Unsteady Natural Convection in an Initially Stratiﬁed Air-Filled Trapezoidal Enclosure Heated from Below

: Natural convection is intensively explored, especially in a valley-shaped trapezoidal enclosure, because of its broad presence in both technical settings and nature. This study deals with a trapezoidal cavity, which is initially ﬁlled with linearly stratiﬁed air. Although the sidewalls remain adiabatic, the bottom wall is heated, and the top wall is cooled. For the stratiﬁed ﬂuid (air), the temperature of the ﬂuid adjacent to the top and the bottom walls is the same as that of the walls. Natural convection in the trapezoidal cavity is simulated in two dimensions using numerical simulations, by varying Rayleigh numbers (Ra) from 10 0 to 10 8 with constant Prandtl number, Pr = 0.71, and aspect ratio, A = 0.5. The numerical results demonstrate that the development of natural convection from the beginning is dependent on the Rayleigh numbers. According to numerical results, the development of transient ﬂow within the enclosure owing to the predeﬁned conditions for the boundary may be categorized into three distinct stages: early, transitional, and steady or unsteady. The ﬂow characteristics at each of the three phases and the impact of the Rayleigh number on the ﬂow’s growth are quantiﬁed. Unsteady natural convection ﬂows in the enclosure are described and validated by numerical results. In addition, heat transfer through the bottom and the top surfaces is described in this study.


Introduction
In an enclosure, natural convection has gained substantial interest among academics since it can be found in a variety of applications and has a big impact on thermal characteristics.Natural convection has been studied inside various shapes of enclosures with numerous boundary conditions to investigate thermal behavior, as well as fluid flow, as natural convection is used in an array of technical applications, ranging from geophysics, geothermal reservoirs, and building insulation to industrial separation processes and so forth.Unsteady natural convection in a differentially heated cavity has attracted extensive attention in the scientific literature.Many researchers [1][2][3][4][5][6] provided comprehensive investigations for regular enclosures (e.g., rectangular, square and triangular) using various numerical models, indicating that multiple investigations have been performed to gain a fundamental understanding of unsteady natural convection flows and heat transfer characteristics in an enclosure.
Due to the importance of attics for occupant thermal comfort in buildings and the resulting energy costs for heating and air conditioning, increased research activities have been carried out on subjects related to heat transfer in attics over the last 40 years.Saha et al. [7,8] examined heat transport through attics under periodic thermal forcing and cooling inclined walls.Natural convection flows within V-shaped triangle enclosures with opposite boundary conditions appeared to be well studied due to their natural presence.Transitions from symmetric steady to asymmetric unsteady flow were studied by Bhowmick et al. [9,10] in a V-shaped triangular enclosure heated from below and cooled from the top for both air and water.Different flow mechanisms were found for the different fluids.Wang et al. [11] has experimented with natural convection in a V-shaped enclosure with the same boundary conditions of [10].
Any triangular, square, or rectangular cavity is inadequate for numerous engineering systems, as well as geophysical circumstances where the enclosure geometry varies or contains extra tending walls.Natural convection in a trapezoidal enclosure is far more difficult to examine than in any regular enclosures because of the sloped walls.This complicated geometry needs a precise and large effect in mesh creation and code development.However, there were a number of studies on natural convection that focused on trapezoidal enclosures.Iyican et al. [12,13] considered trapezoidal cavity with boundary conditions of a heated base wall and parallel cylindrical cooled top wall to investigate natural convection of the cavity, using experimental and computational methods.Lee [14] reported a theoretical and experimental investigation of a nonrectangular enclosure, in which two 45 • inclined sides of a trapezoidal cross-section were chosen, with different heating conditions.Lam et al. [15] found analogous findings for a trapezoidal enclosure with cooled inclined top wall, heated bottom wall, and insulated vertical sidewalls.
Lee [16] numerically examined the fluid flow and the heat transfer passed through a cold chamber of a trapezoidal enclosure, where heated fluid was supposed to flow into one end of the chamber from a depth below the surface and was removed from the other end at a different depth.Lee [17] and Peri [18] showed numerical findings in the case of laminar natural convection within a trapezoidal cavity with inclined sidewalls kept at varying constant temperatures, and adiabatic top and base walls for Ra ≤ 10 6 .Sadat and Salagnac [19] used a finite element based on the control-volume approach to compute the similar geometry for Rayleigh numbers from 10 3 to 2 × 10 5 .Kuyper and Hoogendoorn [20] examined laminar natural convection flow within a trapezoidal cavity in order to see the effect of the flow by the inclination angle, as well as the relationship between the Ra and the average Nu.Moukalled and Darwish [21] looked at how heat transfer was affected by installing baffles upon the top inclined walls inside trapezoidal cavities.Boussaid et al. [22] examined thermal heat transfer inside a trapezoidal chamber in which the base wall was heated and the tending upper half was cooled.The influence of natural convection flow within a trapezoidal cavity was examined by Natarajan et al. [23] under the conditions of a heated base wall, as well as linearly heated and cooled vertical walls, but with no insulation on the top wall.Hammami et al. [24] investigated the fixed heat and mass transport processes within a trapezoidal enclosure using a binary air-water vapor mixture.Later, Natarajan et al. [25] explored natural convection flow inside a trapezoidal cavity in which, on the one hand, the base wall was both consistently and inconsistently warmed, and the upward walls were, through a steady temperature shower, kept cool, where the upper wall again remained insulated.Basak et al. [26] examined natural convection energy fluxes in trapezoidal enclosures in which the top walls were insulated, whereas the bottom walls were heated, and sidewalls were cooled.
For three vertex angles in trapezoidal isosceles, the advancement of natural convection oscillatory flow patterns was examined by Noah and Daniel [27].Mustafa and Ghani [28] explored a natural convection flow inside a trapezoidal cavity with partially heated bottom wall and cooled vertical walls through a constant temperature bath and a well-insulated top wall.By means of the 'element-based finite volume method', Silva et al. [29] studied natural convection inside trapezoidal enclosures.Gholizadeh et al. [30] explored the natural convection inside a trapezoidal enclosure where the right inclined wall was partially heated, by means of the finite difference method.In a porous trapezoidal enclosure saturated through a power-law non-Newtonian fluid, Yazdani et al. [31] considered natural convection, as well as entropy production.To better comprehend the effect of heating length on the active bottom wall, Gowda et al. [32] observed natural convection within the cavity of a trapezoid under the condition that the base wall was partly heated, the upper wall was adiabatic, and the inclined wall remained at a fixed cooled temperature.
Various boundary conditions were used in the trapezoidal cavity by various researchers, according to the above literature studies.However, an initially stratified air-filled trapezoidal enclosure with a cooled top horizontal wall and a heated bottom horizontal wall, along with inclined walls that are kept adiabatic, is still ambiguous, which encouraged this research.It has direct application to environmental fluid dynamics, which deals with the heat transfer and airflow process in the thermal stratification environment.Thus, using two-dimensional numerical simulations for Ra = 10 0 to 10 8 , Pr = 0.71, and A = 0.5, the transitional flow in the trapezoidal cavity is considered in this article.The influence of Ra on fluid flow and heat transfer is thoroughly examined.

Problem Formulations
This study considers a trapezoidal enclosure of height H, as well as a horizontal length of the top, 2L, where L = 2H, i.e., A = H/L = 0.5.Figure 1 illustrates a nondimensional physical model with boundary conditions.A tiny percentage (4% of L) of each top corner was sliced to dispense with the singularity around the position between inclination and upper walls, and cutting walls were subject to an adiabatic thermal state.A fluid in the cavity with Pr = 0.71 was considered, which was, in the beginning, linearly stratified as having the highest T = T h temperature at the bottom and the lowest T = T c temperature at the top.The boundaries were non-slip.
Processes 2022, 10, x FOR PEER REVIEW 3 of 15 [31] considered natural convection, as well as entropy production.To better comprehend the effect of heating length on the active bottom wall, Gowda et al. [32] observed natural convection within the cavity of a trapezoid under the condition that the base wall was partly heated, the upper wall was adiabatic, and the inclined wall remained at a fixed cooled temperature.Various boundary conditions were used in the trapezoidal cavity by various researchers, according to the above literature studies.However, an initially stratified airfilled trapezoidal enclosure with a cooled top horizontal wall and a heated bottom horizontal wall, along with inclined walls that are kept adiabatic, is still ambiguous, which encouraged this research.It has direct application to environmental fluid dynamics, which deals with the heat transfer and airflow process in the thermal stratification environment.Thus, using two-dimensional numerical simulations for Ra = 10 0 to 10 8 , Pr = 0.71, and A = 0.5, the transitional flow in the trapezoidal cavity is considered in this article.The influence of Ra on fluid flow and heat transfer is thoroughly examined.

Problem Formulations
This study considers a trapezoidal enclosure of height H, as well as a horizontal length of the top, 2L, where L = 2H, i.e., A = H/L = 0.5.Figure 1 illustrates a nondimensional physical model with boundary conditions.A tiny percentage (4% of L) of each top corner was sliced to dispense with the singularity around the position between inclination and upper walls, and cutting walls were subject to an adiabatic thermal state.A fluid in the cavity with Pr = 0.71 was considered, which was, in the beginning, linearly stratified as having the highest T = Th temperature at the bottom and the lowest T = Tc temperature at the top.The boundaries were non-slip.In a trapezoidal enclosure, natural convection of stratified air is assumed.The below set of governing equations with the Boussinesq approximation regulates the progress of natural convection flows in the enclosure [7].In a trapezoidal enclosure, natural convection of stratified air is assumed.The below set of governing equations with the Boussinesq approximation regulates the progress of natural convection flows in the enclosure [7].
The followings are the dimensionless variables that were used: ( The three governing parameters, which are aspect ratio (A), Pr, and Ra (see Ref. [33] for details), influence the natural convective flows in the enclosure that can be expressed as follows: After adding the aforementioned dimensionless variables, Equations ( 1)-( 4) become (for details see Ref. [10]) ∂U ∂X

Timestep and Grid Dependency Tests
In this study, ANSYS FLUENT 17.0, a finite-volume-based fluid simulation software, was used to enable the high Rayleigh number flows ).In order to find solutions to the governing Equations ( 7)- (10) and other conditions, the SIMPLE scheme was used.Using the QUICK scheme (see Leonard and Mokhtari [37]), the advection term was discretized.Central differencing, along with second-order accuracy, was used to discretize the diffusion terms.Moreover, a second-order implicit time-marching scheme was employed for the unsteady term.
The grid and timestep dependency tests were also performed for the greatest Rayleigh number, Ra = 10 8 , used in this study.Three symmetrical meshes of 225 × 75, 300 × 100, and 375 × 125 were created nonuniformly using the application ANSYS ICEM, by way of coarser grids in the interior area and finer grids around the edges.From a width of at least 0.002 adjacent to the wall to the width of 0.02 in the interior, the mesh of 300 × 100 was increased at a rate of 3%.At position P 2 (0, 0.4), employing various grids together with timesteps, the temperature timeseries was computed for Ra = 10 8 as depicted in Figure 2. The results evidently show that temperatures predicted with various meshes and timesteps were constant in the initial phases, but somewhat deviated in the fully developed stage.
Table 1 shows the maximum variation of the average temperature at the fully developed stage for the different meshes and timesteps.The highest contrast of the temperature between the coarsest (225 × 75) and finest (300 × 100) meshes was around 1.42%, while the highest difference between the finer meshes (300 × 100) and (375 × 125) was approximately Processes 2022, 10, 1383 5 of 14 0.34%.Therefore, considering the computational cost, a mesh of 300 × 100 and a timestep of 0.01 were used in the numerical simulation.Table 1 shows the maximum variation of the average temperature at the fully developed stage for the different meshes and timesteps.The highest contrast of the temperature between the coarsest (225 × 75) and finest (300 × 100) meshes was around 1.42%, while the highest difference between the finer meshes (300 × 100) and (375 × 125) was approximately 0.34%.Therefore, considering the computational cost, a mesh of 300 × 100 and a timestep of 0.01 were used in the numerical simulation.

Validation
Validation of a model is an integral part of a numerical study.Hence, the numerical results of the present study were compared to those of Basak et al. [26], who used the finite element method to solve the governing equations for laminar natural convection heat transfer in a trapezoidal cavity heated isothermally from below, while maintaining a fixed cold temperature on the other vertical walls and a well-insulated top wall.With Ra = 10 5 , Pr = 0.71, and an inclination angle of φ = 45°, the dimensionless parameters were used to conduct the comparison.The current findings and the numerical findings of Basak et al. [26] for temperature contours in the trapezoidal enclosure exhibited excellent agreement, as shown in Figure 3.

Validation
Validation of a model is an integral part of a numerical study.Hence, the numerical results of the present study were compared to those of Basak et al. [26], who used the finite element method to solve the governing equations for laminar natural convection heat transfer in a trapezoidal cavity heated isothermally from below, while maintaining a fixed cold temperature on the other vertical walls and a well-insulated top wall.With Ra = 10 5 , Pr = 0.71, and an inclination angle of ϕ = 45 • , the dimensionless parameters were used to conduct the comparison.The current findings and the numerical findings of Basak et al. [26] for temperature contours in the trapezoidal enclosure exhibited excellent agreement, as shown in Figure 3. Table 1 shows the maximum variation of the average temperature at the fully developed stage for the different meshes and timesteps.The highest contrast of the temperature between the coarsest (225 × 75) and finest (300 × 100) meshes was around 1.42%, while the highest difference between the finer meshes (300 × 100) and (375 × 125) was approximately 0.34%.Therefore, considering the computational cost, a mesh of 300 × 100 and a timestep of 0.01 were used in the numerical simulation.

Validation
Validation of a model is an integral part of a numerical study.Hence, the numerical results of the present study were compared to those of Basak et al. [26], who used the finite element method to solve the governing equations for laminar natural convection heat transfer in a trapezoidal cavity heated isothermally from below, while maintaining a fixed cold temperature on the other vertical walls and a well-insulated top wall.With Ra = 10 5 , Pr = 0.71, and an inclination angle of φ = 45°, the dimensionless parameters were used to conduct the comparison.The current findings and the numerical findings of Basak et al. [26] for temperature contours in the trapezoidal enclosure exhibited excellent agreement, as shown in Figure 3.

Results and Discussions
For Ra = 10 0 to 10 8 , Pr = 0.71, and A = 0.5, the transient evolution of the flow in an initially stratified air-filled trapezoidal cavity is discussed below in response to consistent heating through the base and similar cooling via the top surfaces using a computational fluid dynamics approach.

Development of the Transient Flow
For Ra = 10 0 to 10 2 , it was found that there was no ascending or descending plume in the flow development, i.e., the flow was always steady under a conduction dominance for those Rayleigh numbers.For the sake of brevity, results are not presented here for Ra = 10 0 to 10 2 .Accordingly, for Ra = 10 3 to 10 8 , the general characteristics of flow development in a trapezoidal enclosure are presented (see Figures 4-6).The development of the flow for these Rayleigh numbers, according to the numerical simulations, may be divided into the following: early stage, transitional stage, and steady or unsteady stage.

Results and Discussions
For Ra = 10 0 to 10 8 , Pr = 0.71, and A = 0.5, the transient evolution of the flow in an initially stratified air-filled trapezoidal cavity is discussed below in response to consistent heating through the base and similar cooling via the top surfaces using a computational fluid dynamics approach.

Development of the Transient Flow
For Ra = 10 0 to 10 2 , it was found that there was no ascending or descending plume in the flow development, i.e., the flow was always steady under a conduction dominance for those Rayleigh numbers.For the sake of brevity, results are not presented here for Ra = 10 0 to 10 2 .Accordingly, for Ra = 10 3 to 10 8 , the general characteristics of flow development in a trapezoidal enclosure are presented (see Figures 4-6).The development of the flow for these Rayleigh numbers, according to the numerical simulations, may be divided into the following: early stage, transitional stage, and steady or unsteady stage.

Flow at the Early Stage
In the trapezoidal enclosure, the air is at first stratified, as indicated in the preceding sections.At the beginning of the numerical studies, the instant conditions for isotherms are created across the surfaces that first cool the cavity via the upper surface and then make it warm through the bottom.A thermal boundary layer forms along all internal surfaces as a result.The cooling thermal boundary layer is near the top wall, whereas the heating thermal boundary layer is alongside the bottom surface.The lower section produces the heating thermal boundary layer, while the top section of the upper layer produces the cooling thermal boundary layer.The progress of the thermal boundary layers through time is depicted in Figure 4, exhibiting isotherms and streamlines (Figure 4a-e) at τ = 6 after startup.
At the early stage, the core's fluid stays isothermal, as shown in Figure 4, in spite of the expansion of the thermal boundary layer, at the initial temperature.Because the bottom part of the enclosure is heated and the top section is cooled, the heated fluid from the bottom travels via the boundary layer toward the upper parts.On the contrary, the boundary layer transports cooled air to the bottom from the top.Both warm and cold

Flow at the Early Stage
In the trapezoidal enclosure, the air is at first stratified, as indicated in the preceding sections.At the beginning of the numerical studies, the instant conditions for isotherms are created across the surfaces that first cool the cavity via the upper surface and then make it warm through the bottom.A thermal boundary layer forms along all internal surfaces as a result.The cooling thermal boundary layer is near the top wall, whereas the heating thermal boundary layer is alongside the bottom surface.The lower section produces the heating thermal boundary layer, while the top section of the upper layer produces the cooling thermal boundary layer.The progress of the thermal boundary layers through time is depicted in Figure 4, exhibiting isotherms and streamlines (Figure 4a-e) at τ = 6 after startup.
At the early stage, the core's fluid stays isothermal, as shown in Figure 4, in spite of the expansion of the thermal boundary layer, at the initial temperature.Because the bottom part of the enclosure is heated and the top section is cooled, the heated fluid from the bottom travels via the boundary layer toward the upper parts.On the contrary, the boundary layer transports cooled air to the bottom from the top.Both warm and cold fluids meet in the middle of the top wall and release into the core.There remains another heated boundary layer near the bottom section that begins to expand, as the thermal forcing begins.At this moment, the isothermal difference reveals the thicknesses of the thermal layer barrier to the center growing with time.The streamlines show that Ra = 10 3 to 10 5 has two weak revolving cells, while Ra = 10 6 to 10 8 has four weak rotating cells.The isotherms and streamlines for different Ra remain symmetric with regard to the cavity's y-axis line at this stage.

Transitional Stage
The formation of convective instabilities marks the flow in the form of ascending and descending plumes at the transitional phase.Through the warming of the bottom portion, the horizontal thermal boundary layer, which has warmed air under colder air and is unstable due to 'Rayleigh-Bénard instabilities', is formed.When the critical conditions are fulfilled, the hot thermal boundary layer becomes unsteady.In this regime, Figures 5 and 6 display the streamlines and isotherms at various periods.It has previously been noted that when air of two different temperatures passes through the boundary layer around the midpoint of inclined surfaces, it invariably travels downward.Subsequently, the heated air plume then moves to the cavity's core, while the cooled air plume goes to the lower portion from the upper layer.Figure 5a,b show that the flow becomes symmetric and steady at τ = 20 for Ra = 10 3 to 10 4 .The flow grows stronger and finally asymmetric as time passes due to pitchfork bifurcation.The figures demonstrate the asymmetric isotherms and streamlines for the higher Rayleigh numbers.It is without a doubt significant that, with a greater Rayleigh number, the flow oscillates for quite a long period.The bifurcation continues to rehash left and close to the symmetric focal line while oscillating.Figure 6c,d depict that the flow becomes symmetric and steady at τ = 100 for Ra = 10 5 to 10 6 .Because of the presence of convective instabilities, the rotating cells that are at the beginning of the growth of the flow are fragmented into several cells, which is demonstrated in Figures 5 and 6 by the outlines of the isotherms and streamlines.
A pitchfork bifurcation starts to happen as a result of Rayleigh-Bénard instability.The x-velocity at point P 3 (0, 0.8) in the Ra-u plane is provided in Figure 7 to explain such a pitchfork bifurcation from the symmetric to asymmetric state at the completely developed stage (τ = 1000).Because the flow is symmetric around the y-axis and point P 3 is on the y-axis, the x-velocity for Ra < 10 5 is close to zero.When the Rayleigh number surpasses or becomes equivalent to 10 5 , the cell inclines to the right side with an increment in the x-velocity, as set apart by the square line, whereas it inclines to the left side with the reduction in x-velocity, as set apart by the blue line (circles) in Figure 7.
the horizontal thermal boundary layer, which has warmed air under colder air and is unstable due to 'Rayleigh-Bénard instabilities', is formed.When the critical conditions are fulfilled, the hot thermal boundary layer becomes unsteady.In this regime, Figures 5 and  6 display the streamlines and isotherms at various periods.It has previously been noted that when air of two different temperatures passes through the boundary layer around the midpoint of inclined surfaces, it invariably travels downward.Subsequently, the heated air plume then moves to the cavity's core, while the cooled air plume goes to the lower portion from the upper layer.Figure 5a,b show that the flow becomes symmetric and steady at τ = 20 for Ra = 10 3 to 10 4 .The flow grows stronger and finally asymmetric as time passes due to pitchfork bifurcation.The figures demonstrate the asymmetric isotherms and streamlines for the higher Rayleigh numbers.It is without a doubt significant that, with a greater Rayleigh number, the flow oscillates for quite a long period.The bifurcation continues to rehash left and close to the symmetric focal line while oscillating.Figure 6c,d depict that the flow becomes symmetric and steady at τ = 100 for Ra = 10 5 to 10 6 .Because of the presence of convective instabilities, the rotating cells that are at the beginning of the growth of the flow are fragmented into several cells, which is demonstrated in Figures 5 and 6 by the outlines of the isotherms and streamlines.
A pitchfork bifurcation starts to happen as a result of Rayleigh-Bénard instability.The x-velocity at point P3 (0, 0.8) in the Ra-u plane is provided in Figure 7 to explain such a pitchfork bifurcation from the symmetric to asymmetric state at the completely developed stage (τ = 1000).Because the flow is symmetric around the y-axis and point P3 is on the y-axis, the x-velocity for Ra < 10 5 is close to zero.When the Rayleigh number surpasses or becomes equivalent to 10 5 , the cell inclines to the right side with an increment in the x-velocity, as set apart by the square line, whereas it inclines to the left side with the reduction in x-velocity, as set apart by the blue line (circles) in Figure 7.

Flow at the Steady or Unsteady Stage
In the late transitional phase, a pitchfork bifurcation occurs, resulting in the formation of an asymmetric flow structure.The pitchfork bifurcation occurs early in the numerical simulation, as previously mentioned.
The convective instabilities alternate on either side of the cavity, and the upwardmoving heated air plumes on the base side appear in the middle at different times, which is a fascinating event as shown in the numerical simulation.During the transitional stage, the flow, on the other hand, has multiple undershoots and overshoots prior to becoming completely stable.Thermal energy travels from the boundary layer on either side of the portion discharge fluid to the cavity's center over time.At the fully developed phase, the fluid inside the enclosure reaches a steady state for Ra = 10 3 to 10 6 (see Figure 8a-d).If Ra ≤ 10 6 , Figure 8a-d reveal that the flow becomes stable under various initial conditions.Furthermore, for Ra ≥ 10 7 , Figure 8e,f depict isotherms and streamlines.Figure 8e represents a few tiny cells forming on the top right and left sides of the larger cell.However, when looking at the numerical data, it can be seen that the two tiny cells alternately emerge, indicating that the flow arrives in an unsteady state at a fully advanced stage for Ra = 10 7 .With the increase in Ra, however, both cells develop in the center of the two biggest cells, as seen in Figure 8f.In Figure 8f, for Ra = 10 8 , the biggest cell in the center also travels between right and left.As a result, the unsteady flow becomes more complicated.
fluid inside the enclosure reaches a steady state for Ra = 10 to 10 (see Figure 8a-d).If Ra ≤ 10 6 , Figure 8a-d reveal that the flow becomes stable under various initial conditions.Furthermore, for Ra ≥ 10 7 , Figure 8e,f depict isotherms and streamlines.Figure 8e represents a few tiny cells forming on the top right and left sides of the larger cell.However, when looking at the numerical data, it can be seen that the two tiny cells alternately emerge, indicating that the flow arrives in an unsteady state at a fully advanced stage for Ra = 10 7 .With the increase in Ra, however, both cells develop in the center of the two biggest cells, as seen in Figure 8f.In Figure 8f, for Ra = 10 8 , the biggest cell in the center also travels between right and left.As a result, the unsteady flow becomes more complicated.To comprehend the unstable flow for greater Rayleigh numbers, a temperature-time series is presented in Figure 9.It is apparent that, at the completely advanced stage, the flow is stable, as shown in Figure 9a, and the flow pattern in Figure 9b concurs with the finding of a Hopf bifurcation from a steady state to periodic condition.With the increase in Ra, the periodic flow fluctuates, while the unstable flow becomes chaotic for Ra = 10 8 .This is illustrated in Figure 9c.To comprehend the unstable flow for greater Rayleigh numbers, a temperature-time series is presented in Figure 9.It is apparent that, at the completely advanced stage, the flow is stable, as shown in Figure 9a, and the flow pattern in Figure 9b concurs with the finding of a Hopf bifurcation from a steady state to periodic condition.With the increase in Ra, the periodic flow fluctuates, while the unstable flow becomes chaotic for Ra = 10 8 .This is illustrated in Figure 9c.To comprehend the Hopf bifurcation, which occurs at the transition from steady to periodic phase, the attractors (τ = 300 to 2000) for Ra = 10 6 and (τ = 1000 to 2000) for Ra = 10 7 at the point P4 (0.4, 0.51) are depicted in Figure 10.In Figure 10a, the curve in the u-θ plane clearly approaches a certain value with the passage of time for Ra = 10 6 .In contrast, To comprehend the Hopf bifurcation, which occurs at the transition from steady to periodic phase, the attractors (τ = 300 to 2000) for Ra = 10 6 and (τ = 1000 to 2000) for Ra = 10 7 at the point P 4 (0.4, 0.51) are depicted in Figure 10.In Figure 10a, the curve in the u-θ plane clearly approaches a certain value with the passage of time for Ra = 10 6 .In contrast, Figure 10b shows a limit cycle for Ra = 10 7 (dense curve).Consequently, a Hopf bifurcation occurs at Ra = 10 7 (referred to [38] for a full description of Hopf bifurcation).To comprehend the Hopf bifurcation, which occurs at the transition from steady to periodic phase, the attractors (τ = 300 to 2000) for Ra = 10 6 and (τ = 1000 to 2000) for Ra = 10 7 at the point P4 (0.4, 0.51) are depicted in Figure 10.In Figure 10a, the curve in the u-θ plane clearly approaches a certain value with the passage of time for Ra = 10 6 .In contrast, Figure 10b shows a limit cycle for Ra = 10 7 (dense curve).Consequently, a Hopf bifurcation occurs at Ra = 10 7 (referred to [38] for a full description of Hopf bifurcation).At point P 3 (0, 0.8) for Ra = 10 7 and 10 8 , the directions of the phase space of the u-θ plane are depicted in Figure 11 with a view to demonstrating the transformation to chaotic from the periodic condition in greater detail.In Figure 11a, the limit cycle can be seen, indicating that the unsteady flow is periodic for Ra = 10 7 , which is compatible with Figure 9.In Figure 11b, the trajectory turns out to be chaotic for Ra = 10 8 , indicating that the periodic flow transforms into chaotic, which happens within Ra = 10 7 and 10 8 .This is referred to [39] for a full description of the phase-space trajectories.
Processes 2022, 10, x FOR PEER REVIEW 11 of 15 At point P3 (0, 0.8) for Ra = 10 7 and 10 8 , the directions of the phase space of the u-θ plane are depicted in Figure 11 with a view to demonstrating the transformation to chaotic from the periodic condition in greater detail.In Figure 11a, the limit cycle can be seen, indicating that the unsteady flow is periodic for Ra = 10 7 , which is compatible with Figure 9.In Figure 11b, the trajectory turns out to be chaotic for Ra = 10 8 , indicating that the periodic flow transforms into chaotic, which happens within Ra = 10 7 and 10 8 .This is referred to [39] for a full description of the phase-space trajectories.

Impact of Rayleigh Numbers on the Progress of the Flow
An array of Rayleigh numbers, ranging from Ra = 10 3 to 10 8 , was used in the simulations.An observation was made in the different transient flow characteristics throughout an array of Rayleigh numbers.For A = 0.5, the isotherms and accompanying streamlines are depicted in Figures 4-7 for different Rayleigh numbers.The numerical findings for the various Rayleigh numbers, as shown in Figure 12, reveal some differences.To begin with, convective flow instabilities can be noticed at the lowermost Rayleigh number.However, with a higher Rayleigh number, the unsteadiness becomes more pronounced, while the corresponding wave number rises.For Ra = 10 3 to 10 4 , the flow is weaker and symmetric behavior is visible, which is expected, i.e., the flow is symmetric and constant.The flow becomes asymmetric for Ra = 10 5 and 10 6 in the transitional stage and becomes steady at the fully developed stage.Lastly, for Ra = 10

Impact of Rayleigh Numbers on the Progress of the Flow
An array of Rayleigh numbers, ranging from Ra = 10 3 to 10 8 , was used in the simulations.An observation was made in the different transient flow characteristics throughout an array of Rayleigh numbers.For A = 0.5, the isotherms and accompanying streamlines are depicted in Figures 4-7 for different Rayleigh numbers.The numerical findings for the various Rayleigh numbers, as shown in Figure 12, reveal some differences.To begin with, convective flow instabilities can be noticed at the lowermost Rayleigh number.However, with a higher Rayleigh number, the unsteadiness becomes more pronounced, while the corresponding wave number rises.For Ra = 10 3 to 10 4 , the flow is weaker and symmetric behavior is visible, which is expected, i.e., the flow is symmetric and constant.The flow becomes asymmetric for Ra = 10 5 and 10 6 in the transitional stage and becomes steady at the fully developed stage.Lastly, for Ra = 10 7 and Ra = 10 8 , the flow becomes periodic and chaotic, respectively.

Heat Transfer
The time series of the averaged Nusselt number (Nu) of the lower and upper surfaces were calculated as demonstrated in Figure 13 in order to measure heat transfer through the cavity's wall.In this study, the Nusselt number was defined as follows (Bhowmick et al. [40] and Cui et al. [41]): The temperature in the internal cavity changes at various periods in the early stages because the fluid in the cavity is initially stratified.As the upper and lower walls are cooled and heated at the same time, due to the initially stratified fluid, no significant distinction in temperature between the fluid observed, and the wall might lead to tiny heat transfer; as a result, a small Nu value is predicted initially.The stratification breaks down with time, and the temperature differential in the interior cavity approaches zero.When the fluid's stratification becomes weaker, then the waviness of Nu is increased with

Heat Transfer
The time series of the averaged Nusselt number (Nu) of the lower and upper surfaces were calculated as demonstrated in Figure 13 in order to measure heat transfer through the cavity's wall.In this study, the Nusselt number was defined as follows (Bhowmick et al. [40] and Cui et al. [41]):

Heat Transfer
The time series of the averaged Nusselt number (Nu) of the lower and upper surfaces were calculated as demonstrated in Figure 13 in order to measure heat transfer through the cavity's wall.In this study, the Nusselt number was defined as follows (Bhowmick et al. [40] and Cui et al. [41]): The temperature in the internal cavity changes at various periods in the early stages because the fluid in the cavity is initially stratified.As the upper and lower walls are cooled and heated at the same time, due to the initially stratified fluid, no significant distinction in temperature between the fluid observed, and the wall might lead to tiny heat transfer; as a result, a small Nu value is predicted initially.The stratification breaks down with time, and the temperature differential in the interior cavity approaches zero.When the fluid's stratification becomes weaker, then the waviness of Nu is increased with increased Ra.In the transitional period, for larger Ra, Nu is oscillatory.It is apparent that The temperature in the internal cavity changes at various periods in the early stages because the fluid in the cavity is initially stratified.As the upper and lower walls are cooled and heated at the same time, due to the initially stratified fluid, no significant distinction in temperature between the fluid observed, and the wall might lead to tiny heat transfer; as a result, a small Nu value is predicted initially.The stratification breaks down with time, and the temperature differential in the interior cavity approaches zero.When the fluid's stratification becomes weaker, then the waviness of Nu is increased with increased Ra.In the transitional period, for larger Ra, Nu is oscillatory.It is apparent that the Nusselt number on the enclosure's hot bottom wall (Figure 13a) is larger during the transition stage.The cool top wall, however, illustrates the opposite situation (see Figure 13b).At the completely developed stage, the Nusselt number is fixed for Ra ≤ 10 6 and oscillatory for Ra ≥ 10 7 .These findings are compatible with those in Figure 12.

Conclusions
This study was concerned with the transient thermal convection in a trapezoidal cavity stuffed with linearly stratified air.Although the inclined walls remained adiabatic, the base wall was warmed, and the top wall was cooled with a specified aspect ratio A = 0.5 with variation of Ra (10 0 to 10 8 ).The finite volume-based FLUENT software was used to conduct the numerical simulation.The key findings of this study are described below.
According to numerical simulation, the development of transient flow within the enclosure owing to the predefined boundary conditions could be categorized into three separate stages: early, transitional, and steady or unsteady, all of which were shown in Figures 4-7.
The flow at the beginning phase was portrayed through the arrangement of thermal boundary layers close toward every internal surface and the commencement of primary circulations.In the energy conditions, whenever the terms of convection and conduction were adjusted, the flow went into the transitional state.In the transitional level, the flow was depicted via the base of convective dangers through ascending and descending thermal plumes, as well as the creation of the cellular flow formations.Furthermore, symmetric flows regarding the geometrically symmetric plane for smaller Ra, as well as for relatively higher Ra, were characterized by pitchfork bifurcation, representing the flow from symmetry to asymmetry.With respect to the variance in Ra, the timescale for the flow development was likewise computed.For the pitchfork bifurcation, the difference in the behavior of symmetric and asymmetric flow was additionally examined.
For the stratified air, the temperatures of the fluid adjacent to the top and base walls were the same as the temperature of the walls.Heat transmission was studied through the enclosure, as well as at the base and top walls, and it was discovered that, initially, the Nu was small for the stratified air, whereas, after stratification breakup, Nu gradually increased in the transitional stage, and it was fixed for Ra ≤ 10 6 and oscillatory for Ra ≥ 10 7 at the completely developed stage.Moreover, on the basis of the numerical results, the heat transmission was measured, and it was shown that the Nusselt number depends on the Rayleigh number.

Figure 7 .Figure 7 .
Figure 7. Pitchfork bifurcation in the Ra-u plane where u is the x-velocity at the point P3 (0, 0.8).Ra

Figure 13 .
Figure 13.Time series of the Nusselt number for various Ra on (a) hot bottom wall and (b) cold top wall.

Figure 13 .
Figure 13.Time series of the Nusselt number for various Ra on (a) hot bottom wall and (b) cold top wall.

Figure 13 .
Figure 13.Time series of the Nusselt number for various Ra on (a) hot bottom wall and (b) cold top wall.