Natural Convection Fluid Flow and Heat Transfer in a Valley-Shaped Cavity

Abstract

The phenomenon of natural convection is the subject of significant research interest due to its widespread occurrence in both natural and industrial contexts. This study focuses on investigating natural convection phenomena within triangular enclosures, specifically emphasizing a valley-shaped configuration. Our research comprehensively analyses unsteady, non-dimensional time-varying convection resulting from natural fluid flow within a valley-shaped cavity, where the inclined walls serve as hot surfaces and the top wall functions as a cold surface. We explore unsteady natural convection flows in this cavity, utilizing air as the operating fluid, considering a range of Rayleigh numbers from Ra = 10⁰ to 10⁸. Additionally, various non-dimensional times τ, spanning from 0 to 5000, are examined, with a fixed Prandtl number (Pr = 0.71) and aspect ratio (A = 0.5). Employing a two-dimensional framework for numerical analysis, our study focuses on identifying unstable flow mechanisms characterized by different non-dimensional times, including symmetric, asymmetric, and unsteady flow patterns. The numerical results reveal that natural convection flows remain steady in the symmetric state for Rayleigh values ranging from 10⁰ to 7 × 10³. Asymmetric flow occurs when the Ra surpasses 7 × 10³. Under the asymmetric condition, flow arrives in an unsteady stage before stabilizing at the fully formed stage for 7 × 10³ < Ra < 10⁷. This study demonstrates that periodic unsteady flows shift into chaotic situations during the transitional stage before transferring to periodic behavior in the developed stage, but the chaotic flow remains predominant in the unsteady regime with larger Rayleigh numbers. Furthermore, we present an analysis of heat transfer within the cavity, discussing and quantifying its dependence on the Rayleigh number.

1. Introduction

The study of natural convection within confined spaces has attracted considerable attention from scientists due to its widespread applicability across various disciplines [1,2]. Researchers have utilized different types of enclosures with diverse boundary conditions as experimental setups to investigate natural convection, aiming to deepen our understanding of heat transfer mechanisms and fluid dynamics. This interest stems from the multitude of applications where natural convection phenomena are integral, spanning fields such as geophysics, building insulation, geothermal reservoir management, and industrial separation processes. However, the complexity of the Earth’s surface, characterized by irregular geometries highly influenced by slanted terrains, poses challenges for traditional geometric configurations. Despite the prevalence of irregular shapes in both natural and industrial settings, natural convection remains a subject of ongoing study owing to its significance in understanding fluid flow behavior. While much research has focused on natural convection within conventional square or rectangular enclosures due to their simplicity [3,4], the investigation of natural convection within triangular-shaped cavities holds particular importance as it contributes significantly to our understanding of phenomena such as Rayleigh–Bénard convection [5,6]. Recently, Cui et al. [7] focused on the study of mixed convection and heat transfer in an arc-shaped cavity with inner heat sources under the conditions of bottom heating and top wall cooling. Chuhan et al. [8] examined the thermal behavior of a power law fluid in a plus-shaped cavity as a result of natural convection, taking into account the Darcy number and magnetohydrodynamics. Sarangi et al. [9] explored the heat loss induced by radiation and persistently laminar natural convection in a solar cooker cavity with a rectangular or trapezoidal cavity.

Enhancing our understanding of natural convection flows holds paramount importance in improving predictions of heat transfer and facilitating flow control mechanisms. Researchers intrigued by this phenomenon can investigate detailed explanations of buoyancy-induced natural convection flows [10]. Natural convection near a thermal boundary is often idealized as flow adjacent to a thermally well-conducted infinite or semi-infinite flat plate, leading to the formation of Rayleigh–Bénard convection. Studies by Manneville and Bodenschatz et al. [11,12] provide comprehensive insights into Rayleigh–Bénard instability findings, whereas Sparrow and Husar [13] focus on exploring Rayleigh–Bénard convection on inclined flat plates. Investigating cavities with vertical and horizontal temperature gradients reveals two distinct natural or free convection flow scenarios. While natural convection is predominantly studied in rectangular or square cavities due to their simplicity, Batchelor’s [14] research offers a significant understanding of natural convection flows within differentially heated cavities, particularly emphasizing the influence of conduction on heat transfer, especially at low Rayleigh numbers. As the Rayleigh number exceeds a threshold, convection controls the flow dynamics. Early studies have primarily examined stable natural convection flows [15,16,17]. The intriguing phenomenon of baroclinity, engendered by the thermal vertical wall, instigates spontaneous convection flows within the interior cavity through viscous shear under symmetric conditions [18]. At lower Rayleigh numbers, the thermal boundary layer proximal to the wall remains steady, while convective instability induces discrete traveling waves as the Rayleigh numbers increase [18]. Recent studies have delved into the dynamics and transient nature of transient natural convection flows, building upon earlier investigations [19,20]. These findings contribute to our understanding of the complex interplay between thermal gradients and fluid dynamics within confined spaces, shedding light on the transient behaviors inherent to natural convection phenomena. Recent research indicates the occurrence of baroclinity-induced spontaneous convection flows and turbulent Rayleigh–Bénard convection [21]. Rahaman et al. [22,23] explore transitional natural convection flows in a trapezoidal enclosure heated from below, supported by numerical investigations, further enriching our comprehension of this complex phenomenon.

Natural convection flows within cavities featuring inclined walls have attracted considerable attention due to their prevalence and ease of observation [24]. Specifically, investigations into the natural convection flows within triangular cavities have revealed their significant enhancement of Rayleigh–Bénard convection, as documented in prior studies [5,6]. Extensive research explores the characteristics and heat transfer mechanisms of natural convection flows within attic-shaped cavities characterized by temperature gradients between the bottom surface and the inclined borders [25,26,27,28,29,30,31]. Notably, studies by Asan and Namli [25,26] and Salman [27] have examined the instability and bifurcations of natural convection flow solutions in triangular cavities, revealing empirical evidence showcasing the profound influence of the aspect ratio and Rayleigh number on both the temperature and flow fields. Analytical findings indicate that as the Rayleigh numbers increase, the aspect ratios decrease, leading to the occurrence of multiple vortex flow patterns. Poulikakos and Bejan [29] investigated heat transmission and natural convection flows within attic spaces during night or winter conditions, establishing scaling relationships and engaging in discussions regarding natural convection flow dynamics [30]. Flack [31] elucidated heat transport dynamics using the Nusselt number–Rayleigh number relationship, while Holtzman et al. [32] observed the transition from symmetric to asymmetric flow patterns with increasing Grashof numbers, identifying a Pitchfork bifurcation based on experimental data. Furthermore, Lei et al. [33] provided evidence of the existence of transient natural convection flows within attic cavities, further enriching our understanding of this complex phenomenon.

The evolution of flow under sudden heating and cooling undergoes three distinct stages: early, transitional, and steady or quasi-steady [33]. It has been observed that lower Rayleigh numbers result in heating across the entire flow zone rather than the splitting of the thermal boundary layer, with layer separation occurring as the Rayleigh numbers increase. Particularly in scenarios of rapid cooling, the thickness of the thermal boundary layer exceeds the vertical distance from the center of the inclined surface to the horizontal bottom, especially at lower Rayleigh numbers. The thermal boundary layer stabilizes prior to cavity cooling as the Rayleigh numbers increase, with fluid passing the sloping surface and contacting the bottom tip before entering the interior. Attic heat transmission has been studied under thermal stimuli by Saha et al. [34,35,36,37,38]. Additionally, researchers have investigated the behavior and heat transfer dynamics of natural convection flows in wedge-shaped cavities, serving as models for various shallow-water bodies with inclined bottom surfaces, such as reservoir sidearms and seashores. As described by Lei and Patterson [39,40,41], convective flow within wedges undergoes three stages from isothermal and stationary states. Mao et al. [42,43,44] argued that the horizontal location of the wedge determines the primary heat transfer mode and flow status for varying Rayleigh numbers, with different flow regimes observed in the shallow littoral zone. Bednarz et al. [45] explored natural convection flows within reservoir-shaped cavities, observing the formation of two heating boundary layers: inflow at the bottom and unstable reflux below the water for different Grashof values. Understanding natural convection flows in V-shaped cavities is crucial, given their prevalence in both natural and industrial systems [46,47,48,49,50,51,52,53,54,55,56,57]. Kenjeres [58] quantified the heat transfer and turbulent natural convection fluxes in V-shaped cavities, noting that upslope flow increases with rising Rayleigh numbers. Kimura et al. [59] utilized angled V-shaped channel open chambers to calculate heat transfer dynamics.

The flow pattern and heat transfer quantities have been meticulously measured in previous studies [30,31]. Despite the detailed examinations conducted, natural convection flows within an attic or a wedge-shaped triangular cavity may not accurately depict the flow dynamics in a V-shaped cavity over time. Given their natural occurrence, natural convection flows within V-shaped triangular enclosures with opposing boundary conditions have been deemed worthy of thorough investigation. Bhowmick et al. [60,61,62] investigated the transition from symmetric steady flow to asymmetric unsteady flow in a V-shaped triangular enclosure, where heating occurred from below and cooling from above. It was observed that different fluids exhibit distinct movement behaviors. Despite witnessing the transition from regular to chaotic flow in the valley-shaped cavity heated from below, the mechanisms driving unsteady flow in such cavities remain unclear. To the best of the authors’ knowledge, no study has examined the unsteady flow structures present at various times in the valley-shaped cavity. Moreover, there is a need for further quantitative analysis of the dynamics of heat transfer in such configurations.

This literature review highlights the significance of investigating unsteady natural convection in V-shaped cavities for air, providing valuable insights into the flow patterns, transition processes, and heat transfer phenomena in such systems. However, a comprehensive understanding of the intricate physics governing unsteady flow mechanisms within V-shaped cavities remains imperative. Analysis of the existing literature reveals a dearth of studies focusing on characterizing unsteady flow structures at different non-dimensional times within V-shaped cavities. To address this gap, the present study aims to examine unsteady flow in a valley-shaped cavity through two-dimensional numerical simulations, considering air with Rayleigh numbers ranging from 10⁰ to 10⁸, a Prandtl number of 0.71, and an aspect ratio of 0.5. The impact of various Ra and τ values on the flow structure, heat transfer, and transient flow characteristics in the fully formed stage of the valley-shaped cavity are covered in this study. This research contributes to existing knowledge by enhancing our understanding of how fluid characteristics affect flow behavior and transition processes. The evolution of flow mechanisms and changes within the cavity are elucidated using the temperature and velocity differences over time. Such insights are crucial for enhancing the energy efficiency of engineering applications and optimizing the design of thermal systems like heat exchangers and cooling systems. Consequently, this study is poised to make a novel contribution to the field, offering distinct and valuable conclusions that will benefit specialists engaged in modeling and experimentation on flow over complex geometries.

2. Numerical Model Formulations

The primary objective of this study is to analyze the characteristics of natural convection flows in a triangular cavity with a valley-shaped configuration. To achieve this, a two-dimensional numerical simulation approach is employed. The physical model and its corresponding boundary conditions are visually represented in Figure 1. In order to rectify the singularity present at the connection between the top and inclined walls, a strategic adjustment is made by removing an appropriate amount of substance from both the top corners. Specifically, 4% of the length is carefully extracted in the form of minuscule points. It should be noted that this minor adjustment does not exhibit any discernible impact on the mechanics of fluid flow and heat transfer, as indicated by previous studies [18,20,21,22,23,24,29,30,31]. The dimensions of the cavity are defined as follows: the horizontal length is 2L, and the height is H, where L = 2H and the ratio A = H/L = 0.5. At time T = T₀, the fluid inside the cavity is initially at a uniform temperature and not in motion. At a given temperature of T_c = T₀ − ΔT/2 and T_h = T₀ + ΔT/2, respectively, the top and inclined walls undergo instantaneous cooling and heating processes. In every given circumstance, all the boundaries are motionless.

This study analyses two-dimensional natural convection flows within a valley-shaped enclosure. The governing equations employed in this investigation are presented below, utilizing the Boussinesq approximation as a simplifying assumption [62].

$$\frac{\partial U}{\partial X}+\frac{\partial V}{\partial Y}=0,$$

(1)

$$\frac{\partial U}{\partial t}+U\frac{\partial U}{\partial X}+V\frac{\partial U}{\partial Y}=-\frac{1}{\rho }\frac{\partial P}{\partial X}+\nu \left(\frac{{\partial }^{2}U}{\partial X^2}+\frac{{\partial }^{2}U}{\partial Y^2}\right),$$

(2)

$$\frac{\partial V}{\partial t}+U\frac{\partial V}{\partial X}+V\frac{\partial V}{\partial Y}=-\frac{1}{\rho }\frac{\partial P}{\partial Y}+\nu \left(\frac{{\partial }^{2}V}{\partial X^2}+\frac{{\partial }^{2}V}{\partial Y^2}\right)+g\beta \left(T-T_0\right),$$

(3)

$$\frac{\partial T}{\partial t}+U\frac{\partial T}{\partial X}+V\frac{\partial T}{\partial Y}=\kappa \left(\frac{{\partial }^{2}T}{\partial X^2}+\frac{{\partial }^{2}T}{\partial Y^2}\right).$$

(4)

U and V in the equations, respectively, represent the horizontal and the vertical flow rates. For a two-dimensional coordinate system:

• P is the pressure;
• ρ is the density;
• t is the time;
• T is the temperature, at time T₀, where T₀ = (T_c + T_h)/2, the fluid medium in the triangular cavity is isothermally stationary.

The dimensionless variables utilized are as follows:

$$x=\frac{X}{H}, y=\frac{Y}{H}, u=\frac{UH}{\kappa {\mathrm{R}\mathrm{a}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}, v=\frac{VH}{\kappa {\mathrm{R}\mathrm{a}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}, p=\frac{P{H}^2}{\rho {\kappa }^2\mathrm{R}\mathrm{a}}, \theta =\frac{T-T_{\infty }}{T_h-T_c},\tau =\frac{t\kappa {\mathrm{R}\mathrm{a}}^2}{H^2}.$$

(5)

In the aforementioned equations, the variables v, x, y, p, τ, and θ represent the normalized counterparts of U, V, X, Y, P, t, and T, respectively. The three governing variables, the aspect ratio (A), Prandtl number (Pr), and Rayleigh number (Ra), have an impact on the enclosure’s natural convectional flows (view [6] for further details), which are best described as follows:

$$A=\frac{H}{L},\hspace{1em}\hspace{1em}\mathrm{P}\mathrm{r}=\frac{\nu }{\kappa },\hspace{1em}\hspace{1em}\mathrm{R}\mathrm{a}=\frac{g\beta \left(T_h-T_c\right)H^3}{\nu \kappa } .$$

(6)

The above dimensionless variables are added, and then Equations (1)–(4) become (for more information, see [61]):

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

(7)

$$\frac{\partial u}{\partial \tau }+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=-\frac{\partial p}{\partial x}+\frac{\mathrm{P}\mathrm{r}}{{\mathrm{R}\mathrm{a}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\left(\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}\right) ,$$

(8)

$$\frac{\partial v}{\partial \tau }+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}=-\frac{\partial p}{\partial y}+\frac{\mathrm{P}\mathrm{r}}{{\mathrm{R}\mathrm{a}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\left(\frac{{\partial }^{2}v}{\partial x^2}+\frac{{\partial }^{2}v}{\partial y^2}\right)+\mathrm{P}\mathrm{r}\theta ,$$

(9)

$$\frac{\partial \theta }{\partial \tau }+u\frac{\partial \theta }{\partial x}+v\frac{\partial \theta }{\partial y}=\frac{\mathrm{P}\mathrm{r}}{{\mathrm{R}\mathrm{a}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\left(\frac{{\partial }^2\theta }{\partial x^2}+\frac{{\partial }^2\theta }{\partial y^2}\right) .$$

(10)

The present study used a finite-volume Navier–Stokes solver [62], Fluent 15.0 in ANSYS, to model natural convection within a valley-shaped cavity. The principles of mass conservation, momentum conservation, and energy conservation are the foundations of the computational model that is used to simulate unsteady natural convection flow in an air cavity. The continuity equation, the Navier–Stokes equations, and the energy equations have all been solved using finite volume methods after the appropriate numerical techniques have been applied. The non-uniform 2D mesh system is created with the help of the commercial software ICEM 15.0, and the numerical results have been displayed graphically with the help of the post-processing program TECPLOT 360.

The governing Equations (7)–(10) are solved by means of a finite volume technique using the SIMPLE method, which is thoroughly explained in reference [63] and Saha [64]. The numerical procedure for the SIMPLE method is shown in Figure 2.

3. Grid Dependency Test

The mesh and time step dependence of the largest Rayleigh number (Ra = 10⁸) for the Prandtl number (Pr = 0.71) and aspect ratio (A = 0.5) was investigated in the present inquiry. Three symmetrical, non-uniform meshes were employed in the test, with dimensions of 600 × 100, 800 × 150, and 1200 × 200. These meshes were designed to have finer grids near the boundaries and coarser grids in the inner zone. The 800 × 150 mesh underwent a 3% expansion, ranging from a minimum width of 0.00025 near the wall to a maximum width of 0.02 in the inside. Figure 3 illustrates the time series of the Nusselt number at the right wall of the cavity. The data were obtained using different meshes and time steps for a Rayleigh number of 10⁸. The Nusselt numbers obtained from the different meshes and time steps show uniformity during the initial stage, along with some divergence during the mature stage. In addition,

Table 1. Nusselt numbers (Nu) for different grids and time steps.

Mesh and Time Step	Average Nu	Difference
600 × 100 and Δτ = 0.0025	121.63	1.94%
800 × 150 and Δτ = 0.00125	122.47	1.27%
800 × 150 and Δτ = 0.0025	124.04	-
1200 × 200 and Δτ = 0.0025	122.95	0.87%

provides a calculation and list of the average Nusselt numbers during the fully developed stage. Based on the analysis conducted, it has been determined that the variations in outcomes observed across the different meshes and time steps are within the acceptable threshold of 2%. Taking this into consideration, the numerical simulation in this study used a mesh size of 800 × 150 and a time step of 0.0025.

4. Validation

Figure 4 compares the laboratory experiment conducted by Holtzman et al. [32] with the current numerical results for additional confirmation. In contrast to the triangular cavity in this study, it is vertically inverted and the Rayleigh numbers have been employed in place of the corresponding Grashof numbers. In comparison to the experimental results in Figure 4b for the Rayleigh numbers Ra = 3.5 × 10³, Figure 4d for Ra = 7 × 10³, and Figure 4f for Ra = 7 × 10⁴, the numerical results depicting the symmetric flow for Ra = 7 × 10³ in Figure 4a and the asymmetric flow for Ra = 1.2 × 10⁴, and Ra = 10⁵ in Figure 4c,e are accurate. As indicated by Xu et al. [18] and Patterson and Armfield [20], the presence of inconsistencies between numerical Ra values and experimental Ra findings gives rise to a notable contradiction: if the numerical findings and experiment results are accurate, the experimental Rayleigh number is approximately 1.5 to 2.5 times the numerical Rayleigh number. The numerical methods utilized in this research can be used to depict a transitional flow in a triangular cavity, as the experimental and computational results are identical.

5. Numerical Results and Discussion

The subsequent section provides an explanation of the primary characteristics of the fluid flows within a V-shaped cavity, wherein the flow is stimulated by heat from the inclined walls and dissipated by cooling from the top wall. The analysis focuses on a range of Rayleigh numbers, spanning from Ra = 10⁰ to 10⁸, a Prandtl number, Pr = 0.71, and an aspect ratio, A = 0.5. Various non-dimensional times are considered to comprehensively examine the flow growth phenomenon. In this work, a 2D numerical simulation has been conducted. According to the numerical simulations, the evolution of the flow for these Rayleigh numbers in different times from τ = 0 to 5000 may be split into symmetric flow, asymmetric flow and unsteady flow.

To produce the simulation results, we used high-performance computing facilities with 64 processors, with which it took 15 days for a small Ra = 10⁵ and 20 days for a large Ra = 10⁸ for non-dimensional time, τ = 5000 with time step size, Δτ = 0.0025.

5.1. Symmetric Flow

It was discovered that for Ra = 10⁰, 10¹ and 10², the transitional flow development with non-dimensional time did not exhibit an increasing or decreasing plume, which indicates that for those Rayleigh numbers, the flow exhibited an ongoing level of stability due to the prevailing influence of conduction dominance. In this study, the results for Ra = 10⁰, 10¹ and 10² at τ = 0, 0.1, 0.5, 1 and 2000 are included in Figure 5. Initially, at τ = 0, the temperature is constant, and at this time, there is no fluid flow in the cavity. Fluid flows develop when the inclined walls are suddenly heated and the top wall is suddenly cooled. In Figure 5, the temperature rapidly changes for a small Ra = 10⁰ and 10¹ at τ = 0.1 and becomes symmetric, but for Ra = 10², it becomes symmetric at τ = 0.5 and this symmetric tendency is observed from τ = 0.5 to 2000. For all the Ra and τ values, the cavity contains a pair of symmetrical cells. Convective flows exhibit relatively low magnitudes, with the cells having a prevailing conduction dominance that provides stability.

5.2. Asymmetric Flow

The flow of different Rayleigh numbers of Ra = 10³, 7 × 10³ and 2 × 10⁴ is shown in Figure 6 on the basis of the time with the displayed streamlines and isotherms. Initially, at τ = 0, the temperature is constant and no fluid moves in the cavity. Fluid flows start when the inclined walls are suddenly heated and the top wall is suddenly cooled. In Figure 6, the temperature gradually changes for Ra = 10³ and 7 × 10³ at τ = 0.1 and becomes symmetric, whereas for Ra = 2 × 10⁴, it becomes symmetric at τ = 1. In the initial stage, when τ = 1, the steady flows for all the Rayleigh numbers are symmetric. The cavity has two symmetric cells. Since the cells have a conduction dominance that keeps them eternally stable, convective flows are really extremely low. The baroclinity near the inclined walls produces viscous shear, which induces natural convection flows in the fundamental symmetric state, but as time passes, the flow for Ra = 7 × 10³ and 2 × 10⁴ starts becoming gradually asymmetric. As the figure shows, in the initial transitional stage at τ = 1, the convection in Ra = 7 × 10³ and 2 × 10⁴ is increased, although it is not sufficiently strong to disrupt the symmetric flow structure (see Figure 7), while the convection in Ra = 10³ remains static. Though there are only two cells when Ra = 10³ and Ra = 7 × 10³, four cells occur in the cavity for Ra = 2 × 10⁴ at τ = 10. That is, aside from symmetric break, the Rayleigh number causes an increase in the number of cells in the cavity. It should be noted that depending on the initial perturbations, any one of the two cells may be bigger and advance into the cavity. The convection of Ra = 10³ is found to be constant and weak in all the subsequent stages of time, like τ = 100, 1000, 2000 and 4000, but the convection in Ra = 7 × 10³ becomes stronger slowly in τ = 100, 1000, 2000 and 4000, though it is not enough to break the symmetric state. On the other hand, for Ra = 2 × 10⁴, the convection grows so strongly that it breaks the symmetrical structure and creates more than two new cells, which is evident at τ = 100, and the flow becomes asymmetric (in Figure 7). Six cells are present for Ra = 2 × 10⁴ at τ = 100 and 1000. Depending on the initial perturbations, one of the two cells at τ = 1000 becomes larger and dominates the other cell in the cavity, moving toward the cavity’s center. At τ = 2000, the number of cells decreases to five, and this trend persists until τ = 4000. Upon careful examination, it is evident that during the further development stage, the asymmetric flow has attained a state of equilibrium commonly referred to as a steady-state situation. The transition from a symmetric to an asymmetric state, occurring within the range of Ra = 7 × 10³ to Ra = 2 × 10⁴, can be understood as a supercritical Pitchfork bifurcation. This transformation is driven by the onset of Rayleigh–Bénard instability. It is seen that there is more circulation of cells and that the flow pattern is becoming continually asymmetric.

Figure 7 depicts the time series of the temperature and velocity for Ra = 10³, Ra = 7 × 10³, and Ra = 2 × 10⁴ at different points P₁, P₂, and P₅. The fluid within the valley-shaped cavity initially exhibits isothermal and stationary behavior. The cooling and heating of the top wall and the inclined walls occur simultaneously, with a non-dimensional temperature denoted as θ_c for the top wall and θ_h for the inclined walls. Figure 7a demonstrates that there is no difference in temperature between any of the points P₁ and P_2, and temperature is different at point P₅ in the early stage for a small Ra = 10³. As a result of the passage of time, the temperature increases and decreases at the three different points in the transitional period. Finally, there is no temperature fluctuation in the completely developed stage, and then the flow becomes steady. For more accuracy of the flow steadiness, here, we have used non-dimensional times from 0 to 5000. In Figure 7d, initially, the velocity is zero; as the time increases, the velocity remains the same at points P₁ and P₂; and at point P₅, the velocity increases and finally becomes constant. Similar consequences have been seen in Figure 7b,e. The figures clearly indicate that the flow within the cavity is in a state of steady symmetry. From this, it would seem that the flow is always steady at symmetric states. Now, in Figure 7c, the temperature decreases suddenly and then increases at points P₁ and P₂ during the transitional period and finally stabilizes at the completely developed stage. At point P₅, there are no changes because of its position. However, the velocity (Figure 7f) clearly overshoots suddenly and then becomes constant in the completely formed stage, and it indicates that the flow becomes an asymmetric steady state for Ra = 2 × 10⁴.

5.2.1. Pitchfork Bifurcation

Figure 8 illustrates the isotherms and streamlines corresponding to a Rayleigh number of Ra = 10⁴. Figure 8a demonstrates the continued presence of clear symmetry in the flow at Ra = 7 × 10³. At Ra = 10⁴ and Ra = 1.3 × 10⁴, the flow shows asymmetry, as depicted in Figure 8b,c. It is to be noted that depending on the initial perturbations, any one of the two cells might grow bigger and move toward the cavity. When the Rayleigh number reaches a value of Ra = 1.3 × 10⁴, as depicted in Figure 8c, an additional cell emerges in the top-right region of the cavity (depending on the initial perturbations), and such an asymmetric flow configuration becomes more clearly evident as being the value of the Rayleigh number increases. That is, aside from the symmetric break, the Rayleigh number is responsible for the growing number of cells within the cavity. For instance, Figure 8b has four cells for Ra = 10⁴, whereas Figure 8c has five cells for Ra = 1.3 × 10⁴. The transition observed around Ra = 10⁴, where a symmetric state transforms to an asymmetric state, can be characterized as a supercritical Pitchfork bifurcation, which happens when Rayleigh–Bénard instability begins to occur.

Table 2. Velocities in the x-direction for different Rayleigh numbers at point P₂ (0, 0.46).

Rayleigh Number	Velocity
101	0
102	0
103	0
7.5 × 103	0
7.6 × 103	±9.47 × 10−5
104	±0.00669
1.2 × 104	±0.0557
1.5 × 104	±0.09
105	±0.096
106	±0.0993

presents the x-velocity at point P₂ (0, 0.46) for various Rayleigh numbers (Ra = 10¹ to 10⁶) to comprehend the transition from a symmetric to an asymmetric state during the fully developed stage (τ = 2000) of the Pitchfork bifurcation. For Ra values less than or equal to 7.5 × 10³, the x-velocity is nearly zero due to the symmetry of the flow. Once the Rayleigh number reaches or exceeds 7.6 × 10³, the cell’s polarity shifts to positive when the x-velocity increases and negative when the x-velocity decreases.

At the initial stage at τ = 0, the temperature is constant, as depicted in Figure 9. The figure indicates that the flows are symmetric and continuous in the early stage at τ = 1 for Ra = 5 × 10⁴, Ra = 10⁵ and Ra = 10⁶. The Pitchfork bifurcation causes the flow to intensify over time and eventually become asymmetric. The graphs show the asymmetric streamlines and isotherms at Rayleigh numbers Ra = 5 × 10⁴, Ra = 10⁵ and Ra = 10⁶ at the transitional stages (τ = 100 and 800). It is undoubtedly interesting that the flow oscillates for a considerable amount of time with a higher Rayleigh number. In this figure, for all the Ra values, the flows in the valley-shaped cavity are asymmetric and steady with the passing of time. For a higher Ra = 10⁶, the flow becomes asymmetric and steady at τ = 800, whereas Ra = 5 × 10⁴ and Ra = 10⁵ take more time to reach the asymmetric steady-state situation.

Clearly, the temperature and velocity effect on the flow in the cavity with time, as has been calculated in Figure 10 for more precision at a range of Rayleigh numbers and various points. Figure 10a,b demonstrate that, at first, the temperatures are uniform at points P₁ and P₂, but point P₅ is located close to the heated wall (see Figure 1), which causes it to heat up more quickly than the other points in the early stage. As time goes on, the temperature in the transitional stage first drops and then rises, ultimately achieving its constant point in the fully developed stage, when there are no longer any temperature variations. Also, from Figure 10d,e, it is clearly seen that the velocity increases with the passing of time. The velocity overshoots in the transitional period before becoming stable in the fully developed stage. Figure 10c shows that the temperature increases in the transitional stage and becomes stable in the fully developed stage. In contrast, the velocity in Figure 10f fluctuates during the transitional period before becoming constant during the last stage. The preceding figures and analyses indicate that in an asymmetric state, the flow becomes unsteady in the transitional period and then steady in the fully developed stages.

5.2.2. Other Bifurcations

For higher Rayleigh numbers, the asymmetric isotherms and streamlines are depicted in Figure 11. It is evident that when the Rayleigh number increases, several types of further bifurcations occur, leading to an increase in the number of cells, as seen in Figure 11. In Figure 11a, there are two cells in the cavity. With the increase in the Ra, we see in Figure 11b that the cell number becomes four; in Figure 11c, the cell number is five. For instance, the cell number increases from six at a Rayleigh number of 5 × 10⁴ in Figure 11d to more than six, as shown in

Table 3. Number of cells with corresponding Rayleigh numbers.

Rayleigh Number	Number of Cells
7 × 103	2
104	4
2 × 104	5
5 × 104	6
105	7
3 × 105	9
5 × 105	10
106	11
5 × 106	13
107	15

. Figure 11 demonstrates that when the Rayleigh number rises, more new tiny cells develop in the left or right side of the cavity. However, a single large cell consistently persists at the center of the cavity. This indicates that the flow configuration within the cavity becomes increasingly complex for an asymmetric steady state when the Rayleigh number rises.

Table 3 shows that the Rayleigh number and the number of cells are linked. The Rayleigh number increased from 7 × 10³ to 10⁷ as the number of bifurcations increased from two to fifteen cells. For Ra values between 10⁴ and 10⁷, it is clear that there is a nearly linear link between the number of cells and the Ra.

5.3. Unsteady Flow

An asymmetric flow structure is created as a consequence of a Pitchfork bifurcation that arises during the later stages of the transitional period. As already noted, the numerical simulation experiences a Pitchfork bifurcation early on. Figure 12 displays the streamlines and isotherms for Ra = 10⁷, Ra = 5 × 10⁷, and Ra = 10⁸ to better study the flow at greater Rayleigh numbers for different times. In Figure 12, initially at τ = 0, the temperature is constant for all the Ra values. The figure depicts that the fluids are symmetric for all the Rayleigh numbers when τ = 1, but as time passes, when τ = 5, they become asymmetric and create two additional tiny cells in the cavity at the right and left upper corners. With increasing cell numbers, all the Ra values at τ = 10 in Figure 12 exhibit the same flow pattern. When τ = 50 for Ra = 10⁷, a tiny cell appears in the top center of the two largest cells in the cavity, but for Ra = 5 × 10⁷ and Ra = 10⁸ at τ = 50, the steady flow breaks and becomes unsteady. In Figure 11, the flow is more convoluted for Ra = 5 × 10⁷ and Ra = 10⁸ at τ = 50, even though it is still constant, and the biggest cell has a few smaller cells at its upper right and left side. This indicates that between Ra = 10⁷ and 5 × 10⁷, there is a Hopf bifurcation (see [19] for details on the bifurcation). When Ra = 10⁷, the asymmetric steady state remains similar at all the subsequent transitional, developed transitional and fully developed stages when τ = 100, 1000, 1500 and 2000, respectively. The observed trends remain consistent during both the developed transitional stage (τ = 1000 and 1500) and the completely developed stage (τ = 2000) for two distinct Rayleigh numbers, namely Ra = 5 × 10⁷ and Ra = 10⁸. But as τ rises, as shown in Figure 12 for Ra = 10⁸ at τ = 2000, both cells grow in the middle of the two largest cells. The largest central cell exhibits a right-to-left movement. The unsteady flow becomes more and more complex due to its lack of stability.

Over time, the temperature series has been followed and analyzed spectrally to gain an understanding of the unsteady flow at higher Rayleigh numbers. Figure 13 displays the temperature variation over time and the power spectral densities according to various Ra values. Figure 13a demonstrates a steady flow throughout the fully established stage for Ra = 10⁷, while Figure 13b exhibits periodic flow for Ra = 5 × 10⁷. Additionally, Figure 13c displays the temperature power spectral density located in Figure 13b. The periodic flow’s fundamental frequency, with harmonic modes, is f_p = 0.397. The periodic flow changes as the Rayleigh number rises. This indicates the occurrence of one more bifurcation for which a periodic solution transforms into another. As the Rayleigh number continues to rise, the unstable flow becomes chaotic, as seen in Figure 13d, which depicts the fully developed stage for Ra = 10⁸. According to Figure 13e, the unique frequency with harmonic modes vanishes for Ra = 10⁸, while the flow is chaotic but unstable at the fully developed stage.

Figure 14 depicts a time series analysis of the temperature as well as the velocity at three distinct points, P1, P2, and P5, with higher Rayleigh numbers, specifically Ra = 10⁷, 5 × 10⁷ and 10⁸. Because the temperature is initially isothermal and constant in Figure 14a–c, it is the same at various stages in the early stage for greater Rayleigh numbers. In the transitional period, the temperature fluctuates with the time for each Ra in Figure 14. In the advanced stage of development, as Figure 14a shows, the temperature becomes constant; that is, there are no changes in the temperature, but in Figure 14b, we see that it becomes periodic. In Figure 14c, with the time increases, the flow becomes chaotic and more complex in the fully developed stage. However, the velocity (Figure 14d–f) starts off at zero and changes as time moves on during the transitional stage. Figure 14d–f all show that the flow is stable in the developed stage, periodic in Figure 14e, and chaotic in Figure 14f. In addition, most importantly, it seems that the flow of a periodic state changes from a periodic one to a chaotic one at a stage of transition. Furthermore, in a chaotic state, the flow is constantly chaotic at the transitional stage and at the developed stages.

5.3.1. Hopf Bifurcation

Figure 15 displays the streamlines and isotherms for Ra = 10⁷ and Ra = 2 × 10⁷ to allow for a more in-depth analysis of the flow in the case of greater Rayleigh numbers. Figure 15a illustrates the more complicated flows throughout the fully developed stage for Ra = 10⁷, even though it is still steady. However, a deeper inspection of the numerical results reveals that more than two tiny cells alternately occur; as a result, the flow becomes unsteady for Ra = 2 × 10⁷ in the fully developed stage. This indicates that there is a Hopf bifurcation between Ra = 10⁷ and 2 × 10⁷.

Figure 16 shows the attractors with values ranging from τ = 300 to 2000 for Ra = 10⁷ and τ = 1000 to 1500 for Ra = 5 × 10⁷ at the defining point P₁ (0, 0.825) in order to facilitate an understanding of the Hopf bifurcation, which takes place during the transition from the steady state to the periodic stage. Figure 16a illustrates that the u-θ plane curve reaches a specific value when Ra = 10⁷. Figure 16b displays a limit cycle for Ra = 5 × 10⁷. As a result, when Ra = 5 × 10⁷, a Hopf bifurcation takes place (for further information on the Hopf bifurcation, see [65]).

5.3.2. Chaotic

Figure 17a illustrates that the two cells in the upper right portion of the biggest cell become bigger precisely as the Rayleigh number increases. The biggest cell in the center likewise travels across the right as well as the left sides when Ra = 10⁸, as seen in Figure 17b. The unsteady flow becomes increasingly complicated, which is known as chaotic.

Figure 18 shows the trajectories through the stage space in the u-θ plane at point P₅ (0.5, 0.255) for Ra = 5 × 10⁷ and 10⁸ in order to more clearly illustrate how the periodic condition changes to a chaotic state. The observed limit cycle depicted in Figure 18a demonstrates the periodic nature of the unsteady flow at a Rayleigh number of Ra = 5 × 10⁷. This finding aligns with the information presented in Figure 16 and further supports the existence of a limit cycle. In Figure 18b, the trajectory for Ra = 10⁸ shows that the periodic flow changes to chaotic, which takes place between Ra = 5 × 10⁷ and 10⁸. For a detailed explanation of the stage-space trajectories, see [66].

5.4. Temperature and Velocity

Figure 19 displays the temperature and velocity at designated points P₁ (0, 0.825) over time across various Rayleigh numbers. This information is provided to help understand the formation of natural convection flow patterns within the cavity in response to sudden heating from the inclined walls and cooling from the top wall. The simulations were conducted using various Rayleigh numbers, ranging from Ra = 10⁰ to 10⁸. An observation was made about the various fluctuating flow properties throughout a range of Rayleigh numbers. Figure 5, Figure 6, Figure 9 and Figure 12 depict the isotherms and corresponding streamlines for various Rayleigh numbers, specifically focusing on the case where A = 0.5. The observed numerical outcomes for the various Ra values, as depicted in Figure 19, exhibit discernible changes. At the lowest Rayleigh number, convective flow instabilities may first be seen. However, the number of waves and the unsteadiness increase with increasing Rayleigh numbers. Based on the symmetry and continuous flow, it is expected that the flow is weaker and displays symmetric behavior at Ra = 10³. During the transitional stage of fluid flow, it is observed that the flow develops asymmetrically for the Rayleigh numbers of Ra = 10⁴, 10⁵, and 10⁶. However, as the flow progresses toward the fully developed stage, it stabilizes. Finally, the flow changes into periodic and chaotic states at Ra = 10⁷ and Ra = 10⁸, respectively, as shown in Figure 19a. As illustrated in Figure 19b, a similar characteristic can also be seen in the x-velocity at the point P₁.

5.5. Heat Transfer

During the transitional stage, convective heat transfer dominates in the V-shaped cavity. Convective heat transfer is enhanced by the irregular fluctuations and vortices that encourage fluid mixing. For air, the Nusselt number that expresses the proportion of convective to conductive heat transfer is examined. The Nusselt number Nu [60,61,62] is defined as:

$$Nu=\frac{1}{ln}{\int }_{ln}^{\frac{\partial \theta }{\partial n}}ds.$$

(11)

In accordance with Figure 20a, a time series representation has been provided to illustrate the average Nusselt number associated with the inclined wall. This particular measurement has been employed to quantify the amount of heat transfer occurring across the cavity wall. It is important to highlight that Figure 20b additionally presents the representation of the Nu after being normalized by Ra^1/4. The presence of a significant temperature difference between the fluid and the wall may contribute to the substantial heat transfer phenomenon. Consequently, it is expected that a large value of the Nusselt number (Nu) is observed. The observed phenomenon can be attributed to the simultaneous application of heating and cooling to both the inclined walls and the top wall. Figure 20b illustrates how, as time moves on, throughout the beginning stage, the Nu dramatically decreases, and throughout the fully devolved stage, it gradually attains a value that is either constant or oscillatory, and this value is determined by the Rayleigh number. This is similar to those in Figure 13, where the Nusselt number oscillates for Ra ≥ 5 × 10⁷ but remains constant for Ra = 10⁷. In contrast to those in Figure 20a, it is obvious that the Nu versus the curves in Figure 20b collapse together. As a result, Nu~Ra^1/4 scaling works effectively, considering the current set of Ra values. Figure 20b demonstrates that NuRa^−1/4 almost maintains a constant value of 1.473 as the Rayleigh number rises, although somewhat decreasingly.

6. Conclusions

The current investigation delves into the numerical exploration of unsteady 2D natural convection flows within a valley-shaped cavity filled with water, where heating occurs along the inclined walls while cooling is facilitated through the top wall. The study encompasses a broad spectrum of Rayleigh numbers ranging from Ra = 10⁰ to 10⁸, under the Prandtl number Pr = 0.71 and the aspect ratio A = 0.5, spanning various non-dimensional times from τ = 0 to τ = 5000. The research delineates the diverse flow structures evolving within the cavity over time and scrutinizes the relationship between heat transfer dynamics and Rayleigh numbers.

Analysis reveals that the progression of natural convection flows manifests in three discernible stages: an initial stage, a transitional stage, and a fully developed stage, following the sudden application of heating through inclined walls and cooling through the top wall. This study examines the symmetric, asymmetric, and unsteady flow patterns characterizing these stages, supported by numerical findings. Specifically, the investigation primarily focuses on elucidating the flow mechanisms across all the stages. It is observed that natural convection flows remain steady at the symmetric state for Rayleigh numbers ranging from Ra = 10⁰ to 7 × 10³. Beyond Ra > 7 × 10³, flows exhibit asymmetry. Furthermore, during the asymmetric state, the flow transitions into an unsteady regime in the transitional stage before stabilizing at the fully developed stage for 7 × 10³ < Ra < 10⁷. This study highlights that periodic unsteady flows evolve into chaotic states during the transitional stage, reverting to periodic behavior in the developed stage at Ra = 5 × 10⁷, while at higher Rayleigh numbers like Ra = 10⁸, the chaotic flow remains predominant in the unsteady regime.

Additionally, the investigation discusses the notable bifurcations observed in the fully developed states. Detailed analyses, including the power spectral density and stage space trajectories for Pr = 0.71, are provided. Numerical studies elucidate the intricacies of heat transfer and demonstrate the influence of the Rayleigh number on both the Nusselt number and flow rate dynamics.