniform flow
Convected vortex
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D. Ricot, S. Marié, P. Sagaut
Grid refinement in LBM based on continuous distribution fun
Continuous distribution functions ?
Grid refinement algorithm
Validations
Grid refinement problem
DVBE and LBE
Numerical verification
Grid refinement problem
• After one standard LB timestep, some distribution functions are missing
• Distribution functions are known to be discontinuous through an
interface between meshes with different grid size :
gαc (•, ◦, t) = gαf (•, ◦, t)
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D. Ricot, S. Marié, P. Sagaut
Grid refinement in LBM based on continuous distribution fun
Continuous distribution functions ?
Grid refinement algorithm
Validations
Grid refinement problem
DVBE and LBE
Numerical verification
Base ideas of Filippova & H anel
(1998)
• Macroscopic properties of the fluid must be conserved (ν, c s )
δx c /δt c = δx f /δt f
τgc = 12 τgf + 14 (the relaxation time is not continuous)
• Macroscopic variables of the flow must be conserved (ρ, u)
gαeq,c = gαeq,f
• The shear stress must be continuous
τc
gαneq,c = m τgf gαneq,f (m =
g
δx c
δx f
= 2)
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D. Ricot, S. Marié, P. Sagaut
Grid refinement in LBM based on continuous distribution fun
Continuous distribution functions ?
Grid refinement algorithm
Validations
Grid refinement problem
DVBE and LBE
Numerical verification
Why the distribution functions of LBM are not continuous ?
• Boltzmann equation
f − f eq
∂f
∂f
+ ci
=−
∂t
∂xi
λ
(BE)
• Quadrature formulae to calculate moments of distribution functions
using a discrete velocity set
´
1`
∂fα
∂fα
+ cα,i
= − fα − fαeq
∂t
∂xi
τ
(DVBE)
• By definition the distribution functions of √
DVBE f α P
are continuous
P in
space and time (and also ν = τ θ 0 , cs =
θ0 , ρ =
fα , ρu =
c α fα )
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D. Ricot, S. Marié, P. Sagaut
Grid refinement in LBM based on continuous distribution fun
Continuous distribution functions ?
Grid refinement algorithm
Validations
Grid refinement problem
DVBE and LBE
Numerical verification
• Integrate DVBE along the characteristic c α for a time interval δt
• The integral of the BGK collision operator is approximated by the
trapezium rule :
δt
{fα (x + cα δt, t + δt)
“ ”
2τ
¯
eq
−fα (x + cα δt, t + δt) + fα (x, t) − fαeq (x, t) + O δt 3
fα (x + cα δt, t + δt) − fα (x, t) = −
• Change of variable ([He et al., 1998][Dellar, 2001])
gα (x, t) = fα (x, t) +
´
δt `
fα (x, t) − fαeq (x, t)
2τ
(1)
• Final Lattice Boltzmann Equation
«
„
δt
δt
gα (x, t)+ fαeq (x, t)
gα (x+cα δt, t +δt) = 1−
τg
τg
(2)
with τg = τ + δt/2.
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D. Ricot, S. Marié, P. Sagaut
Grid refinement in LBM based on continuous distribution fun
Continuous distribution functions ?
Grid refinement algorithm
Validations
Grid refinement problem
DVBE and LBE
Numerical verification
Numerical verification of the link between f α and g α
• LBE and DVBE computations have been performed on the same
uniform grid, for the same flow problems and fluid parameters
• LB model : standard D2Q9, BGK collision operator, equilibrium function
truncated at second order
• DVBE model : D2Q9 velocity model, BGK collision operator, equilibrium
function truncated at second order
6th-order central finite difference scheme for space derivatives
5th-order Runge-Kutta scheme for time integration
• Fluid is air, τ /δt = 0.93, same δt and δx for LBE and DVBE
δt
• For two simple flows, comparison of f α , gα and dα = fα + 2τ
(fα − fαeq )
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D. Ricot, S. Marié, P. Sagaut
Grid refinement in LBM based on continuous distribution fun
Continuous distribution functions ?
Grid refinement algorithm
Validations
Grid refinement problem
DVBE and LBE
Numerical verification
Reference cases
• Acoustic pressure pulse in an uniform flow
8
“`
h
´2 `
´2 ”i
>
< ρ = 1 + aP exp − lnbP2 x1−x10 + x2−x20
u 1 = U0
>
:
u2 = 0
with aP = 0.03 and b P = 5
• Convected vortex
8
ρ =1
>
>
“`
h
<
`
´
´2 `
´2 ”i
u1 = U0 + aT U0 x2−x20 exp − lnbT2 x1−x10 + x2−x20
“`
h
>
`
`
´
´
´ ”i
>
: u2 = −aT U0 x1−x10 exp − ln 2 x1−x10 2 + x2−x20 2
bT
with aT = 0.5 et bT = 25
D. Ricot, S. Marié, P. Sagaut
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Grid refinement in LBM based on continuous distribution fun
Continuous distribution functions ?
Grid refinement algorithm
Validations
Grid refinement problem
DVBE and LBE
Numerical verification
Comparison of the macroscopic variables
Density profile of the pressure pulse
Velocity profile u 2 of the convected vortex
at time t = 50 along the line x 2 = x20
at time t = 100 along the line x 2 = x20
1.0006
0.01
1.0005
0.008
1.0004
0.006
1.0003
0.004
1.0002
0.002
1.0001
0
1
−0.002
0.9999
−0.004
0.9998
−0.006
0.9997
0.9996
−0.008
0
10
20
30
40
50
60
70
80
90
100
−0.01
0
10
20
30
x/δx
40
50
60
70
80
90
100
x/δx
LBM;
DVBE.
• The simulated results are exactly the same in term of macroscopic
variables → g αeq = fαeq
D. Ricot, S. Marié, P. Sagaut
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Grid refinement in LBM based on continuous distribution fun
Continuous distribution functions ?
Grid refinement algorithm
Validations
Grid refinement problem
DVBE and LBE
Numerical verification
Distribution functions at a given point as a function of time
Pressure pulse, α = 6
Convected vortex, α = 1
0.0233
0.139
0.0233
0.138
0.0232
0.137
0.0232
0.136
0.0232
0.135
0.0232
0.0232
0.134
0
5
10
15
20
25
30
35
40
45
50
0.133
0
10
20
30
t/δt
50
60
70
80
90
100
t/δt
(
) : gα
(
) : fα
( ◦ ◦ ◦ ) : dα = fα + δt(fα − fαeq )/2τ
(
(
• This numerical test confirms that g α = fα +
D. Ricot, S. Marié, P. Sagaut
40
δt
2τ
):
):
(fα − fαeq )
gαeq
fαeq
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Grid refinement in LBM based on continuous distribution fun
Continuous distribution functions ?
Grid refinement algorithm
Validations
Rescaling procedure
Overlapping nodes
Algorithm description
• Conversion g α ↔ fα for a given x and t
g α = fα +
´
δt `
2τ
fα − fαeq <=> fα =
2τ
2τ + δt
• By definition f α , τ , ρ =
P
fα , ρu =
“
”
8
c
c
2τ
δt c eq
>
f
g
=
+
f
c
α
α
α
>
2τ
+δt
2τ
>
“
”
>
>
f
f
2τ
δt f eq
>
>
< fα = 2τ +δt f gα + 2τ fα
fαf = fαc
>
>
f
c
>
>
τ = τgf − δt2 = τgc − δt2
>
>
>
:
m=2
P
„
«
δt eq
gα +
fα
2τ
cα fα (→ fαeq ) are continuous
gαf =
1
2τ̃gf +1
gαc =
1
2τ̃gf
==>
`` f ´ c
´
2τ̃g gα + fαeq
´
`` f
´
2τ̃g + 1 gαf − fαeq
τ̃ c
• Remark: these expressions immediately imply that g αneq,c = 2 τ̃gf gαneq,f
g
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D. Ricot, S. Marié, P. Sagaut
Grid refinement in LBM based on continuous distribution fun
Continuous distribution functions ?
Grid refinement algorithm
Validations
Rescaling procedure
Overlapping nodes
Algorithm description
• First approach (implemented with success in our 3D LB code
(L-BEAM))
Conversion g c ↔ fα for needed interface points
α
Spatial and temporal interpolations on f α
Conversion f α ↔ g f
α
• Second approach (preferred in this work)
Conversion g αc ↔ gαf for needed interface points
Spatial and temporal interpolations on g αf
• In both cases
Conversion step needs the equilibrium functions to be known
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D. Ricot, S. Marié, P. Sagaut
Grid refinement in LBM based on continuous distribution fun
Continuous distribution functions ?
Grid refinement algorithm
Validations
Rescaling procedure
Overlapping nodes
Algorithm description
• Fine grid points must also be calculated as coarse grid points
• Conversion g αf out (, t) → gαc out (, t)
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D. Ricot, S. Marié, P. Sagaut
Grid refinement in LBM based on continuous distribution fun
Continuous distribution functions ?
Grid refinement algorithm
Validations
`
´
Rescaling procedure
Overlapping nodes
Algorithm description
`
´
• Conversion g αc in •, t + 2δt f → gαf in •, t + 2δt f can be done because
macroscopic variables can be now calculated at points •
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D. Ricot, S. Marié, P. Sagaut
Grid refinement in LBM based on continuous distribution fun
Continuous distribution functions ?
Grid refinement algorithm
Validations
•
Rescaling procedure
Overlapping nodes
Algorithm description
c
f
Timestep t : all functions g α
and gα
are
known everywhere
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D. Ricot, S. Marié, P. Sagaut
Grid refinement in LBM based on continuous distribution fun
Continuous distribution functions ?
Grid refinement algorithm
Validations
•
•
Rescaling procedure
Overlapping nodes
Algorithm description
c
f
Timestep t : all functions g α
and gα
are
known everywhere
f
c
Stage 1 : convert g α
(, t) → gα
(, t)
out
out
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D. Ricot, S. Marié, P. Sagaut
Grid refinement in LBM based on continuous distribution fun
Continuous distribution functions ?
Grid refinement algorithm
Validations
•
•
•
Rescaling procedure
Overlapping nodes
Algorithm description
c
f
Timestep t : all functions g α
and gα
are
known everywhere
f
c
Stage 1 : convert g α
(, t) → gα
(, t)
out
out
Stage 2 : collide and propagate all points in the
fine and coarse regions (including points )
le-logo
D. Ricot, S. Marié, P. Sagaut
Grid refinement in LBM based on continuous distribution fun
Continuous distribution functions ?
Grid refinement algorithm
Validations
•
•
•
•
Rescaling procedure
Overlapping nodes
Algorithm description
c
f
Timestep t : all functions g α
and gα
are
known everywhere
f
c
Stage 1 : convert g α
(, t) → gα
(, t)
out
out
Stage 2 : collide and propagate all points in the
fine and coarse regions (including points )
Stage“3 : convert”
“
”
c
f
•, t +2δt f → gα
•, t +2δt f
gα
in
in
le-logo
D. Ricot, S. Marié, P. Sagaut
Grid refinement in LBM based on continuous distribution fun
Continuous distribution functions ?
Grid refinement algorithm
Validations
Rescaling procedure
Overlapping nodes
Algorithm description
Injection scheme :
“
”
f
f
gα
•,t +δt f = gα
in
in
“
”
•,t +2δt f
Linear interpolation using
“
”
f
f
•,t +2δt f and gα
gα
in
in
(•,t) (that
must be stored)
•
•
•
•
•
c
f
Timestep t : all functions g α
and gα
are
known everywhere
f
c
Stage 1 : convert g α
(, t) → gα
(, t)
out
out
Stage 2 : collide and propagate all points in the
fine and coarse regions (including points )
Stage“3 : convert”
“
”
c
f
•, t +2δt f → gα
•, t +2δt f
gα
in
in
“
”
f
•,t +δt f
Stage 4 : time interpolation of g α
in
D. Ricot, S. Marié, P. Sagaut
le-logo
Grid refinement in LBM based on continuous distribution fun
Continuous distribution functions ?
Grid refinement algorithm
Validations
Rescaling procedure
Overlapping nodes
Algorithm description
Injection scheme :
“
”
f
f
gα
•,t +δt f = gα
in
in
“
”
•,t +2δt f
Linear interpolation using
“
”
f
f
•,t +2δt f and gα
gα
in
in
(•,t) (that
must be stored)
•
Stage“5 : spatial
” interpolation“of
”
f
f
◦,t +δt f using gα
• − 1,t +δt f
gα
in
“
” in
f
and gα
• + 1,t +δt f
in
•
•
•
•
•
c
f
Timestep t : all functions g α
and gα
are
known everywhere
f
c
Stage 1 : convert g α
(, t) → gα
(, t)
out
out
Stage 2 : collide and propagate all points in the
fine and coarse regions (including points )
Stage“3 : convert”
“
”
c
f
•, t +2δt f → gα
•, t +2δt f
gα
in
in
“
”
f
•,t +δt f
Stage 4 : time interpolation of g α
in
D. Ricot, S. Marié, P. Sagaut
le-logo
Grid refinement in LBM based on continuous distribution fun
Continuous distribution functions ?
Grid refinement algorithm
Validations
Rescaling procedure
Overlapping nodes
Algorithm description
Injection scheme :
“
”
f
f
gα
•,t +δt f = gα
in
in
“
”
•,t +2δt f
Linear interpolation using
“
”
f
f
•,t +2δt f and gα
gα
in
in
(•,t) (that
must be stored)
•
Stage“5 : spatial
” interpolation“of
”
f
f
◦,t +δt f using gα
• − 1,t +δt f
gα
in
“
” in
f
and gα
• + 1,t +δt f
in
•
•
•
•
•
•
f
Timestep t +δt f : all functions g α
are known in
the fine grid region
c
f
Timestep t : all functions g α
and gα
are
known everywhere
f
c
Stage 1 : convert g α
(, t) → gα
(, t)
out
out
Stage 2 : collide and propagate all points in the
fine and coarse regions (including points )
Stage“3 : convert”
“
”
c
f
•, t +2δt f → gα
•, t +2δt f
gα
in
in
“
”
f
•,t +δt f
Stage 4 : time interpolation of g α
in
D. Ricot, S. Marié, P. Sagaut
le-logo
Grid refinement in LBM based on continuous distribution fun
Continuous distribution functions ?
Grid refinement algorithm
Validations
Rescaling procedure
Overlapping nodes
Algorithm description
Injection scheme :
“
”
f
f
gα
•,t +δt f = gα
in
in
“
”
•,t +2δt f
Linear interpolation using
“
”
f
f
•,t +2δt f and gα
gα
in
in
(•,t) (that
must be stored)
•
Stage“5 : spatial
” interpolation“of
”
f
f
◦,t +δt f using gα
• − 1,t +δt f
gα
in
“
” in
f
and gα
• + 1,t +δt f
in
•
•
•
•
•
•
c
f
Timestep t : all functions g α
and gα
are
known everywhere
•
f
Timestep t +δt f : all functions g α
are known in
the fine grid region
Stage 6 : collide and propagate all points in the
fine region
f
c
Stage 1 : convert g α
(, t) → gα
(, t)
out
out
Stage 2 : collide and propagate all points in the
fine and coarse regions (including points )
Stage“3 : convert”
“
”
c
f
•, t +2δt f → gα
•, t +2δt f
gα
in
in
“
”
f
•,t +δt f
Stage 4 : time interpolation of g α
in
D. Ricot, S. Marié, P. Sagaut
le-logo
Grid refinement in LBM based on continuous distribution fun
Continuous distribution functions ?
Grid refinement algorithm
Validations
Rescaling procedure
Overlapping nodes
Algorithm description
Injection scheme :
“
”
f
f
gα
•,t +δt f = gα
in
in
“
”
•,t +2δt f
Linear interpolation using
“
”
f
f
•,t +2δt f and gα
gα
in
in
(•,t) (that
must be stored)
•
Stage“5 : spatial
” interpolation“of
”
f
f
◦,t +δt f using gα
• − 1,t +δt f
gα
in
“
” in
f
and gα
• + 1,t +δt f
in
•
•
•
•
•
•
c
f
Timestep t : all functions g α
and gα
are
known everywhere
f
c
Stage 1 : convert g α
(, t) → gα
(, t)
out
out
Stage 2 : collide and propagate all points in the
fine and coarse regions (including points )
Stage“3 : convert”
“
”
c
f
•, t +2δt f → gα
•, t +2δt f
gα
in
in
“
”
f
•,t +δt f
Stage 4 : time interpolation of g α
in
D. Ricot, S. Marié, P. Sagaut
•
•
f
Timestep t +δt f : all functions g α
are known in
the fine grid region
Stage 6 : collide and propagate all points in the
fine region
“
”
f
•,t +2δt f from
Stage 7 : recover g α
“
” in
c
gα
•,t +2δt f
in
le-logo
Grid refinement in LBM based on continuous distribution fun
Continuous distribution functions ?
Grid refinement algorithm
Validations
Rescaling procedure
Overlapping nodes
Algorithm description
Injection scheme :
“
”
f
f
gα
•,t +δt f = gα
in
in
“
”
•,t +2δt f
Linear interpolation using
“
”
f
f
•,t +2δt f and gα
gα
in
in
(•,t) (that
must be stored)
•
Stage“5 : spatial
” interpolation“of
”
f
f
◦,t +δt f using gα
• − 1,t +δt f
gα
in
“
” in
f
and gα
• + 1,t +δt f
in
•
•
•
•
•
•
c
f
Timestep t : all functions g α
and gα
are
known everywhere
f
c
Stage 1 : convert g α
(, t) → gα
(, t)
out
out
Stage 2 : collide and propagate all points in the
fine and coarse regions (including points )
Stage“3 : convert”
“
”
c
f
•, t +2δt f → gα
•, t +2δt f
gα
in
in
“
”
f
•,t +δt f
Stage 4 : time interpolation of g α
in
D. Ricot, S. Marié, P. Sagaut
•
•
•
f
Timestep t +δt f : all functions g α
are known in
the fine grid region
Stage 6 : collide and propagate all points in the
fine region
“
”
f
•,t +2δt f from
Stage 7 : recover g α
“
” in
c
gα
•,t +2δt f
in
Stage“8 : spatial”interpolation of
f
◦,t +2δt f
gα
in
le-logo
Grid refinement in LBM based on continuous distribution fun
Continuous distribution functions ?
Grid refinement algorithm
Validations
Rescaling procedure
Overlapping nodes
Algorithm description
Injection scheme :
“
”
f
f
gα
•,t +δt f = gα
in
in
“
”
•,t +2δt f
Linear interpolation using
“
”
f
f
•,t +2δt f and gα
gα
in
in
(•,t) (that
must be stored)
•
Stage“5 : spatial
” interpolation“of
”
f
f
◦,t +δt f using gα
• − 1,t +δt f
gα
in
“
” in
f
and gα
• + 1,t +δt f
in
•
•
•
•
•
•
c
f
Timestep t : all functions g α
and gα
are
known everywhere
f
c
Stage 1 : convert g α
(, t) → gα
(, t)
out
out
Stage 2 : collide and propagate all points in the
fine and coarse regions (including points )
Stage“3 : convert”
“
”
c
f
•, t +2δt f → gα
•, t +2δt f
gα
in
in
“
”
f
•,t +δt f
Stage 4 : time interpolation of g α
in
D. Ricot, S. Marié, P. Sagaut
•
•
•
f
Timestep t +δt f : all functions g α
are known in
the fine grid region
Stage 6 : collide and propagate all points in the
fine region
“
”
f
•,t +2δt f from
Stage 7 : recover g α
“
” in
c
gα
•,t +2δt f
in
Stage“8 : spatial”interpolation of
f
◦,t +2δt f
gα
in
•
c
f
Timestep t +2δt f : all functions g α
and gα
are
known everywhere
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Grid refinement in LBM based on continuous distribution fun
Continuous distribution functions ?
Grid refinement algorithm
Validations
Coarse grid : 120x80, fine grid : 60 × 80
Acoustic pulse or vortex is initialized outside or inside the fine grid
M = 0.2, ν = 1.5 × 10 −5 m2 /s, cs = 340 m/s, δx f = 10−2 m
All computations are also done with a uniform coarse grid (reference)
U0
=⇒
80
70
60
y /δ xc
•
•
•
•
Grid parameters
Pulse propagation in uniform flow
Convected vortex
50
40
30
20
10
20
40
60
x/ δ xc
D. Ricot, S. Marié, P. Sagaut
80
100
120
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Grid refinement in LBM based on continuous distribution fun
Continuous distribution functions ?
Grid refinement algorithm
Validations
Grid parameters
Pulse propagation in uniform flow
Convected vortex
Iso-contours of the density fluctuation
Injection (no time interpolation)
Linear interpolation in time
80
70
70
70
60
60
60
50
50
50
y /δ x
40
c
80
y /δ xc
y/ δ xc
Uniform grid
80
40
40
30
30
30
20
20
20
10
10
20
40
60
80
100
120
10
20
40
c
60
80
100
120
c
x/ δ x
x/ δ x
20
40
60
80
100
120
c
x/ δ x
• Small pressure reflexion occurs at the grid interface. The error is
reduced when linear time interpolation is used.
• There is an analytical solution for the propagation of the pulse in
uniform flow → precise error quantification is possible
le-logo
D. Ricot, S. Marié, P. Sagaut
Grid refinement in LBM based on continuous distribution fun
Continuous distribution functions ?
Grid refinement algorithm
Validations
Grid parameters
Pulse propagation in uniform flow
Convected vortex
Simulation with the uniform grid
−3
5
x 10
(
) reference analytical solution
( + + + + ) LB simulation
4
3
ρ / ρ0
2
1
0
−1
−2
−3
0
20
40
60
80
100
120
x/ δ xc
le-logo
D. Ricot, S. Marié, P. Sagaut
Grid refinement in LBM based on continuous distribution fun
Continuous distribution functions ?
Grid refinement algorithm
Validations
Grid parameters
Pulse propagation in uniform flow
Convected vortex
Two-grid simulation with injection (no time interpolation)
−3
5
x 10
(
) reference analytical solution
4
( + + + + ) results on coarse points
3
( ) results on fine points
ρ/ρ
0
2
1
0
−1
−2
−3
0
20
40
60
80
100
120
x/ δ xc
le-logo
D. Ricot, S. Marié, P. Sagaut
Grid refinement in LBM based on continuous distribution fun
Continuous distribution functions ?
Grid refinement algorithm
Validations
Grid parameters
Pulse propagation in uniform flow
Convected vortex
Two-grid simulation with linear interpolation in time
−3
5
x 10
(
) reference analytical solution
4
( + + + + ) results on coarse points
3
( ) results on fine points
ρ/ρ
0
2
1
0
−1
−2
−3
0
20
40
60
80
100
120
x/ δ xc
le-logo
D. Ricot, S. Marié, P. Sagaut
Grid refinement in LBM based on continuous distribution fun
Continuous distribution functions ?
Grid refinement algorithm
Validations
Grid parameters
Pulse propagation in uniform flow
Convected vortex
Convergence rate
• Parameter b p gives the initial width of the pressure pulse, i.e. the spatial
resolution
−6
10
(
) slope -3.5
(
) Uniform grid
−7
10
L
2
( ) Two-grid simulation without time
interpolation (injection)
−8
10
( ◦ ◦ ◦ ◦ ) Two-grid simulation with linear
interpolation in time
−9
10
−10
10
0
1
10
10
b1/2
p
le-logo
D. Ricot, S. Marié, P. Sagaut
Grid refinement in LBM based on continuous distribution fun
Continuous distribution functions ?
Grid refinement algorithm
Validations
Grid parameters
Pulse propagation in uniform flow
Convected vortex
Initial pulse in the fine grid region
Injection (no time interpolation)
Linear interpolation in time
80
70
70
70
60
60
60
50
50
50
y /δ x
40
c
80
y /δ xc
y/ δ xc
Uniform grid
80
40
40
30
30
30
20
20
20
10
10
20
40
60
80
100
120
10
20
40
c
60
80
c
x/ δ x
x/ δ x
100
120
20
40
60
80
100
120
x/ δ xc
le-logo
D. Ricot, S. Marié, P. Sagaut
Grid refinement in LBM based on continuous distribution fun
Continuous distribution functions ?
Grid refinement algorithm
Validations
Grid parameters
Pulse propagation in uniform flow
Convected vortex
Convected vortex (linear interpolation in time)
Linear interpolation in time
Linear interpolation in time
80
70
70
70
60
60
60
50
50
50
y/ δ x
40
c
80
y/ δ xc
y/ δ x
c
Uniform grid
80
40
40
30
30
30
20
20
20
10
10
20
40
60
80
100
120
10
20
40
60
80
100
120
20
40
c
c
x/ δ x
x/ δ x
60
80
100
120
c
x/ δ x
• Fine to coarse grid convection : spurious vorticity appears inside the
fine region due to non-physical reflexion at the grid interface
le-logo
D. Ricot, S. Marié, P. Sagaut
Grid refinement in LBM based on continuous distribution fun
Continuous distribution functions ?
Grid refinement algorithm
Validations
Conclusion
• New theoretical insight in the link between coarse and fine distribution
functions
final rescaling expressions are fully equivalent to the previous
approaches
this new theoretical approach can be useful for other LB models
(other collision operator such as MRT models)
• The proposed grid refinement algorithm minimize the unknown
functions that must be approximated with interpolations
use of overlapping coarse nodes
interpolations are done on the distribution functions
time interpolation is done before spatial interpolation
• Accuracy of the grid refinement scheme can be improved using higher
order interpolation schemes
le-logo
D. Ricot, S. Marié, P. Sagaut
Grid refinement in LBM based on continuous distribution fun