Numerical Investigation of the Flow and Heat Transfer Generated by

Natural Convection and Surface Radiation in an Open Enclosure

Zouhair Charqui

, Mohammed Boukendil

, Lahcen El Moutaouakil, Zaki Zrikem

LMFE, Department of Physics, Cadi Ayyad University, Faculty of Sciences Semlalia, Marrakesh, Morocco

Keywords: Numerical Simulation, Natural Convection, Surface Radiation, Open Enclosure, Finite Volume Method.

Abstract: This work presents a numerical investigation of surface radiation (SR) coupled to natural convection (NC)

within an air-filled open cavity having the same emissivity on its three walls. The control volume method

combined with the algorithm SIMPLE is used to resolve the conservation equations. The radiosity method is

adopted to determine the radiative component of the heat exchange between all the cavity surfaces. This paper

aims to analyze the effects of two parameters controlling the flow and heat transfer, namely the emissivity

and the Rayleigh number. The simulations carried out show that SR makes the streamlines and isotherms very

sensitive to the Rayleigh number. In addition, the convective, radiative, and total heat transfer in the cavity

increase by increasing this parameter. The findings also reveal that the emissivity has almost no effect on the

streamlines for large Rayleigh numbers, while it has a remarkable impact on the isotherms. Furthermore,

increasing this parameter leads to an important increase (slight decrease) in the radiative (convective) flux.

1 INTRODUCTION

Natural convection flows in open cavities have

attracted significant interest from scientific

researchers and engineers during recent decades. This

interest is dictated by the critical role played by this

mode of heat transfer in many industrial applications.

Examples include cooling of electronic components,

aerospace engineering, heating and ventilation of

buildings, heat exchangers, solar thermal receivers,

fire spread in rooms, nuclear reactors, building

insulation, etc.

In the literature, there are many studies, both

numerical and experimental, which describe the flow

and heat transfer caused by NC in open enclosures.

These studies can be classified into two categories:

(a) pure NC, (b) NC coupled with SR.

The first category has been extensively studied in

the last decade. For example, (Bondareva, 2017)

numerically investigated NC and entropy generation

in an open triangular enclosure. They found that the

natural flow and the resulting heat transfer intensify

by increasing the Rayleigh number. On their side,

(Hussein, 2017) investigated NC in a parallelogram-

https://orcid.org/0000-0002-2987-3046

https://orcid.org/0000-0001-7058-2120

https://orcid.org/0000-0002-1786-4310

shaped enclosure, utterly open from the top and filled

with a nanofluid. The obtained results revealed that

increasing the volume fraction of nanoparticles leads

to an intensification of the mean heat transfer. (Öztop,

2017) published a paper intending to study NC in a

triangular open enclosure partially heated from

below. Their results indicate that the heat source cools

less as it gets closer to the opening.

Concerning the second category, the few

publications available in the literature show that NC

coupled to SR in open cavities is poorly documented

and needs more investigative efforts to expand the

fields of application. Among these studies, we can

mention that of (Hinojosa, 2017). They numerically

analyzed the impact of SR on entropy generation

caused by NC in an open enclosure. The simulations

show that SR increases the global entropy generation

from 33% to 560%. On their side, (Shirvan, 2017)

presented a numerical solution of NC combined with

SR in an open cavity of a solar receiver. They

deduced that the Rayleigh number and emissivity

significantly affect the hydrodynamic and thermal

fields.

Charqui, Z., Boukendil, M., El Moutaouakil, L. and Zrikem, Z.

Numerical Investigation of the Flow and Heat Transfer Generated by Natural Convection and Surface Radiation in an Open Enclosure.

DOI: 10.5220/0010734600003101

In Proceedings of the 2nd International Conference on Big Data, Modelling and Machine Learning (BML 2021), pages 371-375

ISBN: 978-989-758-559-3

371

In the present paper, SR coupled to NC in a square

cavity, open from the right and uniformly heated via

its vertical wall is briefly studied. The effects of two

parameters that control the flow and heat transfer in

such a configuration will be discussed, namely the

Rayleigh number and the emissivity of the cavity

surfaces.

2 PROBLEM STATEMENT

The studied configuration is represented in Fig. 1. It

is a bidimensional cavity, of height H and width L,

open on the right side and filled with air (Pr=0.71).

We designate by A=H/L the aspect ratio of the

enclosure, which is taken equal to unity since the

cavity is assumed to be square (H=L=5cm). The

cavity is heated via its vertical wall, which is

isothermal at temperature T

, while the rest of the

walls is kept adiabatic. Beyond the opening, the

ambient air is isothermal at the temperature

∞

=293K. The flow within the cavity is considered

laminar and two-dimensional. The fluid is

Newtonian, incompressible, and transparent to SR.

The internal surfaces of the cavity are considered

grey, diffuse, and have the same emissivity ε. The

aperture is assimilated to a black surface (ε=1)

brought to the temperature T

∞

The thermophysical air properties are considered

constant excluding the density in the thrust term,

which is assumed to follow the Boussinesq

approximation. The dimensionless equations

representing the conservation of mass, momentum,

Figure 1: Studied configuration.

and energy in the fluid medium are expressed as:







(1)

UUUP

τXYX





















(2)

VVVP

τXYY

Pr Ra Pr θ







 













(3)

θθθθθ

τXY





 



(4)

SR is determined by the radiosity method. It

consists of solving the following two systems of

equations. They give respectively the radiosity and

the net radiative flux lost by the i

surface element:



ref

kjjdsds

1JrdF























(5)

 



kjjds-ds

j=1

r,kk kk

-ε J r dFQr=Jr





(6)

Where T

∞

and F

is the form factor

between the i

and j

surface elements.

3 NUMERICAL PROCEDURE

AND VALIDATION

The conservation equations (1-4) have been solved

numerically by the control volume method and the

algorithm SIMPLE (Semi-Implicit Method for

Pressure Linked Equations), using the power-law

scheme. Several tests have been done to study the

influence of the mesh, time step, and convergence

criterion on the accuracy of results. These tests led to

the choice of a non-uniform mesh of 60×60, a

dimensionless time step of 10

-6

, and a convergence

criterion of 10

-4

. The validity of the calculation code

was tested by confronting its results with several

studies well-known in the literature. Table 1 shows an

excellent agreement between the mean Nusselt

numbers (convective and radiative) provided by the

BML 2021 - INTERNATIONAL CONFERENCE ON BIG DATA, MODELLING AND MACHINE LEARNING (BML’21)

372

numerical code and those found by (Singh, 2016),

who studied the interaction of NC and SR in an open

enclosure.

4 RESULTS AND DISCUSSION

4.1 Effect of the Rayleigh Number

Fig. 2 gives the streamlines (a) and isotherms (b)

obtained for four values of the Rayleigh number

(10

≤Ra≤10

). The emissivity of the internal surfaces

of the cavity is kept equal to unity (black surfaces).

This figure shows that the cold fluid penetrates the

cavity from the lower part of the aperture by moving

along the bottom wall. The latter heats up through SR

and transfers heat by convection to the cold fluid.

This one continues to move toward the vertical wall,

and when it becomes sufficiently heated, it rises

upwards under Archimedes' thrust. Thus, as the warm

fluid approaches the top adiabatic wall, it changes its

direction back to the aperture, transferring some of its

energy to this wall by convection before leaving the

cavity.

For low Rayleigh numbers (Ra=10

), the

buoyancy force, which is responsible for NC, is still

low. Thus, the flow structure keeps an almost

symmetrical aspect to the horizontal median of the

cavity. This symmetrical aspect is destroyed as the

Rayleigh number increases, and the streamlines

become denser near the active and upper adiabatic

wall. Consequently, the hydrodynamic boundary

layers developing against these walls become very

thin. This can be explained by the viscous nature of

the working fluid and its fast displacement due to the

increased thrust force.

Fig. 2 also shows that the isotherms exist in the

entire fluid domain for Ra=10

, indicating that the

conduction mode preponderate the heat exchange.

For higher Rayleigh numbers (Ra≥10

), the isotherms

become very tight against the active and upper

adiabatic wall. Thus, thermal gradient in the central

Table 1. Validation in terms of Nusselt number for a

temperature ratio T

=0.8 and ε=1.

Present

wor

(Singh,

2016)

Present

wor

(Singh,

2016)

2.10 2.19 67.95 67.95

3.42 3.54 68.42 68.45

6.46 6.53 69.10 69.09

12.21 12.41 70.05 69.98

part of the enclosure decreases considerably. In other

words, the predominant mode of heat exchange, in

this case, is NC. The isotherms also show that a

thermal boundary layer develops against the active

wall, where a very significant temperature gradient

occurs. Such a gradient becomes stronger by

increasing Ra. This is due to the reduction in the

thickness of the thermal boundary layer. Therefore,

the fluid extracts more heat from the active wall.

For low Rayleigh numbers (10

≤Ra≤10

), Fig. 3

shows that the convective heat exchange is relatively

low. This result is expected because, as already

mentioned, conduction and SR dominate the global

heat exchange in the enclosure. In addition, the

convective Nusselt number increases rapidly for

Ra≥10

because of the intensification of NC with the

difference between the environmental air and the

active wall in terms of temperature, and subsequently

with the Rayleigh number based on this difference.

The thermal radiation between the active wall and the

cavity surfaces also increases with the Rayleigh

number. But less rapidly than the convective heat

(a) (b)

Figure 2: Streamlines (a) and isotherms (b) for

different Ra and ε=1.

Numerical Investigation of the Flow and Heat Transfer Generated by Natural Convection and Surface Radiation in an Open Enclosure

373

Figure 3: Variations of the convective, radiative, and

global Nusselt numbers on the heated wall against Ra

for ε=1.

heat transfer. As an indication, varying Ra from 10

to 10

, the radiative Nusselt number increases by

22.5%. Fig. 3 also shows that for (10

≤Ra≤10

), the

convective component is much less than the radiative

one. This indicates that a conduction~radiation

equilibrium occurs in the enclosure for low Rayleigh

numbers. When Ra≥10

, the radiative and convective

Nusselt numbers are of the same order. In other

words, a convection~radiation equilibrium is

achieved. As for the total Nusselt number, it seems

that it also increases with Ra. Therefore, the higher

this parameter is, the more the active wall is cooled.

4.2 Effect of the Emissivity

Fig. 4 shows the streamlines (a) and isotherms (b)

obtained for different emissivities (0≤ε≤1), keeping

the Rayleigh number at 10

. The streamlines are very

dense near the active and upper adiabatic walls,

indicating a very strong velocity gradient at this

location. Such a density remains almost unchanged

by increasing the emissivity of the internal surfaces

of the cavity. On the other hand, these lines are

relatively scattered in the centre of the cavity and near

the lower adiabatic wall. They slowly approach the

latter by increasing the emissivity. Based on these

findings, it can be concluded that for high Rayleigh

numbers the flow structure in most of the cavity is

insensitive to SR.

The isotherms given in Fig. 4 show that without

radiative heat exchange (ε=0), the thermal boundary

layer expected to be developed against the lower

adiabatic wall is missing, and it only starts to take

place from ε=0.2. This confirms the fact that the

lower adiabatic wall heats up by SR. Furthermore, the

(a) (b)

Figure 4: Streamlines (a) and isotherms (b) for

different ε and Ra=10

isotherms indicate that the thickness of this thermal

boundary layer increases with emissivity. Therefore,

the cold fluid volume in the cavity and the rate of

convective heat exchange decrease.

Fig. 5 illustrates the variations of the mean

convective, radiative, and global Nusselt numbers on

the heated wall against the emissivity. It shows that

the Nu

decreases with emissivity. This decrease,

which is almost linear for ε≥0.2, can go up to 10.9%.

Such a result is predictable because, as mentioned

above, the volume of cold air in the cavity decreases

with emissivity. In contrast, the radiative heat transfer

is an increasing function of this parameter. Such an

increase is also almost linear for ε ≥ 0.2. As for the

total heat transfer, Fig. 5 shows that it increases

linearly with the emissivity although the attenuation

of the convective heat transfer. In other words,

increasing the emissivity promotes the cooling of the

active wall.

BML 2021 - INTERNATIONAL CONFERENCE ON BIG DATA, MODELLING AND MACHINE LEARNING (BML’21)

374

Figure 5: Variations of the convective, radiative, and

global Nusselt numbers on the active wall against ε

for Ra=10

5 CONCLUSION

In the present paper, the authors studied the effects of

the emissivity and the Rayleigh number on the

streamlines, isotherms, and heat exchange in an open

enclosure. The findings show that Ra significantly

affects the velocity and temperature distributions.

However, the emissivity has practically no impact on

the flow structure for high Rayleigh numbers. But it

remarkably affects the isotherms. The numerical

simulations performed also show that increasing Ra

results in an increase in the convective, radiative, and

total heat exchange. While for the emissivity, its

increase leads to an important increase (slight

decrease) in the radiative (convective) flux, and

consequently an increase in the total heat flux

extracted from the active wall.

ACKNOWLEDGEMENTS

The first author is grateful to the National Center for

Scientific and Technical Research CNRST for its

financial support under the Research Excellence

Scholarship Program.

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in fully open parallelogrammic cavity filled with Cu–

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