natural convection flow in the cavity with isofluxboundaries

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

Outlines

Introduction

Objectives

Governing equations and numerical methods

Analytical solution for evenly heated slot

Results

Conclusions

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.

Configurations

Computational domain and coordinate system.

q” q”v

u

y

x

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

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

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.

The development of thermal boundary layers

Streamfunction contour for the isoflux cavity (Ra=5.8x109, Pr=7.5)

The development of thermal boundary layers

Streamfunction contour for the isoflux cavity (Ra=5.8x109, Pr=7.5)

The development of thermal boundary layers

Streamfunction contour for the isoflux cavity (Ra=5.8x109, Pr=7.5)

The development of thermal boundary layers

Streamfunction contour for the isoflux cavity (Ra=5.8x109, Pr=7.5)

The development of thermal boundary layers

Streamfunction contour for the isoflux cavity (Ra=5.8x109, Pr=7.5)

The development of thermal boundary layers

Streamfunction contour for the isoflux cavity (Ra=5.8x109, Pr=7.5)

The development of thermal boundary layers

Streamfunction contour for the isoflux cavity (Ra=5.8x109, Pr=7.5)

The development of thermal boundary layers

Streamfunction contour for the isoflux cavity (Ra=5.8x109, Pr=7.5)

The development of thermal boundary layers

Streamfunction contour for the isoflux cavity (Ra=5.8x109, Pr=7.5)

The development of thermal boundary layers

Streamfunction contour for the isoflux cavity (Ra=5.8x109, Pr=7.5)

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

Isothermal

x

u

vT+∆T T-∆T

y

x

u

y

v q”q”

Isoflux

Fully developed flow

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

Analytical Solution for Evenly Heated Slot

Evenly Heated Slot

v

u

y

q”

x

q”

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

( )( )( )

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

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)

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

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.

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

A Square Cavity, Pr=7.5 Pr = 7.5, Ra = 1x103

Temperature and velocity profiles near the heated wall

Temperature contour Streamfunction contour

T

Temperature and velocity profiles near the heated wall

Temperature contour Streamfunction contour

Pr = 7.5, Ra = 1x104

T

Temperature and velocity profiles near the heated wall

Temperature contour Streamfunction contour

Pr = 7.5, Ra = 1x105

T

Temperature and velocity profiles near the heated wall

Temperature contour Streamfunction contour

Pr = 7.5, Ra = 1x106

T

Temperature and velocity profiles near the heated wall

Temperature contour Streamfunction contour

Pr = 7.5, Ra = 1x107

T

Temperature and velocity profiles near the heated wall

Temperature contour Streamfunction contour

Pr = 7.5, Ra = 1x108

T

Temperature and velocity profiles near the heated wall

Temperature contour Streamfunction contour

Pr = 7.5, Ra = 5.8x109

T

A Square Cavity, Pr=0.7 Pr = 0.7, Ra = 1x103

Temperature and velocity profiles near the heated wall

Temperature contour Streamfunction contour

Temperature and velocity profiles near the heated wall

Temperature contour Streamfunction contour

Pr = 0.7, Ra = 1x104

Temperature and velocity profiles near the heated wall

Temperature contour Streamfunction contour

Pr = 0.7, Ra = 1x105

Temperature and velocity profiles near the heated wall

Temperature contour Streamfunction contour

Pr = 0.7, Ra = 1x106

Temperature and velocity profiles near the heated wall

Temperature contour Streamfunction contour

Pr = 0.7, Ra = 1x107

Temperature and velocity profiles near the heated wall

Temperature contour Streamfunction contour

Pr = 0.7, Ra = 1x108

A = 1 A = 5

Ra=107 Ra=108 Ra=103 Ra=106

Ra=105 Ra=106

Ra=103 Ra=104

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

( )( )( )

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

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

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 =

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

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.