A Computational Study of Chaotic Flow and Heat Transfer within a Trapezoidal Cavity

Abstract

Numerical findings of natural convection flows in a trapezoidal cavity are reported in this study. This study focuses on the shift from symmetric steady to chaotic flow within the cavity. This cavity has a heated bottom wall, a cooled top wall, and adiabatic inclined sidewalls. The unsteady natural convection flows occurring within the cavity are numerically simulated using the finite volume (FV) method. The fluid used in the study is air, and the calculations are performed for different dimensionless parameters, including the Prandtl number (Pr), which is kept constant at 0.71, while varying the Rayleigh numbers (Ra) from 10⁰ to 10⁸ and using a fixed aspect ratio (AR) of 0.5. This study focuses on the effect of the Rayleigh numbers on the transition to chaos. In the transition to chaos, a number of bifurcations occur. The first primary transition is found from the steady symmetric to the steady asymmetric stage, known as a pitchfork bifurcation. The second leading transition is found from a steady asymmetric to an unsteady periodic stage, known as Hopf bifurcation. Another prominent bifurcation happens on the changeover of the unsteady flow from the periodic to the chaotic stage. The attractor bifurcates from a stable fixed point to a limit cycle for the Rayleigh numbers between 4 × 10⁶ and 5 × 10⁶. A spectral analysis and the largest Lyapunov exponents are analyzed to investigate the natural convection flows during the shift from periodic to chaos. Moreover, the cavity’s heat transfers are computed for various regimes. The cavity’s flow phenomena are measured and verified.

1. Introduction

The investigation of natural convection flows within enclosed cavities has gained substantial attention owing to its numerous real-world applications in fields, such as geothermal reservoir fluid flow, nuclear reactor cooling, solar energy collection, sterilization, food processing, electronic device cooling, food drying, double-pane windows, and others. The research works [1,2,3,4] have also addressed practical engineering issues, such as electronic and computer hardware, pollution control, and thermal energy storage systems. Research studies [5,6,7,8,9,10,11,12,13] related to natural convection have focused largely on square and rectangular geometries. Further investigations [14,15,16,17] have explored heat transfer through tending surfaces in an attic space to develop homes that are appropriate for both cold and hot environments. Two thermal boundary conditions have been identified as the primary considerations for modeling attic spaces in building geometry: (i) daytime heating, where the sloped boundaries are heated while the bottom is cooled, as well as (ii) nighttime cooling, where the sloped boundaries are cooled while the bottom is heated.

Earlier studies suggested that the flow within a cavity was symmetrical along its centerline. Holtzman et al. [18] discovered that the natural convection flows within an isosceles triangular cavity changed from symmetric to asymmetric patterns. Their numerical solutions showed that symmetric structures were present at low Grashof numbers, with a single-cell structure on each side of the midline. Once the Grashof number exceeds a critical value, the previously stable symmetric findings turn out to be unstable, and the solutions start exhibiting asymmetrical patterns. Holtzman et al. [18] additionally performed a flow visualization experiment where smoke was gradually injected into the cavity. The flow patterns that they found in the visualization experiment corroborated their numerical solutions, providing evidence for the shift from symmetric to asymmetric as the Grashof number increased. Ridouane and Campo’s [19] numerical calculation further validated this finding. Saha et al. [20] investigated the nature of pitchfork bifurcation under a condition of cyclic thermal forcing on the tending boundaries of an attic space. The study of dynamics and heat transfer inside a V-shaped enclosure was crucial due to their widespread occurrence in both natural and engineering fields. Bhowmick et al. [21,22] conducted a computational analysis to study the shift from steady flow to chaos with a series of bifurcations inside a V-shaped enclosure. Wang et al. [23] experimentally investigated natural convection within a V-shaped enclosure heated from the base and cooled from the top, applying the same boundary conditions as Bhowmick et al. [22].

For many engineering applications, and geological situations, when the enclosure’s geometry differs or has additional sloped surfaces, any triangle, square, or rectangular enclosure is insufficient. Natural convection in enclosed cavities with non-classical geometries, such as trapezoidal shapes, poses a challenge due to the presence of sloped walls and requires accurate code construction and grid generation. Despite this, various studies have been conducted to investigate natural convection flows within trapezoidal cavities. Iyican et al. [24,25] conducted groundbreaking studies of natural convection in a sloped trapezoidal cavity, wherein the upper and base walls were cylindrical and parallel. They studied this phenomenon under various conditions, including different temperatures, adiabatic sidewalls, multiple inclination angles, and Rayleigh numbers. Their experimental findings demonstrated how the angle of inclination affects the natural convection of air in a trapezoidal cavity across an extensive array of Rayleigh numbers. Lam et al. [26] carried out a comprehensive study involving both experimental and computational methods to examine natural convection within a trapezoidal cavity. The cavity had two insulated sidewalls perpendicular to each other, a cold upper wall that sloped downwards, and a hot horizontal base wall. Lee [27] conducted both computational and experimental studies on an incompressible fluid, within a sloped non-rectangular cavity. The findings were reported for a range of Ra = 10² to 10⁵ and Pr = 0.001 to 100. The inclination angles were set at 22.5°, 45°, and 77.5°, while the aspect ratios were 3 and 6. The study found that the heat transfer, as well as fluid flows within the cavity were heavily influenced by both the inclination angle of the cavity as well as the values of the Rayleigh and Prandtl numbers. Perić [28] carried out a similar investigation with thoroughly perfected mashes from 10 × 10 to 160 × 160 and found that the results were independent of the mesh size. Kuyper and Hoogendoorn [29] analyzed laminar natural convection flow inside a trapezoidal enclosure, with a focus on how the tilt angle affects the flow and the extent to which the Nusselt number depends on the Rayleigh number. Natarajan et al. [30] examined the natural convection flow inside a trapezoidal cavity with an evenly heated base wall, a linearly cooled and/or heated perpendicular wall(s), and an insulated top wall. They observed that the flow patterns were symmetrical when the sidewalls were heated linearly, but a secondary circulation was found when the left wall was heated linearly and the right wall was cooled. Varol et al. [31] examined the phenomenon of natural convection inside a trapezoidal cavity that was partially cooled by a sloped wall. Three different scenarios were analyzed, and it was concluded that, as the Rayleigh number raised, both the flow intensity and Nusselt number raised. Additionally, these factors were observed to be highly dependent on the placement of the cooling mechanism. They discovered that the flow intensity was highest when the cooling mechanism was situated near the top wall. The researchers also noted that, in comparison to cases 1 and 2, the local and average Nu values were lower in case 3.

Basak et al. [32] numerically investigated natural convection flows inside a trapezoidal cavity. This cavity had an evenly heated bottom wall, at least one vertical wall that was heated linearly, and an insulated top wall. The findings were shown for various inclination angles of the sidewalls, Pr values (0.7–1000), as well as an array of Ra values (10³–10⁵). The researchers noted that the flow patterns exhibited symmetry when the sidewalls were heated linearly, and they discovered that plots of the Nusselt number displayed a greater rate of heat transfer for an inclination angle of 0°. Lasfer et al. [33] conducted a computational investigation on natural convection inside a trapezoidal enclosure that had a heated left inclined sidewall, a cooled perpendicular sidewall, and two horizontal walls that were insulated. The study investigated how the flow patterns and heat transfer were impacted by the tilt angle of the heated wall, the Rayleigh number, and the aspect ratio. Basak et al. [34] also discovered the natural convection flow in a trapezoidal cavity, utilizing a finite element method, where the left sidewall was linearly heated, and the right sidewall was cooled. They noticed that, regardless of the heating patterns of the sidewalls, the square cavity exhibited greater overall heat transfer rates in comparison to other cavity shapes. Mustafa and Ghani [35] applied the FV method to examine the natural convection flows inside a trapezoidal cavity, which was partially heated from the base and cooled symmetrically from the sides. Four different dimensionless heat source lengths were evaluated in the study. The researchers concluded that there was a proportional increase in the average Nusselt number with the increase in the length of the heat source. Recently, Rahaman et al. [36,37] analyzed the unsteady natural convection flows inside a trapezoidal cavity that was initially filled with stratified air and had a heated base and cooled upper horizontal walls. The inclined walls were adiabatic. The findings of the study hold significance for both atmospheric fluid dynamics and the processes of heat transfer and airflow within a thermally stratified atmosphere.

This literature review suggests that there is a dearth of study on natural convection in trapezoidal cavities, particularly in the case of ambient air when the base wall is heated, the top wall is cooled, and the inclined sidewalls are adiabatic. Thus, this study aims to fill this gap by examining the two-dimensional natural convection flow of ambient air within such a cavity by applying numerical methods and appropriate boundary conditions. The FV method is utilized to simulate the Navier–Stokes and Energy equations, and the study focuses on how the heat and mass transfer varies with the Rayleigh number. This study also investigates the shift from a steady to a chaotic state, which takes place through a series of bifurcations. These include a Pitchfork bifurcation when shifting from a state of symmetry to asymmetry and a Hopf bifurcation when shifting from a steady state to a periodic state. The significance of this study lies in the relevance of natural systems, such as the atmosphere and ocean.

2. Computational Procedure

A Newtonian fluid, specifically air with a Prandtl number of 0.71, occupies the trapezoidal cavity. The fluid within the cavity is stationary and maintains a constant temperature of T₀. The cavity’s inclined sidewalls are adiabatic, while the top and bottom walls are cooled and heated, respectively, at the beginning (t = 0), and the temperature is kept constant thereafter. To investigate the two-dimensional natural convection flows inside a trapezoidal enclosure with an aspect ratio (AR) = 0.5, a numerical analysis is conducted using the FV method. This method was also employed in references [36,37] for a similar geometric model. The physical domain is depicted in Figure 1, with appropriate boundary conditions, and has horizontal and vertical lengths of 2L and H, respectively, where L = 2H (AR = 0.5). To prevent singularities occurring at the points where the inclined and the top walls intersect, the tips of the two top corners are cut by around 4% of L, and the cutting walls are supposed to be adiabatic. It is anticipated that the minor alteration to the domain will not considerably affect the computed flow and heat transfer characteristics, as observed in earlier studies [19,20,38].

The formation of two-dimensional natural convection flows in a trapezoidal cavity is determined by solving the dimensionless Navier–Stokes and energy equations with the Boussinesq approximation [21,37]:

$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0,$$

(1)

$$\frac{\partial u}{\partial \tau }+u\frac{\partial u}{\partial x}+v\frac{\partial v}{\partial y}=-\frac{\partial p}{\partial x}+\frac{Pr}{R{a}^{1/2}}\left(\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}\right),$$

(2)

$$\frac{\partial v}{\partial \tau }+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}=-\frac{\partial p}{\partial y}+\frac{Pr}{R{a}^{1/2}}\left(\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}\right)+Pr\theta ,$$

(3)

$$\frac{\partial \theta }{\partial \tau }+u\frac{\partial \theta }{\partial x}+v\frac{\partial \theta }{\partial y}=\frac{1}{R{a}^{1/2}}\left(\frac{{\partial }^2\theta }{\partial x^2}+\frac{{\partial }^2\theta }{\partial y^2}\right).$$

(4)

The dimensionless parameters are defined in the following manner:

$$x=\frac{X}{H},y=\frac{Y}{H},u=\frac{UH}{\kappa R{a}^{1/2}},v=\frac{VH}{\kappa R{a}^{1/2}},p=\frac{P{H}^2}{\rho {\kappa }^{2}Ra},\theta =\frac{T-T_0}{T_h-T_c},\tau =\frac{t\kappa R{a}^{1/2}}{H^2}.$$

(5)

The objective is to study the natural convection flows within a trapezoidal cavity under varying temperature conditions. The development of the flow within the cavity is characterized by three dimensionless parameters: Rayleigh number (Ra), Prandtl number (Pr), and aspect ratio (AR). These parameters are defined as follows (see references [37,39]):

$$\mathrm{Ra}=\frac{g\beta \left(T_h-T_c\right)H^3}{\nu \kappa },$$

(6)

$$\mathrm{Pr}=\frac{\nu }{\mathsf{\kappa }},$$

(7)

$$AR=\frac{H}{L}.$$

(8)

The dimensionless governing equations are accompanied by the initial and boundary conditions which are described below:

• At the beginning of the simulation, the dimensionless initial temperature of the air is set to θ_i = 0.5, and the air’s velocity is set to zero.
• The dimensionless temperature at the hot bottom wall is θ_h = 1, while the cold top wall is θ_c = 0. Additionally, there is no temperature gradient normal to the sidewalls of the cavity, i.e., $\frac{\partial \theta }{\partial n}$ = 0. The velocity of air at the bottom, top, and sidewalls of the cavity is zero (no-slip, u = v = 0).
• The tiny tips are supposed to be rigid, non-slip, and adiabatic.

The FV method with the SIMPLE scheme is applied to solve the governing Equations (1) to (4) with definite initial and boundary conditions. The linear and viscous elements are discretized by applying a second-order central differencing approach while the advection terms are discretized by using a QUICK scheme, as described in Refs. [20,39]. To solve the discretized equations, the method of under-relaxation factors is applied, and a non-uniform, non-staggered rectangular grid is constructed using the commercial software ICEM 17.0. Rahaman et al. [36,37] effectively utilized this numerical method to investigate the unsteady flows within a 2D trapezoidal cavity. Computational methods were successfully employed to model the natural convection process inside the trapezoidal cavities in studies [24,25,26,27,28,29,30,31,32,33,34,35,36,37]. To solve for pressure, momentum, and energy equations, under-relaxation factors of 0.3, 0.7, and 1.0 are, respectively, applied. The convergence criteria for the successive iterative solution are met when the absolute difference between the solution variables of two consecutive iterations is less than a pre-defined small value, which is set to 10⁻⁵.

3. Mesh-Dependent Test and Model Validation

A mesh-dependent test is carried out to investigate how the mesh size affected the precision of the computational findings. In order to precisely record the actual flow properties in particular areas of the domain, it is crucial to use a substantial quantity of meshes. The sensitivity of the mesh is evaluated at the highest Rayleigh number, and it is anticipated that the mesh chosen for the highest Ra value will also be suitable for situations with lower Ra values. In the current study, the highest Rayleigh number of Ra = 10⁸ is investigated using three distinct mesh sizes: 225 × 75, 300 × 100, and 375 × 125. Figure 2 illustrates the calculation of the temperature time series at point P₂ (0, 0.40), using different meshes to assess the impact of the mesh size on flow phenomena. The results reveal that temperature time series for different meshes are similar in the early stage, but slight variations are observed in the fully developed stage (FDS). In addition, an investigation of time step sensitivity is performed utilizing two distinct dimensionless time steps of 0.01 and 0.005. The temperature sequence at point P₂ is compared for two distinct time steps, and it is observed that small changes occurred only during the occurrence of the oscillations.

The impact of the mesh and time step size on the simulation outcomes are also evaluated in terms of the average u-velocity at P₃ (0, 0.53) in the FDS, and the findings are presented in

Table 1. Quantitative comparison of averaged u-velocity at P₃ (0, 0.53) using various meshes and time steps.

Meshes and Time Steps	Average u-Velocity	Percentage of the Variance
225 × 75 and Δτ = 0.01	0.03716	1.30%
300 × 100 and Δτ = 0.01	0.03765	-
300 × 100 and Δτ = 0.005	0.03774	0.24%
375 × 125 and Δτ = 0.01	0.03782	0.45%

. The table demonstrates that the variance in the average u-velocity computed using distinct meshes is minimal, with a difference of less than 1.4%. This suggests that the size of the mesh has a minimal impact on the simulation outcomes, indicating that the results are not significantly influenced by the mesh size (for more information, see [40]). Additionally, the variation in the average u-velocity obtained using different time steps (0.01 and 0.005) is only about 0.24%, which means that either time step can be used. However, based on computational expense, a mesh size of 300 × 100 and a time step of 0.01 are selected for the current simulation.

To confirm the accuracy of the numerical investigation, model validation is an essential step. Unfortunately, there is an insufficiency of experimental results on cavities with a heated bottom wall, and only a few numerical results can be found for comparison purposes. To validate the current code, the findings are compared with the computational findings of Basak et al. [32] under similar boundary conditions. The streamlines and isotherms show excellent agreement with Basak et al. [32]. The comparison is performed with the same dimensionless parameters: Ra = 10⁵, Pr = 0.7, as well as a trapezoidal angle of φ = 45°. The details of this comparison can be found in the authors’ earlier work [37] and are not included in this paper. Moreover, we have also conducted a comparison between our numerical code outcomes and those obtained by Moukalled and Darwish [41] in the scenario where the trapezoidal cavity does not have any baffles. Please refer to Figure 3 for visual representation. We have observed a favorable agreement between our findings and the outcomes presented in [41] regarding both the streamlines and the Nusselt number.

4. Results and Discussion

The numerical method has been validated and is applied to model the natural convection flows, as well as the heat transfer inside a trapezoidal cavity, where the bottom wall is heated, the top wall is cooled, and the sidewalls are adiabatic. The simulation covers a wide range of Ra values between 10⁰ and 10⁸. The cavity has an AR of 0.5, and the fluid used in the simulation is air, with a Prandtl number of 0.71. As the Ra values increase, a series of bifurcations occurs, causing a shift from a symmetrical flow dominated by conduction at lower Rayleigh numbers to a chaotic flow at higher Rayleigh numbers. The different types of flows have been outlined in the subsequent sections.

4.1. Steady Symmetric Flow

To provide a clear understanding of the reactions happening inside the enclosure, a two-dimensional natural convection flow with streamlines and isotherms is presented in Figure 4. Initially, the air inside the trapezoidal cavity is stationary and isothermal. During the numerical simulation, the bottom boundary is heated, the top boundary is cooled, and the inclined boundaries are adiabatic, resulting in the development of thermal boundary layers (TBL) along the interior boundaries. The TBL adjacent to the bottom boundary is a heating TBL, and the TBL adjacent to the top boundary is a cooling TBL (for details, see [42]). Due to the temperature differences and the gravity field, a flow circulation is generated within the enclosure. The warm air from the lower section travels through the boundary layer (BL) to the cooler upper section, while the cold air from the upper section flows toward the bottom through the BL. Therefore, hot air rises from the bottom and cold air sinks from the top. For small Rayleigh numbers (10⁰ to 10³), the results are not reported due to conciseness. Between the Rayleigh numbers 10⁴ and 10⁶, the isotherms and streamlines indicate the initial flow pattern described by weak rotating convective cells. At Ra = 10⁴, the temperature changes within the cavity follow an almost linear pattern, and conduction dominates the heat transfer mode.

However, with the rise in Ra values, the buoyancy force becomes more dominant than the viscous force, leading to an augmentation of the natural convection impact. As a result, the isotherms deviate from a uniform pattern, indicating the strong circulation in the enclosure. Between Rayleigh numbers 10⁵ and 10⁶, a few secondary cells appear with the two primary cells at the transitional stage. In the FDS, the two symmetric primary cells rotate in opposite directions about the mid-plane. Hence, for Ra values ranging from 10⁰ to 10⁶, the flow exhibits symmetry relative to the cavity’s y-axis. Thus, the flow is steady and symmetric at this stage for Ra ≤ 10⁶.

4.2. Asymmetric Flow

As per earlier studies [19,43], a pitchfork bifurcation is responsible for the shift of the flow from symmetry to asymmetry. This section also illustrates the occurrence of this bifurcation in various ways. The isotherms and streamlines illustrated in Figure 5a indicate that the primary 2D flow is steady and symmetric for Ra = 1.5 × 10⁶ at τ = 250. However, as the Ra is increased to 2 × 10⁶, the flow becomes asymmetrical and the convective cell tilts to one side of the cavity’s symmetry line or the other, which may depend on the initial perturbations. Figure 5b presents the numerical results indicating that when Rayleigh–Bénard (RB) convection begins, an asymmetric flow occurs. The phenomenon where a symmetrical solution transforms into an asymmetric solution is referred to as a pitchfork bifurcation, and further information about pitchfork bifurcations can be found in Refs. [19,42].

To explain the pitchfork bifurcation, two temperature time series are analyzed at two points, P₅ (0.53, 0.51) and P₆ (−0.53, 0.51), for the Rayleigh numbers 1.5 × 10⁶ and 2 × 10⁶. These two points are symmetric with regard to the cavity’s symmetry plane. In Figure 6a, it is evident that the two temperature time series for Ra = 1.5 × 10⁶ coincide for a considerable amount of time, which implies that there is a symmetrical flow with regard to the cavity’s symmetry plane during that time. However, for Ra = 2 × 10⁶, the time series of the temperature at two points start to show discrepancies between τ = 200 and 300, indicating that the pitchfork bifurcation has taken place, and the flow has shifted to an asymmetry state.

In order to obtain a more comprehensive understanding of the pitchfork bifurcation, a bifurcation diagram is produced in the Ra-u plane, depicted in Figure 6b. The diagram shows that, for Rayleigh number Ra ≤ 1.5 × 10⁶, the x-velocity at P₄ (0, 0.80) is approximately zero, which implies that the flow maintains symmetry with regard to the cavity’s symmetry plane. However, for Rayleigh numbers greater than 1.5 × 10⁶, the x-velocity increases to the right side, as indicated by the square black line, and decreases to the left side, as indicated by the circular blue line, as shown in Figure 6b. These changes suggest that the pitchfork bifurcation occurs between the Rayleigh numbers 1.5 × 10⁶ and 2 × 10⁶, leading to the flow shifting to asymmetry for Ra = 2 × 10⁶.

The level of asymmetry in the flow from the trapezoidal cavity is quantified by [19]

$$I=\frac{\int {\left[\theta \left(x, y\right)-\theta \left(-x, y\right)\right]}^{2}dxdy}{4\int {\left[\theta \left(x, y\right)\right]}^{2}dxdy}.$$

(9)

A value of I = 0 corresponds to a steady state that is symmetrically reflective, whereas a non-zero value of I indicates that the steady state is asymmetric. The value of I is computed for different Ra values ranging from 10³ to 10⁸ and is illustrated in Figure 6c. It is apparent that I is roughly zero when Ra is equal to 1.5 × 10⁶. However, for Ra equal to 2 × 10⁶, I substantially increases, implying a transition from symmetry to asymmetry. As depicted in Figure 6c, I continues to increase as the Ra increases.

4.3. Unsteady Flow

As the Ra values rise, the convection flow strengthens, leading to the formation of RB convection, which dominates the heat transfer within the cavity. This rise in the Ra values has a substantial impact on the temperature distribution within the cavity, causing a more pronounced temperature gradient near the hot and cold surfaces.

4.3.1. Hopf Bifurcation

In the FDS for Ra < 2 × 10⁶, as illustrated in Figure 5, the natural convection flow attains an asymmetric steady state. Furthermore, as shown in Figure 7a, the flow also approaches an asymmetric steady state for Ra = 4 × 10⁶ in the FDS. However, for the Ra value of 5 × 10⁶, a couple of tiny cells appear in the upper portions of the cavity, as portrayed in Figure 7b. That is, for Ra = 5 × 10⁶, the flow becomes unsteady in the FDS. This means that Hopf bifurcation occurs between the Ra values of 4 × 10⁶ and 5 × 10⁶.

After the Hopf bifurcation, Figure 8 illustrates the flow patterns with higher Rayleigh numbers, Ra ≥ 5 × 10⁶, which lead to the next findings. The convective flow becomes periodic from Ra = 5 × 10⁶ and chaotic from Ra = 3 × 10⁷ in the FDS. Figure 8a displays that the convective flow is not steady in the FDS for Ra = 5 × 10⁶, characterized by the appearance of small cells in the upper portion of the cavity and the rhythmic oscillation of larger cells in the middle. This indicates that for Ra = 5 × 10⁶ a Hopf bifurcation might occur (see Ref. [44] for more information). Another observation in the FDS is that, when Ra = 2 ×10⁷, two additional tiny cells form in the corner of the cavity, leading to an oscillatory flow (refer to Figure 8b). The flow also becomes more complicated. As the Ra rises, both larger cells in the center cavity migrate continuously toward the left and right sides. In the FDS, a completely chaotic flow develops for Ra = 3 × 10⁷, as depicted in Figure 8c. The chaotic flow gets more complicated as the Rayleigh number rises further.

To understand the unsteady flow, temperature time series are presented in Figure 9. In the FDS, the temperature time series at P₁ (0, 0.27) reaches a steady state for Ra = 4 × 10⁶, as depicted in Figure 9a. Conversely, the time series of the temperature becomes periodic when the Ra is increased to 5 × 10⁶, which indicates a Hopf bifurcation between Ra = 4 × 10⁶ and 5 × 10⁶, as depicted in Figure 9b. Essentially, the occurrence of Hopf bifurcation implies that an oscillating solution arises from an unstable steady solution. Figure 9c illustrates the spectral analysis of the temperature time series, revealing a fundamental peak frequency f_p = 0.345, and separate harmonic modes are present.

To enhance comprehension of the Hopf bifurcation mechanism, Figure 10 displays the state-space trajectories in the v-θ plane at P₃ (0, 0.53) during the time interval τ = 700 and 2000. As shown in Figure 10a, when Ra = 4 × 10⁶, the (v, θ) trajectory converges toward a fixed point (represented by the teal green point), while in Figure 10b, it moves toward a closed limit cycle for Ra = 5 × 10⁶ (represented by the purple orbit). The transition from a limit point to a limit cycle attractor when the Ra changes from 4 × 10⁶ to 5 × 10⁶ indicates the occurrence of a Hopf bifurcation within this range.

4.3.2. Transition to Chaos

The results indicate that a chaotic flow occurs when the Ra value is large enough. The streamlines and isotherms depicted in Figure 7b,c illustrate the transition to chaos. Spectral analysis, phase-space trajectories, and computation of the Lyapunov exponent are performed to demonstrate the generation of complex structures in the flow. Time series of the temperature at P₁ (0, 0.27) for two different Ra values (2 × 10⁷ and 3 × 10⁷) are presented in Figure 11a,c, respectively, along with their corresponding spectral analysis, depicted in Figure 11b,d. As portrayed in Figure 11a,b, the temperature sequence remains periodic for Ra = 2 × 10⁷. The temperature sequence becomes chaotic at Ra = 3 × 10⁷, as displayed in Figure 11c, which is consistent with the findings of the spectral analysis presented in Figure 11d. In fact, the power spectral density for Ra = 3 × 10⁷ does not exhibit a distinct dominant frequency, indicating the onset of chaos in the unsteady flow.

Figure 12 displays phase-space trajectories of v-θ planes at position P₃ (0, 0.53) for Ra values of 5 × 10⁶, 10⁷, 2 × 10⁷, and 3 × 10⁷, in order to facilitate additional observations. It is evident from Figure 12a–c that almost identical limit cycles appear. This indicates that for Ra = 5 × 10⁶, 10⁷, and 2 × 10⁷, periodic flows exist. In Figure 12d, the trajectories degenerate into chaos for Ra = 3 × 10⁷. That suggests that between Ra = 2 × 10⁷ and 3 × 10⁷, another bifurcation occurs between the periodic to the chaotic state (for details, see Ref. [45]).

Earlier studies (e.g., [46,47,48]) have shown that the shift to chaos can be defined by the largest Lyapunov exponent (λ_L). To be precise, ${\epsilon }_0$ refers to the distance between two points in the phase space at the beginning time, while ε(t) denotes the distance after a certain number of time steps, and then the following equation holds true [45,47,49].

$$\epsilon \left(t\right)\cong {\epsilon }_0{e}^{{\lambda }_{L}t}.$$

(10)

If λ_L is greater than zero, then the nonlinear system is classified as chaotic (see Ref. [41]). The most prominent Lyapunov exponent may consider the transition to chaos, which is defined as follows (see also [21]):

$${\lambda }_L=\frac{1}{{\tau }_n-{\tau }_0}{\displaystyle \sum }_{j=1}^n\ln \frac{\epsilon \left(u_j\right)}{\epsilon \left(u_{j-1}\right)}.$$

(11)

where ε(u_j) represents the geometric separation between u_j and u_j−1. In this investigation, the initial time step is τ₀ = 1500, and the time step after the time n (= 5 × 10⁴) is τ_n = 2000, for the fluid with Pr = 0.71. If the exponent is greater than zero, the distance will increase infinitely over time. In other words, a nonlinear system degenerates into chaos when λ_L is greater than zero. Equation (11) was used to calculate the λ_L for various Ra, using the x-velocity at P₃ (0, 0.53). The outcomes are depicted in Figure 13. As depicted in the figure, when the Ra is greater than or equal to 3 × 10⁷, λ_L is greater than zero. However, the figure demonstrates that for Ra = 2 × 10⁷, λ_L is negative. This indicates that the x-velocity becomes chaotic only when the Ra is greater than or equal to 3 × 10⁷. With a further increase in the Ra, λ_L continues to rise, and the flow exhibits greater degrees of chaos, as depicted in Figure 12d.

5. Heat Transfer

The flow visualization and numerical findings presented earlier unambiguously demonstrate that the natural convection flows within the trapezoidal cavity become stronger as the Rayleigh number rises. To be specific, at higher Rayleigh numbers, the convective instabilities are more pronounced. As the Ra increases, the rate of heat transfer across the cavity also increases. To quantify this heat transfer rate, we use the dimensionless Nusselt number (Nu), which is defined as follows (for details [22,23,50]):

$$\mathrm{Nu}=\frac{1}{l}{{\displaystyle \int }}_l^{ }\frac{\partial \theta }{\partial n}ds$$

(12)

Due to the heating and cooling of the bottom and top walls, there is a substantial heat transfer at the beginning of the process, mainly caused by the considerable temperature variance between the fluid and the walls. In the initial phase, the heat transfer rate decreases considerably and ultimately reaches a steady or oscillatory value, subject to the Ra values, in the FDS. It should be emphasized that the Nu is higher at the bottom wall compared to the top wall, as the bottom has a smaller surface area available for heat transfer. For the lower Rayleigh numbers (Ra = 10⁰ to 10⁵), the heat transfer is dominated by conduction. For the sake of brevity, the results are not presented here. The Nusselt number remains constant for Ra ≤ 4 × 10⁶, periodic for 5 × 10⁶ ≤ Ra ≤ 2 ×10⁷, and chaotic for Ra ≥ 3 × 10⁷ in the FDS depicted in Figure 14. From Figure 14b,d, it is illustrated that the lines coincide together, not shown in Figure 14a,c. Which indicates that in the current investigation, the relation of Nu ~ Ra^1/4 is applicable for the Ra values under consideration.

6. Conclusions

The finite volume method is used to numerically simulate the unsteady natural convection inside the trapezoidal cavity, where the bottom wall is heated, the top wall is cooled, and the inclined sidewalls are adiabatic. Calculations were conducted for air utilizing the Prandtl number Pr = 0.71, extensive Ra values ranging from 10⁰ to 10⁸, and an aspect ratio (AR = 0.5).

From the isotherms and streamlines, it is observed that the flow circulation in the trapezoidal cavity is very weak for the lower Ra, because the convection is dominated by conduction. As the Ra rises, there is a distinct change in the shape of the circulating vortices, indicating a significant increase in the dominance of convection over conduction. The convective flow in the cavity undergoes a series of bifurcations that result in a shift from a symmetric to a chaotic state. Between the Ra values 1.5 × 10⁶ and 2 × 10⁶, the first bifurcation happens, which is known as pitchfork bifurcation. With the rise in the Ra values, a Hopf bifurcation happens between Ra = 4 × 10⁶ and Ra = 5 × 10⁶. When the Ra values rise, the flow turns chaotic at Ra = 3 × 10⁷, and as the Ra values continue to rise, the flow becomes even more chaotic. The process of shifting to chaos is characterized by using limit points, limit cycles, attractors, spectral analysis, and the largest Lyapunov exponent.

Additionally, the characteristics of the heat transfer within the cavity have been studied. The Nusselt number has been calculated for both the bottom and top walls, and it has been normalized by using the relation Nu ~ Ra^1/4. It is important to mention that heat transfer is more pronounced at the lower wall compared to the upper wall.