1 Introduction

The investigation of fluid flow and heat transfer has been a focal point of research for the past few decades. Mostly for the purpose of various engineering applications like the design of solar collectors, energy transfer in rooms and buildings, electronic cooling, heat exchangers, boilers, furnace and nuclear equipment due to which the study of combined natural convection and radiation is requisite. However, with improved designs and equipment, enclosures with partitions have become quite popular among researchers. Nagaraja et al [1] in their research work performed a numerical simulation of natural convection inside a square cavity with partitions. The partitions were placed vertically protruding from the horizontal end walls and the height of the partitions was also changed for the study. In their analysis they concluded that the overall heat transfer rate decreases with an increase in partition height and also due to decrease in length of location of partition. Almakhyoul et al [2] studied the effect of staggered non-conductive partitions on natural convection heat transfer in three different shaped cavities rectangular, trapezoidal, and triangular and they performed the numerical simulation using finite difference technique to solve continuity, momentum and energy equations. Similar to the observations of Nagaraja et al and Almakhyoul et al concluded that the increase in the length of the partition decreases the rate of heat transfer and Nusselt number. Another important observation made by them was that for the same length of partition the Nusselt number decreases in the order from rectangular to trapezoidal to triangular.

Parmananda et al [3] performed a numerical computation of combined natural convection and radiation in a cubical enclosure in order to study the effect of partition placed inside the enclosure. The study was carried out by placing the partition at four different positions. Natural convection in a square enclosure with partition having a fluid-saturated porous medium with internal heat generation was studied by Selamat and Ishak Hashim [4]. They considered two thin partitions for the analysis which were fully insulated. As such the effects of the position of the partitions on natural convection was studied. Sathiyamoorthy and Chamka [5] used the finite element method to study the effect of a very thin partition attached at the bottom wall of a square cavity having free convection within the cavity. The square cavity considered was heated from the bottom, had linearly heated side walls and the top wall been adiabatic and the analysis was done to study the effect of thin partition on Rayleigh–Benard instability. Two types of partitions were considered for the analysis, (i) highly conducting and (ii) insulating; with different lengths and positions. Suvash and Gu [6] carried out a numerical simulation for Natural Convection heat transfer in an isosceles triangular enclosure in which a partitioned vertical wall was placed at the centre of the enclosure and the thermal conductivity of the partition was considered to be infinite. In order to study the transient behaviour of the air flow and heat transfer, it was observed by the researchers that the coupled thermal boundary layer development around the partition undergoes several stages like an initial stage, a transition stage, and a steady stage. Yamaguchi et al [7] used a finite volume method to carry out a numerical simulation of natural convection heat transfer inside a three-dimensional enclosure and to study the effect of partition wall on natural convection in a vertical air layer. The partitions considered for the analysis in this research work were of finite thickness and of finite thermal conductivity. Various boundary conditions were considered for the analysis such as conduction, adiabatic and no thickness partitions and the results for all of the different boundary conditions were compared and were presented in the form of overall heat transfer and average Nusselt number.

Bilgen [8] carried out a numerical simulation to study the effect of partial partition in a two-dimensional enclosure. Various geometrical parameters like the aspect ratio, partition position and partition height were varied for the analysis. The researchers concluded that the rate of heat transfer decreases (i) when two partitions were used instead of one, (ii) when the aspect ratio was made smaller and (iii) when the position of partition wall was farther away from hot wall. Khalifa and Khudheyer [9] carried out an experimental investigation in a cavity having an aspect ratio of 0.5 to study natural convection and to study the effect of 14 different arrangements of partition inside the cavity. Air was considered as the working fluid to perform the experiment and the Rayleigh number ranged from 6 × 107 to 1.5 × 108. However, from the previous studies on natural convection inside a differentially heated partitioned enclosures, it can be seen that the radiation effect is often neglected. This could result in over simplification of the considered problem and can lead to a substantial inaccuracy in the results. Ramesh et al [10] used an optical device namely the differential interferometer (DI) to study the effect of partition thermal resistance and also the effect surface radiation and natural convection on total heat transfer between vertical walls of the partition enclosure in an experimental procedure. In this paper, the researchers evaluated the value of average Nusselt number for two different partition materials i.e., pure copper and Teflon for increasing values of Rayleigh number. Karki et al [11] numerically investigated the effect of partition placed parallel to the vertical isothermal walls in a cubic enclosure having free convection. The partitions along with the horizontal walls were assumed to be adiabatic. The width of the opening in the partition was kept constant while the height was varied from 0.25 to 0.75 of the enclosure heights. The results were obtained in the form of average Nusselt number and were also compared with the results obtained from cavities having similar arrangement in two-dimension. The observation made was that the average Nusselt numbers for three-dimensional enclosures are 15–66 percent smaller than those for two-dimensional enclosures with the same relative opening height.

Ciofalo and Karayiannis et al [12] did a numerical and experimental study of natural convection in a rectangular enclosure and presented in their research work, the effect of partitions projecting centrally from the end walls of vertical cavity on heat transfer rates. In their research work they analysed the effects of different thermal boundary conditions at the end walls and at the partition in a non-partitioned and partitioned enclosure. Khatamifar et al [13] studied the conjugate natural convection heat transfer in a square enclosure with a partition of finite thickness and conductivity in the transient state over a Rayleigh number range of 103 to 108. In this research work they observed that the effect of thermal conductivity ratio on heat transfer and fluid flow is insignificant when the thermal conductivity ratio is very large (100 or more) and is effective only when the ratio is between 0.1 and 10. They also concluded that the effect of partition thickness is only significant when the conductivity ratio is small. Nansteel and Grelf [14] studied the phenomenon of natural convection in a two-dimensional rectangular enclosure first without a partition and then with a partition. The partition was projecting vertically from the top adiabatic horizontal wall. The investigation was done experimentally with water being taken as the working fluid. Dye-injection flow visualization and thermocouple probes were used to observe the effect of the partial vertical divisions on the fluid flow and temperature fields respectively. The results in the form of Nusselt number for different Rayleigh number and partition length were presented for both conducting and non-conducting partition materials. Kahveci et al [15] used polynomial based differential quadrature (PDQ) method to study the effect of heat conducting vertical partition placed in an enclosure for natural convection. The results were presented in the form of streamlines, isotherm contours and Nusselt number.

Therefore, the present study deals with the combined effects of laminar natural convection and radiation in a square cavity with partitions of different stature and positions.

1.1 Governing equations

The physics inside the enclosure is governed by the following set of equations (Bejan [16]):

Continuity Equation:

$$ \frac{\partial u}{{\partial x}} + \frac{\partial \nu }{{\partial y}} = 0 $$
(1)

x—Momentum Equation:

$$ \rho \left( {u\frac{ \partial u}{{\partial x}} + \nu \frac{\partial u}{{\partial y}}} \right) = - \frac{\partial p}{{\partial x}} + \mu \left[ {\frac{{\partial^{2} u}}{{\partial x^{2} }} + \frac{{\partial^{2} u}}{{\partial y^{2} }}} \right] $$
(2)

y—Momentum Equation:

$$ \rho \left( {u\frac{\partial v}{{\partial x}} + \nu \frac{\partial \nu }{{\partial y}}} \right) = - \frac{\partial p}{{\partial y}} + \mu \left[ {\frac{{\partial^{2} v}}{{\partial x^{2} }} + \frac{{\partial^{2} \nu }}{{\partial y^{2} }}} \right] + \rho \beta g\Delta T $$
(3)

Energy Equation:

$$ \rho C_{p} \left( {u\frac{\partial T}{{\partial x}} + \nu \frac{\partial T}{{\partial y}}} \right) = K \left[ {\frac{{\partial^{2} T}}{{\partial x^{2} }} + \frac{{\partial^{2} T}}{{\partial y^{2} }}} \right] $$
(4)

1.2 Coupling between Natural Convection and Surface Radiation

The computation domain is enclosed by wall and inbuilt with partitions and therefore, multiple radiation reflections take place from all the surfaces. To evaluate the respective wall temperatures from the radiation model considered in analysis, the local values of respective heat transfer coefficients and fluid temperature are to be obtained from the flow calculations. The coefficient of heat transfer h is evaluated as kf/Δy. The wall surfaces are considered as gray, opaque, and diffuse. Based on these assumptions, the generalized energy balance equation for all the walls is derived as follows (Siegel and Howell [17]):

$$ h_{1} \left( {T_{1} - T_{w} } \right) + h _{2} \left( {T_{2} - T_{w} } \right) + \varepsilon_{1} I_{1} + \varepsilon_{2} I_{2} = \left( {\varepsilon_{1} + \varepsilon_{2} } \right)\sigma T_{w}^{4} $$
(5)

where T1 and T2 are local values of fluid temperature, Tw is wall temperature, h is heat transfer coefficient, ε is emissivity of the wall surface, and I is the irradiation falling on the wall surface. The subscripts 2 and 1 refer to conditions on inner side and outer side of the enclosure. For the isothermal hot wall, h1 is set to a large value with T1 = Th, and ε1 = 0. Similarly, for the isothermal cold wall, h1 is set to a large value with T1 = Tc and ε1 = 0. For these walls the enclosure side heat transfer coefficient, h2 = k/Δxw. For the adiabatic walls, h1 = ε1 = 0 and h2 = k/Δyw. In these, Δxw and Δyw are the distances of the first fluid node from the wall. The irradiation is related to radiosity (B),

$$ B_{i} = \varepsilon_{i} \sigma T_{wi}^{4} + \left( {1 - \varepsilon_{i} } \right) I_{i} $$
(6)

where the subscript i is the index of the elemental segment forming the enclosure. There are N (equal to Nx×Ny) elemental segments in the enclosure, where Nx and Ny are the number of elements in the x and y directions. Substituting for I in terms of B and using the inverse relationship,

$$ I_{i} = \mathop \sum \limits_{j = 1}^{N} B_{ j } F_{ i - j} $$
(7)

and

$$ \mathop \sum \limits_{j = 1}^{N} [\frac{{\delta_{i,j } {-} \left( {1 - \varepsilon_{i } } \right) F_{i - j} }}{{\varepsilon_{i} }}] B_{j} = \sigma T_{i}^{4} $$
(8)

where, Fi-j is shape factor from segment i to segment j.

2 Physical model and problem description

Figure 1 shows the geometry of the physical model schematically. The length of the square cavity is taken as L with a vertical partition placed parallel to the hot and cold walls. The thickness of the partition is considered to be equal to one twentieth of the width of the square enclosure, t = L/20. The distance of the partition from the hot wall is varied as SP L/4, L/2 and 3L/4 as designated by C(a), C(b) and C(c) respectively. Similarly, the height of the partition is changed as in h = L, 3L/4, L/2 and L/4 as shown in figure 1 by C(1), C(2), C(3) and C(4) respectively. The left vertical wall of the square cavity is considered at higher temperature Th and right wall at lower temperature Tc and the top and bottom horizontal walls are assumed to be adiabatic. The temperature difference between the hot and cold wall is taken as 20K. The fluid considered for the analysis is air having a Prandtl number of 0.7. Air is non-emitting and non-absorbing under normal temperature conditions which in turn make it a radiatively non-participating medium. The cavity walls and partitioned surfaces are assumed to be opaque, gray and diffuse emitters and reflectors of radiation with an emissivity ε. The gravity g acts downward normal to the x direction. Certain assumptions are made for the fluid flow for the numerical simulation such as the fluid flow is considered to be two-dimensional, incompressible, steady and laminar. The thermos-physical properties of the fluid in the flow model are assumed to be constant except for the density variation causing a body force term in the momentum equation. The Boussinesq approximation is assumed to be valid. Also, it is assumed that there is no slip wall surface.

Figure 1
figure 1

Computational domain.

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2.1 Solution methodology

The numerical simulation for the present study is done using ANSYS FLUENT 2022R1. The continuity, momentum and energy equations are solved using finite volume method. The pressure velocity coupling is carried out by Coupled algorithm and Second order scheme is used for pressure interpolation. Second Order Upwind is adopted for momentum and energy equation, while First order Upwind is used for radiation equation. Surface radiation is invoked on the walls of the enclosure. The shape factor is evaluated using Hottel’s cross-string method. The value of surface emissivity for all the considered cases is taken as one, i.e., ε = 1.

3 Grid independence test

The Grid Sensitivity Analysis has been carried out for the case 1b (h = L; SP = L/2). Five grid sizes have been chosen for the analysis, namely 50 × 50, 100 × 100, 200 × 200, 400 × 400 and 800 × 800 for Rayleigh number 106. From Table 1 it can be observed that the changes between the results for the grid sizes is very nominal, as such 100 × 100 grid size mesh has been chosen for the analysis.

Table 1 Grid sensitivity study for different grid sizes for Rayleigh number 106.
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4 Results and discussion

In order to enumerate the combined effect of natural convection and radiation in a differentially heated partitioned square cavity, the results are demonstrated with the help of streamline, temperature contours and the total average Nusselt number for the considered range of Rayleigh number i.e., 103–106. The effect of emissivity, Rayleigh number, partition height and its distance from the hot wall are studied extensively. For the purpose of this investigation the partition thickness and thermal conductivity are kept constant. Four different cases 1, 2, 3 and 4 are considered in the present study with their subdivisions as a, b and c taken as shown in figure 1.

For all the cases the boundary conditions of the enclosure remain same i.e., left vertical wall is the hot wall, right wall is the cold wall and the top and bottom walls of the enclosure are adiabatic as described in previous section. In case 1 the height of the partition is taken as h = L. For case 2 the height of the partition is considered as h = 3L/4 maintaining a gap of L/4 between the bottom horizontal wall and the partition. For case 3 two partitions are considered, one protruding from the top horizontal wall and the other from the bottom horizontal wall. The height of the partition attached to the top wall is L/2 while the height of the partition attached to the bottom wall is L/4 keeping a gap between them as L/4. Case 4 corresponds to the inverse situation where the height of the partition attached to top wall as L/4 and height of partition attached to bottom wall as L/2 maintaining a gap of L/4 between the top and bottom partition. In all the cases the subdivisions a, b and c correspond to the situations where the partition distances from the hot wall are L/4, L/2 and 3L/4, respectively.

For the present study, the experimental results obtained by Bajorek [18] on natural convection in partitioned enclosures have been validated. From the graph as shown in figure 2 it can be seen that results of present study confirm with the experimental results of Bajorek.

Figure 2
figure 2

Graph of comparison between experimental results of Bajorek [18] with the numerical simulation results for present study for validation.

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Table 2 demonstrates the comprehensive validation results of average Nusselt number for natural convection for present study which are being compared by the results of Khatamifar et al [13] and Kahveci et al [15]. Foregoing validation results show that the Nusselt number obtained for Rayleigh number 103–106 agree.

Table 2 Comparison of hot wall average Nusselt number for different Rayleigh number range for partitioned square cavity.
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4.1 Pure natural convection

To study the effect of pure natural convection in a partitioned cavity, a particular situation is selected where the Rayleigh number is considered as 104, with different cases 1, 2, 3 and 4 with their sub-categories as a, b and c. Figure 3 represents the streamlines and temperature contours for all the above-mentioned cases. In figure 3a, d, g and j for case 1a, 2a, 3a and 4a, respectively when the partition is near the hot vertical wall, it can be observed from the streamline that the flow field is present at the right side of the partition. The fluid near the right bottom end of the partition rises to the top end then comes in contact with the top horizontal adiabatic wall which then flows in the downward direction along the cooled right vertical wall. Again, fluid moves upward as soon as it comes in contact with right side of the partition which results in the clockwise rotation of the fluid flow. It is also evident from the isotherm contours that are parallel to hot and cold walls and also near the partition however they form smooth curves at the right side of the partition as we move away from the partition and cold walls. Moreover, from the graph in figure 4 it is also clear that the value of average Nusselt number for case 1a, 2a, 3a and 4a for hot wall is close to 1.4.

Figure 3
figure 3

Temperature and Streamline Contours for natural convection for Rayleigh number 104 for all cases of partition height and distance from hot wall.

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Figure 4
figure 4

Variation of Average Nusselt number for natural convection with partition distance from hot wall for all cases of partition height for hot wall for Rayleigh number 104.

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For cases 1b, 2b, 3b and 4b in figure 3b, e, h and k respectively when the partition is present at the centre it is seen from the streamline contours that the fluid in contact with the hot wall rises up from the bottom end then comes in contact with the partition wall which deflects it in the downward direction forming clockwise rotation of fluid. Similar situation occurs on the right side of the partition, which results in flow field on both the sides of the partition wall and only a small interaction occurs between these two fluid flow parts. The temperature contours also clearly depict that the contours are nearly parallel to the hot, cold and partition walls. Similarly, from the graph in figure 4 it is also clear that the value of average Nusselt number for case 1b, 2b, 3b and 4b for hot wall is close to 1.1. The streamline contours for cases 1c, 2c, 3c and 4c in figure 3c, f, i and l when the partition is close to the cold wall shows that the flow field is present on the left side of the partition wall. The fluid present near the bottom end of hot wall rises up along the hot wall to the top end and then moves downward as soon as it comes in contact with the left side of the partition wall. Again, moves upward as it comes in contact to the bottom end of hot wall forming clockwise fluid flow rotation. Also, the temperature contours are parallel to the vertical walls while forming smooth curves at the left side of the partition wall. Likewise, from figure 4 the graph clearly shows that the value of average Nusselt number for cases 1c, 2c, 3c and 4c is nearly equal to 1.4.

Figure 4 shows that the value of Nusselt number for natural convection lies between 1.1 to 1.4, least at the centre when the partition of height h = L is placed at the centre and most when the partitioned is positioned farthest from the hot wall i.e., SP = 3L/4 and the height of upper partition is L/4.

4.2 Combined radiation and natural convection

The combined natural convection and radiation phenomenon within a differentially heated square cavity with partition is presented for Rayleigh number ranging from 103 to 106, Pr = 0.7 and emissivity ε = 1 with help of temperature, streamlines contours and Nusselt number. With the inclusion of surface radiation, the nature of the flow and temperature distribution gets distorted leading to a change in heat transfer phenomenon which can also be observed from figures 9, 10, 11 and 12 where the graph shows that the value of total Nusselt number is more than double that in pure convection.

The temperature and streamline contours for all the considered cases of partition height and distance from hot wall for Rayleigh number 103 is shown in figure 5. From the streamline contours for all the Rayleigh number, it is observed that the flow field is present on the right side of the partition having clockwise rotation of fluid flow for case 1a as can be observed from figures 5, 6, 7 and 8. For Ra = 103 in figure 5a the fluid near the right bottom end of the partition rises up to the top end then comes in contact with the top horizontal adiabatic wall which then flows in the downward direction along the cooled right vertical wall. Again, fluid moves upward as soon as it comes in contact with right side of the partition which results in the clockwise rotation of the fluid flow. It is also evident from the isotherm contours that are nearly parallel to the vertical walls in figure 5a. Even for Ra = 104 in figure 6a the flow field is present on the right side of the partition having a clockwise rotation. Also, the temperature contours are parallel very close to the hot, cold and partition walls and starts forming smooth curve at the centre of the cavity. The flow fields for Ra = 105 as shown in figure 7a shows similar behaviour as for Ra = 103 and 104 as shown in the streamline contours. However, for Ra 106 weak flow field is also present on the left side of the partition along with the flow field on the right side of the partition. The isotherm contours start deforming for Ra = 105 in figure 7a and nearly becomes parallel to the horizontal walls for Ra 106 in figure 8a of the enclosure at the right side of the partition.

Figure 5
figure 5

Streamlines and Temperature Contours for Rayleigh Number 103.

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Figure 6
figure 6

Streamlines and Temperature Contours for Rayleigh Number 104.

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Figure 7
figure 7

Streamlines and Temperature Contours for Rayleigh Number 105.

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Figure 8
figure 8

Streamlines and Temperature Contours for Rayleigh Number 106.

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The temperature and Streamline contours for Case 1b (h = L and SP = L/2) for considered range of Rayleigh number i.e., 103–106 is presented in figures 5–8b respectively. From the streamline contours for Rayleigh number 103 to 106 it is observed that there are two flow fields formed separately on both the sides of the partition. On the left side of the partition the fluid rises from the bottom end of the hot wall, then comes in contact with the top horizontal adiabatic wall and then flows downward along the left side of the partition. Likewise, on the right side of the partition the fluid moves downward along the cold wall from the top end to the bottom end, then comes in contact with the bottom horizontal wall and further moves upward along the right side of the partition. This results in the clockwise rotation of fluid flow on both the sides of the partition. The isotherm contours for Rayleigh number 103 and 104 are parallel to the vertical sides of the cavity as displayed in figures 5–6b respectively. For Ra 105 in figure 7b the contours get distorted near the bottom end and top end of the hot and cold wall and also near the partition wall. As the circulation becomes stronger at the extreme ends when Ra = 106 in figure 8b it is observed that the isotherm contours get distorted which results in the central portion of contours becoming horizontal on both the sides of the partition. Figures 5–8c show the temperature and streamline contours for h = L, SP = 3L/4 for the considered range of Rayleigh number i.e., 103–106 which is exactly similar to that of case 1a as can be referred from figures 5–8a. The only difference being the flow field present on the left side of the partition. Also, the isotherm contours show impeccable similarity with the temperature contours for case 1a.

The streamline and temperature contours for case 2a when the height of the partition is 3L/4 and the distance of partition is L/4 for Rayleigh number 103, 104, 105 and 106 are shown in figures 5–8d. For Ra = 103 and 104 as shown in figures 5–6d the flow fields are present only on the right side of the partition. The fluid rises near the bottom end of the partition, moves along the top horizontal adiabatic wall, gets deflected towards the right vertical cold wall and then again moves along the right side of the partition thus causing clockwise fluid rotation. It is also evident from the isotherm contours for Rayleigh number 103 that are parallel to the vertical sides of the cavity. From figure 6d even for Ra = 104 the contours are parallel very close to the hot, cold and partition walls and starts forming smooth curve at the centre of the cavity. However, for Ra 105 from the streamline contours in figure 7d it is seen that weak currents are also present on the left side of the partition. As the Ra number reaches 106 the circulation currents become stronger resulting in flow fields on both the sides of the partition as shown in figure 8d. For Ra 105 the temperature contours get distorted near the bottom end of the partition. As the circulation becomes stronger at the extreme ends when Ra = 106 it is observed that the isotherm contours get completely distorted which results in the central portion of contours becoming horizontal.

In case 2b when the partition is present at the centre with height being 3L/4, various observations are made from the streamline and isotherm contours as shown in figures 5–8e. For Ra 103 and 104 the fluid rises from the bottom end of the hot wall and moves over it reaching to the top end as shown in figure 5e and figure 6e, respectively. As the partition is projected from the top wall of the enclosure it deflects the fluid motion in the downward direction and again the fluid moves up as soon as it meets the bottom end of the hot wall. Similarly, on the right side of the partition the cold fluid moving along the cold wall in the downward direction, comes in contact with the bottom horizontal adiabatic wall and then moves up along the right side of the partition forming rolls with clockwise rotation on both the sides. Thus, two separate flow fields are formed on both the sides of the partition. It is also clear from the isotherm contours which are nearly parallel to the hot and cold wall for Ra = 103 and 104. Also, the total Nusselt number is greater than when radiation is neglected. For Ra 105 as shown in figure 7e the fluid moves along the hot surface in the upward direction, after coming in contact with the partition moves in the downward direction, then through the gap provided moves in the upward direction along the right side of the partition then flows over the top horizontal adiabatic wall and again after coming in contact with the cold wall moves in the downward direction hence causing a clockwise fluid flow motion inside the cavity. Similar flow fields form for Ra 106 but in this case the flow fields are much stronger as shown in figure 8e. For Ra 105 the temperature contours get distorted near the bottom end of the partition. As the circulations becomes stronger at the extreme ends when Ra = 106 it is observed that the isotherm contours get completely distorted which results in the central portion of contours becoming horizontal on right and left side of the partition respectively as shown in figure 8e.

Figures 5–8f show the streamline and temperature contours for case 2c when the height and distance of partition from hot wall is same i.e., 3 L/4 for Rayleigh number 103, 104, 105 and 106. From the contours a close resemblance is observed for case 2a and 2c, the only difference is the flow field is present at the left side of the partition for Ra 103 and 104 as shown in figure 5 and figure 6f respectively. It is also evident from the isotherm contours for Rayleigh number 103 that are parallel to the vertical sides of the cavity. Even for Ra = 104 the contours are parallel very close to the hot, cold and partition walls and starts forming smooth curve at the centre of the cavity. However, for Ra 105 from the streamline contours in figure 7f it is seen that weak currents are also present on the right side of the partition. As the Ra number reaches 106 the circulation currents become stronger resulting in flow fields on both the sides of the partition as shown in figure 8f. For Ra 105 the temperature contours get distorted near the bottom end of the partition. As the circulation becomes stronger at the extreme ends when Ra = 106 it is observed that the isotherm contours get completely distorted which results in the central portion of contours becoming horizontal on right and left side of the partition respectively.

The streamline and temperature contours for case 3a is shown in figures 5–8g where 2 partitions are considered, one attached to the top horizontal wall and the other to the bottom horizontal wall maintaining a gap of L/4 between them. The height of partition attached to top wall is L/2 while that attached to bottom wall is L/4. The distance between the hot wall and partition is taken as L/4. For Ra 103 as in figure 5g it is observed from the streamline contours that the fluid rises near the corner of the right side of the partition attached to the bottom horizontal wall then flows over the partition attached to the top horizontal wall, then after flowing over the top horizontal wall moves along the cold wall in the downward direction resulting in the clockwise rotation of fluid flow. Similar flow fields are observed in figure 6g for Ra 104. It is also clear from the isotherm contours for Ra 103 which are exactly parallel to the vertical wall. Though, for Ra 104 the isotherms are parallel only near the vertical walls and form smooth curves at the centre of the cavity. In case of Ra 105 in figure 7g it is seen from the streamline contours that the fluid moves along the hot surface in the upward direction, after coming in contact with the left side of the partition attached to top horizontal wall moves in the downward direction, then through the gap provided moves in the upward direction along the right side of the partition, then flows over the top adiabatic horizontal wall and again after coming in contact with the cold wall moves in the downward direction. Hence, after which the fluid flows over the bottom horizontal wall then flows over the partition attached to bottom wall after which moving over the partition flow over the hot wall in the upward direction resulting in the clockwise rotation of fluid flow. However, the flow field on the left side of the partition are very weak for Ra 105 when compared to the flow field for Ra 106. It is also clear from the isotherm contours for Ra 105 the contour gets distorted on the right side of the partition and for 106 the contour gets distorted and becomes parallel to the horizontal walls thus causing thermal stratification on the right side of the partition.

The streamlines and temperature contours for case 3b for the considered range of Rayleigh number i.e., 103–106 respectively is shown in figures 5–8h. The setup is like case 3a except for the partition distance from hot wall which in this case is taken as L/2. For Ra 103 and 104 two flow fields are present on both the side of partitions having a clockwise rotation of fluid flow as referred from figures 5 and 6h respectively. It is also clear from the isotherm contours for Ra 103 which are exactly parallel to the vertical wall. However, for Ra 104 the isotherms are parallel only near the vertical walls and form smooth curves at the centre of the cavity. Unlike for Ra 103 and 104 the flow fields on the right and left side of the partition for Ra 105 and 106 as demonstrated in figure 7h and figure 8h communicates through the gap between the top and bottom partitions. It is also clear from the isotherm for Ra 105 the temperature contours get distorted near the bottom end of the partition. As the circulations becomes stronger at the extreme ends when Ra = 106 it is observed that the isotherm contours get completely distorted which results in the central portion of contours becoming horizontal on right and left side of the partition respectively as shown in figure 8h. For Rayleigh number 103, 104, 105 and 106, figures 5–8i show the streamlines and temperatures contours for case 3c where the arrangement of partitions is similar to that in case 3a, the only difference being the distance of partition from hot wall which in this case is 3 L/4. For Ra 103 from the figure 5i it is observed from the streamline contours that the fluid rises from the bottom end of hot wall to the top end, then flows over the top adiabatic horizontal wall then after flows along both the partitions in the downward direction resulting in the clockwise rotation of fluid flow. Similar flow fields are observed for Ra 104. It is also evident from the isotherm contours for Ra 103 which are exactly parallel to the vertical wall. However, for Ra 104 the isotherms are parallel only near the vertical walls and form smooth curves at the centre of the cavity as shown in figure 6i. In case of Ra 105, it is seen from the streamline contours that the fluid moves along the hot surface in the upward direction, after coming in contact with the left side of the partition attached to top horizontal wall moves in the downward direction, then through the gap provided moves in the upward direction along the right side of the partition, then flows over the top adiabatic horizontal wall and again after coming in contact with the cold wall moves in the downward direction. Hence, after which the fluid flows over the bottom horizontal wall then flows over the partition attached to bottom wall after which moving over the partition flow over the hot wall in the upward direction resulting in the clockwise rotation of fluid flow. However, the flow field on the right side of the partition are very weak for Ra 105 when compared to the flow field for Ra 106 as shown in figure 8i. It is also clear from the isotherm contours for Ra 105 the contour gets distorted on the left side of the partition and for 106 the contour gets distorted and becomes parallel to the horizontal wall.

For case 4a two partitions are considered the height of upper partition being L/4 and that of partition attached to bottom wall is L/2 maintaining a gap of L/4 between them and the distance of the partition from hot wall is L/4, the streamlines and temperature contours are shown in figures 5–8j. The streamline and temperature contours in this case are similar to that of case 3a which can be referred from figures 5–8i. For Ra = 103 and 104 the flow fields are present only on the right side of the partition. It is also clear from the isotherm contours for Ra 103 which are exactly parallel to the vertical wall. However, for Ra 104 the isotherms are parallel only near the vertical walls and form smooth curves at the centre of the cavity. For Ra = 105 and 106 weak flow fields are also present on the left side of the partition. It is also clear from the isotherm contours for Ra 105 the contour gets distorted on the right side of the partition and for 106 the contour gets distorted and becomes parallel to the horizontal walls thus causing thermal stratification on the right side of the partition. Figures 5–8k show the streamline and temperature contours for case 4b which are exactly similar to that of case 3b. Also, after comparing from the graphs in figures 11 and 12 for case 3b and 4b respectively for the values of Nusselt number, it is observed that the values are almost equal.

The streamline and temperature contours for case 4c are presented in figures 5–8 (l). The contours show immense similarity between case 3c and 4c. For Ra = 103 and 104 the flow fields are present only on the left side of the partition. It is also clear from the isotherm contours for Ra 103 which are exactly parallel to the vertical wall. However, for Ra 104 the isotherms are parallel only near the vertical walls and form smooth curves at the centre of the cavity as shown in figure 6(l). For Ra = 105 and 106 weak flow fields are also present on the right side of the partition. It is also clear from the isotherm contours for Ra 105 the contour gets distorted near the hot wall and the left side of the partition and for 106 the contour gets distorted and becomes parallel to the horizontal walls thus causing thermal stratification on the left side of the partition.

The graphs in figure 9 demonstrates the values of convective, radiative, and total Nusselt number for cases 1a, 1b and 1c. For Ra = 103 it is observed that the Nucv is more than the Nurd but from Ra = 104 onwards it is observed that the radiative Nusselt number is higher than the convective Nusselt number. The least value of total Nusselt number is noted for case 1b for all Rayleigh number range when the partition is present at the centre.

Figure 9
figure 9

Variation of convective, radiative and total Nusselt Number for Combined natural convection and radiation for hot wall for ε = 1 in r case 1 for Rayleigh number ranging from 103 to 106.

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The values of radiative, convective and total Nusselt number for Rayleigh number 103, 104, 105 and 106 for cases 2a, 2b and 2c are presented in figure 10. It can be observed that the values of both convective and radiative Nusselt number is increasing as the Rayleigh number increases, Nurd increasing in higher proportion. This results in an increase of overall Nusselt number. For Ra 103 the value of Nucv is greater than Nurd while for rest of the Ra values Nurd is higher. Also, the least value of NuT is achieved when the partition is at the centre.

Figure 10
figure 10

Variation of Total Average Nusselt Number for combined natural convection and radiation for hot wall for ε = 1 in case 2 for Rayleigh number ranging from 103 to 106.

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Figure 11 shows the values of convective, radiative and total Nusselt number for case 3. Similar to above cases, for Ra 103 Nucv is higher than Nurd. For Ra 103 and 104 least values of Nusselt number is observed when the partition is at centre while for 105 and 106 highest values of Nusselt number is observed at this location.

Figure 11
figure 11

Variation of convective, radiative and total Nusselt Number for combined natural convection and radiation for hot wall for ε =1 in a differentially heated partitioned square cavity for case 3 for Rayleigh number ranging from 103 to 106.

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Figure 12 shows the values of convective, radiative and total Nusselt number for case 4. For Ra 103 Nucv is higher than Nurd. For Ra 103, 104 and 105 least values of Nusselt number are observed when the partition is at centre while for 106 highest values of Nusselt number is observed at this location.

Figure 12
figure 12

Variation of radiative, convective and total Nusselt Number for combined natural convection and radiation for hot wall for ε = 1 in case 4 for Rayleigh number ranging from 103 to 106.

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4.3 Effect of Rayleigh number

The effect of Rayleigh number on combined natural convection and radiation inside a differentially heated partitioned square cavity is shown in figure 13. The graph shows the variation of Nusselt number with increasing Rayleigh numbers. The range of Rayleigh number considered is between 103 and 106. To study the effect of Rayleigh number the partition is considered at the centre i.e., SP L/2. Also, the height of the partition is taken as h = L, 3 L/4, L/2 and L/4 respectively. It is clear from the graph that increasing the Rayleigh number increases the overall heat transfer rate which further leads to an increase in Nusselt number. The maximum value of Nusselt number is observed when Ra is 106 while the least being at Ra = 103.

Figure 13
figure 13

Variation of total Nusselt number with Rayleigh Numbers along the hot wall for central position of partition wall and different partition height.

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It is also evident from the streamline contours that when Ra = 103, the values of the flow field is very small and the transfer of heat occurs essentially by conduction. As the Rayleigh number increases to 105 the convection currents and radiation currents become stronger leading to high rate of heat transfer and fluid flow. Further, increasing the Rayleigh number to 106 the effect of radiation dominates the transfer phenomenon which results in an increase in fluid flow. Also, the temperature contours are nearly parallel for Ra = 103 which happens due to weaker fluid flow and only starts forming smooth curve for Ra = 104. However, it is observed from the temperature contours that the fluid gets horizontally stratified when the Rayleigh number increases to 105. When the Rayleigh number reaches to a higher value of 106 it can be observed from the isotherm contours that the central portion of the contour is nearly horizontal except for the case when the partition is placed at the centre resulting in distorted contours near the bottom and top side of the hot and cold surfaces, respectively.

It is also clear from the graphs in figures 9, 10, 11 and 12 that the values of convective, radiative and total Nusselt number are presented for the Rayleigh number ranging from 103 to 106 for Prandtl number 0.7. It can be seen that as the Rayleigh number increases the value of convective, radiative and total Nusselt number increases. This happens because as the Rayleigh number is increased the effect of convection and radiation currents became stronger leading to higher values of Nusselt number.

4.4 Effect of Emissivity

In order to determine the effect of surface emissivity on heat transfer, the value of emissivity is changed as 0.25, 0.5, 0.75 and 1 for Rayleigh number 103 and 106 and the graphs are plotted between Radiative Nusselt number, Total Nusselt number and Emissivity below in figure 14 for cases 2a, 2b and 2c. It can be noticed from the graphs that as the value of emissivity is increased the value of Nurd and NuT increases linearly. This occurs due to higher thermal radiation emitted by surface having higher emissivity.

Figure 14
figure 14

Variation of Radiative Nusselt number and Total average Nusselt number for hot wall with emissivity for Rayleigh number 103 and 106.

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It is also observed that the inclusion of surface radiation decreases the convective Nusselt number. And this decrease in convective Nusselt number is compensated by the increase of radiative Nusselt number. As the emissivity value is increased from 0.25 to 1 it is noticed that the convective Nusselt number decreases to a greater degree for higher Rayleigh number i.e., 106 than for Ra = 103. For an increase in emissivity from 0.25 to 0.5 the total Nusselt number is increased by 17.9% for Ra = 103 and 27.0% for Ra = 106 for combined natural convection and radiation. Again, when the value of emissivity is increased from 0.75 to 1 it is observed that the total Nusselt increase by 14.6% for Ra = 103 and 27.1% for Ra = 106. Thus, we can see that for lower values of emissivity the value of Nusselt number is lowest, and maximum transfer of heat occurs when the value of emissivity is maximum i.e., ε = 1.

It is also evident that for an increase of emissivity from 0.25 to 1, the increase in total Nusselt number is found to be 24.9% and 97.1% respectively when compared with pure convection for Ra = 103. For Rayleigh number 106 the increase in Nusselt number is 32.6% and 170.9% sequentially when compared with pure convection.

4.5 Effect of partition height

Figure 15 shows the variation of total Nusselt number with partition height for Rayleigh number 103 and 106. Four cases of partition height are considered for the analysis. For the case 1 and 2 the height of the partition is h = L and 3L/4 respectively. For the case 3 and 4 two partitions are considered and the gap between them is taken as L/4 keeping 25% of the cavity open. In case 3 the stature of the partition attached to the top horizontal wall is of L/2 and for the one attached to bottom wall is of L/4. For case 4 the heights of upper and lower partitions are reversed, i.e., height of partition projected from top wall is L/4 and height of partition attached at bottom wall is L/2. By comparing the values of total Nusselt number for the hot wall for all the cases, it can be seen that the highest values of NuT are achieved when two partitions are considered having a gap between them as L/4, one partition is positioned at top wall and one at bottom wall i.e., for cases 3 and 4. Which is due to the fact that by decreasing the height of the partition more space is provided for fluid flow, thus causing more heat transfer.

Figure 15
figure 15

Variation of total Nusselt number with partition height for different partition distance for Rayleigh number 103 and 106 respectively.

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The graph shows that the Nusselt number is least when the height of partition is highest and increases as the partition height is decreased for both the values of Rayleigh number. It is observed that when the height of partition is varied from h = L to L/4 for case 1 and 4 respectively, the value of Nusselt number increases to 7.2% for Ra 103 and 30.3% for Ra 106. Also, by comparing the values of total Nusselt number for the hot wall for all the cases as shown in graphs of figures 9, 10, 11 and 12 that the highest values of NuT are achieved when one partition is positioned at top wall and one at bottom wall i.e., for cases 3 and 4.

4.6 Effect of partition distance

In order to determine the effect of partition distance from hot wall on the combined natural convection and radiation phenomenon, three cases are considered ‘a, b and c’ for different partition distance and value of SP for the cases is L/4, L/2 and 3L/4 respectively. In figure 16 variation of total Nusselt number for Ra = 104 and 105 respectively with the distance of partition from hot wall is presented. It is observed from the graph that value of Nusselt number is at a higher value when the partition is near the hot wall, decreases when placed at the centre and then increases when moved further away from hot wall and in closed proximity of cold wall. Again, from figures 9–12, the graphs shows that by comparing the three cases a, b and c it can be observed that the value of Nusselt number is increasing as the partition is moving away from the hot wall and least when placed at centre. Thus, the value of Nusselt number for Ra 103 to 106 is maximum for case ‘c’ when the distance of partition is 3L/4 for almost all the cases.

Figure 16
figure 16

Variation of total Nusselt number with partition height for different partition distance for Rayleigh number 104 and 105.

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5 Conclusion

The combined laminar natural convection and radiation in a differentially heated square cavity with partition of different heights and distance from hot wall protruding on horizontal adiabatic walls are studied with the help of finite volume method. The prime investigation of the present work is to study the effect of radiation, position and stature of the partitions on natural convection and radiation within a square cavity for a range of Rayleigh number. Certain conclusions have been noted from the present research work which could be helpful for future research works. The inclusion of surface radiation effects the total heat transfer phenomenon in an enclosure with partition. The values of total Nusselt number are much higher for combined radiation and natural convection than for pure natural convection as can be observed for Ra 104 where the percentage increase in Nusselt number is 63.2%. It is observed that total Nusselt number increases after surface radiation interaction while the contribution of convective heat transfer decreases. It is also concluded that the average Nusselt number increases linearly as the Rayleigh number increases from 103 to 106 for all considerations of partition height and position. This happens because when Ra = 103, the natural convection flow is significantly low and the transfer of heat occurs essentially by conduction. As the Rayleigh number increases to 105 the convection currents along with interaction with surface radiation significantly enhances the heat transfer characteristics in the cavity. Further, increasing the Rayleigh number to 106 the effect of radiation dominates the heat transfer phenomenon. The lower values of Nusselt number along the hot wall are seen for case 1 and 2 when the height of the partition is L and 3L/4 respectively, as compared to rest of the cases leading to a conclusion that reduction in partition length results in higher rates of heat transfer and fluid flow. It is further observed that the Nusselt number varies with different values of emissivity in the cavity under study. In comparison, the total Nusselt number is achieved most in the case ‘c’ when the partition is farthest from the hot vertical wall, hence forming the conclusion that decreasing the length of partition distance from hot wall decreases the overall transfer of heat. Therefore, Nusselt number for a square cavity with partition placed at the extreme distance i.e., 3L/4 is greater than other partition positions.