Arab J Sci Eng (2014) 39:9187–9204 DOI 10.1007/s13369-014-1422-y

RESEARCH ARTICLE - MECHANICAL ENGINEERING

Numerical Investigation of Mixed Convection Heat Transfer from Block Mounted on a Cavity P. Rajesh Kanna · V. Anbumalar · M. Krishnakumar · A. Ramakrishnan · S. Allwyn Pushparaj · A. V. Santhosh Kumar

Received: 5 March 2014 / Accepted: 31 August 2014 / Published online: 14 November 2014 © King Fahd University of Petroleum and Minerals 2014

Abstract This work aims to numerically investigate the mixed convective flow over a three-dimensional cavity that lies under the horizontal channel with a vertical heat source projecting from the bottom of the cavity. This investigation is progressed by establishing the boundary conditions of the walls of vertical heat source as isothermal, while backward and the forward facing steps as isothermal at ambient temperatures and the rest of the walls are adiabatic. The flow is laminar and incompressible in the cubic geometry. A numerical simulation is undertaken to investigate the flow structure, heat transfer characteristics and the complex interaction between the induced stream flow at ambient temperature and buoyancy-induced flow from the walls of the heated source. The numerical simulations are performed using commercial CFD codes. The numerical codes are validated with previously published benchmark results. The study is performed for three different Reynolds numbers (100, 500 and 1,000) and subsequently for three different Richardson numbers (0.1, 1 and 10) for each Reynolds number. The height of the vertical heat source is kept constant for all the abovementioned cases. A special case is undertaken to study the P. Rajesh Kanna (B) · V. Anbumalar Velammal College of Engineering and Technology, Madurai, India e-mail: [email protected] M. Krishnakumar Department of Mechanical Engineering, Kalasalingam University, Krishnankoil, Tamil Nadu, India A. Ramakrishnan Department of Mechanical Engineering, Bannari Amman Institute of Technology, Sathyamangalam, 638 401, India S. Allwyn Pushparaj · A. V. Santhosh Kumar Department of Mechanical Engineering, Velammal College of engineering and Technology, Madurai, 625009 Tamilnadu, India

flow characteristics and heat transfer for increased depth of the cavity. Keywords Cavity flow · Heated block · Richardson number · Recirculation

List of symbols g Gravity (m/s2 ) Gr Grashof number (gβ H 3 (TH − TC )/ν 2 ) H Height of the channel (m) H Depth of the cavity (m) L Width of the cavity (m) Exit length of the channel (m) Le n Normal vector to the surface Nu Nusselt number (N u = −(∂θ/∂n)s) Num Average Nusselt number P Pressure (N/m2 )

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P Pr Re Ri t T TC TH u, v, w u ∗ , v ∗ , w∗ x, y X, Y

Dimensionless pressure ( p/ρu 2o ) Prandtl number (ν/α) Reynolds number (u oH /ν) Richardson number (Ri = Gr/Re2 ) Time (s) Temperature (K) Inlet flow temperature (K) Heat source temperature (K) Component of velocity (m/s) Dimensionless form of velocity Coordinates Dimensionless coordinates (x/H, y/H )

Greek symbols α Heat diffusivity coefficient (m2 /s) β Thermal expansion coefficient (K−1 ) θ Dimensionless temp. (T − Tc )/(TH − TC ) θfm Average dimensionless temperature μ Fluid viscosity (m2 /s) ν Kinematic fluid viscosity (m2 /s) ρ Density (kg/m3 ) ψ Stream function τ Dimensionless time (u i /H ) Subscripts max Maximum min Minimum

1 Introduction Mixed convection and heat transfer over open cavities have been the subject of extensive research for many years due to their importance in various engineering systems for say, cooling systems in electronic components, industrial processing, and lubrication systems, drying technology and chemical processing units. The study may also assist engineers in designing fluid flow systems involving a cavity as considered in this paper. The study may also assist engineers in designing fluid flow systems involving a cavity with characteristics as considered in this paper. The presence of such type of geometry is interesting because of the convective heat transfer that occurs between the cavity and the forced flow stream of air, i.e., the mainstream. Therefore, a qualitative characterization interaction between natural convection and forced convection is highly important in system design. Selfsustained oscillations inside the cavity generate intense pressure fluctuations that can lead to structural damage and failure of components. Many recent studies have been conducted on open cavities involving thorough investigation of its interaction with mainstream as they are significant in understanding the fluid transport performances. A better understanding of

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the relationship between the cavity and its mainstream may aid in improved fluid–fluid or particle–fluid mixing in fluid systems say, a food processing unit. The added thermal influence by including heated wall within the cavity adds greater value to the study. This paper may also be helpful in studying the Lagrangian coherent structures (LCS). Lagrangian coherent structures are structures which separate dynamically distinct regions in time-varying systems such as fluid flow in mechanics. These actualities necessitate a detailed study to understand the basic mechanisms of fluid flow under such conditions.

2 Literature Review Numerous experimental and numerical studies of flow past open cavity have been reported for forced, natural and mixed convections. Stiriba et al. [1] have numerically studied the flow past open cavity under laminar and incompressible conditions and reported the temperature distribution, flow structures, velocity profiles and heat transfer characteristics from a heated wall. A parametric study was also performed for Grashof number from 103 to 106 and two Reynolds numbers, Re = 100 and 1,000, to investigate the flow structure and the heat transfer. The thermal and flow fields were presented and analyzed in terms of velocity fields, streamlines and temperature distribution. They found that for both high Reynolds and Richardson numbers, the natural convection comes into play and push the recirculating zone further upstream. Stiriba [2] provided additional results by changing the heat source, and it has been reported that for high Richardson number, the mixed convection effects come into play and push the recirculation zone further upstream and the flow may become unstable. Aminossadati and Ghasemib [3] investigated the mixed convection heat transfer in a two-dimensional horizontal channel with an open cavity with a discrete heat source located on one of the walls of the cavity. Three different heating modes were considered viz. the location of the heat source on three different walls (left, right and bottom) of the cavity. An analysis was carried out for a range of Richardson numbers and cavity aspect ratios with the results showing noticeable changes in the different heating modes. Koca [4] has studied the conjugate heat transfer in partially open square cavity with a vertical heat source. The cavity considered has an opening on the top with varying lengths and at three different positions, while the other walls of cavity were assumed as adiabatic. It was found that the ventilation position has a significant effect on heat transfer. Mesalhy et al. [5] have numerically and experimentally studied the flow over a shallow cavity heated with constant heat flux from the bottom side. Boetcher and Sparrow [6] investigated the buoyancyinduced flow in a horizontal open-ended cavity. The investigation was made in three perspectives: (a) assessment of the

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Fig. 1 Schematic of the problem

Fig. 2 a u velocity component, Re = 100. b y velocity component, Re = 100

validity of an existent similarity solution, (b) computational issues relevant to numerical simulation and (c) obtainment and presentation of results of practical utility. Local Nusselt numbers along the walls are reported. Sivakumar et al. [7] have analyzed the mixed convection heat transfer and fluid flow in lid-driven cavities with different lengths of the heating portion and with different locations of it. Results were presented graphically in the form of streamlines, isotherms and velocity profiles. It was concluded that the heat transfer rate enhances when the heating portion was reduced and when the portion is at middle or top of the hot wall of the cavity. Manza et al. [8] numerically investigated the mixed steady-state convection in a two-dimensional horizontal open cavity using a Galerkin finite element method. They obtained results by uniformly heating the three walls separately for Richardson number from 0.1 to 100, for Reynolds number 100–1,000 and for aspect ratios 0.1–1.5. In [9,10], studies were made to analyze the transport process occurring due to the interaction of air streams with buoyancyinduced flow within rectangular enclosures with constant source of heat for different flow parameters. Manca et al. [11]

have investigated experimentally the mixed convection in open cavities with heated inflow wall for Richardson numbers varying from 30 to 110 and for Re = 1,000 and 2,800– 8,700. The flow visualization reveals that for Re = 100, there are two nearly distinct fluid motion such as a parallel flow in the channel and a recirculation flow inside the cavity. Wong and Saeid [12] numerically investigated the opposing mixed convection arising from jet impingement which was intended to cool the heated bottom surface of an open cavity in a horizontal channel filled with porous medium. Pop et al. [13] examined the natural convection in a partially opened enclosure filled with porous media using the Brinkman–Forchheimer model. They found that as the Grashof number increases, due to strengthening buoyancydriven flows, the local Nusselt number from partially heated vertical wall at a given position on this wall also increases. Hinojosa et al. [14] presented numerical results of heat transfer calculations in an open cavity considering natural convection and temperature-dependent fluid properties. The study shows that the thickness of the hydrodynamic boundary

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Fig. 3 Validation of velocity profile Table 1 Summary of values used for simulation Variables

Values simulated

Properties of air

Re

100, 500 and 1,000

ρ = 1.225 kg/m3

Ri

0.1, 1.0 and 10.0

ν = 1.7894e−05 kg/m s

D/L ratio

1, 2

Inlet temperature = 300 K

Fig. 4 a Three-dimensional flow structures, b two-dimensional flow structures at z central plane

Fig. 5 a Three-dimensional flow structures, b two-dimensional flow structures at z central plane

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layer in the vicinity of the hot wall increases as ε (dimensionless temperature difference) increases from 0.03 to 1.6. Nayak et al. [15] have numerically analyzed natural convection in open cavities of three different geometries viz. cubical, spherical and hemispherical. The effect of opening ratios (d/D or a/H ) is also studied. In addition to that, the effect of inclination of the cavity with the natural convection is studied for five inclinations. Xu and Pop [16] investigated fully developed mixed convection flow in a vertical channel filled with nanofluids. The analysis showed that the analytical solution for the opposing flow is only valid for a certain region of the Rayleigh number Ra in physical sense. Besides that, the effects of the nanoparticle volume fraction φ on the temperature and the velocity distributions are studied and their relations were obtained. They have confirmed that the nanoparticle volume fraction φ plays a key role in improving the heat and mass transfer characteristics of the fluids. OvandoChacon et al. [17] published his numerical study on the effect of heater length on the heating of a solid circular obstruction centered in an open cavity in which he observed four types of vortex formations and the increase in obstruction temperature with low Prandtl number with large heater sizes. Khanafer and Aithal [18] studied laminar mixed convection flow and heat transfer characteristics in a grid based on Galerkin method and have obtained optimal heat transfer

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Fig. 6 a Three-dimensional flow structures, b two-dimensional flow structures at z central plane

Fig. 7 a Three-dimensional flow structures, b two-dimensional flow structures at z central plane

Fig. 8 a Three-dimensional flow structures, b two-dimensional flow structures at z central plane

Fig. 9 a Three-dimensional flow structures, b two-dimensional flow structures at z central plane

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Fig. 10 a Three-dimensional flow structures, b two-dimensional flow structures at z central plane

Fig. 11 a Three-dimensional flow structures, b two-dimensional flow structures at z central plane

Fig. 12 a Three-dimensional flow structures, b two-dimensional flow structures at z central plane

results. Results were observed for both adiabatic and isothermal boundary conditions, imposed on the cylinder, which showed that the presence of the cylinder results in an increase in the average Nusselt number compared with the case having no cylinder. The average Nusselt number increases with increase in the Richardson number for all non-dimensional radius of the cylinder considered in this work. Moreover, the optimal heat transfer results were obtained when plac-

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ing the cylinder near the bottom wall for various Richardson numbers. Fontanaa et al. [19] studied three-dimensional analysis of natural convection in a partially open cavity in which he observed four types of vortex formations. He analyzed the work with different Prandtl numbers to consider the effect of the thermal and momentum diffusion properties of the fluid. Kalte et al. [20] published a numerical solution of nanofluid mixed convection heat transfer in a lid-driven

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Fig. 13 a Re = 100 and Ri = 1. b Re = 500 and Ri = 1. c Re = 1,000 and Ri = 1. d Re = 100 and Ri = 0.1

square cavity with a triangular heat source in which he discussed the effects of the volume fraction, nanoparticle diameter and nanoparticle types on the flow and Nusselt number. He also found that suspending the nanoparticles in pure fluid leads to a significant heat transfer increase. Ghasemi and Aminossadati [21] carried out numerical investigation of mixed convection from a rectangle enclosure for discrete heater arrangement. They found that at any Rayleigh number, placing less heat sources on the bottom wall results in a higher Nusselt number and a better cooling performance. Saeid [22] investigated natural convection from porous medium presented in two-dimensional cavity by oscillating wall temperature. When the hot wall temperature oscillates with high amplitude and frequency, it is found that the Nusselt number becomes negative over part of the period for Rayleigh number equals to 103 . This is due to that there will be no enough time to transfer the heat from the hot wall to the cold wall. The two-dimensional version of heat transfer received considerable attention in the past. But, very few results have been obtained for the three-dimensional case which provides motivation for the present study.

3 Mathematical Formulation and Numerical Procedure The three-dimensional domain consists of a three-walled cavity with a closed channel enclosing it on the top thus creating an open cavity domain as shown in Fig. 1. The cavity is of length L and of depth D. The considered cubic cavity is similar to the geometry used in Stiriba et al. [1,2] albeit the vertical heated wall. The transfer length of the bottom phase of the geometry is also of length L. The height of the channel is L/4. A vertical heat source of thickness L/5 projects from the bottom phase to a height of L/2. The projection is perpendicular to the main stream. The height of the projection is kept constant throughout the entire problem. The vertical heat source is kept at constant temperature TH . The immediate backward facing step and the forward facing step walls are isothermal and are maintained at ambient temperature TC < TH . The remaining walls are adiabatic. The incoming fluid is assumed to be at temperature TC . The cooling fluid is considered as air which has Prandtl number (Pr) of 0.71. The flow is considered to be laminar and incompressible. The fluid properties are assumed to be constant in the bouncy terms according to Boussinesq approximation.

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Fig. 14 a Re = 100 and Ri = 1. b Re = 500 and Ri = 1. c Re = 1,000 and Ri = 1. d Re = 100 and Ri = 0.1

Fig. 15 a Temperature contour at different x planes. b Temperature isosurfaces. c Two-dimensional temperature contour at z central plane. d Two-dimensional temperature contour at 25 % of total z distance

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Fig. 16 a Temperature contour at different x planes. b Temperature isosurfaces. c Two-dimensional temperature contour at z central plane. d Two-dimensional temperature contour at 25 % of total z distance

Based on the above-mentioned assumptions, the governing continuity, momentum and energy equations for 3D laminar incompressible fluid can be expressed in dimensional form are as follows. Continuity equation ∂v ∂w ∂u + + =0 ∂x ∂y ∂z

Z-momentum equation   2 ∂w ∂w 1 ∂p ∂ w ∂ 2w ∂ 2w ∂w +v +w =− +ν + + 2 u ∂x ∂y ∂z ρ ∂z ∂x2 ∂ y2 ∂z (4)

(1) Energy equation

X-momentum equation 

u

∂u ∂u ∂u 1 ∂p ∂ 2u ∂ 2u ∂ 2u +v +w =− +ν + + ∂x ∂y ∂z ρ ∂x ∂x2 ∂ y2 ∂z 2

u

(2)

x∗ =

Y-momentum equation

θ= 

u

  2 ∂T ∂T ∂ T ∂2T ∂2T ∂T +v +w =α + + ∂x ∂y ∂z ∂x2 ∂ y2 ∂z 2





∂v ∂v 1 ∂p ∂ 2v ∂ 2v ∂ 2v ∂v +v +w =− +ν + + ∂x ∂y ∂z ρ ∂y ∂x2 ∂ y2 ∂z 2 +βg (T − TO ) (3)

U0 t y ∗ y z , y = , z∗ = ; t = , L L L L T − T∞ TH − T∞

p∗ =

p − p∞ , ρ∞ U02 (5)

By applying the above dimensionless variables, the governing equations from Eqs. 1 to 5 are reduced as follows,

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Fig. 17 a Temperature contour at different x planes. b Temperature isosurfaces. c Two-dimensional temperature contour at z central plane. d Two-dimensional temperature contour at 25 % of total z distance

Continuity equation

z-momentum equation

∂u ∗ ∂u ∗ ∂u ∗ + + =0 ∂x∗ ∂ y∗ ∂z ∗

u∗ (6)

x-momentum equation

∗ ∗ ∂w ∗ ∗ ∂w ∗ ∂w + v + w ∂x∗ ∂ y∗ ∂z ∗   2 ∗ 1 ∂ w ∂p ∂ 2 w∗ ∂ 2 w∗ + =− + + ∂z Re ∂ x ∗2 ∂ y ∗2 ∂z ∗2

(9)

Energy equation ∂u ∗ ∂u ∗ ∂u ∗ + v ∗ ∗ + w∗ ∗ ∗ ∂x ∂y ∂z   2 ∗ 1 ∂ u ∂p ∂ 2u∗ ∂ 2u∗ + =− + + ∂x Re ∂ x ∗2 ∂ y ∗2 ∂z ∗2

∂θ ∂θ ∂θ 1 u +v ∗ ∗ +w ∗ ∗ = ∂x∗ ∂y ∂z Re Pr

u∗

∗

(7)



 ∂ 2θ ∂ 2θ ∂ 2θ + + ∂ x ∗2 ∂ y ∗2 ∂z ∗2 (10)

where p∞ and ρ∞ are reference pressure and density, respectively, TH is the temperature at the heated surface, u i is the is the velocity component, p is the pressure, Re = U oH ν

y-momentum equation

C )L Reynolds number, Gr = gβ(THν−T is the Grashof num2 ber. Here, β, ν and α are the coefficient of thermal expansion, kinematic viscosity and thermal diffusivity, respectively, and properties of air are used for simulation. At the inlet, the flow has uniform velocity U0 in the stream direction. At exit, 3

u∗

∂v ∗

∂v ∗

+ w∗ ∗ ∂ y∗ ∂z   2 ∗ ∗ Gr ∂p 1 ∂ v ∂ 2 v∗ ∂ 2 v∗ =− ∗ + + ∗2 + ∗2 + θ ∗2 ∂y Re ∂ x ∂y ∂z Re2 ∂x∗

+ v∗

∂v ∗

123

(8)

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Fig. 18 a Temperature contour at different x planes. b Temperature isosurfaces. c Two-dimensional temperature contour at z central plane. d Two-dimensional temperature contour at 25 % of total z distance

outflow condition is set. No slip condition is used for the remaining walls.

4 Code Validation and Grid Independence Study In the present work, the above-mentioned physical geometry mesh is created using Gambit software and is processed using fluent CFD package. The code was validated for several benchmark problems like the lid-driven cavity and the L-shaped cavity. Oosterlee et al. [23] reported benchmark results on skewed cavity problems by using FVM method using nonlinear multigrid algorithm. They solved L-shaped cavity for laminar range. In the present validation, we solved their L-shaped cavity problem for Reynolds number equals to 100 and it is compared for geometry mid-plane u and v velocity components and found good agreement with them. Figure 2 shows the comparison of the obtained results with that of [23]. Further to validate the mixed convection mode of heat transfer in open cavity, the numerical results of Stiriba

et al. [1] are verified. They presented numerical results for three-dimensional mixed convection from a cubical cavity for various Richardson number. Their results are simulated for Ri = 10 [1]. The comparison in Fig. 3 for Ri = 10 shows excellent agreement with the present results. The convective terms that appear in the governing equations are discretized by second-order upwind scheme, and diffusion terms are discretized by central difference scheme. The coupled nonlinear governing velocity–temperature equations were solved by SIMPLE algorithm by using fluent CFD package. Constant time step of 0.001 is used for the entire computation to reach the steady-state results. The convergence is checked for the mass residue to reach <10−6 . A grid independent study is carried out similar to Stiriba et al. [1] to check the grid independence of the solution. Three nonuniform different grid sizes (56 × 56 × 56, 81 × 81 × 81 and 101 × 101 × 101) are tested. The average Nusselt number variation for the grids 81 × 81 × 81 with 101 × 101 × 101 is <1.5 %. Therefore, to carry out the entire computations in the present study, 81 × 81 × 81 grid system is chosen.

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Fig. 19 a Temperature contour at different x planes. b Temperature isosurface. c Two-dimensional temperature contour at z central plane. d Two-dimensional temperature contour at 25 % of total z distance

5 Results and Discussion In a regular cavity flow with no blocks, a vortex forms in the cavity as the result of shear force produced by its contact with the fluid flow in channel above the cavity. The eye of this vortex tends to move from the geometric center of cavity at low Reynolds’ numbers toward the edge where the downstream channel and one of the cavities’ wall meet at higher Reynolds’ numbers. In our case, the same tendency exists with vortex formed in primary cavity. This vortex executes a shear force on the fluid present in secondary cavities which results in vortex formations in these secondary cavities as well. However, the vortex characteristics in secondary cavities are not identical with each other though their geometry is symmetric. Its unbalanced nature is found to be influenced predominantly by primary vortex and by the depth of primary cavity, i.e., aspect ratio. The computed thermal fields and flow fields are analyzed in terms of flow structures, temperature distribution for various cases of Reynolds numbers (100, 500 and 1,000) and Richardson numbers (0.1, 1 and 10) for which the flow is

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expected to be in the laminar regime. Heat transfer studies are taken into account in order to study the mixed convection flow. Table 1 shows the summary of parameters considered in this study for simulation. Results presented for streamline contours, isotherms and velocity components from various planes. 5.1 Flow Characteristics Study The impact of mixed convection in flow pattern, recirculating flow structure and streamlines for various cases of Re and Ri numbers is shown in Figs. 4, 5, 6, 7, 8, 9, 10, 11 and 12, while comparison of velocity components is shown in Figs. 13 and 14. The analysis with the cavity of aspect ratio 1 was done for Re 100, 500 and 1,000 with respect to Ri = 0.1, 1 and 10. The Reynold’s number affects the vortex formations in primary cavity which transfers its variations onto the vortices in secondary cavity. The primary vortex center rises diagonally toward the exit as the Reynold’s number increases from Re 100 to Re 1,000. This shift in the primary vortex center, with

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Fig. 20 a Temperature contour at different x planes. b Temperature isosurfaces. c Two-dimensional temperature contour at z central plane. d Two-dimensional temperature contour at 25 % of total z distance

increase in Reynold’s number, increases shear force acting on secondary vortices, thereby increasing the magnitude of secondary vortex. However, Figs. 4, 5, 6, 7, 8, 9, 10, 11 and 12 that suggest the alteration in left and right secondary vortices with Reynold’s numbers are not uniform. The intensity of right secondary vortex is higher than left secondary vortex at higher Reynolds’ numbers. This difference is found to decrease with increase in aspect ratio, and from Fig. 24, it is evident that the secondary vortices are identical at aspect ratio 2. With close observation on the right secondary cavity, we can understand the significance of Reynolds’ number being predominant in the right secondary cavity than in left secondary cavity. This is due to the fact that, as the Reynolds’ number increases, the primary vortex shifts toward right. The primary vortex, which is a clockwise rotation of fluid, pushes the fluid into the right secondary vortex at first instance. This creates a more intense right secondary vortex. The right secondary vortex, now fully grown, redirects the fluid from primary vortex and slides it toward the left secondary vortex. As

the fluid enters horizontally in negative X direction to the left secondary vortex rather than having a direct vertical entrance as it occurs to the right secondary vortex, the intensity of left secondary vortex is less on comparison with right secondary vortex, and as previously discussed, this variation decreases with increase in aspect ratio as Fig. 24 suggests.

5.2 Temperature Profile and Heat Transfer The simulated temperature distributions along different slices of × plane and the two-dimensional temperature contours at various temperature isosurfaces for all the cases of Re and Ri numbers are shown in Figs. 15, 16, 17, 18, 19, 20, 21, 22 and 23. Figure 24 shows temperature contours as well as streamline contours for aspect ratio equals to 2. The average Nusselt number is calculated at both the left and the right cold walls for all the above-mentioned cases. Table 2 represents the Nusselt numbers for both the backward and forward facing step walls.

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Fig. 21 a Temperature contour at different x planes. b Temperature isosurfaces. c Two-dimensional temperature contour at z central plane. d Two-dimensional temperature contour at 25 % of total z distance

The sides of the heated block is supposed to be cooled by secondary vortexes, whereas the top surface by primary vortex. As discussed, the flow characteristics of fluid in left and right secondary cavities vary. This variation in flow physics results in less identical cooling rates in the sides of the heated blocks. The cooling rate is higher in right side of block when compared with that of the left side. However, this discrepancy decreases with reduced Reynolds’ number and increased depth of primary cavity, which when perceived carefully, actually manipulates the position of eye of the primary vortex. At lower Reynolds’ numbers, the secondary vortices are equal in magnitude, and hence, the same cooling rates at both sides of the block are observed. At higher Reynolds’ numbers, as the right secondary vortex increases in magnitude, more than that of the left secondary vortex, the cooling rate at the right wall of the block becomes apparently higher when compared with the cooling rate of left block wall. As previously discussed, this variation in cooling rates

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at left and right block walls decreases with increase in aspect ratio as evident from Fig. 24 of cavity with aspect ratio 2. Table 2 shows the average Nusselt number for various cases investigated. For lower Richardson value, the mean Nusselt number shows down trend when Re is increased from both left and right walls. However, corresponding behavior for higher Richardson in reverse trend is observed. The temperature contours are non-dimensionalized to show a comparative study. Figure 24a–c shows the temperature distribution of geometry with D/L ratio 2. Figure 24b, d shows the streamline flow structures for D/L ratio 2. It is found that at higher aspect ratio, it brings symmetry vortices.

6 Conclusion Mixed convective heat transfer in an open cavity with a heated block is investigated using fluent CFD package. The

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Fig. 22 a Temperature contour at different x planes. b Temperature isosurfaces. c Two-dimensional temperature contour at z central plane. d Two-dimensional temperature contour at 25 % of total z distance

computed thermal fields and flow fields are analyzed in terms of flow structures, temperature distribution for various cases of Reynolds numbers (100, 500 and 1,000) and Richardson numbers(0.1, 1 and 10) for which the flow is expected to be in the laminar regime. Heat transfer studies are taken into account in order to study the mixed convection flow. Development and comparison of velocity components, temperature profile and heat transfer in the geometry along with the special case involving spacing ratios are studied.

It is found that as the Reynold’s number increases, primary vortex center rises diagonally toward the exit. The changes in left and right secondary vortices with Reynold’s numbers are not uniform. The intensity of right secondary vortex increase with increasing Reynold’s numbers; meanwhile, it is found to decrease with increase in aspect ratio. As the flow characteristic in the right and left cavities differs, the heat transfer in right cavity becomes greater than the other. This phenomenon is observed at higher Reynold’s number and at higher aspect ratios.

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Fig. 23 a Temperature contour at different x planes. b Temperature isosurfaces. c Two-dimensional temperature contour at z central plane. d Two-dimensional temperature contour at 25 % of total z distance

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Fig. 24 a Temperature contour for Re = 100 and Ri = 1 at z central plane. b Flow structures for Re = 100 and Ri = 1 at z central plane. c Temperature contour for Re = 100 and Ri = 1 at 25 % of total z distance. d Flow structures for Re = 100 and Ri = 1 at 25 % of total z distance Table 2 Average Nusselt numbers at the walls

Ri

Re = 100 Left wall

Re = 500 Right wall

Left wall

Re = 1,000 Right wall

Left wall

Right wall

Ri = 0.1

2.54

1.84

0.35

0.16

0.45

Ri = 1

2.16

1.45

4.17

5.21

4.64

2.08

Ri = 10

22.45

17.34

41.62

14.52

46.55

18.26

Acknowledgments Authors are gratefully acknowledging the anonymous reviewers to help us to improve the manuscript in the present form.

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2. Stiriba, Y.: Analysis of the flow and heat transfer characteristics for assisting incompressible laminar flow past an open cavity. Int. Commun. Heat Mass Transf. 35, 901–907 (2008) 3. Aminossadati, S.M.; Ghasemib, B.: A numerical study of mixed convection in a horizontal channel with a discrete heat source in an open cavity. Eur. J. Mech. B Fluids 28, 590–598 (1984) 4. Koca, A.: Numerical analysis of conjugate heat transfer in a partially open square cavity with a vertical heat source. Int. Commun. Heat Mass Transf. 35, 1385–1395 (2008) 5. Mesalhy, O.M.; Aziz, S.S.A.; El-Sayed, M.M.: Flow and heat transfer over shallow cavities. Int. J. Therm. Sci. 49, 514–521 (2010)

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