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Natural convection flow in the cavity with isoflux boundaries
S. Jiracheewanun1, G. D. McBain1, S. W. Armfield1 and M. Behnia2
1School of Aerospace, Mechanical and Mechatronic Engineering, University of Sydney, NSW, 2006 Australia
2Dean of Graduate Studies, University of Sydney, NSW, 2006 Australia
CTAC'06 - The 13th Biennial Computational Techniques and Applications
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Outlines
Introduction
Objectives
Governing equations and numerical methods
Analytical solution for evenly heated slot
Results
Conclusions
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IntroductionNatural convection in a rectangular cavity can apply to wide range of engineering applications.The isoflux walls are more appropriate than the isothermal walls for real applications. However, the cavity with isoflux walls has received much less attention than with isothermal walls.
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Configurations
Computational domain and coordinate system.
q” q”v
u
y
x
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x
u
y
vq”q”
The corresponding dimensionless initial and boundary conditions are
,
x
y
T u v x y tT xT y
u v x y
= = == − =
= =
= = = =
0 a t a ll and <0
1 on 0 ,1
0 on 0 ,1
0 on 0 ,1 and 0 ,1
2 2
2 2
2 2
2 2
2 2
2 2
0
1
Pr
Pr
u vx y
u u u p u uu vt x y x x y
v v v p v v Rau v Tt x y y x y
T T T T Tu vt x y x y
∂ ∂+ =
∂ ∂
∂ ∂ ∂ ∂ ∂ ∂+ + = − + + ∂ ∂ ∂ ∂ ∂ ∂
∂ ∂ ∂ ∂ ∂ ∂+ + = − + + + ∂ ∂ ∂ ∂ ∂ ∂
∂ ∂ ∂ ∂ ∂+ + = +∂ ∂ ∂ ∂ ∂
4
Pr
Ra
να
βα ν
g q Lk
=
′′=
where
Governing Equations
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DiscretizationUsing existing codes
direct numerical simulations carried out using a finite volume method.Spatial derivative
standard second-order central differencing used for the viscous, pressure gradient and divergence termsQUICK third-order upwind scheme is used for the advective terms
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DiscretizationUsing existing codes
the time integrationsecond-order Adams-Bashforth scheme for the advective termsCrank-Nicolson scheme for the diffusive term
the momentum and temperature equations are solved using the Biconjugate Gradient Stabilized method. the non-iterative fractional-step pressure correction method is used to construct a Poisson equation, which is solved using the Biconjugate Gradient Stabilized method.
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The development of thermal boundary layers
Streamfunction contour for the isoflux cavity (Ra=5.8x109, Pr=7.5)
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The development of thermal boundary layers
Streamfunction contour for the isoflux cavity (Ra=5.8x109, Pr=7.5)
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The development of thermal boundary layers
Streamfunction contour for the isoflux cavity (Ra=5.8x109, Pr=7.5)
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The development of thermal boundary layers
Streamfunction contour for the isoflux cavity (Ra=5.8x109, Pr=7.5)
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The development of thermal boundary layers
Streamfunction contour for the isoflux cavity (Ra=5.8x109, Pr=7.5)
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The development of thermal boundary layers
Streamfunction contour for the isoflux cavity (Ra=5.8x109, Pr=7.5)
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The development of thermal boundary layers
Streamfunction contour for the isoflux cavity (Ra=5.8x109, Pr=7.5)
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The development of thermal boundary layers
Streamfunction contour for the isoflux cavity (Ra=5.8x109, Pr=7.5)
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The development of thermal boundary layers
Streamfunction contour for the isoflux cavity (Ra=5.8x109, Pr=7.5)
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The development of thermal boundary layers
Streamfunction contour for the isoflux cavity (Ra=5.8x109, Pr=7.5)
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Time traces of temperature at mid-height of the cavity
isothermal case (Ra*=3.28x106, Pr=7.5)(Patterson & Armfield,J.F.M.,1990)
isoflux case (Ra=5.8x109, Pr=7.5) (S. Jiracheewanun et al, 15AFMC, 2004)
Transient flow
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Isothermal
x
u
vT+∆T T-∆T
y
x
u
y
v q”q”
Isoflux
Fully developed flow
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1.05
1.07
1.07
stretching factor
.001
.002
.005
Smallest dx,dy
5x10-7
1x10-6
5x10-5
∆t
66x66102- 104
110x110108- 1011
78x78105- 107
Mesh size(A=1)
Ra
Summary of mesh size and time steps for testing
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Analytical Solution for Evenly Heated Slot
Evenly Heated Slot
v
u
y
q”
x
q”
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Analytical Solution for Evenly Heated Slot
Lietzke (NACA, 1955)
where γ is the stratification parameter which is related to the non-dimensional background stratification Γs, as:
44ΓRasγ
=
( )
( )
sinh 1 sin sinh sin 1( ) 32 sinh sin
cosh 1 cos cosh cos 1( ) +
sinh sin
x x x xL L L LRav x
Pr
x x x xL L L LT x
γ γ γ γ
γ γ γ
γ γ γ γ
γ γ γ
− − − =+
− − − = Γ+ s y
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( )( )( )
292
2 5 9
32 sinh sinsinh sin cosh cos 2 sinh sin
2 ( )
γ γ γ
γ γ γ γ γ γ γ
γ γ
+=
+ − −
= → ∞
Ra
Ra
Desrayaud, G., and Nguyen, T. H. (1989)
Analytical Solution for Evenly Heated Slot
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non-staggered mesh, with stretching factor of 1.07 The smallest grid size, near the boundaries, is 0.005
66x13010
66x1105
66x842
66x661
mesh sizeA
Mesh for testing the analytical solution for evenly heated slot (Ra=1x103)
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Temperature and Velocity Profiles(Ra=1x103,Pr=7.5)
Comparison temperature (a) and velocity (b) profiles at y=A/2 near the heated wall between analytical result and numerical results for Ra=1x103, Pr=7.5, with various A.
T
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Temperature and Velocity Profiles(Ra=1x103,Pr=0.7)
Comparison temperature (a) and velocity (b) profiles at y=A/2 near the heated wall between analytical result and numerical results for Ra=1x103, Pr=0.7, with various A.
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These analytical solutions are valid only for relatively high aspect ratio cavities.Can we apply these solutions to a low aspect ratio cavity? If yes, what is the appropriate Ra?
Analytical Solution for Evenly Heated Slot
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A Square Cavity, Pr=7.5 Pr = 7.5, Ra = 1x103
Temperature and velocity profiles near the heated wall
Temperature contour Streamfunction contour
T
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Temperature and velocity profiles near the heated wall
Temperature contour Streamfunction contour
Pr = 7.5, Ra = 1x104
T
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Temperature and velocity profiles near the heated wall
Temperature contour Streamfunction contour
Pr = 7.5, Ra = 1x105
T
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Temperature and velocity profiles near the heated wall
Temperature contour Streamfunction contour
Pr = 7.5, Ra = 1x106
T
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Temperature and velocity profiles near the heated wall
Temperature contour Streamfunction contour
Pr = 7.5, Ra = 1x107
T
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Temperature and velocity profiles near the heated wall
Temperature contour Streamfunction contour
Pr = 7.5, Ra = 1x108
T
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Temperature and velocity profiles near the heated wall
Temperature contour Streamfunction contour
Pr = 7.5, Ra = 5.8x109
T
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A Square Cavity, Pr=0.7 Pr = 0.7, Ra = 1x103
Temperature and velocity profiles near the heated wall
Temperature contour Streamfunction contour
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Temperature and velocity profiles near the heated wall
Temperature contour Streamfunction contour
Pr = 0.7, Ra = 1x104
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Temperature and velocity profiles near the heated wall
Temperature contour Streamfunction contour
Pr = 0.7, Ra = 1x105
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Temperature and velocity profiles near the heated wall
Temperature contour Streamfunction contour
Pr = 0.7, Ra = 1x106
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Temperature and velocity profiles near the heated wall
Temperature contour Streamfunction contour
Pr = 0.7, Ra = 1x107
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Temperature and velocity profiles near the heated wall
Temperature contour Streamfunction contour
Pr = 0.7, Ra = 1x108
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A = 1 A = 5
Ra=107 Ra=108 Ra=103 Ra=106
Ra=105 Ra=106
Ra=103 Ra=104
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A = 1 A = 5
Ra=107 Ra=108 Ra=103 Ra=106
constant boundary layer thickness and parallel to the side wall
the flow can be considered as one-dimensional flow
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( )( )( )
292 32 sinh sin
sinh sin cosh cos 2 sinh sinγ γ γ
γ γ γ γ γ γ γ
+=
+ − −Ra
Desrayaud, G., and Nguyen, T. H. (1989)
44Γ γ=s Ra
and
Background Stratification
x
y
q”q”
= ΓS
dTdy
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Background Stratification
Background stratification in cavities with various Ra and A, for Pr=7.5 and 0.7
Pr = 7.5
0
0.1
0.2
0.3
0.4
0.5
0.6
1.0E+01 1.0E+03 1.0E+05 1.0E+07Ra
Γ s
A=1
A=2
A=5
A=10
Analytical
Pr = 0.7
0
0.1
0.2
0.3
0.4
0.5
0.6
1.0E+01 1.0E+03 1.0E+05 1.0E+07Ra
Γ s
A=1
A=2
A=5
A=10
Analytical
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Nusselt numberKimura & Bejan (1984)
G.D.McBain (2005)
( )14 / 9 2 / 9
sinh sin2 cosh cos
2
Nu
Nu Ra
γ γ γγ γ
γ−
+=
−
→ ∞:
1/ 92 / 90.34 HNu Ra
L =
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Analytical and numerical heat transfer results
0.1
1
10
100
1.0E+01 1.0E+03 1.0E+05 1.0E+07 1.0E+09 1.0E+11Ra
Nu Kimura&Bejan(1984)
G.D.McBain(2005)
Numerical, Pr=0.7,A=1
Numerical, Pr=7.5,A=1
Nusselt number
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Concluding remarks The analytical solutions are valid with the high aspect ratio cavities (present study, A>5) over the entire range of Rayleigh number .
These solutions also apply to the small aspect ratio cavities with high Rayleigh number, e.g. a square cavity with Ra=1x107, which have a constant boundary layer thickness and parallel to the side walls.
The flow in a square cavity with high Ra can be considered as a one-dimensional flow.