A Numerical Approach for Natural Convection With Curved Obstacles in an Enclosure Using Lattice Boltzmann Method

Natural convection is known to have a lot of industrial and engineering applications as well in solar dryers, cooling of small electronic systems, and battery cooling systems [1,2]. Natural convection in a cavity has been studied in detail using computational fluid dynamics (CFD) techniques by multiple authors. De Vahl Davis [3] provided a benchmark solution for the case with natural convection in a closed cavity with top and bottom walls with adiabatic boundary condition, the left wall at high temperature and the right wall at low temperature. This study is widely used for validating newly developed natural convection codes. Aydin et al. [4] studied steady natural convection in a two-dimensional (2D) enclosure with a cooled ceiling and isothermal heating from one side and found that in shallow enclosures, the Rayleigh number had a greater bearing on heat transfer. Dalal and Das [5] conducted a study of natural convection inside a rectangular cavity with bottom wall having spatially varying temperature and other walls having constant low temperature using the SIMPLE algorithm with a higher order upwind scheme.

Study on natural convection using lattice Boltzmann method (LBM) have been conducted in the past decades due to the various advantages such as suitability of parallel computing and lesser complexity of equations being solved. Dixit and Babu [6] simulated natural convection in a square cavity using LBM with a very fine grid of wall y + < 0.3 for higher Rayleigh numbers. Arumuga Perumal and Dass [7] reviewed the development of lattice Boltzmann method for macro fluid flows and heat transfer problems. Mejri et al. [8] studied the LBM simulation of natural convection in an inclined triangular cavity filled with water, with a horizontal hot wall, vertical cold wall, and inclined insulated wall. Mansouri et al. [9] used MRT-LBM to study natural convection characteristic in a cavity micropolar fluid in it. Abouricha et al. [10] modeled turbulent natural convection in a heated house where the source of heat was placed at the ground. The lattice Boltzmann simulation of incompressible thermal flows in two and three dimensions has been done by Perumal and Dass [11]. Karki et al. [12] studied laminar natural convection and entropy generation in Rayleigh–Benard (R–B) convection with various nanofluids. Bhopalam et al. [13] have used lattice Boltzmann method to compute incompressible flows in a two-sided oscillating lid-driven cavity. Bhopalam et al. [14] have used lattice Boltzmann method to compute flows in double-sided cross-shaped lid-driven cavities.

Natural convection with obstacles has attracted researchers due to its various applications. Kim et al. [15] studied the natural convection for the difference in temperature between a cold outer square enclosure and a hot inner circular cylinder over a range of Rayleigh numbers 10³ to 10⁶. Kim et al. [16] conducted numerical simulations of natural convection in a square enclosure with a cylinder placed inside at a Prandtl number of 0.7 with different bottom wall temperatures using immersed boundary method. They found that a large Rayleigh number caused a significant variation in streamline patterns and isotherms in the enclosure with increase in the bottom temperature. Liao and Lin [17] studied the influence of Prandtl number on instability of natural convection flows within an enclosure with a constant temperature circular cylinder placed at the center for moderate Rayleigh numbers. Karki et al. [18] studied the effects of placing one, two, and four adiabatic square obstacles inside a bounded enclosure at various Rayleigh numbers and studied the influence of the obstacles on Nusselt number at cold wall. Sen et al. [19] have studied mixed convection in a single as well as double-sided lid-driven cavity using lattice Boltzmann method. Mei et al. [20] have used lattice Boltzmann method for three-dimensional flows with curved boundaries. Bararnia et al. [21] have used lattice Boltzmann method to simulate natural convection in a square enclosure having a horizontal elliptical cylinder. Yan and Zu [22] have simulated heat transfer and fluid flow past a rotating isothermal cylinder using lattice Boltzmann method.

A Knudsen number based approach as proposed by Kao et al. [23] is used to select characteristic velocity, thereby eliminating the need to randomly select relaxation times. A modified curved boundary interpolation technique inspired by Yu et al.’s [24] approach is implemented to model the inner boundaries. In the present work, different types of circular and elliptical obstacles are placed inside a cavity. These obstacles are maintained adiabatically, at constant high temperature and at constant low temperature. The adiabatic obstacle helps us understand how the presence of an obstacle influences the flow characteristics within the enclosure without transferring heat to the flow around it. This would provide a case to study purely the effect of geometrical blockage in the domain. The obstacle is then given a constant high and constant low-temperature condition to study the effect of obstacle temperature on the flow and observe how it affects the movement of the fluid on its either side. Physically, contact with a hot wall makes fluid relatively lighter and move upward whereas contact with a cold wall makes the liquid relatively heavier and move downward. Our aim would thus be to search for these flow traits in our results, and check if contact with a hot or cold obstacle would create diagonal asymmetries in flow patterns. The effect of Rayleigh number, obstacle size, and shape on the extent of shift in flow features such as primary vortex is also studied. Laminar flow in steady-state condition is considered for the present study. Results are obtained for various Rayleigh numbers at a fixed Prandtl number of 0.71.

The schematics that have been modeled are shown in Fig. 1. The square enclosure walls are stationary. The top and bottom walls are considered adiabatic. The left wall is considered to be at high temperature T_h, whereas the right wall is maintained at low temperature T_c. In the first case, a square cavity with no obstacle is considered for validation. The results produced by the developed code are compared with results of De Vahl Davis [3], and a grid independence test is also conducted. Second, circles with diameter d = 0.125L and d = 0.25L, where L represents the length which is taken equal to the number of elements to make sure each element is unit size, are placed at the center of the cavity. After that, an ellipse with horizontal major axis a = 0.25L and vertical minor axis b = 0.125L is placed at the center of the domain. Lastly, an ellipse with horizontal minor axis a = 0.125L and vertical major axis b = 0.25L is placed at the center of the domain. All these obstacles are assigned three different types of boundary conditions: adiabatic, hot, and cold boundary conditions.

The aim of the simulation is to reduce the difference between the equilibrium distribution function and probability distribution function for both velocity and temperature. To start the simulation, three important parameters need to be determined:

1. The characteristic velocity for the simulation (u_nat): It is fixed as a function of Knudsen number Kn, Rayleigh number Ra, and Prandtl number Pr and specific heat γ

   $unat=2RaKn2c2πγPr$

   (11)

Here, Kn = 10⁻⁴, Pr = 0.71, and γ = 1.4 yielding a characteristic velocity such that Mach number is in the incompressible region.

1. The momentum due to buoyant flow [25] gives three ways to approximate the gravitational force term of which the following method is chosen:

   $Fi=3wigβT(x,t)ρ(x,t)ciy$

   (12)

   where

   $gβ=unat2H$

   (13)
2. The kinematic viscosity ν: The parameter used to evaluate the relaxation time is calculated using the expression:

   $ν=unat2H2PrRa$

   (14)

The boundary for curved obstacles will not coincide with the nodal points in the domain calling for the need of an interpolation scheme to satisfy appropriate conditions at the boundary. The bounce-back condition for velocity needs to be implemented at the boundary node x_b as shown in Fig. 2 as implemented by Yu et al. [24]. During the collision step, value at x_ff and x_f are computed and during the streaming step, these values are streamed along α direction to x_f and x_b, respectively. Using the values of $fα(xf)$ and $fα(xb)$⁠, the value of $fα(xw)$ is computed as follows:

Figure 4 shows the impact of placing an adiabatic circular obstacle with diameter d = 0.125H. From the figure it can be seen that at low Rayleigh number (10³ − 10⁵), adiabatic obstacle marginally hinder the fluid flow characteristics displayed by a differentially heated block without obstacles (which is shown in Fig. 3). We observe that similar to Fig. 3, the fluid moves up the hot wall and down the cold wall and the obstacle. For all values of Ra, a change in temperature contours is observed surrounding the obstacle and centro-symmetry of velocity streamline patterns can also be seen from the figures. At a higher Rayleigh number of 10⁶, the obstacle causes the development of more than two primary vortices, and a secondary vortex being clearly visible on the right-hand side of the obstacle which can be observed in Fig. 4(g). This is a clear deviation in flow behavior inside the cavity with and without obstacles which strictly forms additional vortices.

Figure 5 shows the impact of placing a cold circular obstacle with diameter d = 0.125H in the enclosure. In Fig. 5(b), when the cold obstacle is placed at low Rayleigh number, Ra = 10³, an extended region of low temperature is formed to the right side of the domain, engulfing the obstacle itself. Unlike the adiabatic case, centro-symmetry of velocity streamlines is not observed for the cold obstacle case. At Ra = 10⁴, the flow field develops a vortex to the left of the obstacle which can be seen in Fig. 5(c). The cold obstacle causes the fluid around it to move downward by increasing its density. This will lead to the formation of a primary vortex on the left side of the circular obstacle as the fluid is moving up along the hot wall. A bounded region of low temperature (T < 0.1) is formed around the circular portion indicating the penetration of slightly higher temperature fields between the obstacle and the cold wall. At Ra = 10⁵, two vortices are noted on either side of the obstacle in Fig. 5(e) but due to the cold nature of the obstacle, the left vortex is larger in size. This occurs as the fluid on the left wall tends to move up the hot wall, whereas the flow close to the obstacle tends to move downward, and thus establishes a more powerful vortex on the left side. This dissimilarity in vortex sizes is not expected in the case of adiabatic obstacles as the obstacle itself just acts as blockage and uniformly changes flow features in both the right half and left half of the domain. The bounded region of low temperature (T < 0.1) as described above are observed for this case in the form of smaller circle. In Fig. 5(g), at Ra = 10⁶, the flow turns chaotic with the development of multiple vortices on both sides of the obstacle. The bounded region of low temperature (T < 0.1) contracts for the obstacle indicating a tendency of higher heat transfer similar to the tendency of compression of isotherms near external walls at constant temperature.

Figure 6 shows the effect of placing a hot circular obstacle with diameter d = 0.125H which is similar to the cold obstacle with certain exceptions. Since the obstacle now has a high temperature boundary condition, it tends to push the fluid upward by reducing its density. The dominant primary vortex will now move to the right side with the fluid being pushed down by the cold obstacle interacting with the fluid being pushed up by the hot obstacle, thus establishing a dominant loop. On the left side, the vortex becomes weaker as the flow direction of the fluid near the cold left wall and the central obstacle are similar in direction, nullifying the strength of the left vortex, and making it a secondary vortex. The bounded regions of low temperature are replaced with bounded regions of high temperature (T > 0.9). At low Rayleigh number the primary vortex is formed toward the right instead of the left as seen in the cold wall case. Also, the streamline patterns and isotherms look like a mirror image along 45 deg line drawn from the lower left end to the top right end of the enclosure, especially at low Rayleigh numbers of 10³ and 10⁴. For isotherms, the value assigned after taking the mirror image should be subtracted from 1 due to the change in type of obstacle.

Figure 7 shows the streamline patterns and isotherms when an adiabatic circular obstacle with diameter d = 0.25H, which is two times the size of the previous obstacle, is placed in the enclosure. The effect is identical to the adiabatic smaller obstacle case where at low Rayleigh numbers (10³ − 10⁵), the obstacle has marginal effect on the fluid flow characteristics shown by a differentially heated block in the absence of obstacles. From the figures, we can also observe centro-symmetry of velocity streamline patterns which was also seen in the smaller circular obstacle case. In Fig. 7(g), for a higher Rayleigh number like 10⁶, the development of more than two primary vortices and a secondary vortex are clearly visible on the right-hand side of the circular obstacle.

Figure 8 shows the impact of placing a cold circular obstacle with diameter d = 0.25H in the enclosure. Similar to the previous cold obstacle case, a vortex on the left side of the larger circular obstacle is formed due to its cold nature which pushes the fluid downward and this combined with the upward motion of the fluid along the hot wall leads to the formation of a primary vortex on the right side. In Fig. 8(b), similar to the smaller circular obstacle at Ra = 10³, a region of low temperature is formed on the right side of the domain. At Ra = 10⁴, the streamline patterns in Fig. 8(c) shows a vortex to the left of the obstacle. Also from the isotherms, it is seen that the obstacle is encapsulated by a constant temperature contour region which becomes smaller in area, indicating higher levels of heat transfer near the top portion of the cold outer wall. At Ra = 10⁵, one vortex is formed to the left of the obstacle. In this case, physically a similar trend is observed with respect to placing the smaller cooled circular obstacle. However, it is noticed that the secondary vortex almost disappears. This can be attributed to the size of the cooled obstacle, which cools more fluid down, and thus on the left side, matches the flowrate of fluid flowing down near the right cold wall. Similar amount of cooling results in nullification of the right vortex. This tendency is in direct contrast to the case with adiabatic obstacles, where two clearly distinguishable vortices of equal size are noticed on the right and left side, which implies the effect is not influenced by geometrical blockage, but by the virtue of a larger sink at the center of the domain. At Ra = 10⁶, the flow becomes chaotic. A notable shift of cold region from being concentrated along the low temperature to being dominant near the horizontal adiabatic wall is noticed for the larger obstacle at Ra = 10⁶ (Fig 8(h)).

Figure 9 shows the effect of placing a hot circular obstacle with diameter d = 0.25H in the enclosure. From the streamline patterns and isotherms, it can be seen that the results obtained are similar to the cold circular obstacle but with some differences. In Fig. 9(b), when Ra = 10³, a region of high temperature is formed on the left side of the domain. The hot nature of the obstacle increases the temperature of the fluid around it making it lighter. This combined with the effect of the cold wall leads to a vortex being formed on the right side of the domain, and similar to the colder case, the secondary vortex is suppressed. At Ra = 10⁴, the streamline patterns show a vortex to the right of the obstacle (Fig. 9(c)). At Ra = 10⁵, one vortex is formed to the right of the obstacle (Fig. 9(e)) and at Ra = 10⁵, the flow becomes chaotic (Fig. 9(g)).

The streamline patterns and isotherms developed, when an adiabatic horizontal elliptical obstacle is placed in the enclosure, is shown in Fig. 10. For low Rayleigh numbers (10³−10⁵), the adiabatic horizontal elliptical obstacle does little to hinder the flow characteristics obtained in differentially heated block without obstacles. For Ra = 10³, the circular vortices which were observed in the circular obstacle case have now become elliptical in shape and like the other adiabatic cases, centro-symmetry of velocity streamline patterns can also be seen. In Fig. 10(g), when Ra = 10⁶, the development of more than two primary vortices is noticed with the secondary vortex on the left-hand side of the obstacle unlike the circular obstacles where the secondary vortex was on the right-hand side.

Figure 11 shows the streamline patterns and isotherms obtained when a cold horizontal elliptical obstacle is placed in the domain. At Ra = 10³, an extended region of low temperature is formed on right side of the obstacle (Fig. 11(b)), engulfing it. Just like the circular obstacle cases, the cold wall of the obstacle makes the surrounding fluid colder and heavier which causes the formation of the vortex on its left side as the fluid is moving up the hot wall. For Ra = 10⁴, development of a vortex to the left of the obstacle is observed (Fig. 11(c)). Similar to the larger circular obstacle, a constant temperature contour region is observed which encloses the obstacle and becomes smaller in area. In Fig. 11(e), at Ra = 10⁵, the horizontal ellipse forms one vortex to the left of the obstacle. The flow turns chaotic at Ra = 10⁶ which is indicated by the development of multiple vortices on either side of the obstacle (Fig. 11(g)).

Figure 12 shows the effect of placing a hot horizontal elliptical obstacle in the enclosure. The hot wall of the obstacle combined with the cold right wall of the enclosure allow the formation of a vortex on the right side of the elliptical obstacle which is a feature also observed in the hot circular obstacles. The bounded regions of low temperature observed in the cold elliptical case are replaced with bounded regions of high temperature (T > 0.9). At low Rayleigh number, the primary vortex is formed toward the right instead of the left as seen in the cold wall case.

Figure 13 shows the streamline patterns and isotherms developed when an adiabatic vertical elliptical obstacle is kept in the enclosure. For low Rayleigh numbers (10³ − 10⁵), just like the other adiabatic obstacles, the adiabatic vertical elliptical obstacle does little to hinder the flow characteristics obtained in differentially heated block in the absence of obstacles and centro-symmetry of velocity streamline patterns is observed for all values of Ra. However in Fig. 13(g), at Ra = 10⁶, unlike the other adiabatic obstacles, the vertical elliptical obstacle does not have a secondary vortex, indicating the decreased tendency with an increased width to height ratio of the object placed.

Figure 14 shows the effect of placing a cold vertical elliptical obstacle in the domain. At Ra = 10³, the obstacle is engulfed by a region of low temperature which is developed on the right side (Fig. 14(b)). Just like the circular and horizontal elliptical obstacle cases, for Ra = 10⁴, a vortex is developed to the left side of the obstacle (Fig. 14(c)) due to the combined effect of the cold obstacle wall which pushes the fluid downward and left hot wall of the enclosure which pushes the fluid upward. Similar to the larger circular obstacle and the horizontal ellipse, a constant temperature contour region is formed which encloses the obstacle and becomes smaller in area.

At Ra = 10⁵, the vertical ellipse forms two vortices on either side of the obstacle (Fig. 14(e)). For Ra = 10⁶, the flow turns chaotic just like the horizontal ellipse case.

Figure 15 shows the effect of placing a hot vertical elliptical obstacle in the enclosure which is similar to the other hot obstacle cases as the vortex is formed on the right side of the elliptical obstacle as the fluid moves downward along the cold wall and upward close to the hot obstacle due to effect of temperature on density of the fluid. From the streamline patterns and isotherms, it can be seen that the results obtained are similar to the cold wall elliptical obstacle with small differences. At low Rayleigh number the primary vortex is formed toward the right instead of the left as observed in the cold wall case.

Comparing the results of horizontal and vertical obstacles at Ra = 10⁵, we see that the horizontal extent of the source or sink plays a more dominant effect in suppression of the secondary vortex in comparison with the vertical extent of the source or sink. This could be attributed to the fact that the distance between a point horizontally furthest from the center of the obstacle, and a vertical wall of the domain are lower in the case of the horizontal ellipse in comparison with the vertical obstacle. This implies that a lowering of this particular distance affects flow closer to the wall, thus suppressing the space available to form a secondary vortex. Another trend that is repeatedly observed is the flow features observed for domains with cold obstacles are a mirror image about a diagonal joining the lower right corner to the upper left corner, of flow features observed for domains with warm obstacles. This is justified as the obstacle produces opposing effects as far as movement of the fluid near the obstacle is concerned.

As for Nusselt number variations as shown in Figs. 16, 17, and 18, there is an increase in the hot wall Nusselt number with increase in Rayleigh number and a decrease in the same with the increase of height as can be implied from the crowding of contour lines near the bottom of the hot wall. The variations between Nusselt number of each obstacle decrease with the increase of Rayleigh number clearly indicating the diminishing importance of type of obstacle on the heat transfer near the hot wall due to the crowding of contour lines irrespective of obstacle. It is noted from the figures that the highest hot wall Nusselt numbers at Ra = 10³, for cold wall and hot/adiabatic walls, are noted for the small circular and large circular obstacles, respectively. The difference between hot wall Nusselt number is least for the adiabatic case followed by the cold and warm obstacles.

The lattice Boltzmann method is used to compute the fluid flow and heat transfer characteristics in a bounded enclosure with curved obstacles of different shapes and sizes. An interpolation method is presented to satisfy the adiabatic and constant temperature boundary conditions. Streamline patterns, isotherms, and Nusselt number variations along hot wall are presented and discussed in detail.

When an adiabatic obstacle is placed within the enclosure, very little difference is observed when compared with the results obtained in an enclosure with no obstacles for low Rayleigh numbers. However at higher Rayleigh numbers, the obstacle causes the formation of more than two primary vortices. A secondary vortex is visible on the right-hand side for circular obstacles of both diameters. In the horizontal ellipse configuration, the secondary vortex is noticed instead on the left-hand side. The vertical ellipse does not have a visible secondary vortex, indicating the decreased tendency with an increased width to height ratio of object placed.

With the cold obstacle placed at low Rayleigh number, Ra = 10³, an extended region of low temperature is formed to the right side of the domain, engulfing the obstacle itself for all cases. The presence of a cold obstacle in the middle of the domain causes the fluid to move downward due to the increase in its density. This combined with the movement of the fluid up along the hot wall leads to the formation of a dominant vortex on the left side of the obstacle. At Ra = 10⁴ the flow field develops a vortex to the left of the obstacle for all cases. At Ra = 10⁵ two vortices are noted on either side of the obstacle for the small circle and vertical ellipse, while the horizontal ellipse and larger circle forms just one vortex to the left of the obstacle. At Ra = 10⁶ the flow turns chaotic for almost all cases with the development of multiple vortices on both sides of the obstacle.

The behavior of enclosure with hot wall obstacle is similar to that of the cold obstacle with certain exceptions. Since the cold temperature of the obstacle is replaced with a constant high temperature, the dominant vortex is formed on the right side of the obstacle instead of the left. The bounded regions of low temperature are replaced with bounded regions of high temperature (T > 0.9). At low Rayleigh number, the primary vortex is formed toward the right instead of the left as seen in the cold wall case.

The study reveals many interesting observations which include the dependence of location of vortices on the type of obstacle boundary used and formation of low or high temperature zones around the obstacle at high Rayleigh numbers.

There are no conflicts of interest. This article does not include research in which human participants were involved. Informed consent not applicable. This article does not include any research in which animal participants were involved.

The authors attest that all data for this study are included in the paper.