## Abstract

Numerical simulations are performed to deduce the effects of slip wall and orientation on entropy generation due to natural convection (NC) in a square cavity for Rayleigh number (Ra) = 105. The laterally insulated square cavity, heated at the bottom wall and cooled at the top wall, is subjected to various orientation angles (ϕ) and slip velocities characterized by the Knudsen number (Kn). The two components of entropy generation, i.e., entropy generation due to heat transfer ($SΘ$) and entropy generation due to fluid friction ($SΨ$), are separately investigated by varying the orientation from 0 deg to 120 deg in steps of 15 deg and Knudsen number from 0 (no-slip) to 1.5 in steps of 0.5. Evidence indicates that, for most cases considered, entropy generation due to fluid friction ($SΨ$) dominates the one due to heat transfer ($SΘ$). It is observed that the slip velocity on the isothermal walls ($us,iso$) has a strong influence on $SΘ$ whereas the variations in $SΨ$ are closely connected to the change in the rate of shear strain. Interestingly, the presence of corner vortices and the secondary circulations near the core of the cavity are also found to affect the variation in entropy generation. The existence of active zones of $SΘ$ in the vicinity of isothermal walls and their elongation and migration while changing the orientation is another unique characteristic noticed in this study. A new parameter called maximum velocity ratio (MVR) is also proposed to highlight the variation in velocity components within the enclosure.

## 1 Introduction

Natural convection (NC) is a thermally induced buoyancy-driven mode of heat transfer. It is the mechanism of heat transfer operating without any requirement of external forces that makes natural convection a sustainable energy transfer phenomenon. The last few decades have observed intensive research on heat transfer by NC due to its relevance in various natural processes as well as real-world applications. Natural convection in an enclosure, particularly a square cavity, has always been a topic of interest due to its geometrical simplicity and richness in features. However, the co-existence of hydrodynamic and thermal boundary layers in the vicinity of the walls and their mutual interaction make the natural convection problems equally exciting and complex. Several articles have already been published owing to the fundamental nature of NC and its versatile applications [14].

It is known that thermodynamic efficiency is the key criterion to be satisfied while modeling new equipment. Since irreversibility is a manifestation of thermodynamic imperfections, the performance of any thermodynamic system is evaluated based on the irreversibility it generates during a process. Though, the thermodynamic efficiency can be increased by effectively utilizing the available heat energy, the irreversibilities associated with the system when it changes state are also a major determining factor. In the literature, these irreversibilities associated with heat transfer and fluid friction are quantified in the form of entropy generated [5]. In this regard, entropy generation minimization (EGM), the terminology proposed by Bejan [6,7], deserves considerable importance. EGM is also known as “thermodynamic optimization” in engineering [7]. Depending on the objective, this optimization method can be utilized for various applications that use heat exchangers, insulation systems, storage systems, solar-thermal power generation, refrigeration, etc., to cite a few [6]. Needless to, say, the analysis of entropy generation is useful in design optimizations meant to enhance the thermal efficiency of a system [8]. Some of the recent investigations further underline the practical importance of employing EGM as an optimization tool in thermal systems [911] with widespread applications in bio-inspired [12], nanofluidic [13], and cryogenic [14] devices.

Following the criteria put forth by Bejan [6], numerous studies have been conducted on the analysis of entropy generation. One of the earliest investigations that gave an insight into entropy generation due to natural convection was conducted by Yilbas et al. [15]. In this report, though the primary objective was to understand the flow and thermal fields in a differentially heated cavity, the authors were also keen on analyzing the aspects of entropy generation due to heat transfer and fluid friction. Erbay et al. [16,17] numerically investigated the two modes of entropy generation; due to heat transfer and fluid friction in a square enclosure with partially/completely heated/cooled vertical walls. These studies explored the influence of Rayleigh number and Prandtl number on the distribution of entropy active spots within the cavity. The authors also found that the contribution of entropy generation due to fluid friction irreversibility was almost negligible in all cases. Hence, the total entropy generation had nearly the same distribution as the entropy generation due to heat transfer. Magherbi et al. [18] used a finite element method to investigate the effects of Rayleigh number and irreversibility distribution ratio on the total entropy generation and Bejan number in a square cavity subjected to a horizontal thermal gradient with top and bottom adiabatic walls. They also studied the evolution of entropy generation with time and reported that the maximum value of total entropy generation increased with the Rayleigh number and irreversibility distribution ratio.

Even though numerous studies have been conducted to analyze the irreversibility effects due to natural convection in square enclosures, only a few discuss the effects of orientation. Following the insights provided by Baytaş [19], Singh et al. [20] conducted a thorough analysis to understand the effects of cavity inclination on entropy generation. They considered a square cavity with a hot bottom wall and cold lateral walls by insulating the top wall. For the range of Rayleigh and Prandtl numbers, the authors reported that the active zones of entropy generation due to heat transfer occurred at the corners due to high thermal gradients. Similarly, active regions of entropy generation due to fluid friction were witnessed in those locations where the circulations interacted with the walls.

Apart from the studies on entropy generation due to natural convection in square enclosures as mentioned above, there are reports on heat transfer and fluid friction irreversibilities on Γ-shaped [21] and inclined wavy enclosures [22]. Similarly, detailed reports are already available on heat transfer and entropy generation in rhombic enclosures [23,24]. Recently, Yıldız et al. [25] modified the square enclosure by introducing a dome-shaped structure at one end of the cavity. In this study, by varying the dome height and orientation, the authors explored the effects of the Rayleigh number on entropy generation and reported that the dome shape effectively increased the circulation within the cavity to enhance the heat transfer characteristics. These findings prove that geometrical variations also affect the entropy generation in internal flows.

For real fluid flows confined by solid walls, the no-slip velocity boundary condition (BC) provides an excellent prediction of the associated pressure and velocity fields. However, when the fluid–solid interaction phenomenon is subjected to a wall-slip where the fluid slips away from the surface; such as in flow over hydrophobic [26], and super-hydrophobic surfaces [27], their interaction modifies the boundary layer formation and the associated flow dynamics. It is also reported that wall-slip can lead to considerable drag reduction [28]. However, reports on the influence of these boundary conditions on natural convection and the associated irreversibilities are scarce even though there are extensive applications, as in microscale heat exchangers [29].

Note that there are other extensive studies on the influence of Rayleigh number and Prandtl number [3032] on the irreversibilities that are also performed by varying the aspect ratio of the enclosure and irreversibility distribution ratio [33,34]. However, apart from the findings by Singh et al. [20], no organized attempt to the best of the authors' knowledge has been made to understand the effects of inclination on entropy generation in square enclosures. The authors also find it necessary to understand the influence of slip boundary conditions owing to the participation of hydrophobic surfaces in thermal management [35]. Hence, it is evident from the discussions that apart from heat addition, though orientation and slip are the two mechanisms that result in entropy generation, their influence on the modulation of irreversibilities has not been discussed in the previous literature. It is, therefore, necessary to conduct studies on entropy generation due to heat transfer and fluid friction in a square cavity under the combined influence of wall slip and orientation, considering its range of applications in thermal management. It is expected that the introduction of slip boundary conditions will further extend the possibility of EGM as an effective optimization tool in engineering applications. The objective is to find out the configurations wherein the heat transfer is maximized, and entropy generation is minimized. Table 1 summarizes the data on available literature on entropy generation in square enclosures alone. Note that reports on entropy generation considering the porosity effects have not been included in this table.

Table 1

Review of existing literature pertaining to entropy generation due to natural convection in square enclosures in the chronological order of reporting. The left, right, top, and bottom indicate the four walls of the square enclosure.

ReferencesParametersTemperature BCs
Baytaş [19]Ra = 104$107$, Pr = 0.71Left:hot, right:cold
Yilbas et al. [15]Ra = (1.32–3.96) × 105Left:hot, right:cold
Erbay et al. [16]Ra = 102–108Left:hot/partially hot
Pr = 0.01, 1.0Right:cold, top, bottom:adiabatic
Magherbi et al. [18]Ra = 103–105, Pr = 0.71Left:hot, right:cold
χ = $10−4$$10−1$Top, bottom:adiabatic
Kaluri and Basak [30]Ra = 103–105Left, right:cold/discrete heating
Pr = 0.015, 0.7, 10, and 1000Top:adiabatic, bottom: hot/discrete heating
Soleimani et al. [31]Ra = 103–105, Pr = 0.7Left:hot, right:cold, top, bottom:adiabatic
Kaluri and Basak [32]Ra = 103–105Left, right:discrete heating
Pr = 0.015, 0.7, 10, and 1000Top:adiabatic, bottom: discrete heating
Ilis et al. [33]Ra = 102–105, Pr = 0.71Left:hot, right:cold
χ = $10−4$$10−2$, AR=1–16Top,bottom:adiabatic
Oliveski et al. [34]Ra = 103$107$, Pr = 0.71Left:hot, right:cold
χ = $10−5$$10−2$, AR=0.25, 0.5, 1, 2, and 4Top,bottom:adiabatic
Singh et al. [20]Ra = 103–105, Pr = 0.71Left:cold, right:cold
ϕ = 15 deg, 45 deg, and 75 degTop: adiabatic, bottom:hot
ReferencesParametersTemperature BCs
Baytaş [19]Ra = 104$107$, Pr = 0.71Left:hot, right:cold
Yilbas et al. [15]Ra = (1.32–3.96) × 105Left:hot, right:cold
Erbay et al. [16]Ra = 102–108Left:hot/partially hot
Pr = 0.01, 1.0Right:cold, top, bottom:adiabatic
Magherbi et al. [18]Ra = 103–105, Pr = 0.71Left:hot, right:cold
χ = $10−4$$10−1$Top, bottom:adiabatic
Kaluri and Basak [30]Ra = 103–105Left, right:cold/discrete heating
Pr = 0.015, 0.7, 10, and 1000Top:adiabatic, bottom: hot/discrete heating
Soleimani et al. [31]Ra = 103–105, Pr = 0.7Left:hot, right:cold, top, bottom:adiabatic
Kaluri and Basak [32]Ra = 103–105Left, right:discrete heating
Pr = 0.015, 0.7, 10, and 1000Top:adiabatic, bottom: discrete heating
Ilis et al. [33]Ra = 102–105, Pr = 0.71Left:hot, right:cold
χ = $10−4$$10−2$, AR=1–16Top,bottom:adiabatic
Oliveski et al. [34]Ra = 103$107$, Pr = 0.71Left:hot, right:cold
χ = $10−5$$10−2$, AR=0.25, 0.5, 1, 2, and 4Top,bottom:adiabatic
Singh et al. [20]Ra = 103–105, Pr = 0.71Left:cold, right:cold
ϕ = 15 deg, 45 deg, and 75 degTop: adiabatic, bottom:hot

The layout of the presentation of our findings is as follows. Section 2 describes the problem statement followed by a schematic representation of the computational domain with boundary conditions. Section 3 explains the fundamental equations and numerical method used in this study. The equations that quantify the heat transfer and entropy generation characteristics are illustrated in Sec. 4. Section 5 details the sensitivity studies and validation of the computational code used in the present investigation, followed by discussions on heat transfer and entropy generation characteristics. Conclusions in Sec. 6 summarize the observations and findings of this study.

## 2 Problem Definition

This numerical study is devoted to understanding the influence of orientation and wall-slip on entropy generation due to natural convection in a square enclosure. The Prandtl number (Pr) of the fluid inside the cavity is 0.71, and the entire analysis is performed by keeping the Rayleigh number constant at Ra = 105. The orientation of the cavity (ϕ), defined as the angle subtended by the isothermal hot wall with the horizontal axis and measured in the clockwise direction, is varied from 0 deg to 120 deg in steps of 15 deg. The wall-slip parameter is given using Knudsen number (Kn) as defined in Sec. 2.2, and for each case of orientation, the value of Kn is varied from 0 to 1.5 to analyze the variation in heat transfer and entropy generation characteristics.

### 2.1 Computational Domain.

Figures 1(a) and 1(b) illustrate the schematics of the computational domain, which is a square cavity of size L. At the base configuration (Fig. 1(a)), the bottom and top walls are, respectively, isothermally heated and cooled, and the lateral walls are kept insulated. Figure 1(b) represents the schematic of the domain when the cavity is oriented by angle, ϕ.

Fig. 1
Fig. 1
Close modal

### 2.2 Boundary Conditions.

The temperature and velocity BCs imposed on the walls are tabulated in Table 2. It should be noted that the BCs are defined for ϕ = 0 and is consistent for all other values of ϕ used in this study.

Table 2

Boundary conditions

Temperature BCVelocity BC
Adiabatic left wall$∂Θ∂x=0$$us$ = Kn$∂u∂n$
Isothermal top wall (cold)Θ = 0
Adiabatic right wall$∂Θ∂x=0$
Isothermal bottom wall (hot)Θ = 1
Temperature BCVelocity BC
Adiabatic left wall$∂Θ∂x=0$$us$ = Kn$∂u∂n$
Isothermal top wall (cold)Θ = 0
Adiabatic right wall$∂Θ∂x=0$
Isothermal bottom wall (hot)Θ = 1
The slip boundary condition at the walls is introduced using slip length (λ), defined as the fictitious distance beyond the rigid boundary where the velocity is extrapolated to zero [28]. It can also be described as the equivalent distance outside the solid boundary where the no-slip boundary condition is satisfied [36]. Here, the slip length is nondimensionalized using the size of the enclosure (L) to introduce a new variable named Knudsen number (Kn), which is reproduced from Legendre et al. Ref. [28] and given in the following equation:
$Kn=λL$
(1)
Thus, the nondimensional slip velocity (us) can be described as given in the following equation which is also similar to the one used by You and Moin [37]:
$us=Kn∂u∂n|wall$
(2)

where $∂u/∂n$ is the gradient of velocity normal to the wall.

## 3 Numerical Methodology

The 2D incompressible laminar flow under consideration is governed by the continuity, momentum, and energy equations in Cartesian coordinates as given below in the nondimensional form under the Boussinesq assumption
$Continuity:∂u∂x+∂v∂y=0$
(3)
$x−momentum:u∂u∂x+v∂u∂y=−∂p∂x+Pr(∂2u∂x2+∂2u∂y2)$
(4)
$y−momentum:u∂v∂x+v∂v∂y=−∂p∂y+Pr(∂2v∂x2+∂2v∂y2)+RaPrΘ$
(5)
$Energy:u∂Θ∂x+v∂Θ∂y=∂2Θ∂x2+∂2Θ∂y2$
(6)
The dimensionless quantities in the above equations are defined as follows:
$x=XL, y=YL, u=ULα, v=VLα, Θ=T−TcTh−Tc, p=PL2ρα2$
(7)

Here, x and y represent the nondimensional horizontal and vertical Cartesian coordinates in the X and Y directions. u and v are the nondimensional velocities corresponding to U and V, respectively, in the x and y directions. Θ is the nondimensional temperature and p is the dimensionless form of pressure. Tc is the cold wall temperature and Th is the temperature of the hot wall. α represents the thermal diffusivity, L denotes the size of the enclosure, and ρ is the density of the fluid.

Rayleigh number based on the size of the enclosure and Prandtl number in the momentum equations are defined as
$Ra=gβL3(Th−Tc)να$
(8)
$Pr=να$
(9)

where $ν$ is the kinematic viscosity of the fluid, g is the acceleration due to gravity acting downward in the negative y direction, and β is the coefficient of thermal expansion of the medium.

A second-order finite difference algorithm is employed to solve the governing equations in Cartesian coordinates. The velocity and pressure fields are decomposed using the projection algorithm, which is based on the fractional step method [38]. In this method, the momentum equations are integrated over time without considering the pressure field, which eventually produces an intermediate velocity field. The next step is the spatial projection of this intermediate velocity field to the divergence-free vector fields to update velocity and pressure [39]. The time marching is subjected to the CFL criterion [40]. The computations are continued till steady-state solutions are reached, and the L2 norms of u, v, and Θ are monitored to reduce below $10−6$ [40]. The continuity of mass is also checked below $10−6$. Note that similar studies based on the projection method have been conducted and reported earlier [4042].

## 4 Formulation of Heat Transfer and Entropy Generation

The heat transfer characteristics are quantified using the Nusselt number and entropy generation due to heat transfer and fluid friction irreversibilities using the EGM principle put forth by Bejan [6]. The respective formulations are as given below.

### 4.1 Nusselt Number.

The rate of heat transfer in the steady flow is computed at each wall and is expressed in terms of the local Nusselt number (Nul) and average Nusselt number (Nuavg)
$Nul=−∂Θ∂n|wall$
(10)
$Nuavg=1L∫0LNul dL$
(11)

where n is the unit normal to the small elemental length, dL.

### 4.2 Entropy Generation.

The dimensionless local entropy generation due to heat transfer and fluid friction for a 2D system can be written as
$SΘ=(∂Θ∂x)2+(∂Θ∂y)2$
(12)
$SΨ=χ{2[(∂u∂x)2+(∂v∂y)2]+(∂u∂y+∂v∂x)2}$
(13)
where $χ$ is known as the irreversibility distribution ratio and is given by the following equation:
$χ=μTk(αLΔT)2$
(14)
Here, in this study, the value of χ is taken as $10−4$ following Ilis et al. [33], and Anandalakshmi and Basak [24]. The total entropy, also known as entropy generation number, is given by the sum of entropy generation due to heat transfer and fluid friction
$Stotal=SΘ+SΨ$
(15)
The global quantities of entropy generation can be obtained by integrating the respective local quantities over the entire domain Ω; which gives
$〈SΘ〉=∫ΩSΘ dΩ$
(16)
$〈SΨ〉=∫ΩSψ dΩ$
(17)
The total entropy integrated over the domain is
$〈Stotal〉=〈SΘ〉+〈SΨ〉$
(18)
The relative strength of entropy generation due to heat transfer irreversibility is given by the local Bejan number (Be) which is defined as
$Be=SΘStotal$
(19)

## 5 Results and Discussion

The numerical code is thoroughly validated for its accuracy, and sensitivity studies have been performed to filter out the influence of mesh density on the results and to optimize the computational time. Numerical simulations are performed for Ra = 105 by varying the wall-slip (0 $≤$ Kn $≤$ 1.5) and orientation (0 $≤ϕ≤$ 120) of the cavity and steady-state solutions are obtained for all combinations in the parametric space. The authors have also simulated and analyzed the results for Ra = 104 and 5 × 105. It is observed that the heat transfer characteristics follow a trend similar to that of Ra = 105, and the entropy generation contours preserve a qualitative similarity as well. Hence, the observations at Ra = 104 and 5 × 105 have been omitted to maintain the brevity of the paper.

Sections 5.1 and 5.2 elaborate on the sensitivity studies and code validation followed by the discussion of flow patterns, heat transfer, and entropy generation characteristics from Secs. 5.35.8 for all the cases considered in this study.

### 5.1 Grid Sensitivity Studies.

The domain under consideration is divided into quadrilateral grids, and eight different mesh configurations, as displayed in Table 3, are considered for the grid sensitivity studies. At each configuration, the average Nusselt number (Nuavg) on the hot wall is computed for ϕ = 90 deg, Kn=0 and Ra = 105 and compared with that of the preceding configuration in terms of absolute percentage deviation. Following this comparison, a mesh size of 121 × 121 is chosen for the entire analysis presented in this paper.

Table 3

Grid sensitivity study: convergence of Nuavg values (case: ϕ = 90 deg, Kn=0 at Ra = 105). Absolute percentage deviation of the computed values at each configuration from that of the preceding configuration is given in brackets.

Grid size21 × 2141 × 4161 × 6181 × 81101 × 101121 × 121141 × 141161 × 161
Nuavg4.8764.7154.6174.5534.5234.5064.5054.504
(-)(3.31)(2.08)(1.37)(0.66)(0.39)(0.03)(0.01)
Grid size21 × 2141 × 4161 × 6181 × 81101 × 101121 × 121141 × 141161 × 161
Nuavg4.8764.7154.6174.5534.5234.5064.5054.504
(-)(3.31)(2.08)(1.37)(0.66)(0.39)(0.03)(0.01)

### 5.2 Validation.

The computational code developed is validated by comparing the present results with the values available in the literature. The Nuavg values on the hot wall (case: ϕ = 90 deg, Kn=0, and Ra = 105) obtained using the present code and tabulated in Table 4 are in agreement with the existing values from the previous studies.

Table 4

Comparison of Nuavg values on the hot wall (case: ϕ = 90 deg, Kn=0 at Ra = 105) obtained using the present code with the existing literature. Absolute percentage deviation of present values from the previously reported values is given in brackets.

$Nuavg$
ReferencesRa = 104Ra = 105
Present study2.252 (-)4.506 (-)
De Vahl Davis [1]2.243 (0.41)4.519 (0.29)
Wan et al. [2]2.254 (0.08)4.598 (2.00)
Tang and Tsang [43]2.250 (0.10)4.520 (0.31)
Hortmann et al. [44]2.245 (0.33)4.522 (0.35)
Quere and de Roquefort [45]2.238 (0.65)4.519 (0.29)
$Nuavg$
ReferencesRa = 104Ra = 105
Present study2.252 (-)4.506 (-)
De Vahl Davis [1]2.243 (0.41)4.519 (0.29)
Wan et al. [2]2.254 (0.08)4.598 (2.00)
Tang and Tsang [43]2.250 (0.10)4.520 (0.31)
Hortmann et al. [44]2.245 (0.33)4.522 (0.35)
Quere and de Roquefort [45]2.238 (0.65)4.519 (0.29)

### 5.3 Evolution of Flow Patterns Inside the Cavity.

Figure 2 shows the evolution of streamlines for different orientations (ϕ) and Knudsen numbers (Kn) at Ra = 105. For the base configuration (ϕ = 0 deg, Kn=0), a primary circulation in the clockwise direction is seen to occupy majority of the domain. In addition, two tiny secondary vortices having opposite spinning characteristics to that of the primary vortex inhabit the top left and bottom right corners of the cavity. A closer visual inspection reveals that with an increase in Kn, the primary circulation gets intensified, which could be due to the increase in slip velocity on the cavity walls, which is discussed in detail in Sec. 5.5.2. Additionally, for ϕ = 0 deg, the flow patterns are nearly invariant of Kn; however, they get strongly influenced by the orientation.

Fig. 2
Fig. 2
Close modal

With an increase in orientation (ϕ = 30 deg), the secondary vortices vanish, and the entire cavity can be seen occupied by the primary vortex. Beyond ϕ = 30 deg, the visualization of streamlines (at ϕ = 60 deg) witnesses the inception of two additional circulations near to the center of the cavity. With further increase in orientation (ϕ = 90 deg), these inner circulations begin to move away from the core of the cavity and align themselves toward the isothermal walls (see the last two rows of Fig. 2). Increase in cavity orientation (ϕ = 120 deg) further enhances the movement of inner circulations toward the hot and cold walls. The movement of these vortices also pushes the flow lines of the primary circulation toward the walls, thereby promoting the occurrence of strong velocity gradients near the walls. Moreover, the enlargement of inner circulations further obstructs the flow lines of the primary circulation, which eventually weakens the convection currents within the cavity. This could be the reason why the rate of heat transfer from the hot wall deteriorates beyond $ϕ=75 deg$, which is discussed in Sec. 5.4.

### 5.4 Heat Transfer Characteristics.

The heat convected at the hot wall is quantified using the Nusselt number and is evaluated using Eqs. (10) and (11). Figure 3 elucidates the variation in average Nusselt number (Nuavg) on the hot wall for various values of wall slip (Kn) and inclination (ϕ). It should be noted that, for a system in a steady-state, the Nuavg values on both hot and cold walls are the same, and we have considered the Nuavg on the hot wall throughout the analysis. It is observed that for a particular value of Kn, Nuavg initially increases as the cavity is inclined from $ϕ=0 deg$ to $ϕ=75deg$. At $ϕ=75deg$, Nuavg attains its peak value, indicating a maximum rate of heat transfer. Beyond this value of ϕ, Nuavg monotonically decreases. It is noticed that these occurrences repeat for all values of Kn considered in this study. Additionally, at any particular inclination angle, the Nuavg increases with an increase in Kn. With an increase in Kn, the slip velocity at the cavity walls increases, thereby enhancing convection within the cavity, which is accompanied by an increase in the Nuavg.

Fig. 3
Fig. 3
Close modal

Our investigations also ascertain some interesting facts on the heat transfer characteristics. As mentioned earlier, after $ϕ=75deg$, there is a change in trend in the Nuavgϕ curve for all values of Kn. It is inferred that up to $ϕ=75deg$, the increase in orientation favors an enhancement in heat transfer due to convection. The evolution of flow patterns, as discussed in Sec. 5.3, also favors the augmentation of heat transfer characteristics in this range of ϕ. The existence of three different patterns of streamlines for Kn=1.5, respectively, at ϕ = 0 deg, 30 deg, and 90 deg that are shown embedded in Fig. 3 demonstrates how the development of flow lines influence the rate of heat transfer. As illustrated, the secondary circulations at the top left and bottom right corners (see the streamlines at ϕ = 0 deg shown in Fig. 3) get completely suppressed as the primary circulation intensifies with increase in orientation (see the streamlines at ϕ = 30 deg shown in Fig. 3) which is reflected by an enhancement in Nuavg. The intensification of primary circulation is observed to strengthen the convection currents up to $ϕ=75deg$; however, beyond this value of ϕ, the cavity inclination and modification in flow lines begin to affect the mode of heat transfer. The convection currents cease to strengthen further, and conduction heat transfer gets gradually augmented thereafter, due to which there is a reduction in Nuavg after $ϕ=75 deg$. The enlargement of inner circulations, as discussed in Sec. 5.3 (also illustrated in Fig. 2), is found to attenuate the convection currents. Even though the inner circulations exist for ϕ = 60 deg, their presence become influential only after ϕ = 75 deg (see the streamlines at ϕ = 90 deg shown in Fig. 3). As such, the entire range of orientation can be classified into three different zones, namely, (i) convection augmentation zone where the rate of heat transfer continuously increases, (ii) transition zone where the convective heat transfer begins to decline gradually, and (iii) convection attenuation zone in which the heat transfer due to convection drastically reduces together with the emergence of conduction mode of heat transfer (see Fig. 3). The fluid flow attains stable density stratification, and convection discontinues after ϕ = 120 deg which eventually weakens heat transfer from the hot wall. This study specifically focuses on entropy generation mechanisms due to convective heat transfer and is the reason to evade the region of ϕ > 120 deg where heat transfer due to convection becomes considerably less significant.

### 5.5 Entropy Generation.

The heat transfer characteristics discussed in Sec. 5.4 have strong interconnection with the entropy generation within the cavity. The components that contribute to the total entropy (Stotal) are the entropy generation due to heat transfer ($SΘ$) and entropy generation due to fluid friction ($SΨ$) [32] hence, these parameters have been estimated using Eqs. (12) and (13) and analyzed separately to understand the relative significance of one another. $〈SΘ〉$ and $〈SΨ〉$ indicate the respective magnitudes of entropy generation due to heat transfer and fluid friction integrated over the domain and evaluated using Eqs. (16) and (17). $〈Stotal〉$ is the total entropy generation integrated over the domain which is the sum of $〈SΘ〉$ and $〈SΨ〉$.

The authors would like to take readers' attention to the case: ϕ = 90 deg, Ra = 105 that has been reported earlier by other investigators and is also a configuration widely used for benchmarking the results. As given in Fig. 4, the contours of entropy generation ($SΘ, SΨ$, and Stotal) and Bejan number (Be) computed using the present code are consistent with the findings of Soleimani et al. [31]. These results provide additional endorsement to our observations and also reiterate the accuracy of the code used in this study.

Fig. 4
Fig. 4
Close modal

Figures 5(a)5(c) illustrate $〈SΘ〉, 〈SΨ〉$, and $〈Stotal〉$ plotted for various orientations (ϕ) and Knudsen numbers (Kn). At a particular value of Kn, as evidenced in Fig. 5(a), $〈SΘ〉$ monotonically increases up to $ϕ=75deg$ and thereafter it decreases. The same trend is observed for all values of Kn. Interestingly, this is exactly the same variation pattern observed in the Nuavgϕ curves (Fig. 3) discussed in Sec. 5.4. It can also be seen that for a particular orientation, $〈SΘ〉$ increases with an increase in Kn, which is guaranteed for all values of ϕ.

Fig. 5
Fig. 5
Close modal

Figure 5(b) demonstrates the variation of integral quantity $〈SΨ〉$ with orientation and Knudsen number. It can be seen that $〈SΨ〉$ initially increases when the cavity is tilted up to an angle of 30 deg, and thereafter there is a monotonic reduction observed for the rest of the ϕ values. The $〈SΨ〉$ values registered at ϕ = 30 deg is also the maximum for all values of Kn. Interestingly, at a particular inclination angle, $〈SΨ〉$ decreases with an increase in Knudsen number, which is opposite to the variation observed for the $〈SΘ〉$ values (Fig. 5(a)). This could be due to the reason that, with an increase in Kn, the velocity gradients at the walls reduce, which in turn will limit the entropy generation due to fluid friction irreversibility. Note that, at ϕ = 120 deg, even though $〈SΨ〉$ decreases with Kn, the effect of Knudsen number on entropy generation due to fluid friction is found to be the least significant among all the cavity orientations considered in this study.

The total entropy generation (calculated using Eq. (18)) as illustrated in Fig. 5(c) strictly follows the variation trend as that of the entropy generation due to fluid friction; $〈SΨ〉$. This is because, at a particular Ra, when convection is the dominant mode of heat transfer, $〈SΨ〉$ is the major contributor to the total entropy generation. However, even though the Ra is kept constant, $〈SΘ〉$ does have an influence in the total entropy generation with its contribution ranging between 12% (at ϕ = 0 deg, Kn=0) and 47% (at ϕ = 120 deg, Kn=1.5). The total entropy generation increases with cavity orientation up to ϕ = 30 deg, and thereafter it monotonically reduces for the rest of the values of ϕ. Even though $〈Stotal〉$ is observed to reduce with increase in Kn at a particular orientation, the variation becomes marginal at the highest orientation considered in this study; ϕ = 120 deg. To summarize, the total entropy generation reaches its maximum when the cavity is oriented by an angle, ϕ = 30 deg whereas, the combined irreversibility effects get minimized at the highest orientation considered in this study; ϕ = 120 deg.

As reported by Kaluri and Basak [30,32], higher magnitudes of $SΘ$ are observed near the hot–cold junctions where heat exchange is maximized due to high thermal gradients. In this study, the isothermal walls are in direct contact with the adiabatic walls. Due to the existence of these thermally inactive walls between the source (hot wall) and the sink (cold wall), the direct interaction between the corresponding thermal boundary layers is restricted, which is the reason why the values of $〈SΘ〉$ are significantly less compared to that of $〈SΨ〉$ for most of the cases. It is also reported that, at higher Ra, the entropy generation due to fluid friction is substantially high [33,34]. Due to the strengthening of fluid movement, at higher Ra, the thermal energy transport is intensified. These reports further substantiate our findings that $〈SΨ〉$ is more than $〈SΘ〉$.

Sections 5.5.1 and 5.5.2 will discuss the reason behind the variation in entropy generation due to heat transfer and fluid friction while changing the cavity orientation and Knudsen number.

#### 5.5.1 Strain Rate Effects.

Having realized the significance of entropy generation due to fluid friction and its contribution to the total entropy in Sec. 5.5, it is now essential to understand the reason behind the variations in $〈SΨ〉$ while changing the Knudsen number and cavity orientation.

Let us now revisit Eq. (13) which represents the entropy generation due to fluid friction. Careful observation reveals an interesting fact that the terms on the R.H.S. of Eq. (13) have close resemblance with various terms in the 2D strain rate tensor in Eq. (20) which is adapted from Batchelor Ref. [46]
$(ϵxxϵxyϵyxϵyy)=(∂u∂x12(∂u∂y+∂v∂x)12(∂v∂x+∂u∂y)∂v∂y)$
(20)

Here, ϵxx and ϵyy are the components of the rate of linear strain and ϵxy and ϵyx are the rate of shear strain components.

Now, let us separate Eq. (13) mathematically to rewrite it as
$SΨ=χ{2[(∂u∂x)2+(∂v∂y)2]}︸A+χ{(∂u∂y+∂v∂x)2}︸B$
(21)
Here, A and B can be designated as the components of linear and shear strain rates manifested as follows:
$A=χ[2(ϵxx2+ϵyy2)]=SΨ,linear$
(22)
$B=χ[4(ϵxx2)]=SΨ,shear$
(23)

The above equations enable the authors to conclude that component A of the entropy generation due to fluid friction is proportional to the sum of the squares of rates of linear strain in the x and y directions, respectively whereas, the component B of entropy generation due to fluid friction is directly proportional to the square of the shear strain rate.

Figure 6 illustrates separately the variations in the linear ($〈SΨ,linear〉$) and shear ($〈SΨ,shear〉$) components of entropy generation due to fluid friction while changing the Knudsen number and orientation. The $〈SΨ〉$ϕ curves illustrated in Fig. 5(b) have been reproduced here (shown in dotted lines in Fig. 6) to understand which component of $SΨ$ is more influential in driving the entropy generation due to fluid friction. It can be seen that, irrespective of Kn, $〈SΨ,linear〉$ increases with orientation up to ϕ = 15 deg and thereafter it decreases for the rest of the inclination angles. Also, at a particular orientation, the magnitude of $〈SΨ,linear〉$ increases with increase in Kn. It is also noticed that the influence of Kn is more pronounced for cavity orientations in the range 30 $≤ϕ≤$ 60, and note that, when ϕ approaches 120 deg, the influence of Knudsen number on $〈SΨ,linear〉$ is found to be negligible. The effect of the shear component ($〈SΨ,shear〉$) on entropy generation due to fluid friction is significant. As evidenced in Fig. 6, the magnitude of $〈SΨ,shear〉$ increases up to ϕ = 30 deg, attains a maximum value and thereafter it decreases for the rest of the orientations. It is pertinent to note that, Fig. 5(b), which illustrates the variation in entropy generation due to fluid friction, has closer reflections in trend when compared to the $〈SΨ,shear〉$ϕ curves in Fig. 6 which is an indication that the rate of shear strain drives the variations in $SΨ$. Also, at a particular orientation, the magnitude of $〈SΨ,shear〉$ decreases with an increase in Knudsen number, which is also true with the $〈SΨ〉$ϕ curves.

Fig. 6
Fig. 6
Close modal

#### 5.5.2 Effect of Slip Velocity.

It is known that the introduction of slip conditions influences the boundary layer formation, which eventually modifies the vortical structures formed inside the enclosure, as seen in Fig. 2, thereby altering the overall fluid flow and heat transfer dynamics. As we have seen earlier how flow dynamics can contribute to the heat transfer and entropy generation characteristics, this section is devoted to examining the influence of slip walls on irreversibilities due to heat transfer and fluid friction. For this, the slip velocities (us) at the isothermal and adiabatic walls are evaluated. Note that us at the corresponding locations of hot and cold isothermal walls are identical, and for our discussions, we have considered us at the hot wall.

Figures 7(a) and 7(b) demonstrate the variation in slip velocities, respectively, at the midpoint of the isothermal ($us,iso$) and adiabatic walls ($us,adi$) plotted against cavity orientation for all values of Knudsen numbers at Ra = 105. The selection of midpoint for the evaluation of slip velocity is random, and during our investigations, it is observed that irrespective of the location, the slip velocities follow the same trend when plotted against the orientation angles.

Fig. 7
Fig. 7
Close modal

It can be noticed that, with an increase in Kn, the relative velocity between the cavity walls and adjacent fluid layer increases, which in turn enhances the convection currents. As observed in Fig. 7, where the slip velocity at the midpoint of isothermal walls is plotted against the cavity orientation, the orientation effects are seen to operate in harmony with the Knudsen number to increase the slip velocity up to ϕ = 75 deg. Interestingly, beyond this angle, the slip velocity exhibits a monotonically decreasing trend for the rest of the orientation angles. Note that, for all values of Kn, the trend exhibited by the $us,iso$ϕ curves is the same as shown by the $〈SΘ〉$ϕ curves in Fig. 5(a). This is conclusive evidence that the slip velocity at isothermal walls is essentially a parameter that can influence entropy generation due to heat transfer.

Similarly, the slip velocity at the midpoint of adiabatic walls ($us,adi$) is a strong indicator of the variation in the magnitude of the linear component of entropy generation due to fluid friction ($〈SΨ,linear〉$). As illustrated in Fig. 7(b), with increase in orientation, $us,adi$ increases up to ϕ = 15 deg, attains a maximum value and thereafter it monotonically reduces. Close observations reveal that the variations observed in $〈SΨ,linear〉$ϕ curves presented in Fig. 6 share strong reflections with that of Fig. 7(b).

### 5.6 Distribution of Entropy Generation Due to Heat Transfer and Fluid Friction.

Figures 8 and 9 illustrate the respective distributions of entropy generation due to heat transfer ($SΘ$) and fluid friction ($SΨ$) for various values of Kn and ϕ considered in this study. These contours are useful in spotting the entropy active regions in the enclosure and how the variation in slip and orientation influence the irreversibility distribution within the domain. We have used arrows to indicate the regions, where the entropy generation is maximum, otherwise known as the entropy active zones. Since, it is established earlier in this discussion (Sec. 5.5) that the total entropy generation has the same characteristics as that of the entropy generation due to fluid friction, the former has not been further discussed here.

Fig. 8
Fig. 8
Close modal
Fig. 9
Fig. 9
Close modal

As evidenced in Fig. 8, the magnitude of $SΘ$ is maximum near the isothermal hot and cold walls. It can also be seen that, for any orientation, with an increase in Kn, $SΘ$ gets intensified within the domain, which can be attributed to the increase in convective currents and enhancement in the rate of heat transfer from the hot isothermal wall. This is also reflected in Fig. 5(a) where the value of $〈SΘ〉$ increases with Kn for all orientations. Note that the heat transfer characteristics discussed in Sec. 5.4 that is observed to share quantitative similarity with $SΘ$ further substantiate the distribution of entropy generation due to heat transfer. It is also worth noting that for ϕ = 0 deg, the entropy generation is maximum near the junctions (top-left and bottom-right) of isothermal–adiabatic walls, which are diagonally connected to each other in the square cavity. With the increase in orientation up to ϕ = 60 deg, these spots elongate and move closer to the isothermal–adiabatic junction. Higher orientation angles (ϕ = 90 deg and ϕ = 120 deg) however, restrict the elongation. This needs to be correlated with the diminishing trend observed both in Nuavg (Fig. 3) and $〈SΘ〉$ (Fig. 5(a)). With further increase in orientation (ϕ = 120 deg), the active spots of entropy generation due to heat transfer are observed to inhabit in the closest proximity to the junction of isothermal and adiabatic walls. Also, for all cases considered, most of the enclosure is dormant as far as the entropy generation is concerned, and the irreversibilities are produced only near the walls. This is because, the rate of heat transfer is maximum near the isothermal walls where the major heat transportation takes place, and the rest of the enclosure is subjected only to the distribution of heat energy.

The contours of entropy generation due to fluid friction depicted in Fig. 9 reveal several interesting aspects. At ϕ = 0 deg, the active spots of $SΨ$ are located in the vicinity of all four walls, among which the ones near the adiabatic walls are more pronounced. Note that no active spots of entropy generation due to heat transfer ($SΘ$) could be observed near adiabatic walls (Fig. 8) since heat transfer is absent across them. With the increase in orientation, the entropy active regions near the isothermal walls elongate, whereas those near the adiabatic walls gradually diminish. At ϕ = 90 deg and 120 deg, the active spots in the proximity of adiabatic walls vanish entirely. Also, for any orientation, with an increase in the value of Kn, the active regions of entropy generation due to fluid friction reduce in size and magnitude. This is also reflected in Fig. 5(b) where the magnitude of $〈SΨ〉$ reduces with Kn for all orientations. Similar to the observations made with respect to the contours of entropy generation due to heat transfer (Fig. 8), except for the walls, the majority of the enclosure is inhabited by entropy dormant regions. This is because frictional effects are concentrated only in the vicinity of the boundaries. Even though the fluid layers interact with each other during their movements, there is negligible entropy generation due to such occurrences. The loss of available energy due to these canonical interactions is observed to be less significant compared to the entropy generation due to wall–fluid interactions.

### 5.7 Effect of Knudsen Number and Orientation on Entropy Active Regions.

Figure 10 illustrates the contours of the Bejan number (Be) for all cases considered in this study. As the Bejan number is the ratio of $SΘ$ over Stotal and is used to predict the relative dominance of $SΘ$ (or $SΨ$) [32], we have used these contours to differentiate the regions of entropy generation due to heat transfer from that due to fluid friction. It should be noted that in the previous section (Sec. 5.6), we have discussed the evolution of entropy generation considering the absolute values of $SΘ$ and $SΨ$. Here, we have used the fractional value, defined as Bejan number (Be), and evaluated using Eq. (19) to distinguish the regions of entropy generation where $SΘ$ (or $SΨ$) is prevalent. Previous literature have reported on the average values of Be [30,33]. Here, in Fig. 10, we have plotted the Be contours to identify the location and type of irreversibility dominant within the cavity.

Fig. 10
Fig. 10
Close modal

As evidenced in Fig. 10, for ϕ = 0 deg, the regions of entropy generation due to heat transfer can be seen to occupy all the four corners of the cavity of which, the ones at the top-left and bottom-right are more pronounced. These are the regions of secondary circulations as illustrated in Fig. 2, and these circulations enhance convection, thereby increasing the contribution of $SΘ$ in these locations. Within the cavity, except for the scattered tiny pockets of $SΘ$, irreversibility due to fluid friction is undoubtedly the dominant mode of entropy generation. Even though visual inspection of Bejan number contours does not truly reveal the variation in entropy generation with Knudsen number for a particular orientation, the increase in $SΘ$ and reduction in $SΨ$ with Kn can be accurately predicted by the estimation of average entropy generation over the entire domain as discussed already in the previous section (Sec. 5.5). With increase in orientation (ϕ = 30 deg and 60 deg), the dominance of $SΘ$ at the top-left and bottom-right corners gradually shrink. At the same time, some of the minute pockets of $SΘ$ spotted within the cavity move toward the isothermal walls, stretch and begin to inhabit there. This could be due to the influence of the components of buoyancy force along the hot and cold walls, which activate only when the cavity is inclined. Singh et al. [20] have also mentioned the significance of buoyancy components in their report. However, further investigations are required to understand the effect of these buoyancy components on localizing the regions of entropy generation. Further, an increase in orientation (ϕ = 90 deg) witnesses the coalescence of the regions of $SΘ$ observed near the center of the cavity, which also promotes the emergence of $SΘ$ dominant regions seen aligned with the isothermal walls. Meanwhile, the total entropy keeps reducing. For higher orientation angles, the space occupied by the regions of entropy generation due to fluid friction shrinks as, at these orientations, the influence of convection as the dominant mode of heat transfer gradually weakens. It can also be seen that up to ϕ = 75 deg, $SΨ$ dominates over $SΘ$ after which the contribution of irreversibility due to fluid friction becomes nearly equal to that of the one due to heat transfer (ϕ = 120 deg).

### 5.8 A Note on the Ratio of Maximum Velocity Components Within the Cavity.

Even though our investigations have revolved around the analysis of entropy generation, we have found some interesting occurrences in the velocity magnitudes within the cavity. To record our observations, we propose a new parameter called maximum velocity ratio (MVR), which is defined as the ratio of maximum vertical (vmax) and horizontal (umax) velocities in the enclosure (MVR = $vmax/umax$). As illustrated in Fig. 11, in which MVR is plotted against orientation (ϕ), it is pertinent to note that, in the convection augmentation zone and transition zone, at any particular value of ϕ, MVR is nearly a constant and does not vary with Kn. The curves perfectly match in these zones, whereas, in the convection attenuation zone, MVR is slightly influenced by the variation in slip. This could be due to the emergence of conduction, as we have also observed in Fig. 3. Among the inclination angles considered in this study, is also observed that up to ϕ = 45 deg, vmax exceeds umax. At ϕ = 60 deg, vmax is nearly equal to umax and for the rest of the configurations, vmax is lesser than umax.

Fig. 11
Fig. 11
Close modal

## 6 Conclusions

Using a second-order accurate, massively parallel in-house code that works on the projection algorithm, this study numerically investigates the effect of slip wall (introduced using Knudsen number, Kn) and orientation (ϕ) on the entropy generation due to natural convection in a square enclosure for Rayleigh number (Ra) = 105. The Knudsen number (Kn) is varied from 0 (no-slip) to 1.5 in steps of 0.5 and the orientation is varied from 0 deg to 120 deg with increments of $15deg$. The bottom and top walls of the domain are isothermally heated and cooled by keeping the lateral walls adiabatic, and the two components of entropy generation, i.e., entropy generation due to heat transfer ($SΘ$) and entropy generation due to fluid friction ($SΨ$) are separately analyzed. While observing the influence of Kn and ϕ on the heat transfer and entropy generation characteristics, we have come across compelling evidence that points toward the following facts.

1. The introduction of slip and orientation produces three different zones within the enclosure, namely,

2. (i) Convection augmentation zone, where heat transfer due to convection enhances with Kn and ϕ,

3. (ii) Transition zone, where the convective heat transfer begins to decline gradually and

4. (iii) Convection attenuation zone, which is distinguished by the sharp decline in convection together with the emergence of conduction.

5. The presence of corner vortices and inner circulations in the vicinity of the center of the enclosure is observed to play a critical role in the distribution of heat transfer and entropy generation within the cavity.

6. Even though Ra is kept constant, our observations reveal that entropy generation due to heat transfer ($SΘ$) is a function of slip and cavity orientation. It is also noticed that the variation in $SΘ$ shares quantitative similarity with Nuavg.

7. The shear component of entropy generation due to fluid friction ($SΨ,shear$) plays a significant role in driving the variations in $SΨ$. For most of the cases considered, $SΨ$ dominates over $SΘ$ for the same value of Kn and ϕ. Following the concept of EGM, among all configurations considered, the least values of $SΘ, SΨ$, and Stotal are observed at ϕ = 120 deg.

8. The slip velocity on the isothermal wall ($us,iso$) is truly an influential parameter that can predict the variation in $SΘ$ whereas it is the slip velocity on the adiabatic walls ($us,adi$) that contribute to the change in one of the components of the entropy generation due to friction, $SΨ,linear$, which is influenced by the change in linear strain rate.

9. Visualization of the contours of entropy generation due to heat transfer and fluid friction reveal the presence of entropy active zones in the vicinity of isothermal hot and cold walls. While varying the orientation, these active zones of $SΘ$ appearing as clusters of closely spaced contours elongate and migrate toward the junction of isothermal–adiabatic walls.

10. A new parameter called MVR is proposed to highlight the variation in the velocity components while changing the values of Kn and ϕ. It is observed that MVR is nearly invariant in the convection augmentation and transition zones. With the emergence of conduction, MVR is slightly influenced by the slip in the convection attenuation zone.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

## Nomenclature

Be =

Bejan number

g =

acceleration due to gravity, $m/s2$

k =

thermal conductivity, W/m K

Kn =

Knudsen number

L =

size of the enclosure, m

Nuavg =

average Nusselt number

Nul =

local Nusselt number

p =

nondimensional static pressure

P =

static pressure, $N/m2$

Pr =

Prandtl number

Ra =

Rayleigh number

Stotal =

total entropy generation

$SΘ$ =

entropy generation due to heat transfer

$SΨ$ =

entropy generation due to fluid friction

T =

temperature, K

Tc =

cold wall temperature, K

Th =

hot wall temperature, K

Tref =

reference temperature, K

u, v =

nondimensional horizontal and vertical velocities

U, V =

horizontal and vertical velocities, m/s

umax =

maximum horizontal velocity in the cavity

us =

nondimensional slip velocity

$us,adi$ =

nondimensional slip velocity at the midpoint of adiabatic walls

$us,iso$ =

nondimensional slip velocity at the midpoint of isothermal walls

vmax =

maximum vertical velocity in the cavity

x, y =

nondimensional horizontal and vertical coordinates

X, Y =

horizontal and vertical coordinates, m

### Greek Symbols

Greek Symbols
α =

thermal diffusivity, $m2/s$

β =

coefficient of thermal expansion, $K−1$

$ΔT$ =

temperature difference, K

Θ =

nondimensional temperature

λ =

slip length, m

μ =

kinematic viscosity, $Ns/m2$

ν =

dynamic viscosity, $m2/s$

ρ =

density of the fluid, $kg/m3$

ϕ =

cavity orientation, deg

χ =

irreversibility distribution ratio

### Abbreviations

Abbreviations
BC =

boundary condition

EGM =

entropy generation minimization

MVR =

maximum velocity ratio

NC =

natural convection

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