**ABSTRACT**

A numerical analysis is performed to study the influence of the amplitude b of the sinusoidal undulation, of bottom wall, of an horizontal enclosure on the heat transfers by two-dimensional natural laminar convection. The hot wavy bottom wall and the cold straight top wall are kept isothermal at temperature equal, respectively to T

_{h}and T

_{c}. And the two vertical straight walls (right and left) are considered symmetrical. Results are presented in the form of local and global Nusselt numbers distribution for a selected range of Raleigh (10

^{3}-10

^{6}) and for four values (0.00, 0.15, 0.25, 0.35) of aspect ratios A = 2.b/a. Air of Pr = 0.71 in an enclosure of non dimensional length As = L/H equal to 4 is chosen as the flow model to examine the influences of the aspect ratios at various Raleigh Number. The natural convection equations are discretised, using an implicit finite difference method, based on the finite volume approach. The SIMPLE algorithm, assumes the linkage between velocities and pressure fields. The variation of the Raleigh Number has allows the obtaining of several types of flows and several forks between these flows. The flow in enclosure is characterized with recirculation zones over summits and in hollows, where the local Nusselt numbers is always minimal. The numerical results show that the flow and the heat transfer are strongly affected by the amplitude b. In a general the transfers developed within an enclosure by not flat topography are lower than those obtained in a horizontal and uniform enclosure of the same length.

PDF Abstract XML References Citation

####
**How to cite this article**

*Journal of Applied Sciences, 7: 334-341.*

**DOI:**10.3923/jas.2007.334.341

**URL:**https://scialert.net/abstract/?doi=jas.2007.334.341

**INTRODUCTION**

Natural convection **heat transfer** wall-fluid has many significant engineering applications experimentally and analytically; for example, double-wall thermal insulation, solar-collectors, electric machinery, cooling systems of micro-electronic devises, natural circulation in the atmosphere, etc. A considerable number of published articles are available that deal with flow characteristics, heat transfer, flow and **heat transfer** instability, design aspects, etc. On the other hand, studies dealing with convection problems inside more complex geometries have been rather limited. Complex geometry covers different types of geometries configurations, namely L-shaped cavities (Mahmud, 2002), trapezoïdal cavities (Peric, 1993), arc shaped enclosures (Chen and Cheng, 2002), wavy cavities (Mahmud *et al*., 2002), etc. The complex fluid dynamic behaviour as well as their geometrical complexity and its effect on flow phenomenon have motivates many researchers to perform experimental or numerical work on this topic. In present study (Mahmud *et al*., 2002), have shown the effect of surface waviness on natural convection **heat transfer** and fluid flow inside a vertical wavy walled enclosed for a range of wave ratio (0.00≤λ≤0.4) and aspect ratio (1.0≤A≤2.0). They observed that aspect ratio is the most important parameter for heat and fluid flows and higher **heat transfer** is obtained at a lower aspect ratio for a certain value of Grashof number. Das and Mahmud (2003) analyzed the free convection inside both the bottom and the ceiling, wavy and the isothermal enclosure. They indicated that, only at the lower Grashof number, the **heat transfer** rate rises when the amplitude wave length ratio changes from zero to other values. Mahmud and Islam (2003) solved the laminar free convection and entropy generation inside an inclined wavy enclosure using SIP (Strongly Implicit Procedure) solver on a non-staggered grid arrangement-They obtained that the inclination angle of cavity affects the entropy generation due to the **heat transfer** and fluid friction. Among them, Kumar (2000), presented the parametric results of flow and thermal fields inside a vertical wavy enclosure with porous media. Hadjadj and Kyal (1999) numerically examined the effect of sinusoidal protuberances on fluid flow and **heat transfer** inside an annular space using a non-orthogonal coordinate transformation. We have reported that both the local and average **heat transfer** increase with the increase of protuberance amplitude and Rayleigh number and decreasing Prandtl number. Kumar and Shalini (2003) investigated the effects of surface undulations on natural convection in porous enclosure with global cumulative heat flux boundary conditions for different undulation numbers and thermal stratification level. They indicated that the local Nusselt numbers are very sensitive to thermal stratification. Kumar *et al*. (1998) solved the free convection problem in an enclosure with bottom wave wall heated and cooled from the ceiling using **finite element** method. He indicated that separation zones starts around Ra = 50 in the single wave case and around Ra = 25 in the case with six waves per unit length. Murthy *et al*. (1997) analyzed the effect of the surface undulations on the natural convection **heat transfer** from an isothermal surface in a Darcian fluid-saturated porous enclosure by using the **finite element** method on a graded non-uniform mesh System. They found that the flow-driving Rayleigh number *Ra *together with geometrical parameters of wave amplitude, wave phase and the number of waves considered in the horizontal dimensions of the cavity influenced the flow and **heat transfer** process in the enclosure. Gao *et al*. (2000) solved the natural convection inside the wavy and inclined solar collector but they did not interest in flow behavior. Adjiout *et al*. (2002) performed and solved a similar problem with Gao and others, but in their case, the left vertical wall is flat and cold, whereas the right wall is wavy and hot. They indicated that the mean Nusselt number is decreased when it is compared with the square cavity. Dalal and Das (2005) made a numerical solution to investigate the inclined right wall wavy enclosure with spatially variable temperature boundary conditions. Oztop (2005) applied the elliptic grid generation to obtained sinusoidal duct geometry to enhance the forced convection heat transfer. Varol and Oztop (2006) investigated the effects of inclination angle on the laminar natural convection **heat transfer** and fluid flow in a wavy solar collector in steady state régime. They observed that the inclination angle is the most important and effective parameter on heal transfer which can be used to control the **heat transfer** inside the collector. Saidi *et al*. (1987) presented numerical and experimental results of flow over and **heat transfer** from, a sinusoidal cavity. They reported that the total heat exchange between the wavy wall of the cavity and the flowing fluid was reduced by the presence of vortex. The vortex plays the role of a thermal screen, which creates a large region of uniform temperature in the bottom of the cavity. Wang and Vanka (1995) presented **heat transfer** and flow characteristics inside a wavy walled charnel. Nishimura *et al*. (1984) investigated flow characteristics such as flow pattern, pressure drop and wall shear stress in a channel with symmetry; sinusoidal wavy wall. Asako and Faghri (1987) gave a finite-volume prediction of **heat transfer** and fluid flow characteristics inside a wavy walled duct and tube, respectively. Aydin *et al*. (1999), Eisherbiny (1996), Hamady *et al*. (1989) and Ozoe *et al*. (1975) presented results of **heat transfer** characteristics inside the rectangular enclosures at different aspect ratios and orientations without surface waviness.

The main purpose of the present study is to provide the effect of the amplitude of the sinusoidal oundulation, of bottom wall, of a shallow and horizontal wavy enclosure on the heat transfers by two-dimensional natural laminar convection natural. To the best of the authors' knowledge; free convection in the shallow wavy enclosure has not yet been investigated.

The rate of **heat transfer** in terms of local and global Nusselt numbers are presented for different aspect ratios and Rayleigh numbers; respectively. Flow and thermal fields are analyzed by parametric presentation of streamlines and isothermal lines.

**PROBLEM STATEMENT**

Consider a two-dimensional wavy wall horizontal enclosure of length L and height H as shown in Fig. 1. The hot wavy bottom wall and the cold straight top wall are kept isothermal at temperature equal respectively to T_{h} and T_{c} and the two vertical straight walls (right and left) are considered symmetrical. The fluid is assumed to be of constant properties and the Boussinesq approximation is employed for the gravity terms. Stress-work are neglected. In the present study, a, a^{2}/α and α/a are used, respectively, as the length, time and velocity scales, where α denotes the thermal diffusivity. The dimension less temperature function is defined as θ = (T-T_{r})/ΔT, in which ΔT = (T_{h}-T_{c}) stands for the temperature difference of the two isothermal walls and T_{r} = T_{c} is the fluid temperature at reference state.

The wavy wall function and governing equations in dimension less form can be cast into the following form respectively.

(1) |

Fig. 1: | Schematic representation of the computational geometry under consideration |

(2) |

(3) |

(4) |

(5) |

where A = (2.b/a)v is the aspect ratio of the enclosure; Ra = g.β.a^{3}.(T_{h}-T_{c})/v.α is the Rayleigh number, Pr = v/α is the Prandtl number, β is the thermal expansion coefficient and v is the kinematics viscosity. The two opposite walls (bottom and top) are kept at the uniform temperatures T_{h} and T_{c}, respectively. The boundary conditions for the system of Eq. 2-5 are

U = ∂V/∂X = ∂θ/∂X = 0; at two side walls

U = V 0, θ = 1; at hot wavy bottom wall

U = V = θ = 0; at cold straight top wall

**HEAT TRANSFER**

Heat transfer rate is measured by local (Nux) and average (Nu) Nusselt numbers. Following equations arc used to calculate the local and average Nusselt numbers:

(6) |

(7) |

Where the gradient term in (∂θ/∂N)_{w} Eq. (6) is the temperature gradient normal to the hot wavy wall.

**NUMERICAL PROCEDURE**

The system of Eq. (2-5) with the boundary conditions stated above is solved by using finite volume method and the S1MPLE algorithm, Patanker (1980). Since the flow fields for the parameter range considered lie in the steady flow regime. The value of Δτ = 10^{-3} is used through the course of the computation.

Fig. 2: | Streamlines (right) and isotherms (left) for Ra=3800, Pr = 5580 and As = 1, a) Present prediction b) Soong et al. (1996) prediction |

Fig. 3: | Streamlines (bottom) and isotherms (top) for Ra = 5000, Pr = 0.71 and As = 4, a) Present prediction b) Soong et al. (1996) prediction |

Fig. 4: | Streamlines (bottom) and isotherms (top) for Ra = 20000, Pr = 0.71 and As = 4, a) Present prediction b) Soong et al. (1996) prediction |

As. the maximum relative deviation of the mean Nusselt numbers between two successive time-steps is less than the value of 10^{-4} or the maximum value of the relative deviation of' the temperature less than 10^{-6} the procedure is regarded as converged.

To cheek the validity of the present numerical procedure, thermally driven flows in a square and rectangular cavity were solved.

For code validation; our numerical resultants (for aspect ratio of the enclosure A = 0) are compared with the resultants of Soong *et al*. (1996) solutions (for inclination angle γ = 0). Comparisons arc lines in Fig. 2-4. Present prediction shows a very good agreement with the result of Soong *et al*. (1996).

**ACCURACY**

In the present study, five combinations (30x113; 46x177; 62x241; 74x289 and 82x321) of control volumes are used to test the effect of grid size on the accuracy of the predicted results. Figure 5 shows the distribution of average Nusselt numbers of the hot wall as a function of grid sizes for four different Rayleigh numbers.

Fig. 5: | Variation of average Nusselt number as a Function of control volume for A = 0.25 and As = 4 |

It is clear from the figure that at lower Rayleigh number, average Nusselt number is almost independent of grid sizes. At higher Rayleigh numbers, the two higher grid sizes (74x289) and 82x321) give almost the same result (Fig. 5). It is well known that the high mesh refinement always provides better result in the finite-volume method. The main disadvantage in taking higher mesh number is the increase in calculation time, which can be reduced by using a higher speed of Pentium processor. Our goal was to get better results. Thus; throughout this study, the results are presented for 74x289 CVs' for better accuracy.

**RESULTS AND DISCUSSION**

**Flow and thermal field:** The flow pattern inside the wavy enclosure and the temperature profile are presented in terms of streamlines and isothermal lines in Fig. 6(a-d) for A = 0.25; As = 4 at four selected Rayleigh numbers. At low Rayleigh number when convection current inside the cavity is comparatively weak fluid stream near the hot wall tends to move towards the centerline or crest of the cavity were two streams from the opposite direction mix and rise upwards.

Two symmetric counter-rotating vortices are observed at top and bottom of the wave wall on the flow field due to the uniform temperature gradient. At low Rayleigh number Fig. 6(a, b) circulation inside the cavity is very weak. Convection is less prominent at this Rayleigh number and heat transfer is mainly dominated by conduction. Isothermal lines are nearly parallel to each other and follow the geometry of the wavy surfaces. A further increase of Rayleigh numbers increase the circulation strength inside the enclosure.

Fig. 6: | Streamlines (bottom) and isotherms (top) for different Rayleight number at A = 0.25; As = 4 and Pr = 0.71 |

Fig. 7: | Streamlines (bottom) and isotherms (top) for different aspect ratio at As = 4; Pr = 0.71 and Ra = 10^{5} |

Here, uniform temperature profile is changed and three high gradient spot is observed at top and bottom of the hot wall due to rapid circulation of fluid inside the cavity. Isotherms turn up (convective distortion) towards die cold wall due to the strong influence of the convection current.

Figure 7 shows the effect of aspect ratio (A) on flow field at constant Rayleigh number (Ra = 1.10^{5}). Whatever the value of A, flow inside the cavity is characterized by same number of cell multi-cellular pattern. An increase in the value of A decreases the size and the maximum values of vortices respectively.

Fig. 8: | Variation of local Nusselt number along the wave wall; for As = 4; Pr = 0.71 and for different aspect ratios at different Rayleigh number |

**Local heat transfer distribution:** Local Nusselt number distribution at the bottom wall is shown in Fig. 8(a-g) for As = 4 and A = 0.15; 0.25 and 0.35 at seven selected Rayleigh numbers. The profile of the distribution of the local Nusselt number present a symmetry with regard to hollows and a periodicity with regard to summits.

For the low values of the Rayleigh number (Ra≤10^{4}) the thermal exchange makes essentially by conduction, the distribution of the local Nusselt number possesses maximums (peaks) at the level of summits (Fig. 8a, b). For Ra>10^{4}, the thermal exchange by convection is dominant, the peaks observed at summits ease with the increase of the Rayleigh number (Fig. 8c-h).

**Average heat transfer distribution:** The effect of aspect ratio of the enclosure on average heat rate transfer is shown in Table 1.

Fig. 9: | Variation of average Nusselt number for As = 4; Pr = 0.71 and for different aspect ratios at different Rayleigh number |

Table 1: | Average Nusselt number and maximum values of ψ (ψ_{max}) at different Rayleigh number for different aspect ratios |

At lower Rayleigh number the effect of aspect ratio is significant. However, at higher Rayleigh number this effect is very small. At higher aspect ratio **heat transfer** rate is higher at lower Rayleigh number when aspect ratio increase from zero to other values and after then, further increases of aspect ratio shows a negligible effect on average **heat transfer** rate (Fig. 9).

**CONCLUSIONS**

Laminar steady natural convection **heat transfer** and fluid flow in shallow wavy enclosure are investigated numerically. The main conclusions that are drawn the present study are provided below.

• | Two circulation cells are obtained at the enclosure cavity of wave in different directions especially at the small aspect ratio. |

• | Flow and thermal fields are affected by geometrical parameters and Rayleigh number |

• | Heat transfer is increased with the increasing of aspect ratio. |

• | Local Nusselt numbers show wavy variation and maximum Nusselt number values are obtained on the top of the wave at low Rayleigh number (Ra = 1.10^{4}). For Ra>10^{4}, the peaks observed at summits ease with the increase of the Rayleigh number. |

**ACKNOWLEDGMENTS**

This study was realized in the Institute of Mechanical Engineering of Béchar University, in collaboration with the L.T.P.M.P Laboratory, Faculty of Mechanical and Processes Engineering, Houari Boumediene University of Sciences and Technology. Algiers, Algeria.

**NOMENCLATURE**

A | aspect ratio, A = 2.b/a (-) |

a | wave length of the wavy wall; (m) |

As | non dimensional length, As = L/h (-) |

b | amplitude height of the wavy wall; (m) |

g | gravitational acceleration, (m sec^{-2}) |

H | height of the enclosure; (m) |

h | heat transfer coefficient, (W m^{-2}.K) |

L | length of the enclosure, (m) |

Nu | mean Nusselt number |

Nux | local Nusselt number |

p, P | dimensional pressure and pressure (N m^{-2}) |

Pr | Prandtl number, Pr = v/α |

Ra | Rayieigh number, Ra = g.β (T_{p} - T_{o}).h^{3}/α.v |

t | time, (sec) |

u, v | dimensional velocity components, (m sec^{-1}) |

U, V | dimensionless velocity components |

x, y | cartesian coordinates; (m) |

X, Y | dimensionless coordinates |

**Greek symbols**

α | thermal diffusivity.(m^{2} sec) |

β | thermal expansion coefficient,(K^{-1}) |

λ | Thermal conductivity, (w.m.^{1} K.^{1}) |

v | Kinemtic viscosity, (m^{2} sec^{-1}) |

θ | Dimensionless temperature function θ = (T-T _{c})/(T_{h}-T_{c}) |

ρ | Density of the fluid(kg m^{-3}) |

Ψ | Dimensionless stream function |

τ | Dimensionless time, τ = α.t/a^{2} |

**Subscripts**

h | Hot |

c | Cold |

####
**REFERENCES**

- Adjiout, L., O. Imine, A. Azzi and M. Beikadi, 2002. Laminar natural convection in an inclined cavity with a wavy wall. Int. J. Heat Mass Transfer, 45: 2141-2152.

Direct Link - Aydin, O., A. Unal and T. Ayhan, 1999. A numerical study on buoyancy-driven flow in an inclined square enclosure heated and cooled on adjacent walls. Numer. Heat Transfer, Part A, Appl., 36: 585-899.

Direct Link - Chen, C.L. and C.H. Cheng, 2002. Buoyancy-induced flow and convective heat transfer in an inclined arc-shape enclosure. Int. J. Heat Fluid Flaw, 23: 823-830.

Direct Link - Dalal, A. and M.K. Das, 2005. Laminar natural convection in a inclined complicated cavity with spatially variable wall temperature. Int. J. Heat Mass Transfer, 48: 3833-3854.

Direct Link - Das, P.K. and S. Mahmud, 2003. Numerical investigation of natural convection inside a wavy enclosure. Int. J. Them. Sci., 42: 397-406.

Direct Link - Eisherbiny, S.M., 1996. Free convection in inclined air layers heated from above. Int. J. Heat Mass Transfer, 39: 3925-3930.

Direct Link - Gao, W., Lin and E. Lu, 2000. Numerical study on natural convection inside the channel between the flat-plate cover and sine-wave absorber of a cross-corrugated solar air heater. Energy Convers. Manage., 41: 145-151.

Direct Link - Hadjadj, A. and M.E. Kyal, 1999. Effect of two sinusoidal protuberances on natural convection in a vertical annulus. Numer. Heat Transfer, 36: 273-289.

Direct Link - Kumar, B.V.R., P.V.S.N. Munhy and P. Singh, 1998. Free convection heat transfer from an isothermal wavy surface in a porous enclosure. Int. J. Numer. Methods Fluids, 28: 633-661.

Direct Link - Kumar, B.V.R., 2000. A study of free convection induced by a vertical wavy surface with heat flux in a porous enclosure. Numer. Heat Transfer, Part A, Appl., 37: 493-510.

Direct Link - Kumar, B.V.R. and Shalini, 2003. Natural convection in a thermally stratified wavy vertical porous enclosure. Numer. Heat Transfer Applied, 43: 753-776.

Direct Link - Mahmud, S., 2002. Free convection inside an L-shaped enclosure. Int. Comm. Heat Muss Transfer, 29: 1005-1013.

Direct Link - Mahmud, S., P.K. Das, N. Hyder and A.K.M.S. Islam, 2002. Free convection in an enclosure with vertical wavy walls. Int. J. Thermal. Sci., 41: 440-446.

Direct Link - Mahmud, S. and A.K.M.S. Islam, 2003. Laminar free convection and entropy generation inside an inclined wavy enclosure. Int. J. Them. Sci., 42: 1003-1012.

Direct Link - Murthy, P.V.S.N., B.V.R. Kumar and P. Singh, 1997. Natural convection heat transfer from a horizontal wavy surface in a porous enclosure. Numer. Heat Transfer Applied, 31: 207-221.

Direct Link - Oztop, H.F., 2005. Numerical study of flow and heat transfer in curvilinear ducts: Applications of elliptic grid generation. Applied Math. Comp., 168: 1449-1460.

Direct Link - Saidi, C., F. Legay and B. Pruent, 1987. Laminar flow past a sinusoidal cavity. Int. J. Heat Mass Transfer, 30: 649-661.

Direct Link - Soong, C.Y., P.Y. Tzeng D.C. Chiang and T.S. Sheu, 1996. Numerical study on mode-transition of natural convection in differentially heated inclined enclosures. Int. J. Heat Mass Transfer, 39: 2869-3882.

Direct Link - Wang, G. and S.P. Vanka, 1995. Convective heat transfer in periodic wavy passages. Int. J. Heat Mass Transfer, 38: 3219-3230.

Direct Link