ORIGINAL

Natural convection in inclined two dimensionalrectangular cavities

Lyes Khezzar • Dennis Siginer • Igor Vinogradov

Received: 9 May 2010 / Accepted: 25 July 2011

� Springer-Verlag 2011

Abstract Steady two-dimensional natural convection in

fluid filled cavities is numerically investigated. The chan-

nel is heated from below and cooled from the top with

insulated side walls and the inclination angle is varied. The

field equations for a Newtonian Boussinesq fluid are solved

numerically for three cavity height based Rayleigh num-

bers, Ra = 104, 105 and 106, and several aspect ratios. The

calculations are in excellent agreement with previously

published benchmark results. The effect of the inclination

of the cavity to the horizontal with the angle varying from

0� to 180� and the effect of the startup conditions on the

flow pattern, temperature distribution and the heat transfer

rates have been investigated. Flow admits different con-

figurations at different angles as the angle of inclination is

increased depending on the initial conditions. Regardless of

the initial conditions Nusselt number Nu exhibits discon-

tinuities triggered by gradual transition from multiple cell

to a single cell configuration. The critical angle of incli-

nation at which the discontinuity occurs is strongly influ-

enced by the assumed startup field. The hysteresis effect

previously reported is not always present when the

calculations are reversed from 90� to 0�. A comprehensive

study of the flow structure, the Nu variation with varying

angle of inclination, the effect of the initial conditions and

the hysteresis effect are presented.

List of symbols

A Surface area (m2)

AR Aspect ratio (=L/H)

g Gravitational acceleration (m/s2)

H Height of cavity

L Width of cavity

Nu Nusselt number

p Pressure (N/m2)

Pr Prandtl number

Ra Rayleigh number based on the height H of the cavity

TH Hot wall temperature

TC Cold wall temperature

T Fluid temperature

u, v Flow velocity components in x and y directions

respectively (m/s)

Greek symbols

b Fluid expansion coefficient (K-1)

/ Angle of inclination (deg)

j Thermal diffusivity of fluid (m2/s)

m Kinematic viscosity (m2/s)

q Fluid density (kg/m3)

w Stream function

1 Introduction

Natural convection in enclosures is important in applica-

tions such as materials processing, solar energy systems,

inclined flat plate solar collectors, nuclear energy systems,

L. Khezzar (&)

Department of Mechanical Engineering, Petroleum Institute,

Abu Dhabi, United Arab Emirates

e-mail: lkhezzar@pi.ac.ae

D. Siginer

Centro de Investigaciones en Creatividad y Educacion Superior,

Universidad de Santiago de Chile, Santiago, Chile

I. Vinogradov

Department of Mechanics and Aerospace Engineering,

Peking University, Beijing 100871, People’s Republic of China

123

Heat Mass Transfer

DOI 10.1007/s00231-011-0876-7

double-glazed window heat transfer, electronic cooling and

food sterilization. Literature for free convection in rectan-

gular enclosures has been reviewed by Gebhart et al. [1],

Ostrach [2] and recently Khalifa [3]. The work of

Ayyaswamy and Catton [4] and Clever [5] both published

the same year in 1973 are the first investigations to our

knowledge of this flow configuration. Ozoe et al. [6]

experimentally investigated and numerically computed

values of the Nusselt number for natural convection heat

transfer in a square channel with two inclined sides

maintained isothermally hot and cold and the other two

sides insulated. Complex flow patterns appear as the angle

of inclination decreases to less than 10� which suggests the

possibility of multiple, stationary flow modes. However, as

the inclination approaches zero degree a series of two-

dimensional roll-cells with their axes horizontal and per-

pendicular to the axis of the channel are eventually attained

as the stable flow mode. The maximum rate of heat transfer

is obtained both theoretically and experimentally at about

50� of inclination, whereas the minimum heat transfer rate

was experimentally found to occur at an inclination of

about 10� from the horizontal.

Ozoe et al. [7] presented both numerical and experi-

mental studies in two dimensional rectangular channels

with four different aspect ratios ranging from 1 to 4. The

maximum rate of heat transfer occurs at an angle of

inclination of 60� (bottom heated). The minimum rate of

heat transfer occurs at a small angle of inclination, and it is

a moderate function of aspect ratio and a slight function of

Ra. A change in the mode of circulation was observed

when the heat transfer rate reaches the minimum value.

The circulation pattern for angles of inclination below this

point of transition is observed to be a series of oblique roll-

cells. The computed values of average Nu showed general

agreement with experimental results except at small angles

of inclination for which a different mode of circulation

prevailed. Experimental measurements of heat transfer

rates were reported by Ozoe and Sayama [8] for laminar

natural convection of silicone oil and air in a long rect-

angular channel with aspect ratios of 1, 2, 3, 4.2, 8.4, 15.5,

and Rayleigh numbers ranging from 3 9 103 to 105 and

findings similar to [7].

Catton et al. [9] and Arnold et al. [10] investigated

experimentally and numerically heat transfer in inclined

cavities for a range of aspect ratios, Rayleigh numbers and

angles of inclinations. Heat transfer rates due to natural

convection in rectangular bottom heated regions inclined at

various angles from 0� to 180� with four different aspect

ratios from 1 to 12 were measured for Rayleigh numbers

between 103 and 106 by Arnold et al. [10] and a simple

scaling law valid for angles of inclination from 0� to 90�defining the relationship of Nu with the angle of inclination

and Ra was derived. Karayiannis and Tarasuk [11] studied

natural convection inside a rectangular cavity with differ-

ent temperature boundary conditions on the cold top plate

using a Mach–Zehnder interferometer. Correlation equa-

tions for coupled and non-coupled average heat transfer

rates are presented.

The transition of flow modes was also studied numeri-

cally by Soong et al. [12] who note the influence of initial

conditions on the flow pattern formation in a cavity of

aspect ratio of 4 and for a range of Rayleigh numbers from

1,500 to 20,000. For a horizontal cavity heated from below

of aspect ratio equal to 3, the flow pattern was found to be

strongly influenced by the initial conditions; however the

average Nusselt number remained unaffected. Corcione

[13] with horizontal cavities of several aspect ratios and

using numerical techniques considers the effect of bi-

directional differential heating at Rayleigh numbers

between 103 and 106 and conjectures that the increase in

the number of roll cells as the aspect ratio increases may be

explained through the progressive breakdown of the den-

sity stratification in the fluid layers adjacent to the top and

bottom walls affecting the formation of hot and cold fluid

streams moving upward and downward across the cavity

and thereby shaping the temperature distribution. Flow

mode transition and hysteresis phenomena for Rayleigh

numbers greater than 3,000 were demonstrated in [12] as

well as by Wang and Hamed [14] who conducted a sys-

tematic numerical study of the variation of the Nusselt

number with angle of inclination for a range of Rayleigh

numbers from 103 up to 104 and a single aspect ratio of 4.

They considered in addition to adiabatic side walls, the

effect of bi-directional temperature gradients on the side

walls and concluded that these types of flows could have

dual or multiple solutions and attributed them to initial

condition effects. The discontinuity in the Nusselt number

variation with angle of inclination and the hysteresis phe-

nomenon mentioned previously were observed for steady

as well as unsteady calculations under the Boussinesq

approximation [12–14].

Clearly for differentially heated two-dimensional

enclosures in one direction and adiabatic side walls, the

heat transfer characteristics are influenced by the inclina-

tion of the cavity with respect to the horizontal plane, the

thermal boundary conditions and the Rayleigh number

based on the height of the cavity. However, despite the

studies conducted so far on rectangular cavity flows with

varying Rayleigh and Prandtl numbers and aspect ratios,

several aspects of the phenomenon remain to be clarified,

for instance the maximum Nusselt number with varying

angle of inclination usually thought to be around 50�, the

presence and location of the Nusselt number discontinuity

related to the flow structure and the ensuing circulation

mode change, as well as the hysteresis phenomenon and the

effect of the initial conditions.

Heat Mass Transfer

123

Against this background a comprehensive numerical

investigation of the steady buoyancy induced flow in an

inclined two dimensional rectangular cavity of a Newto-

nian fluid with linear Boussinesq density-temperature

dependence is conducted in this paper. Constant surface

temperatures on two opposite walls are maintained at a

fixed differential value with the other two walls insulated.

The interest is focused on the dynamic and thermal aspects

of a two-dimensional cavity heated from below and cooled

from the top with the other two sides being insulated when

the cavity is inclined over the horizontal from 0� to 180�, in

other words the heating starts from below at 0� and ends

from above when the angle of inclination is 180�. Several

cavity aspect ratios are investigated, with three Rayleigh

numbers of 104, 105 and 106 for a linear fluid with a Prandtl

number of 10. Cavity flows are known to depend on initial

assumed fields, Hart [15] and Soong et al. [12], even in

steady flow computations and thus the investigation of the

combined effects of initial start up field conditions and

angle of inclination on the flow structure and the heat

transfer together with the hysteresis effect is the subject of

the present work. In particular, the present work expands

on the combination of aspect ratios and Rayleigh numbers

considered thus far and focuses on the idea of the effect of

initial conditions to consider for each angle of inclination a

zero start up field in addition to the previously used

approach of taking the solution at the preceding angle as

the start up field in seeking the solution for the subsequent

angle. Whenever possible and in order to provide the

necessary confidence, the computational results are com-

pared with a number of previously published numerical and

experimental results with the latter confined to the average

Nusselt number.

2 Mathematical formulation

A two dimensional rectangular cavity filled with a New-

tonian fluid is considered. The assumption of two dimen-

sional flow is adequate in most cases to simulate the flow

and heat transfer even in three dimensional cavities as edge

effects remain small, Bairi et al. [16]. The inclination angle

of the cavity / varies between 0� B / B 180�. The aspect

ratio is AR = L/H, the ratio of the length L of the iso-

thermal walls to the length H of the adiabatic walls. The

top (cold) and the bottom (hot) surfaces of the cavity are

maintained at constant temperatures Tc and Th while the

two side walls are kept adiabatic as shown in Fig. 1. Flow

in the cavity is assumed laminar, steady and two-dimen-

sional. The steady form of the conservation equations for

this type of flow has been previously used [13, 14, 16]. It is

known that for horizontal (zero inclination) cavities the

onset of convection is expected around the critical

Rayleigh number of 1708, Holland and Raithby [17].

Beyond this critical value the flow may quite possibly

exhibit multiple solutions around the same Ra. However,

grid independence tests and solutions obtained with

increasing angles starting at a finite 1� angle rather than

zero for AR = 3 and Ra = 104 confirm the approach

adopted (see discussion at the end of Sect. 3). Boussinesq

approximation holds, viscous dissipation is assumed to be

negligible, and all other fluid properties are assumed con-

stant. The buoyancy force is caused only by the density

gradient, thus:

qq0

¼ 1� b T � T0ð Þ ð1Þ

where b is the coefficient of thermal expansion,

b = 7.0 9 10-5 (�K)-1 for water, q is the fluid density

at temperature T, q0 and T0 are the corresponding reference

values, respectively. The field conservation equations of

mass, momentum and energy are given by:

ou

oxþ ov

oy¼ 0 ð2Þ

uou

oxþ v

ou

oy¼ � 1

qop

oxþ m

o2u

ox2þ o2u

oy2

� �þ gb T � T0ð Þsin/

uov

oxþ v

ov

oy¼ � 1

qop

oyþ m

o2v

ox2þ o2v

oy2

� �þ gb T � T0ð Þcos/

ð3Þ

uoT

oxþ v

oT

oy¼ j

o2T

ox2þ o2T

oy2

� �ð4Þ

The area-averaged Nusselt number on the conducting

walls is calculated using:

Nu ¼ 1

A

ZA

0

oT

oy

������wall

dx ð5Þ

The velocity vector is expressed in terms of its Cartesian

components (u, v) along the x and y directions of the

coordinate system shown in Fig. 1; p, m, j and g represent

the pressure, the kinematic viscosity, the thermal

diffusivity and the acceleration of gravity, respectively.

The Rayleigh and Prandtl numbers are defined as,

y x

HL

φ

TC

Th

g

Fig. 1 Flow geometry

Heat Mass Transfer

123

Ra ¼ gb Th � Tcð ÞH3

mj; Pr ¼ m

jð6Þ

The set of Eqs. 2–4 is solved numerically using the

finite volume technique and the Fluent code on a Cartesian

grid with grid sizes that ensure grid-independent solutions.

The grid independence issue is explored in depth in the

next section with important observations. Convective terms

are discretized using the Quick Scheme, see [18], since for

structured meshes this scheme has greater formal accuracy

and central differencing is used for the diffusion terms

discretization. The SIMPLE algorithm is used for velocity–

pressure coupling and since a co-location scheme is used

PRESTO is adopted for pressure interpolation at cell faces

and is known to perform well for high Rayleigh number

natural convection flows [19]. On the walls zero slip

conditions are prescribed for the momentum equations.

Constant temperatures are prescribed for heated and cooled

walls and zero heat flux for the side adiabatic walls.

Convergence was assumed when the normalized residuals

reached a value of 10-4 in monitoring appropriate field

variables. For some configurations near zero angle of

inclination of the cavity no convergence was obtained and

therefore for these angles solutions are not reported.

Grid independence tests were performed using three

grids for the square cavity for the three Rayleigh numbers

adopted and for the angle of inclination of 90� as shown in

Table 1. The intermediate grid in each case was used since

the solution obtained is very close to the solution for the

finest grid based on a comparison of the Nusselt numbers

and stream function values. In addition, the benchmark

results for the square cavity with side walls at constant

temperatures and the top and bottom walls insulated of De

Vahl Davis [20] were used to test the calculations for a

Prandtl number of 0.7 and for Ra = 104, 105 and 106. The

maximum deviation of the calculated Nusselt numbers from

the benchmark values was less than 1%. For the rectangular

configurations again three grids were chosen as shown in

Tables 2 and 3 for two Rayleigh numbers Ra = 105 and 106

and an angle of inclination of 25� and 90�. The intermediate

angle of 25� was chosen because at this angle the flow

contains a multi-cell structure and hence represents a more

stringent case for grid independence. It is assumed that

the grid used for Ra = 105 will also be valid for Ra = 104.

The intermediate grids in Tables 2 and 3 were used for the

calculations and are appropriate for the purpose of the

present work based on the comparisons of the Nusselt

numbers and stream function values. The results are also

compared when possible with experimental and numerical

results found in the literature for the rectangular and square

cavities as will be shown in the next section.

For each cavity aspect ratio and Rayleigh number,

calculations are carried varying the angle of inclination in

three ways. In the first set, the computations are started

from zero velocity field conditions and reference tem-

perature T0 at zero angle of inclination and the solution

obtained is used as a subsequent initial condition for the

next angle and so on. The step increase in the angles is 5�between 0� and 90�, larger step angles are used beyond

90�. In the second set of calculations, a solution is sought

for each angle starting from an initial zero field condition

and reference temperature T0, while the third set of cal-

culations starts from an angle of 90� and proceeds back-

wards by decreasing the angle to reach the horizontal

position. In the latter in a similar fashion to the first set,

the results of each angle are used as start up conditions for

the next angle.

3 Results and discussion

Calculations were performed for Pr = 10 and three Ray-

leigh numbers of 104, 105 and 106 for several aspect ratios

and the results reported are discussed considering the effect

of the Rayleigh number, aspect ratio, the initial conditions

and the hysteresis effect on the average Nusselt number on

the active walls and the flow structure.

Figures 2, 3, 4, and 5 show the variation of the average

Nusselt number on the conducting walls with angle of

inclination and Rayleigh number for four aspect ratios

AR = 1, 3, 6 and 12. Each figure shows three sets of data,

the filled dots are the results obtained assuming zero field

start up conditions for each angle and the square dots show

the result of calculations done using the computed solution

at the previous angle of inclination as a startup field. The

third set of data shown by the empty triangles illustrate the

results of calculations obtained using the solution at 90� as

an initial start-up field and reversing the sequence of cal-

culations by decreasing the angle from 90� until the cavity

regains its horizontal 0� position.

Figure 2 shows the variation of Nu with angle of incli-

nation for AR = 1 the square cavity. Nu increases mono-

tonically to reach a maximum value of around 2.44, 4.92

and 9.71 with increasing Ra at an angle between 65 and

70�. There is no difference between computational results

the two sets of initial conditions described above yield. For

Ra = 106 convergence is elusive for inclinations below 3�.

The residuals oscillate around a value without ever reach-

ing the convergence levels mentioned above. The absolute

maximum of Nu occurs at about 60� for AR in the neigh-

borhood of AR = 1. At any aspect ratio AR the absolute

maximum of Nu grows with growing Ra and tends to occur

at an angle slightly larger at increasingly larger Ra. How-

ever at the same Ra with increasing AR the Numax shows

considerable decrease, Figs. 2, 3, 4, and 5. The variation of

Numax is more strongly dependent on Ra than it is on AR.

Heat Mass Transfer

123

Conversely the dependence on AR is weaker as compared

to the dependence on Ra.

For AR = 3, 6 and 12, Figs. 3, 4, and 5, and when

Ra = 104 and 105 the dependence of Nu on / is quite

different. Nu shows an aspect ratio and Ra dependent dis-

continuity and computations using different sets of initial

conditions yield different behavior and discontinuities that

take place at different angles of inclination. The compu-

tational results with either set are in complete agreement

before and after the discontinuities. The agreement past the

latest discontinuity is clearly due to the flow configuration

in this region which is single cell for calculations corre-

sponding to either set of start-up conditions. With zero

start-up conditions the flow configuration switches at a

smaller inclination angle to a single cell configuration and

with continuous initial conditions the single vortex gets

established at a greater angle of inclination, i.e it is

delayed, Figs. 6, 8, 9, 11. The critical angle at which the

vortices merge to become a single cell is Ra and AR

dependent. Keeping Ra constant the critical angle grows

with growing aspect ratio AR regardless of the initial

conditions used to compute the field. With both initial data

set and with increasing / Nusselt number Nu reaches a

local maximum after which it drops suddenly to a lower

value. For Ra = 104 the step change in Nu for continuous

flow computations takes place at angles 40�, 45� and 50�whereas for Ra = 105 it occurs at 40�, 50� and 47� for

AR = 3, 6 and 12 respectively. For the zero initial field

Table 1 Grid independence tests AR = 1 and / = 90�

Ra Grid Nu (% Dev.) wmax (% Dev.)

104 100 9 100 2.2680 3.2% 0.00078436 8.2%

150 9 150 2.2081 0.51% 0.00072036 -0.60%

300 9 300 2.1970 (-) 0.00072473 (-)

105 100 9 100 4.5550 -0.09% 0.00156345 1.0%

150 9 150 4.5597 0.02% 0.00154844 0.06%

300 9 300 4.5590 (-) 0.00154753 (-)

106 150 9 150 9.1000 0.14% 0.00284406 1.4%

200 9 200 9.1330 0.51% 0.00278972 -0.57%

300 9 300 9.0870 (-) 0.00280562 (-)

Table 2 Grid independence

tests Ra = 105 and / = 90� and

/ = 25�

AR Grid Nu (% Dev.) wmax (% Dev.)

Angle = 90�3 150 9 150 4.0980 -0.36% 0.003486 0.11%

250 9 100 4.0970 -0.39% 0.003484 0.06%

300 9 300 4.1130 (-) 0.003482 (-)

6 150 9 150 3.5769 -0.30% 0.006182 0.00%

250 9 100 3.5765 -0.31% 0.006185 0.05%

300 9 300 3.5876 (-) 0.006182 (-)

12 200 9 160 3.0737 -0.50% 0.010234 0.11%

250 9 150 3.0800 -0.30% 0.010225 0.02%

500 9 300 3.0892 (-) 0.010223 (-)

Angle = 25�3 150 9 150 4.3 0.77% 0.00713 -0.97%

250 9 100 4.31 1.01% 0.00727 0.97%

300 9 300 4.267 (-) 0.0072 (-)

6 150 9 150 4.3607 3.90% 0.00737 -2.25%

250 9 100 4.1906 -0.15% 0.0076 0.80%

300 9 300 4.197 (-) 0.00754 (-)

12 200 9 160 4.352 2.16% 0.00758 -4.44%

250 9 150 4.26018 0.00% 0.00776 -2.17%

500 9 300 4.26 (-) 0.007932 (-)

Heat Mass Transfer

123

Table 3 Grid independence

tests Ra = 106 and / = 90� and

/ = 25�

AR Grid Nu (% Dev.) wmax (% Dev.)

Angle = 90�3 150 9 100 7.6281 0.31% 0.006383 0.19%

200 9 100 7.6291 0.32% 0.006385 0.22%

300 9 200 7.6048 (-) 0.006371 (-)

6 300 9 50 6.6232 1.67% 0.01096 1.28%

200 9 100 6.5445 0.46% 0.010848 0.25%

400 9 200 6.5146 (-) 0.010821 (-)

12 200 9 70 3.0737 -0.50% 0.01835 0.27%

250 9 100 3.0800 -0.30% 0.0183 0.00%

400 9 140 3.0892 (-) 0.0183 (-)

Angle = 25�3 150 9 100 6.6960 1.27% 0.00934 -2.20%

200 9 100 6.6490 0.56% 0.00968 1.36%

300 9 200 6.6120 (-) 0.00955 (-)

6 300 9 50 5.6647 0.51% 0.01096 -4.70%

200 9 100 5.69338 1.02% 0.0114 -0.87%

400 9 200 5.636 (-) 0.0115 (-)

12 200 9 150 4.7079 0.17% 0.0168 -1.18%

250 9 100 4.6896 -0.22% 0.01677 -1.35%

500 9 300 4.7000 (-) 0.017 (-)

Fig. 2 Variation of Nu when

AR = 1 with / (open squarenon-zero startup field, filledcircle zero startup field)

Heat Mass Transfer

123

Fig. 3 Variation of Nu when

AR = 3 with / (open squarenon-zero startup field, filledcircle zero startup field)

triangles indicate decreasing

angle calculations from 90� to

0�

Fig. 4 Variation of Nu when

AR = 6 with / (open squarenon-zero startup field, filledcircle zero startup field),

triangles indicate decreasing

angle calculations from 90� to

0�

Heat Mass Transfer

123

computations, the step change for Ra = 104 occurs at an

approximate angle of 12� for AR = 3 and 6� and 15� for

AR = 12, whereas for Ra = 105 the discontinuity takes

place at 9.5�, 11� and 12� for AR = 3, 6 and 12. Consid-

ering only one set of initial conditions results, it can be

seen that as Ra is changed its effect on angle of mode

transition is barely noticeable. For all aspect ratios the drop

in Nu seems to be much attenuated for continuous field

startup conditions with the size of the steep drop in Nu

about 1/3 of the size of the earlier occurring drop resulting

from the zero initial condition computations. For all aspect

ratios when Ra = 106 and in particular for AR = 3 and 12

the step change in Nu moves closer to 0� with increased

size. For AR = 12, Ra = 106 there is a steep drop in Nu

from a value of 6–4 starting in the very neighborhood of 0�with zero initial condition computations whereas continu-

ous field initial condition computations seem to indicate a

rise in Nu from 6 to 6.5 and then dropping steeply to 4.75 at

about 15�. This behavior is in line with previous findings

[10, 12]. In all cases the maximum Nu gets smaller with

decreasing Ra at a given AR and the discontinuity in Nu

seems to move closer to 0�. For Ra = 106, except for

AR = 12 convergence problems were experienced for

angles less or equal than 4� and thus results for these angles

are not shown.

The sudden drop of Nu with angle of orientation has

been observed previously in [10] and [12] and is due to the

transition of the flow cell structure from multiple cells to a

single cell. The multi-cell structure which prevails at 0�(Benard flow) is gradually modified with increasing /through a series of cascading configurations with a smaller

number of cells culminating in a final single cell in the

enclosure. Stream function and temperature contours nor-

malized by the local corresponding maximum values for

Ra = 105 and AR = 6 and AR = 3 at different angles of

inclination for continuous start-up conditions are illustrated

in Figs. 6 and 9. For AR = 6, as the angle is increased the

initial multi-cell pattern of 6 cells observed at 0� gradually

shifts to a lower number of cells in stages, a structure with

5 cells between 20� and 35� followed by a three cell con-

figuration at about 45�. The multiple cell structure further

bifurcates into a single-cell around 55� associated with a

discontinuity in the Nu. The single cell mode prevails

thereafter until the angle of inclination reaches 180�whereby the mode of heat transfer is conduction dominated

as implied by the isotherms for 170�, Fig. 7. Figure 8

shows the stream function evolution for the same angles as

those of Fig. 6 for zero initial conditions. The discontinuity

in Nu and consequently the gradual evolution of the flow

structure from multiple cell to single cell occurs at a much

earlier angle of inclination around 12�. The flow patterns at

55� and 170� are very much similar. For AR = 3 the cell

pattern starts with 4 unequal cells shifting to 3 cells

between 20� and 40� and at 45� the pattern bifurcates to a

Fig. 5 Variation of Nu when

AR = 12 with / (open squarenon-zero startup field, filledcircle zero startup field);

triangles indicate decreasing

angle calculations from 90� to

0�

Heat Mass Transfer

123

single cell which prevails until the inclination reaches

180�, Fig. 9. The temperature contours of Fig. 10 associ-

ated with the multi cell structures are similar to the ones

depicted in Fig. 7 for AR = 6. The particular angle at

which a flow mode transition occurs depends upon the

Rayleigh number as evidenced by Figs. 3, 4, and 5. The

evolution of the field with increasing angle / with zero

initial conditions for the same aspect ratio is shown in

Fig. 11. The flow structure from multiple cell to single cell

occurs at much earlier inclination of 12� as compared to the

computations with continuous start-up conditions, Figs. 9

and 10.

In all the cases investigated the initial number of cells

depends upon the Rayleigh number and aspect ratio in line

with the findings of Corcione [13]. A single cell pattern

prevails after the last transition until 180�. Flow and tem-

perature fields are symmetric about the central vertical

plane for 0� and about the central horizontal plane when

the angle is 90�. Multiple cell flow configurations create

upward and downward convecting cold and hot streams.

The temperature contours of Figs. 7 and 9 show alternating

compression and spreading of isotherms near the bottom

and top conducting walls implying a varying heat flux in a

wavy manner on these walls.

Calculations were performed in reverse, starting with

the calculated field at 90� and gradually decreasing the

angle of inclination to reach the horizontal position of 0�.

The results of these computations are illustrated in Figs. 3,

4, and 5 for AR = 3 to AR = 12. These calculations were

conducted to investigate the hysteresis phenomenon

observed in [12, 14] and thought to be related to the multi

solution nature of this flow due to the effects of the starting

flow fields during the numerical integration of the conser-

vation equations. The hysteresis phenomenon was not

observed for AR = 3 for all Ra and for AR = 6 for

Ra = 105 and 106. In these cases where it is absent, the

Fig. 6 Stream function contours at inclination / for AR = 6 and

Ra = 105; non-zero startup field, Nx = 250 and Ny = 100Fig. 7 Temperature contours at inclination / for AR = 6 and

Ra = 105; non-zero startup field, Nx = 250 and Ny = 100

Heat Mass Transfer

123

Nusselt number decreased monotonically until 0� was

reached without showing any discontinuity, and the single

cell flow structure obtained at 90� persists all the way to

angle 0�. When hysteresis was present Nu decreased with

decreasing angle and exhibited a discontinuity to reach a

lower value at 0� than the one calculated with the

increasing angle computations. The jump in Nu is associ-

ated with the change in the flow mode with the appearance

of multiple roll cells. For both cases, the variation of Nu

coincided exactly with the values obtained with increasing

angle computations between 90� and the angle where either

discontinuity took place. The reasons behind this behavior

are difficult to ascertain since it is not clear what exactly

causes the transition between one flow mode and the other.

The variation of Nu with aspect ratio and Ra for an

inclination of 90� that is with vertical active walls and

horizontal adiabatic walls is shown in Fig. 12. Nu for each

Ra increases to a maximum value for an aspect ratio around

one and thereafter decreases monotonically with AR. As

expected Nu is a strong function of Ra for this orientation,

with higher Ra achieving higher heat transfer rates. The

calculated Nu values agree very well with the experimental

values of Arnold et al. [10] when AR = 1, 6 and 12 and the

Fig. 8 Stream function contours at inclination / for AR = 6 and

Ra = 105; zero startup field, Nx = 250 and Ny = 100

Fig. 9 Stream function contours at inclination / for AR = 3 and

Ra = 105; non-zero startup field, Nx = 250 and Ny = 100

Heat Mass Transfer

123

computed values of Catton et al. [9]. This serves to confirm

the reliability of the present computational results. On the

same graph the experimental correlations of Catton [21].

Equations 7–9 are also plotted. It can be seen that there is

very good agreement between Eq. 8 and the calculation

results when 2 \ AR \ 10 with an average error of about

2%. For AR = 1 the results reported for the three Rayleigh

numbers considered agree to within 1–2% with the calcu-

lations of Bairi et al. [16]. When AR is between 1 and 2 the

agreement is still good between the calculations and Eq. 7

for Ra = 104 and Ra = 105 but the calculations over pre-

dict the correlation values for the third Rayleigh number,

the average error in this case is around 4%. Beyond

Fig. 10 Temperature contours at inclination / for AR = 3 and

Ra = 105; non-zero startup field, Nx = 250 and Ny = 100

Fig. 11 Stream function contours at inclination / for AR = 3 and

Ra = 105; zero startup field, Nx = 250 and Ny = 100

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123

AR = 10 the numerical calculations continue to exhibit a

decrease of the Nusselt number with AR and under predict

the correlation values that show an increase beyond

AR = 10 with an average error of about 15%.

Nu ¼ 0:18Pr

0:2þ PrRa

� �0:29 1\ LH \2

10�3\Pr\105

103\ Ra Pr0:2þPr

8<: ð7Þ

Nu ¼ 0:22Pr

0:2þ PrRa

� �0:28L

H

� ��0:25

2\ LH \10

Pr\105

103\Ra\1010

8<:

ð8Þ

Nu ¼ 0:42Ra0:25Pr0:012 L

H

� ��0:3 10\ LH \40

1\Pr\2� 104

104\Ra\107

8<:

ð9Þ

Figure 13 shows the variation of Nu with aspect ratio for

Ra = 104 and 105 when the angle of orientation is 0�. Nu

increases rapidly between AR = 0 and 1 and thereafter

gradually to tend asymptotically to a constant value in a

similar fashion to horizontal air layers equal to the unique

value reported in the numerical simulations of Corcione

[13] for Ra = 105. The actual fit with the correlation Nu ¼0:21 ARð Þ0:09

proposed by Corcione [13] and developed for

aspect ratios between 0.66 and 8 and a range of Rayleigh

numbers between 104 and 106 is very good specially for

Ra = 104 and with a maximum local discrepancy for

Ra = 105 less than 5%. It can hence be extrapolated with

confidence to aspect ratios up to 12.

4 Conclusions

The buoyancy induced flow with linear density-tempera-

ture dependence in a two dimensional rectangular cavity is

investigated numerically for angles of inclinations between

0� and 180�. Flow configuration and heat transfer behavior

due to natural convection in enclosures of aspect ratios

1, 3, 6 and 12 for Ra = 104, 105 and 106 and Pr = 10 are

determined.

The results are consistent and in very good agreement

with experimental and simulation data in the literature.

However this investigation goes far beyond the existing

studies and probes aspects of this flow configuration hith-

erto unexplored. The behavior in Nu and in particular the

sudden decrease in average heat transfer as the angle is

increased in the first quadrant is associated with the cas-

cading flow changes leading from a multiple cell configu-

ration in the neighborhood of 0� to configurations with

gradually decreasing number of cells ending up with a

single cell structure thereafter leading to more and more

conduction dominated heat transfer mode as the angle of

inclination approaches 180�. The critical angle of inclina-

tion at which the discontinuity occurs is strongly influenced

by the assumed startup field. It was found that previously

reported hysteresis phenomenon is not always present.

The computations also revealed that the transition to a

single cell structure associated with the sudden drop in the

Nu depends upon the assumed startup field. Assuming a

zero startup field leads to an earlier transition towards a

single cell structure. However, the solutions obtained with

different initial conditions are in full agreement when both

reach the single cell mode. Although the results are

repeatable, it is somehow difficult to explain this behavior

Fig. 12 Variation of Nu with AR when / = 90�, lines represent

correlations of Eqs. 7–9 and empty symbols are taken from Ref. [8, 9]

Fig. 13 Variation of Nu with AR when / = 0�, lines indicate

correlation Nu = 0.21(AR)0.09 of [13]

Heat Mass Transfer

123

except to stress that by its very nature, the multi cell flow is

complex not stable and sensitive to orientation of the cavity

and hence would tend to show sensitivity to the initial

conditions in the process of changing from a multi cell to a

single cell structure.

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