Numerical simulations of
Rayleigh-Bénard systems with complex
boundary conditions
XXXIII Convegno di Fisica Teorica - Cortona
Patrizio Ripesi
Iniziativa specifica TV62 “Particelle e campi in fluidi complessi”
Department of Physics, INFN University of Rome “Tor Vergata”
In collaboration with Luca Biferale & Mauro Sbragaglia
Outline
• Complex Rayleigh-Bénard convection: why ?
• Transition to steady convection (theory and numerics)
• Kinetic theory and Lattice Boltzmann model (LBM)
• Turbulent regimes with mixed boundary conditions
• Conclusions and perspectives
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1
“Classic” Rayleigh-Bénard systems
A Rayleigh-Bénard system is a layer of fluid subject to an external gravity field
placed between two plates, heated from below and cooled from above. The
dynamic behavior is determined by the geometry, the temperature difference
and the physical properties of the fluid.
Bénard cells
Tup
H
g
L
[Chandrasekhar, 1961]
uzT x - k¶z T
a gH 3DT
Nu =
Ra =
k ( DT / H )
kn
Tdown
x
ΔT=Tdown-Tup , α=thermal expansion coefficient
ν=viscosity, κ=thermal diffusivity
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“Classic” Rayleigh-Bénard systems
uzT x - k¶z T
a gH 3DT
What is the dependence of Nu on Ra? Ra =
Nu =
kn
k ( DT / H )
x
Rac
Convective state
Conductive state
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Turbulent convection
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[Lathrop et al, 2000]
3
“Classic” Rayleigh-Bénard systems
uzT x - k¶z T
a gH 3DT
What is the dependence of Nu on Ra? Ra =
Nu =
kn
k ( DT / H )
Convective
plumes
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[Sugiyama et al.,2007]
Turbulent convection
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x
[Lathrop et al, 2000]
4
“Complex” Rayleigh-Bénard systems
Considering a Rayleigh-Bénard system with an insulating lid
on the upper boundary.
What happens into the bulk region?
¶z T = 0
Tup
Heat transfer mechanism
from bottom to up?
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5
Tup
Tdown
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“Complex” Rayleigh-Bénard systems: why?
Ice-insulating effect on the Deep
Water formation (part of the
thermohaline circulation)
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Continental-insulating effect on
the Earth Mantle Convection
6
The static solution: analytical approach
The equations for a slightly compressible flows ( ρ ≈ const ) are described by
Solving for the static case, we need to solve the problem
ì ¶2T (x, z) ¶2T (x, z)
+
=0
ï
2
2
¶z
ï ¶x
L
ï
T
(x,
H
)
=
T
L
<
x
<
up
1
ï
2
í
ï ¶T (x, H ) = 0
0 < x < L1
ï ¶z
ï
L
ïT (x, 0) = Tdown
0<x<
î
2
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L
ξ=2L1/L
insulating fraction
2L1
T = Tup
H
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¶z T = 0
T = Tup
z
x
T = Tdown
7
The static solution: analytical approach
Looking for a solution of the form T(x, z) = T0 (z) +
T
(
-
down
- Tup )
where T0 (z) = Tdown
H
function along x, we have
z = Tdown - b z and
DTL
Q(x, z)
2
2p H
Q(x, z) is a periodic
z
(z-H )
ù
-4 p j ö +2 p j
æ
ö
DTL é a0 z ¥ æ
2
p
j
L
T(x, z) = Tdown - b z +
ê
+ å a j ç1- e L ÷e
cos ç
x ÷ú
è L øúû
2p H êë 2 H j=1 è
ø
where the aj are fixed by the boundary conditions
Fourier series
ì ¶ Q(x, z) ¶ Q(x, z)
+
=0
ï
2
2
¶x
¶z
ï
jH ö
¥
ìa
æ
-4 p
æ 2p j ö
0
L
ï
L
+
a
1e
cos
x÷ = 0
ï
ç
÷
ç
j
L1 < x <
ïQ(x, H ) = 0
è
ø
2
L
ï
è
ø
2
í
j=1
í
ï ¶Q(x, H ) 2 p H
jH ö
¥
æ
-4 p
æ 2p j ö
ï
L
ï
=
0 < x < L1
L
a
+
ja
1+
e
cos
x÷ = 1
ç
÷
ç
j
¶z
L
ï 0 4p H
ï
è
ø
L
è
ø
î
j=1
ï
îQ(x, 0) = 0
Dual Series problem
2
2
å
å
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L1 < x <
L
2
0 < x < L1
[Sneddon, 1966]
8
Why numerical simulations?
• All data are available for each time step
• Fine resolution between motion scales
[Ahlers, 2008]
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Numerics: a little bit of Kinetic theory…
The main feature of the Kinetic theory is the formulation of an equation (called the
Boltzmann equation) which describe the evolution for the single particle distribution
function (pdf) f(ξ,x,t)
Collision operator
into the BGK
approximation
¶f
1é
+ x × Ñf + g × Ñx f = - ë f - f (eq) ùû
¶t
t
f
(eq)
(x ; x, t) =
r
( 2p T )
D/2
e
2
æ
ö
ç - x -u /2T ÷
è
ø
Local equilibrium
distribution
The momenta (in velocity spaces) of the pdf give to the hydrodynamical fields:
r (x, t) =
u(x, t) =
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ò
ò
R
D
fd Dx
f x d x velocity
D
R
D
density
T(x, t) =
ò
2
R
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D
f x - u d Dx temperature
11
The Lattice Boltzmann Model
Discretized BGK Boltzmann equation
Dramatic reduction of number
of degrees of freedom
From this equation, it can be shown that by using a Chapman-Enskog expansion of
the distribution function (fl = fl (0) + εfl (1)+ ε2fl (2)+…) where ε<<1, we can recover the
thermo-hydrodynamical equations
mean free path
e»
lKn
LH
<< 1
hydro scale
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perfect gas equation of state
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Numeric (LBM) vs Theory for the static case
Tup=0.5, Tdown=1.5
ξ = 0.4
0.2
T(x,z)
0.4
0.2
Numerics
Theory
z/H=1.0
0.4
0.6
0.6
0.8
z/H=0.7 0.8
1
1
ξ = 0.8
Numerics
Theory
z/H=0.4
1.2
1.2
1.4
z/H=0.1 1.4
1.6
1.6
-0.4
-0.2
0
0.2
0.4
-0.4
x/L
-0.2
0
0.2
0.4
x/L
• Perfect agreement between static dual series solution and LBM
• Deeper penetration as ξ 1
T(x, z) = T0 (z) +
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DTL
Q(x, z)
2
2p H
ξ1
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T(x, z) = Tdown
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Transition to the convection in the limit L<<H
Limiting temperature profile for the case L<<H:
Basic temperature profile
renormalized by mixed
boundary conditions
DT æ
a0 L ö
T(z) » Tdown ç1÷ z = Tdown + b ¢z
è
H
4p H ø
Linear stability analysis with a renormalized basic temperature profile
provides a new estimate for the critical Rayleigh number:
Ra
c
1800
1790
1780
1770
1760
1750
1740
1730
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L/H=0.2 analytical
L/H=0.1 analytical
L/H=0.2 numerical (LBM)
L/H=0.1 numerical (LBM)
x
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0.1 0.2 0.3 0.4 0.5 0.6 0.7
Rac (x = 0)
0 £ x £1
æ
ö
æ xp ö
L
1+
log ç cos ç ÷÷
pH
è è 2 øø
ξ = insulating fraction
1720
1710
1700
0
Rac (x ) =
15
Turbulent regime
Numerical simulation (on massively parallel computers of CINECA&CASPUR) using
LBM on a 2D domain (2080x1040) at Ra ≈ 5x108 for various λ at ξ=0.5.
λ = number of cells of length L
70
Homogeneous
l=1
l=13
l=26
l=52
l=208
Nu(z)
60
50
40
¶t T(x, z) x +¶z ( UzT(x, z) x - k¶z T(x, z)
x
)=0
30
20
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Nusselt number
(Nu)must be a
constant in
stationary system
200
Nu =
uzT
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A
- k¶z T
k ( DT / H )
400
600
800
1000
z
A
16
Turbulent regime
Numerical simulation using LBM on a 2D domain (2080x1040) at Ra ≈ 5x108 for
various λ at ξ=0.5.
λ = number
of(t)
cells of length L
<T>
x,z
0.006
Homogeneous
l=1
0.005
l=13
l=52
l=208
0.004
0.003
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0.002
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0.001
Inhomogeneity on the upper boundary
causes the average temperature of the
fluid to increase with time
)=0
0
x
-0.001
300 400 500 600 700 800 900 1000 1100 1200
time
¶t T(x, z) x +¶z ( UzT(x, z) x - k¶z T(x, z)
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Turbulent regime
λ=1, 2080x140
T
0.08
0.06
0.04
0.02
0
-0.02
0
38
-0.04
-0.06
-0.08
40
0.025
Time Average 3 x 10 5 £ t £ 8 x 105
Time Average 4 x 10 6 £ t £ 11 x 106
20
0.02
36
40
<T>x,t(z)
0.015
0.01
34
32
60
0.045
y
<T>x,y(t)
0.075
Nu(z)
0.015
-0.015
80
30
-0.045
Increasing of temperature
localized in the central region
-0.075
70
105
140
26
x=200 t=2x1066
x=1040 t=2x10
x=1040 t=1x107
x=200 t=1.1x10 77
x=1040 t=1x10
z
28
120
35
100
0
0.005
0
0
2e+06
4e+06
6e+06
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8e+06
1e+07
1.2e+07
140
time
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20
40
60
80
100
120
z
18
Conclusions & Perspectives
• Insulating lid on cold boundary can alter the classic RB
convection, leading to an increasing of the bulk temperature
of the fluid depending on size (ξ) and wave-number (λ) of the
lids
• How the global heat transfer (Nu) is affected by changing ξ
and λ for different Ra ? Ongoing work
• 3D numerical simulations of a case of geophysical interest (
like ice and Deep-Water formation) Planned
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Thanks for your
attention!!!!
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References
•Ahlers G., Grossmann S., Lohse D. “Heat transfer and large scale dynamics in
turbulent Rayleigh-Bénard convection”. Rev. Mod. Phys., 81, 503-537, (2008).
•Chandrasekhar S.“Hydrodynamic and Hydromagnetic Stability”. Dover Pub., (1961).
•Duffy DG. “Mixed Boundary value problems”. Chapman & Hall/CRC, (2008).
•Ripesi P., Biferale L., Sbragaglia M. “High resolution numerical study of turbulent
Rayleigh-Bénard convection with non-homogeneous boundary conditions, using a
Lattice Boltzmann Method”. in preparation.
•Sneddon I. “Mixed boundary value problems in potential theory”. North-Holland Pub.
Co., (1966).
•Sugiyama K., Calzavarini E., Grossmann S., Lohse D. “Non-Oberbeck-Boussinesq
effects in two-dimensional Rayleigh-Bénard convection in glycerol”. Europhys. Lett.,
80,(2007).
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