J. Basic. Appl. Sci. Res., 3(11)189-200, 2013

© 2013, TextRoad Publication

ISSN 2090-4304 Journal of Basic and Applied

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*Corresponding Author: Ahmed Kadhim Hussein, Department of Mechanical Engineering ,College of Engineering , Babylon University, Babylon City, Iraq. ahmedkadhim7474@gmail.com

Numerical Study of Natural Convection around an Adiabatic Circular Cylinder Located Inside a Square Open Cavity

Ahmed Kadhim Husseina,*, Sumon Sahab, Waqar Ahmed Khanc

aDepartment of Mechanical Engineering ,College of Engineering , Babylon University, Babylon City, Iraq

bDepartment of Mechanical Engineering, The University of Melbourne, Parkville, VIC 3010, Australia c, * National University of Sciences and Technology - Department of Engineering Sciences - PN Engineering College PNS

Jauhar, Karachi 75350, Pakistan Received: August 3 2013

Accepted: September 12 2013

ABSTRACT

A numerical investigation is performed to analyze the steady state laminar natural convection phenomena inside a square open cavity. An adiabatic circular cylinder is placed at the center of the cavity while the sidewall in front of the breathing space is heated by a constant heat flux. The top and bottom walls are kept at the ambient constant temperature. Two-dimensional forms of Navier-Stokes equations along with the energy equations are solved using Galerkin finite element method. Results are obtained for a range of Grashof number from 103 to 106 and the diameter ratios ranging from 0.2 to 0.4 at Pr = 0.71 with constant physical properties. The parametric studies for a wide range of governing parameters show consistent performance of the present numerical approach to obtain stream functions and temperature profiles. The computational results indicate that both the flow and the thermal fields are influenced by the effect of Grashof number and diameter ratio. KEYWORDS: Open cavity, Natural convection, Finite element, Circular Cylinder

INTRODUCTION

Natural convection in cavities has been extensively studied both experimentally and numerically and being of considerable interest in many engineering and science applications. Such type of flow has a wide range of applications, for example, multi-pane windows, effective cooling of electronic components, solar thermal central receiver design and operation of nuclear reactors. Recently, sloped windows and skylights have been more and more frequently applied in buildings, which makes it necessary to gain more understanding on the natural convection in cavities. Most of researches in this field are substantially directed toward the study of rectangular cavities where the heat flow is typically unidirectional, i.e., the buoyancy is induced by imposing a heating either from the side (for conventional convection) or from below (for thermal instabilities) [1]. Many literature was available which deal with the study of natural convection in enclosures (Chadwick et al. [2] , Fusegi et al. [3] , Saha et al. [4] and Sivasankaran [5]) with either vertical or horizontal imposed heat flux or temperature difference. Hadjisophocleous et al. [6] solved the natural convection of a square cavity problem by non-orthogonal boundary fitted coordinate system. Chan and Tien [7] studied numerically two-dimensional laminar natural convection in a shallow open cavity. They found that for a square open cavity having an isothermal vertical side facing the opening and two adjoining adiabatic horizontal sides a satisfactory heat transfer results could be obtained especially at high Rayleigh numbers. Xia and Zhou [8] studied natural convection in an externally heated partially open cavity with an internal heat source. They found that the opening was advantageous to the flow and heat transfer in the cavity while the characteristics of flow and heat transfer changed with heat source location, external and internal Rayleigh number, and opening size. Sezai and Mohamad [9] analyzed numerically natural buoyant flow and heat transfer in a cubic cavity with side opening. Vertical wall of the cavity was at a higher temperature than the ambient, while other walls were assumed to be adiabatic. Numerical results were presented for Rayleigh numbers of 103 to 106 for a fluid having a Prandtl number of 0.71.The results indicated that as Rayleigh number increased the difference between two and three dimensional predictions increased. Also, it was found that this difference was greater for the flow field than for the rate of heat transfer. Elsayed and Chakroun [10] investigated experimentally the effect of aperture geometry on heat transfer between the cavity and surrounding air in tilted partially open cavities. They examined different geometrical arrangements, different opening ratios and tilt angles. Polat and Bilgen [11] numerically investigated laminar natural convection in inclined open shallow cavities for Rayleigh numbers ranging from 103 to 107 and cavity aspect ratio ranging from 1 to 0.125. Nateghi and Armfield [12] investigated natural convection flow in an inclined open cavity for Rayleigh numbers varying from 105 to 1010 with Prandtl number 0.7 and the inclination angles from 10o to 90o. Flow patterns and isotherms were shown in order to give a better understanding of the heat transfer and the flow mechanisms inside the cavity. The critical Rayleigh numbers for all angles had been obtained and showed that the critical Rayleigh number decreased when the inclination angle increased. Bilgen and Oztop [13] studied numerically natural convection heat transfer in a partially open inclined square cavities. They investigated numerically the steady-state heat transfer by laminar natural convection in a two dimensional partially open cavity. Mariani and Coelho [14] investigated numerically steady heat transfer and flow phenomena of natural convection of air in partially open enclosures, with three aspect ratios (H/W = 1, 2, and 4), within which there was a local heat source on

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the bottom wall at three different positions. Numerical simulations were performed for several values of the Rayleigh number ranging between 103 and 106, while the intensity of the two effects – the difference in temperature on the vertical walls and the local heat source – was evaluated based on the Rai/Rae ratio in the range between 0 and 2500. Results showed the presence of different flow patterns in the enclosures studied and the flow and heat transfer could be controlled by external heating and local heat source. The purpose of the present work is to investigate numerically natural convection heat transfer around an adiabatic circular cylinder located inside a square open cavity.

PHYSICAL MODEL, ASSUMPTIONS AND MATHEMATICAL MODEL

Figure 1: Schematic diagram of the square open cavity.

The flow and thermal fields in a two-dimensional open square cavity of length ( L ) is considered, as shown in the schematic diagram of figure 1. The opposite wall to the aperture is first kept to constant heat flux ( q ) , while the surrounding fluid interacting with the aperture is maintained to an ambient temperature (θ∞). The top and bottom walls are kept to constant temperature (Tc) while the remaining cylinder of diameter (d) is assumed to be adiabatic. The fluid is assumed to be air (Pr = 0.71) , Newtonian, incompressible, steady and the fluid flow is considered to be laminar. The properties of the fluid are assumed to be constant. Natural convection is governed by differential equations expressing conservation of mass, momentum and energy. The viscous dissipation term in the energy equation is neglected. The Boussinesq approximation is invoked for the fluid properties to relate density changes to temperature changes, and to couple in this way the temperature field to the flow field. The governing equations in non-dimensional form are written as follows:

0YV

XU

(1)

2

2

2

21YU

XU

GrXP

YUV

XUU (2)

2

2

2

21YV

XV

GrYP

YVV

XVU (3)

2

2

2

2

Pr1

YXGrYV

XU (4)

Equations (1-4) are converted to dimensionless form using the following dimensionless parameters:

2c

ro o o

T Tp px y d u v q LX ,Y ,d ,U ,V ,P , , t ,L L L U U t kU

3

2g TLPr ,Gr ,Ra Gr Pr .

(5)

The reference velocity (Uo) is related to the buoyancy force term and is defined as

tLgUo (6)The boundary conditions in non-dimensional form are written as follows:

0Y,0U1,XU0,XU ,

x

y

g q

Tc

Tc

Tc d

L

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0,01,0, YVXVXV ,

01,XYT0,X

YT

1,0 YX

0,1 Y if U < 0 and 0,1 YX

if U > 0

0sin5.0,cos5.0 drdr

S

The Nusselt number (Nu) is one of the important dimensionless parameters to be computed for heat transfer analysis in natural convection flow. The local Nusselt number ( Nux) can be obtained from the temperature field by applying

Y,0T

1xNu (7)

While the average or overall Nusselt number (Nu) is calculated by integrating the temperature gradient over the heated wall as

1

0dY

Y,0T1Nu (8)

FINITE ELEMENT FORMULATION

Finite Element Method (FEM) is a method to solve numerically the partial differential equations which can be applied to many problems in engineering. The method has been extended to solve problems in various fields such as in the field of heat transfer [15-18]. In spite of the great success of the method in this field, its application still under intensive research. This is due to the fact that the governing differential equations for general flow problems consist of several coupled equations that are inherently non-linear. Accurate numerical solutions thus require a large amount of computer time and data storage. One-way to minimize the large amount of computer time and data storage used is to employ an adaptive meshing technique [19]. This technique places small elements in the regions of large change in the solution gradients to increase solution accuracy and at the same time, uses large elements in the other regions to reduce the computational time and computer memory. The velocity and thermal energy equations (1)-(8) result in a set of non-linear coupled equations for which an iterative scheme is adopted. To ensure convergence of the numerical algorithm the following criteria is applied to all dependent variables over the solution domain

5101mij

mij

where () represents a dependent variable U, V, P and T; the indexes i, j indicate a grid point; and the index (m) is the current iteration at the grid level. The six node triangular element is used in this work for the development of the finite element equations. All six nodes are associated with velocities as well as temperature; only the corner nodes are associated with pressure. The velocity component and the temperature distributions and linear interpolation for the pressure distribution according to their highest derivative orders in the differential equations ( Eq.(1-4) ) can be written as

UNY,XU (9) VNY,XV (10)

NYX , (11)

PHY,XP (12) where α = 1, 2, … …, 6; λ= 1, 2, 3; Nα are the element interpolation functions for the velocity components and the

temperature, while Hλ are the element interpolation functions for the pressure. To derive the finite element equations, the method of weighted residuals is applied to the continuity equation (Eq. (1)) , the momentum equations (Eqs. (2)-(3)), and the energy equation ( Eq. (4)) , as follows

0dAA YV

XUN

(13)

dAYU

XUN

Gr

dAXPHdA

YUV

XUUN

A

AA

2

2

2

21

(14)

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dANdAYV

XVN

Gr

dAYPHdA

YVV

XVUN

AA

AA

2

2

2

21 (15)

dAYX

NGr

dAY

VX

UNAA

2

2

2

2

Pr1

(16) where (A) is the element area. Gauss’s theorem is then applied to Eqs. (14)-(16) to generate the boundary integral terms associated with the surface tractions and heat flux. Then Eqs (14)-(16) become,

00

1

S x

A

AA

dSSN

dAYU

YN

XU

XN

GrdA

XPHdA

YUV

XUUN

(17)

00

1

S yA

A

AA

dSSNdAN

dAYV

YN

XV

XN

Gr

dAYPHdA

YVV

XVUN

(18)

wS wwA

A

dSqNdAYY

NXX

NGr

dAY

VX

UN

Pr

1

(19)

Here Eqs.(14)-(15) specifying surface tractions (Sx, Sy) along outflow boundary S0 while Eq.(16) specifying velocity components and fluid temperature or heat flux that flows into or out from domain along wall boundary Sw. By substituting the element velocity component distributions, the temperature distribution, and the pressure distribution from Eqs. (9)-(12), the finite element equations can be written in the form,

0VKUK yx (20)

uyyxx

xyx

QUSSGr

PMUVKUUK

1 (21)

vyyxx

yyx

QKVSSGr

PMVVKVUK

1 (22)

Tyyxx

yx

QSSGr

VKUK

Pr1 (23)

where the coefficients in element matrices are in the form of the integrals over the element area and along the element edges S0 and Sw as,

dAx,NA NK x , (24a)

dAy,NA NK y , (24b)

dAx,NNA NK x , (24c)

dAy,NNA NK y , (24d)

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dANA NK , (24e)

dAx,NA x,NS xx , (24f)

dAy,NA y,NS yy , (24g)

dAx,HA HM x , (24h)

dAy,HA HM y , (24i)

0u S 0dSxSNQ , (24j)

0

v S 0dSySNQ , (24k)

w

T S wdSwqNQ . (24l)

These element matrices are evaluated in closed-form ready for numerical simulation. Details of the derivation for these element matrices are deleted here for brevity. COMPUTATIONAL PROCEDURE

The derived finite element equations, Eqs.(20)-(23), are non-linear. These non-linear algebraic equations are solved by applying the Newton-Raphson iteration technique [15] by first writing the unbalanced values from the set of the finite element equations (Eqs. (20)-(23)) as follows,

VKUKF yxp (25a)

uyyxx

xyxu

QUSSGr

PMUVKUUKF

)(1 (25b)

vyyxx

yyxv

QKVSSGr

PMVVKVUKF

)(1 (25c)

Tyyxx

yxT

QSSGr

VKUKF

Pr1 (25d)

This leads to a group of algebraic equations with the incremental unknowns of the element nodal velocity components, temperatures, and pressures in the form,

p

v

u

pvpu

v

vpvvvu

upuuvuu

FFFF

p

vu

KKKKK

KKKKKKKK

u

v

000 (26)

where

yyxx

yxx

SSGr

VKUKUKK uu

1

UKuvK y ,

KsinuTK ,

xMupK ,

VKvuK x ,

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yyxx

yyx

SSGr

VKVKUKK vv

1

KK v cos ,

yMvpK ,

xKK u ,

yKK v

yyxx

yx

SSGr

VKUKK

Pr1

0pK ,

xKpuK

,

yKpvK and

ppp KK 0 . The iteration process is terminated if the percentage of the overall change compared to the previous iteration is less than the specified value. GRID SENSITIVITY CHECK

1141315642

13686

12356103657963

4818

5.435

5.436

5.437

5.438

5.439

5.440

5.441

4000 6000 8000 10000 12000 14000 16000Element Numbers

Nu

Figure 2: Convergence of average Nusselt number with grid refinement for Gr = 106 and dr = 0.2 In order to obtain grid independent solution, a grid refinement study is performed for a square open cavity with Gr = 106 and dr = 0.2. Figure 2 shows the convergence of the average Nusselt number, (Nu) at the heated surface with grid refinement. It is observed that grid independence is achieved with 13686 elements where there is insignificant change in the average Nusselt number, with further increase of mesh elements.

RESULTS AND DISCUSSION A numerical investigation on natural convection phenomena around an adiabatic circular cylinder in an open cavity has

been performed by using finite element method. The left vertical wall is maintained at constant heat flux, as shown in Figure 1, while the top and bottom walls are kept at the ambient constant temperature. In order to validate the numerical code, pure natural convection with Pr = 0.71 in a square open cavity was solved, and the results were compared with those reported by Hinojosa et al. [20]. In Table 1, a comparison between the average Nusselt numbers is presented.

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Table 1: Comparison of the results for the average Nusselt numbers with Pr = 0.71.

Gr Nu The present work Hinojosa et al. [20] difference (%)

103 1.32 1.30 1.54 104 3.45 3.44 0.29 105 7.41 7.44 0.40 106 14.44 14.51 0.48

The streamlines and isotherms at various Grashof number ranging from 103 to 106 and the diameter ratios ranging

from dr = 0.2 to dr = 0.4 are shown in Figures (2-4) , when the left sidewall is maintained at constant heat flux while the right sidewall is considered open to air. The mechanism of natural convection begins when the cold air enters into the open square cavity from the bottom of the aperture, rotates in a clockwise direction following the shape of the cavity , then changes its direction by the hot left sidewall and leaves toward the upper part of the aperture. The flowing air becomes lighter to move at the top isothermal wall and then leaves through the top-right part of the cavity. The temperature of air adjacent the hot left sidewall of the cavity is greater than the temperature of air adjacent the top cold upper wall due to constant heat flux effect , so the air adjacent the left sidewall have a lower density than those near the cold upper one. For this reason, the air adjacent the hot left sidewall begins to move in upward direction and then fallen downward near the cold upper wall producing a rotating vortices in the cavity where the vortices core located at the center of the cavity around an adiabatic circular cylinder. The figures explain that the streamline contours are very similar when the Grashof number is low, but the air moves faster for Gr = 104. Therefore, viscous forces are more dominant than the buoyancy forces when the Grashof number is low. Furthermore, for Gr = 105 and 106, the streamline contours are similar also , but the boundary layer becomes thinner and faster. The velocity of the air flow moving toward the aperture increases while the area that is occupied by the leaving hot air decreases compared with that of the entering air. At high Grashof number when the intensity of convection increases significantly, the core of the rotating vortices moves up indicating that buoyancy forces are more dominant than the viscous forces. The air circulation around an adiabatic circular cylinder is strongly dependent on Grashof number where a circulation flow region of high intensity can be observed close to the hot left sidewall due to constant heat flux effect and pushes the air towards the cavity center. Also, a clear deformation in streamlines can be observed when the Grashof number increases. With respect to isotherms, when the values of Grashof number are low ( i.e., when Gr = 103 and 104 ) , the isotherms are in general smooth and approximately parallel which covering all the open cavity size .In this case, the heat is transferred by conduction and the temperature distribution is like to that with stationary fluid. At high Grashof number ( i.e., when Gr = 105 and 106 ) , the isotherm patterns changes significantly indicating that the convection is the dominating heat transfer mechanism in the cavity. In this case, the isotherms begin to deform and move towards the lower part of the cavity. Moreover, the thermal boundary layer decreases. From the other hand , at low Grashof number, the existence of the adiabatic circular cylinder does not make a significant change in the flow field. But , as the Grashof number increases, a clear deformation occurs in the flow field adjacent the top of the adiabatic circular cylinder. Figures (2-4) also show the effect of increasing the diameter ratio of the adiabatic circular cylinder on the flow and thermal fields. It can be noticed that , the flow volume within the cavity diminishes as the diameter ratio increases. Therefore, the quantity of air entering the open cavity decreases as the diameter ratio increases. This is because, the increase of the diameter ratio of the adiabatic circular cylinder causes to increase the size of the cylinder and this prevents the movement of entering and leaving air. Also, the thermal boundary layer increases as the diameter ratio increases. Figure 5 explain the average Nusselt number variations for various Grashof number and diameter ratio. From this figure, it is seen that the average Nusselt number increases significantly with Grashof number and there is a clear change from a conduction-dominant region at low Grashof number to a convection-dominant region at high Grashof number. The reason of this behavior, because when the Grashof number increases, the air moves faster due to increase the buoyancy force effects. Moreover, the average Nusselt number decreases when the diameter ratio increases CONCLUSION From the present work results, the following conclusions are found :-

1- For low Grashof number (103 and 104), the convection intensity around an adiabatic circular cylinder is very weak. As the Grashof number increases (105 and 106), the intensity of convection increases significantly, indicating that buoyancy forces effect becomes dominant. 2-At low Grashof number (103 and 104), conduction is the dominating heat transfer mechanism whereas at high Grashof number (105 and 106) , convection is the dominating heat transfer mechanism. 3- A significant deformation occurs in streamlines and isotherms when the Grashof number increases while the thermal boundary layer decreases. 4- At low Grashof number, the presence of the adiabatic circular cylinder does not make a significant change in the flow field. But , as the Grashof number increases, a clear deformation can be observed in the flow field. 5- When the diameter ratio of the adiabatic circular cylinder increases, the flow field inside the open cavity diminishes while the thermal boundary layer increases. 6- The average Nusselt number increases significantly with the Grashof number, while it decreases when the diameter ratio increases.

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Gr = 103

Gr = 104

Gr = 105

Gr = 106

Isotherms Streamlines

Fig 2. Isotherms and streamlines patterns for dr = 0.2

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Gr = 103

Gr = 104

Gr = 105

Gr = 106

Isotherms Streamlines

Fig 3. Isotherms and streamlines patterns for dr = 0.3

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Gr = 103

Gr = 104

Gr = 105

Gr = 106

Isotherms Streamlines

Fig 4. Isotherms and streamlines patterns for dr = 0.4

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Fig 5: Average Nusselt number as a function of Grashof number for different diameter ratio NOMENCLATURE

d Adiabatic circular cylinder diameter (m) dr Adiabatic circular cylinder diameter ratio g Gravitational acceleration (m/s2) Gr Grashof number

k Fluid thermal conductivity (W/m K) L Height and width of the cavity (m) Nu Nusselt number p Pressure (N/m2) P Non-dimensional pressure

Pr Prandtl number

q Heat flux (W/m2) Ra Rayleigh number T Temperature (K) u, v Velocity components (m/s) U, V Non-dimensional velocity components x, y Cartesian coordinates (m) X, Y non-dimensional Cartesian coordinates Greek symbols α Thermal diffusivity (m2/s) β Thermal expansion coefficient (K–1) ρ Fluid density (kg/m 3) υ Fluid Kinematic viscosity (m2/s) θ Non-dimensional temperature Abbreviations FEM Finite Element Method ITHS Isothermal heat source IFHS Isoflux heat source Subscripts

c Cold o Reference Ambient

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