Page 1

Stability analysis of natural convection in porous cavities

through integral transforms

L.S. de B. Alvesa, R.M. Cottaa,*, J. Pontesb

aLaborat? o orio de Transmiss~ a ao e Tecnologia do Calor (LTTC),

Programa de Engenharia Mec^ a anica, EE, COPPE, Universidade Federal do Rio de Janeiro,

Cidade Universit? a aria, Cx. Postal 68503, Rio de Janeiro, RJ, 21945-970, Brazil

bPrograma de Engenharia Metal? u urgica e de Materiais, EE, COPPE, Universidade Federal do Rio de Janeiro,

Cidade Universit? a aria, Cx. Postal 68503, Rio de Janeiro, RJ, 21945-970, Brazil

Received 27 December 2000; received in revised form 6 July 2001

Abstract

The onset of convection and chaos related to natural convection inside a porous cavity heated from below is in-

vestigated using the generalized integral transform technique (GITT). This eigenfunction expansion approach generates

an ordinary differential system that is adequately truncated in order to be handled by linear stability analysis (LSA) as

well as in full nonlinear form through the Mathematica software system built-in solvers. Lorenz’s system is generated

from the transformed equations by using the steady-state solution to scale the potentials. Systems with higher trun-

cation orders are solved in order to obtain more accurate results for the Rayleigh number at onset of convection, and

the influence of aspect ratio and Rayleigh number on the cell pattern transition from n to n þ 2 cells (n ¼ 1;3;5;...) is

analyzed from both local and average Nusselt number behaviors. The qualitative dependence of the Rayleigh number at

onset of chaos on the transient behavior and aspect ratio is presented for a low dimensional system (Lorenz equations)

and its convergence behavior for increasing expansion orders is investigated. ? 2002 Published by Elsevier Science

Ltd.

1. Introduction

Transport phenomenon in porous media represents

an important segment of the heat and mass transfer field

and have a variety of applications in engineering. The

effect of free convection as a result of the gravitational

body force is of particular interest from both practical

and theoretical points of view. Engineering applications

include among others thermal insulation, radioactive

waste disposal, solar energy collectors and geothermal

energy analysis. Other applications of the porous me-

dium modeling are discussed by Kakac ? et al. [1] and

more recently by Kaviany [2].

The present work is concerned with the stability

analysis of transient natural convection within a two-

dimensional porous cavity heated from below, based

on a modified Darcy law that includes the velocity

time derivative [3,4]. While steady-state behaviors

have been almost fully determined and understood, a

substantial amount of research is yet to be under-

taken in the analysis of transient nonlinear phenom-

enon related to natural convection within porous

media.

Within the last two decades, the ideas in the so-called

generalized integral transform technique (GITT) [5–7]

were progressively advanced towards the establishment

of an alternative hybrid numerical–analytical approach,

based on the formal analytical principles in the classical

integral transform method [8], for the solution of a

priori nontransformable diffusion and convection–dif-

fusion problems. In the last few years, the analytical

steps inherent in the integral transformation procedure

were markedly improved by packages that allow for

mixed symbolic and numerical computations such as the

Mathematica system [9,10].

International Journal of Heat and Mass Transfer 45 (2002)1185–1195

www.elsevier.com/locate/ijhmt

*Corresponding author. Tel.: +55-21-562-8368; fax: +55-21-

290-6626.

E-mail address: cotta@serv.com.ufrj.br (R.M. Cotta).

0017-9310/02/$ - see front matter ? 2002 Published by Elsevier Science Ltd.

PII: S0017-9310(01)00231-9

Page 2

In the present study, the integral transformation

procedure generates an ordinary differential system of

equations, which is adequately truncated in order to

generate Lorenz equations, and other low dimensional

systems, generally employed in linear stability analysis

(LSA) [11]. Thereafter, systems with higher truncation

orders are generated and a more accurate correlation for

the Rayleigh number at onset of convection [12–14] as a

function of the aspect ratio is obtained and compared

with the results obtained from a direct numerical solu-

tion of the full nonlinear system. The influence of the

aspect ratio on the cell pattern transition into multiple

cells is observed through the steady-state Nusselt num-

ber behavior. This transition is also governed by the

Rayleigh number and its influence is observed through

the steady-state local Nusselt number behavior.

The qualitative behavior of the Rayleigh number at

onset of chaos is investigated using a correlation ob-

tained from applying LSA to Lorenz system [3]. This

critical parameter behavior, as the truncation order of

the nonlinear ordinary differential system is increased, is

shown for a specific case and ideas to improve its con-

vergence behavior are discussed.

2. Problem formulation

Consider a two-dimensional fluid saturated porous

layer enclosed by two isothermal horizontal walls at

temperatures Th and Tc, ThPTc, and two adiabatic

vertical walls as presented in Fig. 1, with the walls of

the cavity assumed to be impermeable. In the porous

Nomenclature

A;B;C

cp

D

Da

Ei

Eo

f;f?

g

h

H

ke

matrix coefficients

specific heat (J/kg K)

matrix coefficient

Darcy number (¼ j=l2)

energy input

energy output

vector coefficients

gravity (m=s2)

height of the porous cavity (m)

aspect ratio, equals (¼ h=l)

effective thermal conductivity of the

porous medium (W/m K)

length of the porous cavity (m)

number of terms in the temperature

expansion

number of cell structures

number of terms in the stream function

expansion

total number of equations (Nt ¼ N þ M)

local Nusselt number

averaged Nusselt number

steady-state local Nusselt number

Lorenz constants

Prandtl number (¼ mðqcpÞe=ke)

Rayleigh number (¼ gbðqcpÞfDTlj=mke)

critical conduction Rayleigh number

critical convection Rayleigh number

scaled Rayleigh number (¼ Ra=p2)

re-scaled dimensionless time

temperature (K)

temperature difference (¼ Th? Tc) (K)

cold wall temperature (K)

hot wall temperature (K)

dimensionless velocity components

(¼ uðqcpÞfl=km, vðqcpÞfl=km)

l

M

n

N

Nt

Nu

Nuav

Nuloc

p1;p2;p3

Pr

Ra

Racond

Raconv

Ra0

t

T

DT

Tc

Th

U;V

X

Y1;Y2;Y3

Z

dimensionless vertical coordinate (¼ x=l)

Lorenz potentials

dimensionless horizontal coordinate

(¼ z=l)

Greek symbols

ai

stream function eigenvalue in X-direction

b

isobaric coefficient of thermal expansion

of fluid (K?1)

bj

stream function eigenvalue in Z-direction

cm

temperature eigenvalue in X-direction

h

dimensionless temperature (¼ ðT ? TcÞ=

ðTh? TcÞ)

H

dimensionless filtered temperature

(¼ h ? X)

~?h h?h hm;n

dimensionless transformed temperature

j

permeability of porous medium (m2)

kn

temperature eigenvalue in Z-direction

m

kinematic viscosity of fluid (m2=s)

q

density (kg=m3)

s

dimensionless time (¼ tkm=ðqCpÞel2)

U

porosity

v

re-scaled coefficient (¼/Pr=Da)

w

dimensionless stream function

~?w w?w wi;j

dimensionless transformed stream

function

C;C?

temperature eigenfunctions

U;U?

stream function eigenfunctions

Subscripts

av

c

e

f

h

i;j;k

m;n;o

averaged parameter

related to cold wall

related to effective property

related to fluid property

related to hot wall

stream function expansion indices

temperature expansion indices

1186

L.S. de B. Alves et al. / International Journal of Heat and Mass Transfer 45 (2002)1185–1195

Page 3

medium, Darcy law, modified to include the time de-

rivative term, is assumed to hold, and the fluid is as-

sumed to be Newtonian and the flow to be within

Boussinesq approximation limits.

With these assumptions, the dimensionless conser-

vation equations for momentum and energy, trans-

formed to the stream function formulation, for a

two-dimensional transient flow in an isotropic porous

medium, are [3,4]:

1

v

o

os

o2w

oX2

?

þo2w

oZ2

?

þo2w

oX2þo2w

oZ2¼ ?Raoh

oZ;

ð1Þ

oh

osþow

oZ

oh

oX?ow

oX

oh

oZ¼o2h

oX2þo2h

oZ2:

ð2Þ

The related hydrodynamic and thermal boundary

conditions are

w ¼ 0; h ¼ 0

w ¼ 0; h ¼ 1

at X ¼ 0;

at X ¼ 1;

ð3Þ

ð4Þ

w ¼ 0;

oh

oZ¼ 0

oh

oZ¼ 0

at Z ¼ 0;

ð5Þ

w ¼ 0;

at Z ¼ H:

ð6Þ

The stream function and temperature initial condi-

tions are

w ¼ w0

h ¼ h0

where the stream function distribution is defined from

at s ¼ 0;

at s ¼ 0;

ð7Þ

ð8Þ

U ¼ow

oZ;

V ¼ ?ow

oX

ð9Þ

and the dimensionless parameters used are

Pr ¼m qcp

H ¼h

l

??

e

ke

;

Da ¼j

Ra ¼gb qcp

l2;

?

v ¼ /Pr

?

Da;

and

fDTlj

mke

:

ð10Þ

3. Transformed equations

Following the formalism in the integral transform

approach, we choose auxiliary eigenvalue problems for

the stream function in the X- and Z-directions, respec-

tively,

d2UiðXÞ

dX2

þ c2

iUiðXÞ ¼ 0;

06X 61;

ð11Þ

Uið0Þ ¼ 0; Uið1Þ ¼ 0;

d2U?

dZ2

U?

i ¼ 1;2;...;

ð12Þ

jðZÞ

þ k2

jU?

jðZÞ ¼ 0;

06Z 6H;

ð13Þ

jð0Þ ¼ 0; U?

jðHÞ ¼ 0;

j ¼ 1;2;...;

ð14Þ

where the associated normalized eigenfunctions and

eigenvalues are

p

sinðciXÞ;

ffiffiffiffi

Following the same procedure, the auxiliary eigen-

value problems for temperature in the X- and Z-direc-

tions are

UiðXÞ ¼

ffiffiffi

2

ci¼ ip;

kj¼jp

ð15Þ

U?

jðZÞ ¼

2

H

r

sinðkjZÞ;

H:

ð16Þ

d2CmðXÞ

dX2

þ a2

mCmðXÞ ¼ 0;

06X 61;

ð17Þ

Cmð0Þ ¼ 0; Cmð1Þ ¼ 0;

d2C?

dZ2

m ¼ 1;2;...;

ð18Þ

nðZÞ

þ b2

nC?

nðZÞ ¼ 0;

dC?

06Z 6H;

ð19Þ

dC?

nðZÞ

dZ

????

Z¼0

¼ 0;

nðZÞ

dZ

????

Z¼H

¼ 0;

n ¼ 0;1;2;...;

ð20Þ

where the associated normalized eigenfunctions and

eigenvalues are

p

sinðamXÞ;

1ffiffiffiffi

CmðXÞ ¼

ffiffiffi

2

am¼ mp;

ffiffiffiffi

ð21Þ

C?

0ðZÞ ¼

H

p

;

C?

nðZÞ ¼

2

H

r

cosðbnZÞ;

bn¼np

H:

ð22Þ

The eigenvalue problems (11)–(16) and (17)–(22) are

of the Sturm–Liouville type and allow definition of the

following integral transform pairs:

?w wiðZ;sÞ ¼

Z1

0

UiðXÞwðX;Z;sÞdX

ðtransform in X-directionÞ;

ð23Þ

~?w w?w wi;jðsÞ ¼

ZH

0

U?

jðZÞ?w wiðZ;sÞdZ

ðtransform in Z-directionÞ;

ð24Þ

Fig. 1. Representation of two-dimensional horizontal cavity

filled with porous material.

L.S. de B. Alves et al. / International Journal of Heat and Mass Transfer 45 (2002)1185–1195

1187

Page 4

wðX;Z;sÞ ¼

X

1

i¼1

X

1

j¼1

UiðXÞU?

jðZÞ~?w w?w wi;jðsÞðinverseÞ;

ð25Þ

?H HmðZ;sÞ ¼

Z1

0

CmðXÞHðX;Z;sÞdX

ðtransform in X-directionÞ;

ð26Þ

~ ?H H?H Hm;nðsÞ ¼

ZH

0

C?

nðZÞ?H HmðZ;sÞdZ

ðtransform in Z-directionÞ;

1

X

ð27Þ

hðX;Z;sÞ ¼ X þ

X

m¼1

1

n¼0

CmðXÞC?

nðZÞ~ ?H H?H Hm;nðsÞðinverseÞ;

ð28Þ

where a filtering solution was employed to make the

temperature boundary conditions homogeneous. Ap-

plying the principles of the generalized integral trans-

form technique [5,7] to Eqs. (1)–(6), we obtain the

transformed equations shown below:

d~?w w?w wi;jðsÞ

ds

¼ v

Ra

iþ k2

ðc2

jÞ

X

1

m¼1

X

1

n¼0

Ai;j;m;n~ ?H H?H Hm;nðsÞ

?~?w w?w wi;jðsÞ

!

ð29Þ

;

d~ ?H H?H Hm;nðsÞ

ds

¼?

X

X

X

1

i¼1

1

X

X

X

mþb2

1

j¼1

1

Bm;n;i;j~?w w?w wi;jðsÞ

?

i¼1

1

j¼1

1

X

X

nÞ~ ?H H?H Hm;nðsÞ

1

o¼1

1

X

X

1

p¼0

1

Cm;n;i;j;o;p~?w w?w wi;jðsÞ~ ?H H?H Ho;pðsÞ

þ

i¼1

j¼1

o¼1

p¼0

Dm;n;i;j;o;p~?w w?w wi;jðsÞ~ ?H H?H Ho;pðsÞ

?ða2

ð30Þ

and the integral transformation of the initial conditions

(7) and (8) yields

~?w w?w wi;j¼ w0fi;j

~ ?H H?H Hm;n¼ f?

and f, f?, A, B, C and D represent the integral coeffi-

cients obtained throughout the integral transformation

and are given by

at s ¼ 0;

ð31Þ

m;n

at s ¼ 0

ð32Þ

fi;j¼

Z1

Z1

0

UiðXÞdX

ZH

0

U?

jðZÞdZ;

ð33Þ

f?

m;n¼

0

ðh0? XÞCmðXÞdX

ZH

ZH

0

C?

nðZÞdZ;

ð34Þ

Ai;j;m;n¼

Z1

0

UiðXÞCmðXÞdX

0

U?

jðZÞdC?

nðZÞ

dZ

dZ;

ð35Þ

Bm;n;i;j¼

Z1

0

CmðXÞUiðXÞdX

ZH

0

C?

nðZÞdU?

jðZÞ

dZ

dZ;

ð36Þ

Cm;n;i;j;o;p¼

Z1

?

0

CmðXÞUiðXÞdCoðXÞ

ZH

Z1

ZH

System (29)–(32) defines the set of coupled nonlinear

ODEs to be numerically solved for the transformed

potentials, at any user requested value of dimensionless

time. Using the inversion formulae (25) and (28) with a

suitable reordering scheme [6,7], one can obtain the

single series expansions shown below:

dX

dX

0

C?

nðZÞdU?

jðZÞ

dZ

C?

pðZÞdZ;

ð37Þ

Dm;n;i;j;o;p¼

0

CmðXÞdUiðXÞ

dX

CoðXÞdX

jðZÞdC?

?

0

C?

nðZÞU?

pðZÞ

dZ

dZ:

ð38Þ

wðX;Z;sÞ ¼

X

X

1

i¼1

N

X

1

j¼1

UiðXÞU?

jðZÞ~?w w?w wi;jðsÞ

¼

k¼1

UiðkÞðXÞU?

jðkÞðZÞ~?w w?w wiðkÞ;jðkÞðsÞ;

ð39Þ

hðX;Z;sÞ ¼ X þ

X

X

jðZÞ, CmðXÞ and C?

1

m¼1

M

X

CmðlÞðXÞC?

1

n¼0

CmðXÞC?

nðZÞ~ ?H H?H Hm;nðsÞ

¼ X þ

l¼1

nðlÞðZÞ~ ?H H?H HmðkÞ;nðkÞðsÞ;

ð40Þ

where UiðXÞ, U?

functions obtained from the eigenproblem (11)–(22)

used as the basis for the integral transformation.

Local Nusselt number results used in this analysis

were obtained from

nðZÞ are the eigen-

NuðZ;sÞ ¼oh

oX

????

X¼1

ð41Þ

and average Nusselt number results were obtained from

NuavðsÞ ¼1

H

ZH

0

oh

oX

????

X¼1

dZ:

ð42Þ

By properly truncating the expansions in Eqs. (29)

and (30) or (25) and (28), one can for instance generate

the system of three equations below:

d~?w w?w w1;1ðsÞ

ds

¼ ?v

RaH

ðH2þ 1Þp

þ1

H2

p

~ ?H H?H H1;1ðsÞ

?

?

þ~?w w?w w1;1ðsÞ

?

;

ð43Þ

d~ ?H H?H H1;1ðsÞ

ds

¼ ? 1

?

ffiffiffi

p2 ~ ?H H?H H1;1ðsÞ

?

p

H

?

2

p2

H3=2

~ ?H H?H H2;0ðsÞ

!

~?w w?w w1;1ðsÞ;

ð44Þ

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L.S. de B. Alves et al. / International Journal of Heat and Mass Transfer 45 (2002)1185–1195

Page 5

d~ ?H H?H H2;0ðsÞ

ds

¼ ?4p2 ~ ?H H?H H2;0ðsÞ ?

ffiffiffi

2

p

H3=2

p2

~ ?H H?H H1;1ðsÞ~?w w?w w1;1ðsÞ;

ð45Þ

where the respective initial conditions are

p

p2

~?w w?w w1;1ð0Þ ¼8

ffiffiffiffi

H

w0;

ð46Þ

~ ?H H?H H1;1ð0Þ ¼ 0;

~ ?H H?H H2;0ð0Þ ¼1

ð47Þ

p

ffiffiffiffi

2

H

r

:

ð48Þ

The steady-state solutions of Eqs. (43)–(45) are

~?w w?w w1;1ð1Þ ¼ 0;

~?w w?w w1;1ð1Þ ¼1

~ ?H H?H H1;1ð1Þ ¼ 0;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

~ ?H H?H H2;0ð1Þ ¼H2Ra ? ð1 þ H2Þ2p2

ffiffiffi

~?w w?w w1;1ð1Þ ¼ ?1

p

~ ?H H?H H2;0ð1Þ ¼ 0;

ð49Þ

p

2H3Ra

1 þ H2? 2Hð1 þ H2Þp2

2H3Ra

1 þ H2? 2Hð1 þ H2Þp2

p

H3=2pRa

r

;

~ ?H H?H H1;1ð1Þ ¼ ?1 þ H2

H Ra

r

;

2

;

ð50Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2H3Ra

1 þ H2? 2Hð1 þ H2Þp2

2H3Ra

1 þ H2? 2Hð1 þ H2Þp2

p

H3=2pRa

r

;

~ ?H H?H H1;1ð1Þ ¼1 þ H2

~ ?H H?H H2;0ð1Þ ¼H2Ra ? ð1 þ H2Þ2p2

H Ra

r

;

ffiffiffi

2

;

ð51Þ

where Eq. (49) represents the conduction steady-state

solution and Eqs. (50) and (51) represent two possible

convection steady-state solutions.

Eqs. (43)–(45) can be transformed into the so-called

Lorenz equations by using one of the convection fixed

point solutions (50) and (51) to re-scale the transformed

potentials and then re-scale the time variable as well, to

obtain

dY1ðtÞ

dt

¼ p1ð?Y1ðtÞ þ Y2ðtÞÞ;

ð52Þ

dY2ðtÞ

dt

dY3ðtÞ

dt

¼ ?Y2ðtÞ þ Y1ðtÞðp2þ ð1 ? p2ÞY3ðtÞÞ;

ð53Þ

¼ 4p3ðY1ðtÞY2ðtÞ ? Y3ðtÞÞ;

ð54Þ

where the constants and the time variable from the

above equations are defined as

p1¼

vH2

ð1 þ H2Þp2;

H2

1 þ H2

p2¼

H2Ra

ð1 þ H2Þ2p2;

H2

ð1 þ H2Þp2t:

p3¼

and

s ¼

ð55Þ

Applying the same scaling to the initial conditions

(46)–(48) we find

p

w0

p2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Y2ð0Þ ¼ 0;

Y3ð0Þ ¼

Y1ð0Þ ¼

4

ffiffiffi

2

ð1 þ H2Þðp2? 1Þ

p

;

ð56Þ

ð57Þ

p2

p2? 1:

ð58Þ

The initial conditions above are those to be used with

Lorenz equations whenever they are derived from a

problem formulation such as in (29) and (30), in order to

maintainthe validity of the problemphysics. Other initial

conditions have been used [3,14,15] in order to demon-

strate the solution behavior around the fixed points.

4. Linear stability analysis (LSA)

The nature of the dynamics about the fixed points

(49)–(51), of the dissipative system (43)–(45) (or, in a

more general form, Eqs. (29) and (30)), is then investi-

gated. When such systems can be re-written in the more

compact form

dX

dt¼ FðX;kÞ;

ð59Þ

where X represents the potentials of the system and k its

control parameters, their equilibrium points are defined

by

FðXs;kÞ ¼ 0:

The stability matrix is established by evaluating the

Jacobian J,

ð60Þ

Ji;j¼oFi

oXj

????

Xs

ð61Þ

at the fixed point of interest Xs.

Based on this matrix, one can use the principle of

linearized stability [11] to compare the stability proper-

ties of the fully nonlinear problem and the auxiliary

linearized one in which higher-order terms are omitted.

Considering x the eigenvalues of the stability matrix, we

can summarize the basic ideas of this principle on the

two statements below:

1. If Rekxk < 0, the reference state Xsis asymptotically

stable.

If Rekxk > 0, the reference state Xsis unstable.

2. If the transversality condition is satisfied (dRekxk=

dkjk¼k06¼ 0; k0? critical parameter) and x is an

eigenvalue of odd multiplicity, one can say that:

• After bifurcation, the solutions will be stationary

if Imkxk ¼ 0,

• After bifurcation, the solutions will be periodic if

Imkxk 6¼ 0.

L.S. de B. Alves et al. / International Journal of Heat and Mass Transfer 45 (2002)1185–1195

1189

Page 6

These statements set up a quantitative criterion to

establish if the steady-state solutions are stable or not

and to distinguish two important critical parameters, the

Rayleigh numbers at onset of convection and at onset of

chaos. (or, more precisely, periodic or non-periodic

regimer)

The results obtained from this linearized analysis are

compared with those generated from a direct numerical

simulation of the full nonlinear system, using the well-

established numericalalgorithms

NDSolve of the Mathematica software system [10] to

solve the Ntth order truncated system formed by (29)

and (30) and initial conditions (31) and (32).

built-infunction

5. Results and discussion

All numerical solutions were obtained using the same

initial conditions, which were selected as w0¼ 0:1,

h0¼ 0:5 and for v ¼ 1000 [16]. Eqs. (43)–(45) were used

instead of the Lorenz equations form (52)–(54) to make

it easier to compare their results with those from higher

dimensional systems generated from Eqs. (29) and (30).

Critical Rayleigh numbers and Nusselt numbers de-

pendence on the aspect ratio, obtained from higher di-

mensional systems, are then investigated.

Using the system of three Eqs. (43)–(45), the LSA

returns

Ra0

cond1¼Racond

p2

¼ð1 þ H2Þ2

H2

;

ð62Þ

where Racond is the Rayleigh number at onset of con-

vection, as a function of the cavity aspect ratio. How-

ever, when we increase the order of the truncated system

to 21 equations, the LSA then returns

Ra0

cond2¼ð9 þ H2Þ2

9H2

:

ð63Þ

Using a system of 21 equations, the obtained Ray-

leigh number at onset of convection (Racond) goes

through a sudden change in its behavior as a function

of the aspect ratio, H, which does not occur to the

system of three equations, passing through a minimum

and then increasing again. The comparison of Eqs. (62)

and (63) with these numerical results generated is pre-

sented in Fig. 2, where one may see a good agreement

between results for the linear and nonlinear ap-

proaches.

When the number of equations numerically solved is

gradually increased to 136, other peaks in the Racond

versus H behavior can be observed. The comparison of

the obtained numerical results is presented in Fig. 3,

where systems with Nt ¼ 3; 21; 55; 105 and 136 equa-

tions are shown and demonstrate the numerical con-

vergence of the critical parameter solution.

The peaks that one can observe in Figs. 2 and Fig. 3

appear due to the transition of the pattern structure

from 1 to 3 cells, from 3 to 5 cells;...;from n to n þ 2, up

to the parallel plate cell structure at H ! 1. Stream

function and temperature isolines right before and after

the transition points are shown in Fig. 4, where the

pattern modification from 1 to 3 cells appears in Fig.

4(a), from 3 to 5 cells in Fig. 4(b), and from 5 to 7 in Fig.

4(c). In this way, formula (62) provides the critical

Rayleigh number when the cavity presents a one-roll cell

structure, in the same way as formula (63) provides the

value of this parameter when the cavity presents a three-

roll cell structure. The difference between the two for-

mulae relies on the number of rolls within the structure,

allowing the generalization of Eqs. (62) and (63) valid

for any number of rolls as

Ra0

cond!ð1 þ H2Þ2

H2

!ð9 þ H2Þ2

9H2

! ??? !ðn2þ H2Þ2

n2H2

;

ð64Þ

where n is the number of rolls in the cavity cell structure.

Expression (64) is similar to the one presented by

Caltagirone [13] and through it one may find the various

Fig. 2. Ra0

results.

condversus H: linear stability analysis and numerical

Fig. 3. Ra0

136 equations.

condversus H: numerical results for 3, 21, 55, 105 and

1190

L.S. de B. Alves et al. / International Journal of Heat and Mass Transfer 45 (2002)1185–1195

Page 7

values of the aspect ratio and Rayleigh number at onset

of convection when the transitions occur. These results

are presented in Table 1, where numerical and analytical

solutions are compared, showing a good agreement.

Knowing that the number of cells in each structure is

odd, one may find the values of H when the singularities

appear by equating Eq. (64) for an n cell structure to Eq.

(64) for an n þ 2 cell structure. Doing so, one may find

Fig. 4. Stream function and temperature transition of the cell structures: (a) 1–3 cells, (b) 3–5 cells and (c) 5–7 cells.

L.S. de B. Alves et al. / International Journal of Heat and Mass Transfer 45 (2002)1185–1195

1191

Page 8

the formula that generated the values that appear in

Table 1,

Hsing¼ ðn2ðn þ 2Þ2Þ1=4

for n ¼ 1ð1 ! 3Þ;3ð3 ! 5Þ;5ð5 ! 7Þ;...

and in the same way, the formula used to find the values

of Ra0

given by

ð65Þ

condat the singularity points presented in Table 1 is

Ra0

cond;sing¼4ðn þ 1Þ2

for n ¼ 1ð1 ! 3Þ;3ð3 ! 5Þ;5ð5 ! 7Þ;...

Using the values provided by Eqs. (65) and (66), the

numerical results obtained from the system of 136

equations can be compared with Eq. (64) in Fig. 5,

showing a very good agreement between the LSA and

the GITT hybrid solution of the full nonlinear system.

Another group of points can be found from expres-

sion (64), associated with the cell symmetry. At the

values of H and Ra0

number has its lowest local value. However, at the

points of minimum energy dissipation, the perturbation

introduced by the cell breakdown is minimum, causing

the cell structure to be symmetric. There is more than

one point of minimum due to the cell structure modifi-

cation caused by the geometry variation and some of the

points of symmetry can be observed in Fig. 6. These

points can be found by equating the derivative of

Eq. (64) with respect to H to 0, which gives H ¼ n as a

nðn þ 2Þ

ð66Þ

condgiven by Eq. (64), Nusselt

solution (n ¼ 1;3;5;...). Using this solution at Eq. (64),

one may find Racond¼ 4p2[3,13]. Nevertheless, these

same points, when obtained by direct numerical simu-

lation are lower than 4p2, as one may see in Fig. 5. Also,

when H ! 1 we have that n ! H, which makes it

possible to find the parallel plate Rayleigh number at

onset of convection from Eq. (66), 4p2.

Fig. 7 presents the steady-state average Nusselt

number variation with aspect ratio for Ra ¼ 50; 55 and

100, where the influence of the aspect ratio on the cell

pattern transition can be observed. One may see that for

Ra ¼ 50, Nuavre-enters the conduction region (Nuav¼ 1)

between 0 < H < 0:6 and 1:6 < H < 1:8. For Ra ¼ 55,

average Nusselt numbers are out of the conduction re-

gion for H > 0:6. This figure also shows the influence of

the Rayleigh number on the cell pattern transition since

the peaks occur at lower aspect ratio values for Ra ¼ 100

in comparison to Ra ¼ 55. This influence can be better

observed in Fig. 8, where steady-state local Nusselt

numbers of a squarecavity are

Ra ¼ 50; 100 and 200 there is a one-roll cell structure,

but at Ra ¼ 300 a three-roll cell structure appears.

Following the LSA procedure, but now using the

convection steady-state solution as the reference state,

the Rayleigh number at onset of chaos may be found.

Using Lorenz system one may find the following ex-

pression:

Raconv¼ð1 þ H2Þvðð3 þ 7H2Þp2þ H2vÞ

H2v ? ð1 þ 5H2Þp2

which is equivalent to the one found by Vadasz and

Olek [3].

The comparison of Eq. (67) with numerical results

for v ¼ 1000 and 10000 is shown in Fig. 9. Despite the

fact that Eq. (67) has been derived from a system of

three equations, it allows a qualitative interpretation of

this critical parameter. The time derivative coefficient

from the momentum equation, v, has a significant in-

fluence on the Rayleigh number at onset of chaos. The

critical parameter expression relative to the classical

Darcy model can be obtained by taking v ! 1,

Raconv¼ ð1 þ H2Þv;

which shows that the classical Darcy model without

transient term in the momentum equations, eliminates

the chaotic region by extending the convection region to

infinity.

presented. For

ð67Þ

ð68Þ

Fig. 5. Ra0

numerical results.

condversus H: generalized analytical solution and

Table 1

Comparison of analytical and numerical results of the cell transition points

n

1 ! 3

ffiffiffi

16/3 (5.33)a

5.19

3 ! 5

ffiffiffiffiffi

64/15 (4.27)

4.18

5 ! 7

ffiffiffiffiffi

144/35 (4.11)

4.04

7 ! 9

3

8.06

256/63 (4.06)

3.96

Hanalytical

Hnumerical

Racond;analytical

Racond;numerical

3

p

1.75

ð1:73Þa

15

p

3.88

ð3:87Þ

35

p

6.00

ð5:92Þ

ffiffiffi

7

p

ð7:94Þ

aRounded values.

1192

L.S. de B. Alves et al. / International Journal of Heat and Mass Transfer 45 (2002)1185–1195

Page 9

The analysis of the Rayleigh number at onset of

chaos through the full nonlinear system solution proved

to be much more computationally involved than for the

case at onset of convection, since much larger truncation

orders were required for full convergence to be obtained

by the GITT algorithm. Fig. 10 shows the behavior of

Fig. 6. Stream function and temperature isolines at local minima: (a) H ¼ 1, (b) H ¼ 3 and (c) H ¼ 5:25.

L.S. de B. Alves et al. / International Journal of Heat and Mass Transfer 45 (2002)1185–1195

1193

Page 10

Ra0

equations Nt ¼ N þ M used in the direct numerical

simulation is increased. LSA loses adherence when Nt is

increased, which was expected since the nonlinear terms

in the system are increased in number and importance,

and presentation of such results is here avoided.

conv, for H ¼ 1 and v ¼ 100 when the total number of

6. Conclusions

Cell pattern transition and aspect ratio govern the

behavior of the Rayleigh number at onset of convection.

Nusselt numbers, as well as the cell transition points, are

determined by Rayleigh number and aspect ratio. The

structure change from n to n þ 2 (n ¼ 1;3;5;...) cells

causes a sharp variation in both Nusselt and Rayleigh

number at onset of convection.

The importance of the time derivative term, usually

excluded from the Darcy model, was shown to be vital

for the proper understanding of natural convection in

cavities heated from below. Even though the critical

Rayleigh number at onset of convection shows no de-

pendence on v, the parameter gauging the transient be-

havior in the system, the critical Rayleigh number at

onset of chaos qualitative behavior is clearly dependent

on the modeling of the flow history, which is not prop-

erly taken into account in the classic Darcy model.

The GITT proved to be quite useful in this analysis

since it was capable of capturing the Rayleigh number

influence on the cell transition point, which does not

necessarily occur when purely numerical methods are

employed [17]. In order to reduce convergence oscilla-

tions in the results for the Rayleigh number at onset of

chaos, and thus reduce computational costs, GITT must

be used together with recently advanced optimization

schemes such as local-instantaneous filtering and dy-

namical reordering [6,7].

The symbolic manipulation allowed by the Math-

ematiaca software system, besides the fact that it fits per-

fectly with analytic-based approaches such as the present

integral transform method [6], improves the reliability in

the analytical steps needed in the stability analysis.

References

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Fig. 8. Square cavity steady-state local Nusselt number:

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Fig. 7. Steady-state average Nusselt number against aspect

ratio: Ra ¼ 50; 55 and 100.

Fig. 10. Ra0

convversus Nt: v ¼ 100 and H ¼ 1.

Fig. 9. Ra0

convversus H: v ¼ 1000 and 10000.

1194

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