Transp Porous MedDOI 10.1007/s11242-014-0347-y

Unsteady Natural Convection Within a Porous Enclosureof Sinusoidal Corrugated Side Walls

Atta Sojoudi · Suvash C. Saha · Marzieh Khezerloo ·Y. T. Gu

Received: 17 March 2014 / Accepted: 22 May 2014© Springer Science+Business Media Dordrecht 2014

Abstract Numerically investigation of free convection within a porous cavity with differen-tial heating has been performed using modified corrugated side walls. Sinusoidal hot left andcold right walls are assumed to receive sudden differentially heating where top and bottomwalls are insulated. Air is considered as working fluid and is quiescent, initially. Numericalexperiments reveal 3 distinct stages of developing pattern including initial stage, oscilla-tory intermediate, and finally steady-state condition. Implicit Finite Volume Method withTDMA solver is used to solve the governing equations. This study has been performed forthe Rayleigh numbers ranging from 100 to 10,000. Outcomes have been reported in terms ofisotherms, streamline, velocity and temperature plots and average Nusselt number for variousRa, corrugation frequency, and corrugation amplitude (CA). The effects of sudden differen-tial heating and its resultant transient behavior on fluid flow and heat transfer characteristicshave been shown for the range of governing parameters. The present results show that thetransient phenomena are enormously influenced by the variation of the Rayleigh Numberwith CA and frequency.

Keywords Natural convection · Transient behavior · Differentially heating ·Porous enclosure · Sinusoidal corrugation

A. Sojoudi (B)Department of Mechanical Engineering, College of Engineering, University of Tehran,P.O. Box: 11155-4563, Tehran, Irane-mail: meoiotu@yahoo.com; sojoudi@ut.ac.ir

S. C. Saha · Y. T. GuSchool of Chemistry, Physics and Mechanical Engineering, Queensland University of Technology,George St., GPO Box 2434, Brisbane, QLD 4001, Australia

M. KhezerlooDepartment of Mechanical Engineering, Tabriz University, Tabriz, Iran

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List of Symbols

CA Corrugation amplitudeCF Corrugation frequencyg Gravitational acceleration (m/s2)h Convective heat transfer coefficient (W/m2K)H Height of the enclosure (m)k Thermal conductivity (W/mK)K Permeability of porous mediaNu Average Nusselt number, Eq. (7)Pr Prandtl numberRa Rayleigh Numbert Dimensional time (s)T Temperature (K)Tc Temperature of the cold surface (K)Th Temperature of the hot surface (K)ΔT Temperature difference of cold and hot walls (K)u Velocity component in x, y-direction (m/s)U Dimensionless velocity component in X- direction, u H/αv Velocity component in y-direction (m/s)V Dimensionless velocity component in Y- direction, vH/αx, y Cartesian coordinates (m)X, Y dimensionless Cartesian coordinates, (x, y)/H

Greek Symbols

Γ dummy variableσ ratio of composite material heat capacity to convective fluid heat capacityτ dimensionless time, αt/H2

k Thermal conductivity of fluid (W/mk)α Thermal diffusivity (m2/s)β Coefficient of volumetric expansion (1/K)θs local dimensionless surface temperatureθ Dimensionless temperatureυ kinematic viscosity (m2/s)ψ Dimensionless stream function Dimensional stream function (m2/s)

Subscripts

h Cold temperature (K)c Hot temperature (K)s Surface

1 Introduction

Convective heat transfer in porous enclosures has attracted considerable concern of inves-tigators as the transport phenomenon in a fluid where the motion is driven merely by the

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interaction between density variation and the gravitational field. Water desalination devises,geothermal energy extraction, oil recovery, food processing, thermal insulation of build-ings, air conditioners filters, cosmetic applicators, petroleum resources, packed bed reac-tors of solar systems, sensible heat storage beds, dispersion of chemical contaminants invarious processes in the chemical industry, and many others are huge practical applica-tions of porous media. Due to large amount of applications, it requires detailed analysisof convective heat transfer in different industrial operations. Therefore, natural convectionappears inherently in several fields, where the heat to be dissipated is low enough and anattractive system in thermal control because of its low cost, reliability, and simplicity ofuse.

Due to the above applications, convective heat transfer in the rectangular/square saturatedporous enclosures has received extensive attention in past decades. A thorough review andextensive bibliography on natural convection in porous cavities can be found in Nield andBejan (2013), Nield (2012) and Ingham and Pop (2005) chipped into an extensive overview ofthis important area of heat transfer in porous media. There are many published studies relatedto natural convection in rectangular porous enclosures Baytas and Pop (1999), Moya et al.(1987), Bejan (1979), Saeid (2005), Lai and Kulacki (1988), Manole and Lage (1993), Varolet al. (2008a,b) are available in the literature. Bejan (2003) has outlined a complete reviewof scale analysis for convection heat transfer in porous media and revealed that the averageNuesselt number for the natural convection boundary layer near a vertical impermeable wallembedded in a saturated porous medium at a constant temperature is proportional to second-order root of wall Darcy Rayleigh number. He could anticipate the exact results within the23 % of those obtained using scale up analysis. Magyari et al. (2004) derived analytic solutionsfor two of the similarity cases identified by Johnson and Ping (1978) for the unsteady freeconvection boundary layer flow over an impermeable vertical flat plate adjacent to a fluidsaturated porous medium. They are the solutions corresponding to an exponential and a powerlaw variation of the surface temperature, respectively.

In numerous technical applications and industry, it is urgent to evaluate the natural convec-tion heat transfer that is time dependent, i.e., unsteady problem. As an earlier study, Pattersonand Imberger (1980) discussed the features of unsteady free convection in a air-filled cavitysubjected to sudden heating and cooling based on scaling analysis. A thermal boundary layerflow next to the sidewall, a horizontal intrusion flow beneath the ceiling or above the bottomand the flow in the core region were the transient phenomena and stages analyzed by them.The transient characteristics of natural convection Goldstein and Briggs (1964), Ivey (1984),Patterson and Armfield (1990), Schöpf and Patterson (1996) and Siegel (1958) have beenstudied in a cavity widely using different methods.

When a differentially heating condition is imposed to an enclosure where fluid inside theenclosure is at stationary, two separate fluid flows are developed near the side walls. Thesetwo flows diffuse into the core region. At the time when the flow approaches a steady orquasi-steady state, the inside fluid of the core region becomes stratified, as mentioned byEckertf and Carlson (1961), and a double-layer structure of the vertical thermal boundarylayer is developed, Xu et al. (2005). Studies of Schöpf and Patterson (1996) and Xu et al.(2005) showed the existence of a convective instability in the boundary layer flow. From theresults of Xu et al. (2005, 2008, 2010), during start-up to a steady state, the transition of theboundary layer adjacent to the sidewall was divided into three stages: (1) initial growth, (2)an entrainment development, and (3) steady stage. The early stage includes the growth ofthe boundary layer. At the entrainment development stage, horizontal intrusion is dominatedby the vertical boundary layer which lasts for a relatively long period. Finally, at the steadystage, a double layer structure of the vertical boundary layer is formed with stratified interior

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fluid at the center. This is due to the transient behavior of sudden heating in natural convectionof a cavity fill with air.

The available literature on the both steady and unsteady natural convection within airfilled and porous saturated enclosures is performed within simpler geometry such as therectangular, square, triangular etc. Some extensive investigations have been documented inDe Vahl Davis (1983), Varol et al. (2009), Campo et al. (1988), Saha and Khan (2011),Saha (2011a,b), Saha et al. (2010) and Varol (2010). For past decades, natural convection ofair in modified enclosures with complex configurations has also been researched. The mostdistinctive works of Morsi and Das (2003), Saha et al. (2007), Noorshahi et al. (1996), Yao(1983), Mahmud et al. (2002) and Hasan et al. (2008) and Hasan et al. (2009) on air naturalconvection in complex enclosures are available in the literature which describe the flow patternwithin various geometries. One of the famous side wall corrugation is sinusoidal edge whichhas been reported in some investigations for air natural convection Das and Mahmud (2003),Saidi et al. (1987), Wang and Vanka (1995), Nishimura et al. (1984), Kumari and Nath (2009),Asako and Faghri (1987) and Hasan et al. (2012). But most of the related studies are devotedto air-filled cavity and the study of saturated porous cavity of complex geometry is still rare.

Saeid and Pop (2004) studied transient-free convection in a two-dimensional square cavityfilled with a porous medium. Left and right vertical flat walls are suddenly heated to aconstant hot and cold temperatures. It is observed that the average Nusselt number indicatingan undershoot during the transient period and that the time needed to reach the steady-statestage is longer for low Rayleigh number and shorter for high Rayleigh number. The mainobjective of this work is to extend the work done in Saeid and Pop (2004) to modify the sidevertical walls where the corrugation of the walls is sinusoidal.

By the variation of corrugation amplitude (CA) and frequency, the region that remainsunchanged throughout the consequent transient process or the core region and adjacent to thewall are affected significantly. For a saturated porous media, there exists scarce informationof wavy surfaces.

The aim of the present study is to have a thorough analysis of transient natural convectionwith a range of Rayleigh numbers, CA, and corrugation frequency (CF) of sinusoidal sidewalls. The analysis is based on streamlines, isotherms, temperature plots, average Nusseltnumbers, traveling waves, temperature, and velocity.

2 Problem Formulation

Consider the flow of air within porous enclosure with height Has shown in Fig. 1 In the currentstudy, unsteady natural convection within a modified square porous saturated enclosure, withsinusoidal corrugated side walls, has been considered. The left and right walls are consideredto be differentially heated and the flat top and bottom walls are assumed as adiabatic. Theschematic diagram of the investigating enclosure has been shown in the Fig. 1. Rangesof investigating parameters for thermal and flow behavior and heat transfer features are102 ≤ Ra ≤ 104, 0.1 ≤ CA ≤ 0.5 and 1 ≤ CF ≤ 5.

The governing equations are obtained considering following assumptions: (a) The entireenclosure is filled with porous material completely which is isotropic in thermal conductivity,(b) Darcy’s law is applicable, (c) Normal Boussinesq incompressible fluid assumption isconsidered, and (d) Inertia effects are ignorable.

Under the above assumption, the non-dimensional governing equations in terms of thestream function (ψ) and temperature (θ ) are:

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Fig. 1 Physical model and coordinate system

∂2ψ

∂X2 + ∂2ψ

∂Y 2 = −Ra∂θ

∂X(2.1)

∂θ

∂τ+ ∂ψ

∂Y

∂θ

∂X− ∂ψ

∂X

∂θ

∂Y= ∂2θ

∂X2 + ∂2θ

∂Y 2 (2.2)

where dimensionless variables are defined by:

X = x

H, Y = y

H, τ = t

H2/α, U = u

α/H

V = v

α/H, ψ =

α, θ = T − TC

TH − TC, Ra = gβΔT H K

αυ(2.3)

The non-dimensional stream function, ψ , satisfies the following equations

U = ∂ψ

∂Y, V = −∂ψ

∂X(2.4)

The equation of vertical surfaces is:

y = CA sin

[C F

(2π

Hx

)](2.5)

Equations (2.1) and (2.2) are subject to the following boundary conditions:Bottom surface

U = 0, V = 0,∂θ

∂Y= 0 (2.6a)

Top surface

U = 0, V = 0,∂θ

∂Y= 0 (2.6b)

Left surface

U = 0, V = 0, θ = 1 (2.6c)

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Fig. 2 A sample grid showing the major features of the structured meshes

Right surface

U = 0, V = 0, θ = 0 (2.6d)

The working fluid has been chosen as Pr = 0.71. The Darcy Rayleigh number, Ra isvaried from 102 to 104. Flow and temperature fields are presented in terms of streamline andisotherm contours, respectively. The physical quantities of interest in this problem are theaverage Nusselt number along the hot wall, defined by

Nu = −H∫

0

hy

kdy = −

1∫0

y

H

(∂θs(Y )

∂X

)X=0.1

dY = −1∫

0

Y

(∂θs(Y )

∂X

)X=0.1

dY (2.7)

3 Numerical Modeling

The governing normalized coupled Eqs. of (2.1) and (2.2) with the boundary conditions ofEq. (2.6) were solved using Finite Volume Method with TDMA solver. The Poisson likemomentum Eq. (2.1) and the Energy Eq. (2.2) are discretised using the central difference, butthe time derivative is discretised employing the three points backward difference relationshipto ensure the second-order accuracy in both time and space. The computational domain hasbeen discretized with Nonuniform grids of rectangular elements with denser grids clusteringin regions near the heated and cooled walls (see Fig. 2).

The convergence of the solution is assumed when the relative error for each variablebetween the iterations is reported through the below convergence criterion:

∣∣∣∣�n+1 − �n

�n

∣∣∣∣ ≤ 10−6 (3.1)

To evaluate grid independence of the current solution, many numerical runs are carried outfor different Rayleigh numbers. These experiments suggest that a structured spaced grid of

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Table 1 Comparison of Nu atsteady state with previousnumerical results

Author Nu

Ra = 100 Ra = 1000 Ra = 10, 000

Bejan (1979) 4.200 15.800 50.800

Manole and Lage (1993) 3.118 13.637 48.117

Baytas and Pop (1993) 3.160 14.060 48.330

Saeid and Pop (2004) 3.002 13.762 43.953

Present study 3.105 13.019 43.948

0

10

20

30

40

50

60

70

0 0.01 0.02 0.03 0.04 0.05

Nu

Ra=1900, Aldabbagh et al (2007)

Ra=1900, Present Results

Ra=870.5, Present Results

Ra=870.5, Aldabbagh et al. (2007)

A=0.67

Fig. 3 Comparison of the transient behavior of the average Nusselt number at the hot wall for aspect ratio of0.67 with experimental results of Aldabbagh and Manesh (2007)

40000 elements is adequate to capture correctly the flow and heat transfer process inside theenclosure (see Fig. 2).

The numerical method used in present study was compared with the classical naturalconvection heat transfer problem in a differentially heated flat right and left square porousenclosure. The obtained numerical values of the average Nusselt number along the hot wallof the cavity at the steady-state flow for Ra = 102 to 104 for CA = 0 (flat vertical side walls)are compared with those given by different researchers as tabulated in Table 1. As can be seenfrom the table, the obtained result shows good agreement with the results of literature. Forfurther validation of the numerical solution, experimental results of Aldabbagh and Manesh(2007) were compared with present results of numerical method. The transient behavior of theaverage Nusselt number at the hot wall for aspect ratio of 0.67 is shown in Fig. 3 which showsa good agreement with experimental data. Therefore, these results provide great confidenceto the accuracy of the present numerical model.

4 Results and Discussion

The working fluid of the present study is assumed to be air with Pr being 0.71. The consideredvalues of Ra are 100, 1000, and 10000 Saeid and Pop (2004), values of CA are 0.1, 0.3, and

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0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

0.6

0.65

0.7

0.75

0.8

0.85

0.9

1

0.95

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

Initial Rising

Transitional Stage

Steady State

(a)

(b)

Fig. 4 Dimensionless temperature, θ , versus dimensionless time, τ , plot at point (0.01, 0.5) for a) Ra = 1000b) Ra = 100, 1000, and 10,000 at CA = 0.1 and CF = 1

0.5 and values of CF are 1, 3, and 5. To observe the effect of specified parameter, other valuesare kept constant to capture the related parameter influence.

This section discusses about the change in dimensionless temperature for a specified pointadjacent to the hot wall within the porous enclosure versus dimensionless time for various Ra.Figure 4a, illustrates the dimensionless temperature, θ , for a very close point to the hot leftwall, (0.01,0.5) at Ra = 1000 from instantaneous differential heating of the wall, τ = 0, to asteady-state condition. It can be concluded that after the sudden heating, the dimensionlesstemperature of the point, suddenly rises up to a transitional region. This region, initial rising,is due to the pure conduction of early development of the boundary layer. After rapid risingof the temperature, the intermediate transitional stage develops. The transitional stage isdominated by the formation of several overshoots and undershoots. Figure 4b shows that thetemperature at the considered point is increased for higher Ra. Moreover, it takes less timefor higher Ra to reach the steady-state condition. Dimensionless time, τ , is almost 0.026 forRa = 100 which is higher than others.

Figure 5 depicts the development of streamlines from the sudden left wall heating up tothe steady-state condition for Ra = 100,CF = 1, and CA = 0.1. It is seen that, at the earlystages of the dimensionless time, 4 separated cells are formed adjacent to the concave and

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τ = 0.0016 τ = 0.0032

τ = 0.0048 τ = 0.0064

τ = 0.008 τ = 0.0096

Fig. 5 Development of streamline within porous cavity at different dimensionless times, τ , for Ra =100,C A = 0.1, and CF = 1

convex walls. The heated layers intrude along the upper wall and the cold layers intrudethough the bottom wall. Then each two separated cells gets closer forming a single elongatedcell through the side walls. These two cells tend to travel to the enclosure core, making asingle cell within the cavity which occurs for higher Ra.

Thermal and flow behavior of the fluid in porous media inside the differentially heatedenclosure with sinusoidally corrugated side walls under steady-state condition are presentedin this section. The isotherms and streamlines analyzed in this section are obtained once thesystem assuages the stable and steady state.

The isotherm and streamlines plots for different Ra are indicated in Fig. 6. To observe theeffects of Ra variation, values of CA and CF are kept constant as 0.1 and 1, respectively, at

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Ra=100

Ra=1000

Ra=10,000

Fig. 6 Streamline (left) and isotherm (right) plot for different Rayleigh numbers at steady state for CA = 0.1and CF = 1

the steady state. It is seen that for Ra = 100, isotherms are approximately congested besidesthe side walls, and the intrusion at the top and bottom wall seems to be gentler. At the coreregion, the isotherms are curvy compared to the high Rayleigh numbers. As Ra increases,due to the enhanced buoyancy, the isotherms tend to be more packed and the intrusive wavesseem to have greater extend. The isotherms at the core region gradually become more andmore flat with sharp bends at both ends with increase of Ra. For the streamlines, it is observedthat increase in Ra results stronger flow and it is just as it supposed to be as Ra increases thebuoyancy effect. For All values of Ra, it is seen that the streamlines are comparatively denseat the adjacent area of the walls especially the sidewalls and the streamlines are less dense atthe core region.

Figure 7 illustrates the isotherms and streamlines of the different CA, CA = 0.1, 0.3,and 0.5 for constant Ra of 1000 and CF of 1. Values of Ra and CF are kept as 1000 and 1,respectively to observe the effect of CA. From this figure, it can be concluded that an increasein CA decreases the region that unaltered throughout the consequent transient process. Thereis a narrow section at the core region that remains somewhat unaffected by the resultanttransient phenomena, i.e., through the initial rise stage, transitional stage and at steady state

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CA=0.1

CA=0.3

CA=0.5

Fig. 7 Streamline (left) and isotherm (right) plot for different CA at steady state for Ra = 1000 and CF = 1

that narrow segment is more or less unaffected by the traveling waves and goes through notemperature change. Increase in CA results in reduction of that narrow section. It can easilybe understood from the isotherm plots that how the narrow segment at core region reducesfrom CA = 0.1 to CA = 0.3 and CA = 0.3 to CA = 0.5.

Figure 8 shows the effect of CF on the streamlines and isotherms of the flow within thesinusoidally corrugated porous enclosure. Increase in CF has less significant effects at the coreregion unlike the affects of CA. The thermal and flow patterns adjacent to the boundary aremainly affected by the variation of CF. As CF increases, the isotherms near the boundary seemto have a gradual inward shift. This is because, due to the increase in CF, the fluid adjacent tothe crested segment of the curved hotter or cooler side walls have lesser temperature gradient.This inward shift of isotherms also reduces the traveling distance and changes impact on theopposite boundary layers.

Temperature and velocity profiles at the mid section are presented in Fig. 9. The figureindicates the temperature and velocity Y = 0.5 for 0 ≤ X ≤ 0.5. The half of the distancehas the same and horizontally flipped profile for the other half.

Figure 9a shows the velocity profiles along the same as midsection half line. It is seenthat the velocity near the boundary is lower which gradually increases and reaches the peakvelocity which is not very far from the wall. This is obvious as no slip condition has beenassumed for all boundaries. After the peak velocity point, the velocity gradually descendsdownwardly to zero at the core region. As τ increases, such as for 0.001 ≤ τ ≤ 0.01,the pattern remains unchanged, but the magnitude of the maximum velocity increases withincrease of τ .

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CF=1

CF=3

CF=5

Fig. 8 Isotherm and streamline plots for different CA at steady state for Ra = 1000 and CA = 0.1

Figure 9b shows the development of thermal boundary along the same as for velocityprofiles. The temperature decreases gradually from the heated surface to the core region.For the early dimensionless time τ = 0.0001, there is a larger quiescent region and thetemperature behavior shows the pure conduction adjacent to the hot left wall. As τ increases,core region senses the conducted heat from hot wall. For the τ = 0.005, undershootingtemperature decay is seen. Finally, all thermal boundary layers experience steady temperaturereduction.

In this section, the heat transfer characteristics of heated wall are discussed. Figure 10shows the average Nusselt number, Nu plot for various cases for the analysis of heat transfercharacteristics. Figure 10a shows the change of Nu for 100 ≤ Ra ≤ 10, 000,CA = 0.1and CF = 1. Obviously, higher Ra generates larger temperature gradient at right and leftwalls resulting in higher heat transfer rate and Nu. At the early stage, Nu has larger amountdue to less improvement of boundary layer. As the τ increases Nu descends rapidly to anintermediate region. Then Nu assuages to a steady condition. There can be a beneficialcomparison of Table 1 and Fig. 10a. It can be concluded that corrugation of heated and cold

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-5

15

35

55

75

95

115

0 0.1 0.2 0.3 0.4 0.5 0.6

V

X

0

0.2

0.4

0.6

0.8

1

1.2

0 0.1 0.2 0.3 0.4 0.5 0.6

(a)

(b)

Fig. 9 a Dimensionless velocity and b dimensionless temperature along 0 ≤ X ≤ 0.5 at Y = 0.5 for differ-ent dimensionless times, τ for Rayleigh number, Ra = 103, corrugation amplitude, CA = 0.1, corrugationfrequency, CF = 1

walls is going to increase the average Nu. For higher Rayleigh number, Ra = 10, 000, 52.86%increased Nu is predicted where it is lower for Ra = 1000. Even though, it is seen that theamount of average Nu is reduced for Ra = 100 of corrugated walls.

Figure 10b shows the effect of CA on Nu. It is seen that higher CA results in larger Nu.Transient behavior is the same as discussed above. Figure 10c depicts effect of CF on averageNu. A slight increase in Nu is seen for higher frequency of corrugation. It can totally be saidthat the frequency did not affect the flow field at small CA = 0.1. Increment of frequencyfor above CF = 5 may not seem to be economic, because the rate of arising Nu is reducedfrom CF = 3 to CF = 5.

5 Conclusion

A numerical study has been performed on transient natural convection within a corrugatedporous cavity with differentially heated walls. Sinusoidal modification is employed for rightand left walls. The main objective of the present work was to observe the influence of suddenheating and transient behavior in terms of isotherms, streamlines, temperature and velocityplots, average Nusselt number, and various CA and CF. Main outcomes of the study are asfollows:

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Fig. 10 Average Nusselt number, Nu along the hot left wall versus dimensionless time, τ for a different Ra, atCA = 0.1 and CF = 1, b different CA, at Ra = 103 and CF = 1, c different CF, at Ra = 103 and CA = 0.1

• The numerical results reveal that for the range of Ra studied, the flow undergoes 3 distinctstages including initial stage, oscillatory intermediate, and finally steady stage.

• It takes less time for larger Ra to assuage a steady condition.• The increase in CA affects the flow pattern of the core region, whereas the increase in

CF affects the flow pattern at the boundary.

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• CF has no significant affects on flow pattern unlike the CA.• Increasing Ra enhances the heat transfer, presented by the increment of average Nusselt

number.• Enhancement of average Nu for higher CA is much more than the increase of CF.

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