Heat Transfer Correlations and Flow-Mode Transitions in Partitioned Cavities for Efficient Thermal Management

Abstract

Partitioned cavities are widely used in passive, compact thermal management systems (data-center liquid cooling, cryogenic hydrogen/LNG storage, and battery modules) where geometric confinement governs natural convection and heat transfer. This study examines buoyancy-driven convection using a two-dimensional steady laminar model with adiabatic partitions under the Boussinesq approximation over Ra = 10³ to 10⁶, partition heights H = 0.1–0.9, and partition numbers N = 0–7. The model is validated against benchmark data. Flow fields are categorized into four modes—single circulation, corner vortices, secondary vortices, and stagnant flow—and their combinations, yielding an integrated flow-mode map that captures regimes and transitions. Two transition mechanisms are identified: slot-scale transitions driven by nonlinear changes in localized vortices and partition-dominated transitions that reorganize the primary circulation. Thermal-field analysis shows how partitions reshape temperature stratification, while the dependence of the Nusselt number on flow modes and geometric parameters is quantitatively analyzed. Quantitatively, strong confinement (H = 0.9, N ≥ 6) reduces global heat transfer by 75–85%, reaching 98% at Ra = 10⁶. Intermediate partitions (H ≈ 0.5, N = 3–4) yield 40–60% reduction. Shallow partitions (H ≤ 0.3) cause <20% loss even at high Ra. The framework links confinement, flow modes, and heat-transfer suppression for design. By unifying partition-induced flow modes and quantifying heat-transfer suppression, this study provides a framework for confined convection.

1. Introduction

Recent advances in high-density data centers, artificial intelligence computing clusters, and sustainable-energy systems have renewed interest in passive- and compact thermal management strategies. In these applications, natural convection within confined or partitioned enclosures governs the overall cooling efficiency and heat-transfer performance [1], directly affecting the operation of electronic racks [2,3], building energy-efficiency optimization [4], hydrogen, LNG storage tanks [5,6,7], and battery modules [8,9]. Because buoyancy-driven transport interacts strongly with geometric constraints, even modest geometric modifications—such as the insertion of partitions—can drastically reorganize flow structures and alter global heat-transfer pathways. The differentially heated square cavity has long served as a canonical benchmark for fundamental studies [10,11]. Extending this paradigm to partitioned geometries transforms it into a case-relevant model for next-generation cooling architectures and energy-efficient thermal systems.

The classical work of Davis [10] established benchmark solutions for ${10}^3\le Ra\le {10}^6$, providing high–accuracy predictions of flow fields and Nusselt numbers that remain the standard for code validation. Subsequent studies extended the problem to higher $Ra$ and different boundary conditions [11,12,13,14,15,16,17], establishing the fundamental framework for partition-free cavities. In addition to such benchmark configurations, several studies introduced internal obstacles or heat sources to explore geometric effects on buoyancy-driven flow [18,19,20,21,22,23], demonstrating that geometric placement significantly alters vortex interactions and global heat transfer. These findings provide a conceptual bridge toward the present study, which examines partitioned cavities as an extended configuration of geometrically constrained enclosures.

Beyond the benchmark cavity without partitions, extensive attention has been devoted to cavities incorporating a single internal partition or baffle under the classical condition of differentially heated vertical walls with insulated horizontal boundaries [24,25,26,27,28,29,30,31,32,33,34]. Sun et al. [24] examined conjugate natural convection with a conductive wall and a single vertical baffle for ${10}^3\le Ra\le {10}^7$, showing strong wall–baffle coupling. Tansim et al. [25] numerically studied cavities with one vertical partition for ${10}^3\le Ra\le {10}^6$ and reported that its height and location reorganize the flow into distinct upper- and lower cells. Selimefendigil et al. [26] analyzed cavities filled with nanofluids and a single partition (${10}^3\le Gr\le {10}^6$), showing that nanoparticle concentration and partition conductivity strongly affect both local and average Nusselt numbers. Khatamifar et al. [27] considered one finite-thickness partition for ${10}^5\le Ra\le {10}^9$, finding that heat transfer rises with $Ra$ but decreases with thickness. Later studies extended the physics while still focusing on single partitions: Yasuri et al. [28] examined magnetic fields, and Zontul et al. [29] showed that periodic fields outperform uniform ones for ${10}^3\le Ra\le {10}^7$. Recent years have seen continued investigations of natural convection in enclosed cavities, with growing attention to geometric effects, internal structures, and complex physical interactions [30,31,32,33].

Partitioned cavities have also been explored under modified boundary conditions. Zimmerman et al. [34] investigated a conducting vertical baffle with adiabatic or conducting end walls for ${10}^4\le Ra\le {7.1\times 10}^5$, while Fu et al. [35] examined transient convection with uniform heat flux for ${10}^4\le Ra\le {10}^9$, highlighting the strong but time-dependent influence of partitions. Additional variations introduced localized or non-uniform heating [36,37,38]. Kandaswamy et al. [36] emphasized the role of thin heated plates (${10}^3\le Gr\le {10}^5$), Mahmoudi et al. [38] reported $Ra$- and nanoparticle-dependent enhancement with horizontal heaters (${10}^3\le Ra\le {10}^6$), and Sathiyamoorthy et al. [38] demonstrated that thin bottom partitions modify Rayleigh–Bénard structures in ${10}^3\le Ra\le {10}^5$.

Recent works have further incorporated multi physics effects into partitioned or baffled cavities. Raisi et al. [39] analyzed fluid–structure interaction for ${10}^3\le Ra\le {10}^6$, showing that higher $Ra$ intensifies both natural convection and structural deformation. Hussein et al. [40] and Al-Farhany et al. [41] studied nanofluid-filled baffled cavities in the same range, reporting that the Nusselt number increases with $Ra$ and nanoparticle fraction but is suppressed by long baffles or sinusoidal heating. Eshaghi et al. [42] optimized the baffle angle and Lewis number in double-diffusive convection, while Chen et al. [43] demonstrated that partition thickness and arrangement strongly influence inverse estimation of heat-transfer coefficients. Alsayegh [44] investigated open cavities with hot baffles filled with hybrid nanoparticles for $5\times {10}^3\le {Ra}_E\le 5\times {10}^4$, showing up to 58% enhancement in average $Nu$ and a distinctive V-shaped trend with the Marangoni number.

In contrast, the effect of introducing multiple partitions has only recently been examined [45,46,47,48,49,50,51,52,53,54]. Under the standard configuration of isothermal vertical walls with insulated horizontal boundaries, Han et al. [45] showed that for ${10}^3\le Ra\le {10}^6$, increasing the number of partitions progressively suppresses circulation and reduces the average Nusselt number, with conduction-dominated regions expanding at higher $Ra$. Saravanan et al. [46] examined two heat-generating baffles at $Gr={10}^7$, $Pr=0.71$, reporting that narrow spacing lowers $Nu$, whereas wall-adjacent placement enables partial recovery of convective transport. Costa [47] studied natural convection in square cavities with two adiabatic partitions for ${10}^4\le Ra\le {10}^6$, showing that partition geometry strongly alters flow organization and heat transfer.

In addition to studies on multiple partitions under classical boundary conditions, recent research has examined configurations where several partitions are combined with non-uniform or unconventional thermal boundaries [48,49]. Dagtekin et al. [48] investigated enclosures with two bottom-heated partitions for ${10}^4\le Ra\le {10}^6$, showing that taller partitions suppress convection, enhance conduction, and significantly influence vortex formation and global transport. Kalidasan et al. [49] analyzed ventilated cavities with a vertical partition and a cold transverse baffle for ${10}^3\le Ra\le {10}^5$, finding that the baffle enhances heat transfer by reshaping vortices, but blockage dominates when it is placed adjacent to the partition.

Further extensions have considered differentially heated vertical walls with thin baffles attached [50,51,52,53,54,55]. Fontana et al. [50] studied trapezoidal cavities with one or two adiabatic baffles for ${10}^3\le Ra\le {10}^6$, showing that adding a second baffle or increasing height suppresses convection, enhances stratification, and reduces the average Nusselt numbers. Silva et al. [51] examined trapezoidal cavities with two baffles for ${10}^3\le Ra\le {10}^6$, highlighting the roles of baffle height and top-wall inclination and proposing $Nu$ correlations with $Ra, Pr$, and geometry. Jabbar et al. [52] investigated two thin sidewall-mounted baffles over ${10}^4\le Ra\le {10}^8$, reporting that longer baffles split circulation into dual vortices and reduce the average $Nu$. El Hamri et al. [53] studied horizontal solid partitions for ${10}^3\le Ra\le {10}^6$, finding that extending the partition length enhances heat transfer, with inline layouts minimizing entropy generation. Hansda et al. [54] investigated thermosolutal convection of Casson fluids with porous baffles under sinusoidal heating for ${10}^4\le Ra\le {10}^6$, demonstrating that larger Casson parameters improve heat and mass transfer while reducing entropy generation. Kim et al. [55] conducted a systematic numerical investigation on natural convection control using a single vertical partition over the range of ${10}^3\le Ra\le {10}^6$, demonstrating that partition height is a key parameter governing convective flow suppression and heat-transfer reduction.

As reviewed above, prior studies have mainly focused on partition-free cavities or on enclosures with a single internal partition or baffle under limited thermal and geometric conditions. More recent efforts have begun to examine multiple partitions, but these remain fragmented and lack a unified framework that systematically captures the combined effects of partition height, number, and buoyancy strength on flow organization and heat transfer.

In this context, internal partitions are often employed in confined thermal systems to regulate buoyancy-driven flow structures and associated heat-transfer characteristics. From this perspective, the present study provides physical insight that may be useful for interpreting partitioned enclosures, such as enclosed electronic systems and other passive thermal-management configurations. In particular, partition height and number are interpreted as generic design parameters that, respectively, control the suppression of buoyancy-driven convection and the degree of conductive isolation in confined enclosures.

The present work therefore conducts a systematic numerical and parametric investigation of natural convection in square cavities containing multiple vertical partitions of varying height ($0.1\le H\le 0.9$) and number ($0 \le N\le 7$) across a Rayleigh-number range of ${10}^3$ to ${10}^6$. The goal is to establish, for the first time, a unified framework that links partition geometry and buoyancy strength to flow-mode transitions, thermal stratification, and global heat-transfer suppression. Unlike earlier studies restricted to single partitions or limited parameter ranges, the present work develops a comprehensive flow-mode map and quantifies how geometric confinement reorganizes convection toward conduction-dominated states.

The novelty of this study lies in providing (i) a systematic parametric analysis that identifies key governing parameters, (ii) a unified classification of flow modes and transition mechanisms, and (iii) quantitative trends and relationships for global heat-transfer reduction. Beyond its academic significance, the results yield engineering guidance for efficient thermal management, including passive electronic cooling, building energy optimization [4], and natural convection control in storage tanks.

2. Numerical Details

2.1. Physical and Numerical Formulation

This study considers steady, two-dimensional, laminar natural convection in a square cavity of side length ($L$), as illustrated in Figure 1. The cavity is differentially heated such that the left wall is maintained at a uniform hot temperature $T_h$ and the right wall at a uniform cold temperature $T_c$, while the horizontal walls are thermally insulated. Inside the cavity, vertical adiabatic partitions are mounted on the bottom wall. Each partition is attached to the bottom wall at its root, has a uniform thickness of $w=0.02$, and extends upward with a prescribed height, where all geometric parameters are defined relative to the cavity length $L$. Six representative heights are examined, namely $H=0.1,\hspace{0.17em}0.3,\hspace{0.17em}0.5,\hspace{0.17em}0.7,\hspace{0.17em}0.8$, and $0.9$.

The number of partitions is varied from $N=0$ (no partition) to $N=7$. For $N\ge 2$, the partitions are uniformly distributed with equal spacing across the cavity width, thereby generating systematically subdivided flow domains. For example, Figure 1a presents an example of an even-numbered partition configuration with two partitions ($H=0.9, N=2$), whereas Figure 1b illustrates the corresponding odd-numbered configuration with multiple partitions ($H=0.9, N=7$) uniformly distributed across the cavity.

The working fluid is assumed to be Newtonian and incompressible. Density variations are neglected except in the buoyancy term of the momentum equations through the Boussinesq approximation. Within the considered Rayleigh number range, the flow remains laminar. Although natural convection in square cavities may exhibit weak unsteadiness or three-dimensional flow structures as $Ra$ approaches ${10}^6$, benchmark and numerical studies on differentially heated square cavities have shown that two-dimensional steady laminar solutions remain a valid approximation up to $Ra\approx {10}^6$ for $Pr=0.71$ [10,11,12,13,14,15,16,17].

To enable nondimensional analysis and comparison with previous studies, the governing equations are expressed in dimensionless form using the following scaling:

$$x={\displaystyle \frac{x^{\ast }}{L^{\ast }}}, y={\displaystyle \frac{y^{\ast }}{L^{\ast }}}, u={\displaystyle \frac{u^{\ast }{L}^{\ast }}{\alpha }}, v={\displaystyle \frac{v^{\ast }{L}^{\ast }}{\alpha }}, P={\displaystyle \frac{P^{\ast }{L}^{\ast 2}}{{\rho }^{*}{\mathsf{\alpha }}^2}}, \theta ={\displaystyle \frac{T^{\ast }-T_c^{\ast }}{T_h^{\ast }-T_c^{\ast }}}+{\displaystyle \frac{\partial v}{\partial y}}=0$$

(1)

here, $x,y$ denote the nondimensional spatial coordinates, while $u,v$ correspond to the velocity components in the horizontal and vertical directions, respectively. The variable $P$ represents the nondimensional pressure, and $\theta$ indicates the scaled temperature field. On this basis, the governing equations for conservation of mass, momentum, and energy can be formulated in the following dimensionless form.

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

(2)

$$u{\displaystyle \frac{\partial u}{\partial x}}+v{\displaystyle \frac{\partial u}{\partial y}}=-{\displaystyle \frac{\partial P}{\partial x}}+{\displaystyle \frac{1}{Pr}}\left({\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}+{\displaystyle \frac{{\partial }^{2}u}{\partial y^2}}\right)$$

(3)

$$u{\displaystyle \frac{\partial v}{\partial x}}+v{\displaystyle \frac{\partial v}{\partial y}}=-{\displaystyle \frac{\partial P}{\partial y}}+{\displaystyle \frac{1}{Pr}}\left({\displaystyle \frac{{\partial }^{2}v}{\partial x^2}}+{\displaystyle \frac{{\partial }^{2}v}{\partial y^2}}\right)+RaPr\theta$$

(4)

$$u{\displaystyle \frac{\partial \theta }{\partial x}}+v{\displaystyle \frac{\partial \theta }{\partial y}}={\displaystyle \frac{{\partial }^2\theta }{\partial x^2}}+{\displaystyle \frac{{\partial }^2\theta }{\partial y^2}}$$

(5)

The flow behavior is characterized by two nondimensional parameters: the Rayleigh number $Ra$ and the Prandtl number $Pr$. They are given as

$$Ra={\displaystyle \frac{g^{\ast }\beta \left(T_h^{\ast }-T_c^{\ast }\right)L^{\ast 3}}{v\alpha }}, Pr={\displaystyle \frac{v}{\alpha }}$$

(6)

where $g^{\ast }$ denotes the gravitational acceleration, $\beta$ the thermal expansion coefficient, $v$ the kinematic viscosity, and $\alpha$ the thermal diffusivity of the working fluid. In the present simulations, the Rayleigh number is systematically varied from ${10}^3$ to ${10}^6$, covering regimes from conduction-dominated to strongly buoyancy-driven laminar convection. The Prandtl number is fixed at $Pr=0.71$.

The boundary conditions are shown in Figure 1. All solid surfaces are impermeable and satisfy the no-slip condition ($u=v=0$). The hot and cold vertical walls are kept at fixed temperatures corresponding to $\theta =1$ and $\theta =0$, respectively, while the horizontal walls and vertical partitions are adiabatic (${\displaystyle \frac{\partial \theta }{\partial n}}=0$). For the pressure field, homogeneous Neumann boundary conditions (${\displaystyle \frac{\partial P}{\partial n}}=0$) are applied.

The equations are discretized using the finite volume method on a non-uniform Cartesian mesh with refinement near hot/cold walls and partitions. Convective terms are treated with a second-order central difference scheme and diffusive terms with second-order central differencing. Pressure–velocity coupling is handled by the SIMPLE algorithm. Iterations continue until residuals fall below ${10}^{-6}$, ensuring steady-state convergence. A segregated solver under steady conditions is employed.

The heat transfer characteristics are quantified using the local and surface-averaged Nusselt numbers. The local Nusselt number along the heated wall is defined as

$$Nu(y)=-{\left.{\displaystyle \frac{\partial \theta }{\partial n}}\right|}_{\mathrm{w}\mathrm{a}\mathrm{l}\mathrm{l}},$$

(7)

where $\theta$ is the dimensionless temperature and $n$ denotes the coordinate normal to the wall. The surface-averaged Nusselt number, representing the overall heat transfer rate through the heated wall, is obtained by integrating the local values along the wall height:

$$\overline{Nu}={\int }_0^{1}Nu(y)dy$$

(8)

Here, the wall height has been nondimensionalized by the cavity length. Grid independence is demonstrated in Section 2.2, and validation against benchmark solutions is presented in Section 2.2.

2.2. Grid Dependency Study and Validations

The grid configuration adopted in this study is illustrated in Figure 2, where (a) shows a typical even-partition arrangement and (b) an odd-partition arrangement, both based on the medium grid (202 × 202). Local clustering is applied near the hot and cold sidewalls as well as around the vertical partitions to accurately capture steep boundary-layer gradients. To verify grid independence, three meshes were tested: a coarse grid (101 × 101), a medium grid (202 × 202), and a fine grid (303 × 303). The assessment was performed at $Ra$ = ${10}^6$ for two representative configurations: Case 1 with a relatively low partition height ($H=0.1,N=2$) and Case 2 with strong confinement ($H=0.9,N=7$). The surface-averaged Nusselt numbers from these tests are summarized in

Table 1. Surface-averaged Nusselt number ($\overline{Nu}$) and relative difference with respect to the fine grid at $Ra={10}^6$, Case 1 for $H=0.1, N=2$, Case 2 for $H=0.9, N=7$.

Grid Case	Grid Resolution	$\overline{\mathit{Nu}}$	Relative Difference vs. Fine (%)
Case1 ($H=0.1, N=2$)	101 × 101 (coarse)	8.794	0.57
	202 × 202 (medium)	8.744	0.10
	303 × 303 (fine)	8.735	0
Case2 ($H=0.9, N=7$)	101 × 101 (coarse)	0.131	1.55
	202 × 202 (medium)	0.129	0
	303 × 303 (fine)	0.129	0

. When compared with the fine grid, the medium grid shows only a 0.10% deviation in Case 1 and 0% in Case 2, confirming their close agreement. In contrast, the coarse grid differs from the medium grid by 0.57% in Case 1 and 1.55% in Case 2. Accordingly, the medium grid (202 × 202) was adopted for all subsequent simulations, as it provides sufficient accuracy while avoiding excessive computational expense.

To ensure the reliability of the numerical methodology, a two-step validation procedure was conducted. In the first step, simulations were benchmarked against the classical square cavity natural-convection problem. The surface-averaged Nusselt number ($\overline{Nu}$) along the heated wall was evaluated over the Rayleigh number range ${Ra=10}^3$ to ${10}^6$, and compared with the reference data of Davis [10] and Markatos et al. [11]. As summarized in

Table 2. Validation of the present surface-averaged Nusselt number ($\overline{\mathit{Nu}}$) for both pure cavity and the partitioned cavity. Results are compared with benchmark data from Davis [10] and Marakos et al. [11] for the classical square cavity, and with Costa [47] for partitioned cavity configurations.

			Pure Cavity			Partitioned Cavity
$\mathit{R}\mathit{a}$	Present	Davis [10]	Difference (%)	Markatos et al. [11]	Difference (%)	Present	Costa [47]	Difference (%)
${10}^3$	1.118	1.118	0.000	1.108	0.903	-	-	-
${10}^4$	2.246	2.243	0.134	2.201	2.045	2.07	2.06	0.49
${10}^5$	4.533	4.519	0.310	4.430	2.325	4.17	4.17	0.00
${10}^6$	8.896	8.799	1.102	8.754	1.622	8.19	8.17	0.24

, the present results exhibit excellent agreement, with deviations limited to 0.00–2.33% across the tested Rayleigh numbers. This confirms that the adopted discretization and solution procedure are sufficiently accurate for the canonical cavity problem.

In the second stage, validation was extended to partitioned cavity configurations by comparison with the work of Costa [47], who investigated natural convection in square enclosures containing two vertical adiabatic partitions in the range $Ra={10}^4$ to ${10}^6$. As shown in Table 2, the present simulations closely reproduce Costa [47]’s results, with discrepancies confined within 0.49–0.24%. This strong agreement demonstrates that the present numerical model can reliably capture partition-induced modifications to natural convection. Collectively, these validation efforts confirm that the present approach yields accurate and robust predictions for both classical and partitioned cavity problems across a broad range of Rayleigh numbers.

3. Results and Discussion

3.1. Definition of Convection Flow Modes

In this section, the natural-convection flow fields driven by the temperature difference between the two vertical walls are systematically classified into distinct flow modes according to the Rayleigh number ($Ra$), partition height ($H$), and number of partitions ($N$). This mode-based framework enables diverse flow phenomena to be interpreted consistently and clarifies how structural transitions induced by geometry and buoyancy strength relate to the heat-transfer characteristics discussed in the following sections.

As shown in Figure 3, the fundamental configuration in which the cavity is dominated by a single large-scale circulation is referred to as single circulation (SC), which may include embedded inner vortices. This mode represents the baseline structure observed under most conditions. Although the number of inner vortices varies with $Ra$, $H$, and $N$, the overall loop remains unchanged: heated fluid rises along the left hot wall, moves rightward across the top, descends the cold wall, and returns along the bottom, forming a closed circulation. The SC mode typically occurs in the absence of partitions, as shown in Figure 3a, or when the number of partitions is very low, particularly at relatively high partition heights, as shown in Figure 3b. In some cases, partitions displace the primary vortex without breaking it apart, leaving the SC mode intact. This behavior is consistent with previous reports of single-cell circulation in partition-free cavities [10,11,12,25,40].

In addition to the dominant SC mode, three partition-induced modes emerge as the partition height and number increase, with confinement effects progressively altering and ultimately breaking down the primary vortex [37]. As shown in Figure 4, three additional modes arise in the presence of partitions. The corner-vortex (CV) mode develops when descending or ascending streams collide with partition tips and cavity corners, deflecting the flow, suppressing lower circulation, and generating localized recirculation cells, as shown in Figure 4a. The secondary-vortex (SV) mode occurs when part of the primary loop detaches to form additional small-scale circulations that coexist with the main vortex, typically when partitions restrict inflow into the lower region, as shown in Figure 4b. The stagnant-flow (SF) mode emerges when tall partitions weaken or block momentum exchange, preventing the descending cold stream from penetrating into the lower cavity and leaving conduction-dominated stagnant zones between partitions, as shown in Figure 4c. Together with the baseline single-circulation (SC) mode, these partition-induced modes constitute the four fundamental flow patterns, with all observed fields interpretable as their combinations or transitions. The emergence and transition of these flow modes can be interpreted as the result of competing effects between buoyancy-driven jets and partition-induced geometric confinement, which control vortex formation, detachment, and circulation suppression.

For clarity and reproducibility, the classification of flow modes is based on explicit criteria derived from streamline topology and vortex connectivity.

The single-circulation (SC) mode is identified by the presence of a single continuous large-scale circulation spanning the cavity height, regardless of embedded inner vortices. The corner-vortex (CV) mode is characterized by localized recirculation cells forming near partition tips or cavity corners without detachment of the primary loop. The secondary-vortex (SV) mode is distinguished by the formation of additional detached vortices coexisting with the main circulation. The stagnant-flow (SF) mode is identified when the lower cavity region exhibits negligible circulation and is dominated by conduction, indicating suppression of convective penetration.

3.2. Classification and Map of Convection Flow Modes

3.2.1. Representative Streamline Patterns

As defined in Figure 3 and Figure 4, the convection flow field can be classified into four fundamental modes—single circulation (SC), corner vortices (CV), secondary vortices (SV), and stagnant flow (SF)—and their combinations. These modes serve as the building blocks for interpreting all observed flow structures. Figure 5 presents representative streamline patterns illustrating how these modes evolve with $H, N$ and $Ra$. Each panel corresponds to a specific ($H, N, Ra$) condition, thereby linking the schematic definitions in Figure 3 and Figure 4 to the physical realizations of convection flow modes. The observed transitions are primarily governed by two coupled mechanisms: (i) geometric confinement, which reduces the available circulation path and promotes localized recirculation, and (ii) buoyancy strength, which dictates the stability and persistence of vortical structures.

At $H=0.1$, short partitions cause only minor disturbance. For $N=0$, the cavity sustains the global SC loop across all $Ra$. At $N=1$, SC persists at $Ra={10}^3$, but as $Ra$ increases, a corner vortex forms near the partition tip and cavity corner, producing SC–CV. At $N=3$, the impinging cold jet strengthens this vortex, yielding a clearer SC–CV state, consistent with earlier reports of corner- and secondary vortices. Similar emergence of corner- and secondary vortices under comparable conditions has been reported [25]. At $N=5$, confinement reduces momentum transfer into the lower cavity, weakening CV while inducing mid-region recirculation that generates SV, leading to SC–SV. With $Ra$ raised from ${10}^5$ to ${10}^6$, buoyancy further energizes the jet, amplifying SV and driving earlier transitions to multi-vortex states, in agreement with previous observations of buoyancy intensification [16].

At $H=0.5$, the mid-height partitions introduce stronger confinement, narrowing the effective flow passage. For $N=0–1$, SC is preserved at $Ra={10}^3$ and ${10}^4$. At $N=3$, the descending jet is partially blocked, forcing localized recirculation in the mid-domain and producing SC–SV. At $N=5$, the jet is redirected toward the partition tip, reinforcing CV that coexists with SV, resulting in SC–CV–SV. This coexistence of vortices is consistent with earlier partition studies [25,34]. As $Ra$ increases further, buoyancy alters these interactions: at $Ra={10}^3 \mathrm{t}\mathrm{o} {10}^4$, CV is weaker and prone to suppression, whereas at $Ra={10}^5$, CV persists and strengthens, yielding robust SC–CV–SV coexistence. At $Ra={10}^6$, buoyancy is sufficiently strong that additional small-scale vortices appear, destabilizing the main loop and intensifying the complexity of the SC–CV–SV state. This regime clearly highlights how geometric confinement and buoyancy jointly dictate vortex persistence and interaction.

At $H=0.8$, tall partitions nearly split the cavity: SC dominates for $N=0–3$, but at $N=5$ the descending jet loses momentum and a conduction-dominated stagnant zone forms below (SF), while SC or SV persists above (SC–SF/SC–SV–SF). Circulation suppression with increasing partition length is well-documented [37,53]. Even as $Ra$ increases, the upper loop strengthens but the lower stagnation remains, showing geometric confinement outweighs buoyancy.

Overall, Figure 5 shows that short partitions mainly trigger CV and SV, mid-height partitions promote multi-vortex coexistence, and tall partitions suppress lower circulation to form stagnant zones. As $N$ increases, geometric confinement intensifies, progressively narrowing flow space and amplifying localized recirculation. Meanwhile, higher $Ra$ amplifies SV and accelerates the breakdown of SC, driving transitions to complex multi-vortex states.

3.2.2. Slot- and Partition-Level Transitions

Figure 6 and Figure 7 highlight two distinct regimes of confinement-driven transition. In the slot-scale regime (Figure 6: $H=0.1,N=3$), the global SC remains intact, but local slot vortices undergo sensitive nonlinear shifts between CV and SC or transitions from SC to SV, demonstrating how small-scale recirculations respond to buoyancy and geometry. In contrast, the partition-dominated regime (Figure 7: $H=0.7,N=4$) destabilizes the SC backbone itself, with the upper loop compressed, fragmented, and ultimately reorganized into a composite SC–CV–SV–SF state. This contrast establishes a clear hierarchy: local confinement reshapes secondary vortices, whereas strong confinement restructures the global circulation. These pathways connect geometry and buoyancy to the multi-mode convection map in Figure 8.

Figure 6 illustrates the detailed evolution of flow structures in a partitioned cavity ($H=0.1,N=3$) as $Ra$ increases from ${10}^3$ to ${10}^6$. The figure highlights how local recirculation forms, vanish, or transform within distinct wall–partition slots, thereby revealing the mechanisms by which buoyancy strength and geometric confinement interact to reorganize the circulation. In particular, Figure 6a–d demonstrate that global SC persists as the backbone of the cavity flow as $Ra$ increases, while slot-level vortices undergo distinct nonlinear or unidirectional transitions depending on their location relative to the main loop.

In the left wall–first partition slot (L-slot), the sequence CV → SC → SC → CV is observed with increasing $Ra$. At $Ra={10}^3$, a corner-attached vortex (CV) is present, as shown in Figure 6a. At $Ra={10}^4$, the vortex disappears and a clear SC through-flow develops, as shown in Figure 6b. The SC persists at $Ra={10}^5$ in Figure 6c, but at $Ra={10}^6$, the corner vortex reattaches, completing a nonlinear back-and-forth transition, as shown in Figure 6d. This cyclic response reflects the sensitivity of the L-slot to the competition between buoyancy-driven ascent and partition-induced confinement. Importantly, these slot-level reversals are closely tied to the location of the SC vortex center: when the primary circulation shifts upward at intermediate $Ra$, the L-slot is flushed clear; when it shifts back downward at high $Ra$, the ascending branch re-energizes the corner, allowing CV to reappear.

In the right wall–first partition slot (R-slot), the progression follows CV → CV → SC → SC, constituting a unidirectional transition. At $Ra={10}^3$ and ${10}^4$, the descending cold jet sustains a corner vortex, as shown in Figure 6a,b. At $Ra={10}^5$, the vortex vanishes and the SC state dominates, which persists at $Ra={10}^6,$ as shown in Figure 6c,d. Unlike the L-slot, the R-slot shows monotonic suppression of recirculation, with no return to CV. This trend is consistent with the compression of the SC vortex center toward the hot wall, which reduces kinetic energy delivery to the right-lower corner and permanently suppresses corner circulation. Between the first- and second right partitions (P12-slot), the sequence evolves as CV → CV → CV → SV. Corner vortices remain stable from $Ra={10}^3$ to ${10}^5$, as depicted in Figure 6a–c, but at $Ra={10}^6$ the intensified descending jet and strong inter-partition shear destabilize the vortex, yielding a detached secondary vortex (SV), as shown in Figure 6d. This again represents a unidirectional transition. Here, the appearance of SV correlates with the increase in the number of internal vortices within SC: as $Ra$ increases, SC fragments into multiple cells, one of which projects into the partition gap and seeds the detached SV.

Overall, the slot-dependent pathways reveal two classes of transitions. The L-slot undergoes a nonlinear back-and-forth transition (CV → SC → SC → CV), whereas the R-slot and P12-slot show unidirectional behavior, with vortices either suppressed into SC or transformed into SV. These outcomes are not isolated effects but are directly linked to the position and multiplicity of SC internal vortices. Shifts in the vortex center determine whether slot vortices reattach or vanish, while the proliferation of SC cells at high $Ra$ promotes the emergence of detached SV in partition gaps.

Figure 7 illustrates flow evolution in a cavity with tall partitions ($H=0.7,N=4$) as $Ra$ increases from ${10}^3$ to ${10}^6$. The results emphasize how geometric confinement governs the onset of composite convection states. Unlike the slot-scale case of Figure 6, where transitions hinge on shifts in the SC vortex center and its sub-vortices, Figure 7 shows that tall partitions reorganize the SC itself, compressing and destabilizing the main loop until a composite onset emerges. Across $Ra={10}^3 \mathrm{t}\mathrm{o} {10}^5$, as shown in Figure 7a–c, the upper domain is dominated by a large-scale SC, while the lower cavity remains stagnant (SF). This SC–SF coexistence reflects the overwhelming influence of partition-induced blockage: the cold-wall downflow is unable to penetrate into the lower cavity, and buoyancy effects alone cannot overcome confinement. As $Ra$ increases, the SC itself is progressively compressed upward, and its internal structure begins to reorganize. At $Ra={10}^4$ in Figure 7b, the single loop elongates horizontally. While at $Ra={10}^5$ in Figure 7c, weak cell-like vortices emerge within the upper SC, signaling the sensitivity of the main circulation to inter-partition shear. The presence, number, and positions of these sub-vortices are directly conditioned by the partition slots: narrow gaps constrain circulation pathways and dictate where the SC center shifts or where localized recirculation nucleates. Thus, the partitions sculpt not only the lower-cavity suppression but also the internal topology of the surviving SC.

At $Ra={10}^6$ in Figure 7d, buoyancy finally rivals geometric confinement, driving a sharp reorganization of the flow. The impinging cold jet destabilizes the SC, leading to the simultaneous onset of CV, SV within partition gaps, and persistent SF regions below. The result is a fully composite SC–CV–SV–SF state. Notably, the SC no longer persists as a monolithic single loop but fragments into slot-aligned vortices, demonstrating how partition geometry governs both the persistence of stagnation and the architecture of the upper circulation, with buoyancy acting as the trigger for its breakdown into a multi-mode state at high $Ra$.

3.2.3. Parametric Maps of Mode Occurrence

The flow modes across partition height $H$, partition number $N$, and Rayleigh number $Ra$, showing how single circulation evolves into dual and composite states are presented in Figure 8a, Figure 8b, Figure 8c and Figure 8d, for $Ra={10}^3$, ${10}^4$, ${10}^5$, and ${10}^6$, respectively. The transition hierarchy proceeds from single modes, through dual-mode perturbations, to multi-mode composites, with buoyancy accelerating confinement-induced breakdown.

I.

Single Mode (SC).

At all Rayleigh numbers, the single circulation (SC) mode appears as the baseline state, typically at $N=0$ and persisting into small $N$ when $H$ is low to moderate. At $Ra={10}^3$ (Figure 8a), SC spans a wide range of $H$ up to $N=2–3$. With stronger buoyancy (Figure 8b–d), SC remains at $N=0–1$, but its stability range contracts, surviving only up to $N=2$ at $Ra={10}^6$. The defining feature is that, despite small internal vortices, a single large-scale loop governs the cavity.

II.

Dual Mode Transitions (SC–CV, SC–SV, SC–SF).

As confinement increases, SC undergoes localized perturbations that generate dual-mode states. The SC–CV state consistently appears at low $H$ and small $N$ (e.g., $H=0.1$, $N=2$), where the descending jet interacts with the partition tip and lower corner to form a corner vortex. With increasing $Ra$, this corner vortex develops earlier and with greater strength. The SC–SV state is observed at intermediate heights such as $H=0.5, N=3$, where geometric restriction forces part of the main loop to recirculate in the mid-region, forming a secondary vortex alongside the dominant circulation. Finally, tall partitions promote the SC–SF state, in which the lower cavity becomes conduction-dominated while SC persists above; this is typical at $H\ge 0.8$ with $N\ge 5$ (Figure 8c,d).

III.

Composite Multi-Mode States (SC–CV–SV, SC–SV–SF, SC–CV–SV–SF).

At larger $N$ or stronger buoyancy, these dual modes merge into more complex multi-mode composites. For example, at $H=0.5, N=5$, the diverted jet simultaneously reinforces corner vortices and induces mid-region recirculation, producing an SC–CV–SV configuration (Figure 8b,c). Under taller partitions ($H=0.7–0.9$) and larger $N$, the lower cavity stagnates while secondary vortices persist above, giving rise to SC–SV–SF (Figure 8b–d). As the partition length increases, the lower region becomes stagnant while vortices continue to persist in the upper region [37,53]. In the most confined cases—such as $H=0.5, N=6$, or $H=0.8, N=7$—all three mechanisms coexist: corner vortices at partition tips, secondary vortices in the mid-region, and stagnation in the lower cavity. This yields the most complex composite mode, SC–CV–SV–SF, observed particularly at $Ra={10}^3$ to ${10}^5$ (Figure 8a–c).

IV.

Transition Hierarchy and Buoyancy Effects.

The overall sequence of transitions follows a clear hierarchy: SC (single mode) dominates at low confinement, dual-mode states (SC–CV, SC–SV, SC–SF) emerge as partitions obstruct the circulation, and multi-mode composites (SC–CV–SV, SC–SV–SF, SC–CV–SV–SF) develop under stronger geometric confinement and buoyancy. As $Ra$ increases, transitions occur at smaller $N$, with CV and SV strengthening while SF expands in tall-partition cases. Buoyancy thus amplifies the geometric effects, accelerating the breakdown of SC into composite vortex structures.

3.3. Mode-Based Analysis of Thermal Fields and Heat Transfer

3.3.1. Organization of Temperature Fields and Thermal Stratification

Figure 9 presents representative isotherm distributions for selected ($H,N$) conditions across $Ra={10}^3$ to ${10}^6$, classified according to the convection flow modes defined in Figure 3 and Figure 4. In the canonical single circulation (SC) case shown in Figure 9a, the thermal field exhibits the typical buoyancy-driven organization, previously reported in [9,10,11,12,34,35]. Along the heated wall, isotherms are densely packed near the bottom and gradually relax toward the top, while the cooled wall shows the opposite trend. This vertical asymmetry reflects the strong lower-wall gradient and its attenuation with height. With increasing Ra, the isotherms become progressively compressed and tilted, signaling intensified convective transport and enhanced vertical heat removal by the main loop. Such compression and tilting trends have also been documented in [10,11,12,13,23,27,29].

A modified SC pattern appears in tall, few-partition configurations (e.g., $H=0.7,N=2$) in Figure 9b. Here, the global single loop persists, but the lower hot wall becomes nearly isothermal, revealing localized conduction-dominated behavior by partition-induced blockage. Thus, convection is limited to the upper cavity, where dense, tilted isotherms dominate. This transitional behavior indicates that partition geometry can induce stratification even within the SC regime, suppressing lower circulation while preserving the global loop. It has been reported that the lower circulation can be partially suppressed while the overall loop structure is still preserved [37,53].

In the transitional modes SC–CV and SC–SV [Figure 9c,d], the thermal field departs from canonical SC through localized distortions. In SC–CV, corner vortices compress isotherms near the mid-height cold wall, shifting heat transfer upward. In SC–SV, secondary recirculations bend isotherms in the mid-cavity, weakening lower hot-wall gradients and concentrating convective transport above. These patterns show how moderate confinement redirects buoyancy-driven transport from lower boundary layers into upper regions.

When the cavity enters the stagnant flow (SF) configurations in Figure 9e, the lower domain becomes almost completely conduction dominated. The isotherms here are widely spaced and nearly horizontal, highlighting the suppression of convection. Heat transfer is then confined to the upper boundary of the cavity, where only weak circulation remains.

At larger partition heights and numbers, composite modes emerge [Figure 9f–h]. In SC–CV–SV [Figure 9f], the main loop persists above the partition but is modified by corner vortices and secondary recirculations, producing dense isotherm clusters and irregular tilting. In SC–SV–SF [Figure 9g], the lower cavity stagnates while the upper domain retains circulation with secondary vortices, yielding a two-layered structure. The most complex state, SC–CV–SV–SF [Figure 9h], combines these features: the lower region remains stagnant, while above the partitions the isotherms are simultaneously distorted by corner vortices, bent by secondary recirculations, and steepened by the overarching loop. The result is a patchwork of dense and sparse zones, reflecting the superposition of multiple competing mechanisms. Similar patterns of intensified stratification and flow distortion in the upper region have also been documented in previous studies [39,49,50].

Together, Figure 9c–h reveal how confinement reshapes buoyancy-driven transport: SC–CV shifts heat transfer upward through corner-vortex distortions, SC–SV redirects convection to the mid-cavity via secondary recirculations, SF suppresses lower hot-wall gradients into conduction, and composite states (SC–CV–SV–SF) superimpose all mechanisms into a layered, patchwork thermal field. This progression shows that confinement does not simply weaken convection, but redistributes and fragments it into localized modes.

The wall heat transfer characteristics are most clearly interpreted by linking the basic flow mode definitions in Figure 3 and Figure 4 with the thermal fields in Figure 9 and the corresponding local Nusselt number profiles in Figure 10. These $Nu$ distributions reflect the underlying flow modes, illustrating how buoyancy and geometric confinement jointly govern wall heat transfer.

3.3.2. Pure-Mode Nusselt Distributions

In SC mode, as defined in Figure 3 and illustrated in the thermal field of Figure 9a, the large-scale loop generates strong boundary-layer gradients along both vertical walls. This produces monotonic $Nu$ distributions, as shown in Figure 10a,b, with the largest values near the bottom of the hot wall and the top of the cold wall. At $H=0.0,N=0,Nu$ peaks sharply near the hot-wall bottom ($\approx 3$ at $Ra={10}^3$, rising to $\approx 8$ at $Ra={10}^5$), consistent with buoyancy-driven enhancement of boundary-layer transport.

In CV mode, introduced in Figure 4a and represented thermally in Figure 9c, descending flow impinges on the partition tip and generates localized vortices. These vortices redistribute the thermal gradients near mid-height, weakening the smooth monotonic profile of SC. The effect appears in Figure 10c,d, where $Nu$ distributions deviate from monotonicity and show local enhancements, especially along the cold wall.

The SV mode, defined in Figure 4b and illustrated in Figure 9d, shifts active circulation upward, bending the isotherms and reducing gradients near the lower hot wall. Consequently, as shown in Figure 10e,f, the $Nu$ distributions are no longer monotonic: suppressed values appear at the bottom, while enhanced peaks arise at mid-to-upper heights. This redistribution reflects the upward displacement of convective activity by secondary vortices.

In SF mode, illustrated in Figure 4c and in the thermal field of Figure 9e, the lower cavity becomes nearly conduction dominated. Isotherms flatten, and only weak circulation persists above the partitions. This is directly mirrored in the $Nu$ distributions of Figure 10g,h, which are flattened and of very low magnitude across both walls. Even at $Ra={10}^5$, the hot wall $Nu$ remains close to unity in the lower region, indicating the near absence of convection.

Together, these results show a clear hierarchy: SC yields strong monotonic gradients, CV introduces local deviations, SV shifts enhancement upward, and SF nearly suppresses convection. While $Ra$ raises overall $Nu$ levels, the spatial distribution remains governed by the flow mode, consistently linking streamlines (Figure 3 and Figure 4), thermal fields (Figure 9), and wall heat transfer (Figure 10).

3.3.3. Composite-Mode Correlations in Nusselt Distributions

The influence of composite flow modes on wall heat transfer can be interpreted by linking the streamline classifications in Figure 3 and Figure 4, the thermal fields in Figure 9, and the corresponding $Nu$ distributions in Figure 10 and Figure 11. While single modes show distinct and predictable patterns—monotonic in SC, localized in CV, bent in SV, and flat in SF—their superposition produces irregular and multi-peaked profiles that reflect the simultaneous action of multiple vortical mechanisms.

In the SC–CV–SV mode, as shown in Figure 11a,b, the large-scale loop persists above the partitions, but the addition of corner vortices and secondary recirculation distorts the wall heat transfer. $Nu$ profiles no longer follow the smooth monotonic shape observed in SC (Figure 10a,b); instead, multiple regions of local enhancement emerge, particularly along the cold wall, where corner-vortex impingement intensifies gradients. These peaks grow stronger as $Ra$ increases, confirming that vortex–wall interactions redistribute heat transfer away from the lower boundary layers.

A stronger departure appears in the SC–SV–SF configuration, illustrated in Figure 11c,d. Here, the lower domain becomes nearly conduction dominated, resembling the flat $Nu$ response of the SF mode in Figure 10g,h, while the upper cavity retains active secondary vortices that generate sharp peaks in mid-height regions. At low $Ra$, the $Nu$ values near the hot-wall bottom remain close to unity, but at higher $Ra$ distinct maxima exceeding 10 develop around the mid-height, showing how stratification enforces a two-layer structure: conduction-dominated at the base and convection-enhanced above.

The most complex behavior arises in the SC–CV–SV–SF case, shown in Figure 11e,f. In this configuration, the $Nu$ distributions lose any single defining shape and become irregular, with multiple peaks distributed along the vertical walls. Local maxima at different heights correspond to the competing contributions of the main circulation, corner vortices, and secondary vortices, while the stagnant lower region continues to suppress bottom-wall heat transfer. Compared with the clear monotonic SC distributions in Figure 10a,b or the bent SV profiles in Figure 9e,f, these irregular patterns highlight the instability and complexity of multi-mode interactions.

Collectively, Figure 10 and Figure 11 reveal a clear progression: pure modes yield orderly $Nu$ profiles tied to their dominant circulation structures, whereas composite modes disrupt this order through vortical superposition and stagnation. Although $Ra$ amplifies $Nu$ magnitude in all cases, the vertical distribution is governed by partition geometry and flow mode, confirming that buoyancy–confinement coupling controls both the organization of the thermal field and the detailed pattern of wall heat transfer.

3.3.4. Parameter-Dependent Global Heat-Transfer Trends and Design Implications

The variation in the surface-averaged Nusselt number ($\overline{Nu}$) with respect to the number of partitions $N$ and the partition height $H$ is presented in Figure 12a–d for Rayleigh numbers ranging from $Ra={10}^3$ to ${10}^6$. As shown in Figure 12a–d, a monotonic decrease in $\overline{Nu}$ with increasing $N$ is consistently observed across all Rayleigh numbers. The insertion of additional partitions progressively obstructs the primary convective circulation, promoting conduction-dominated zones in the lower part of the cavity and thereby reducing the global heat transfer.

The effect of the buoyancy intensity is clearly visible when comparing the subfigures. At $Ra={10}^3$ (Figure 12a), the maximum $\overline{Nu}$ is only about 1.2, whereas at $Ra={10}^4$ (Figure 12b), it rises to approximately 2.5. For $Ra={10}^5$ (Figure 12c), the baseline $\overline{Nu}$ level exceeds four. At the highest Rayleigh number, $Ra={10}^6$ (Figure 12d), the baseline value without partitions reaches nearly nine (as also validated in Figure 3), and the subsequent decrease with $N$ reflects the strongest suppression by partitions. This systematic upward shift confirms that stronger buoyancy sustains higher heat transfer capacity, even under increasingly restrictive partitioning.

The influence of partition height is also distinct across all cases. For shallow partitions ($H=0.1$ or $0.3$), $\overline{Nu}$ values remain relatively large, and the decrease with $N$ is gradual. In contrast, for tall partitions ($H=0.7$ or $0.9$), a sharp drop occurs even for small $N$, and the curves rapidly converge toward their minimum values. This indicates that tall partitions are highly effective in suppressing convective transport and enforcing conduction-dominated behavior, even when only a few dividers are present.

Finally, the separation between the curves corresponding to different $H$ becomes particularly pronounced at higher Rayleigh numbers (Figure 12c,d). Partition height therefore emerges as the primary governing parameter in reducing the heat transfer, while the number of partitions mainly acts to accelerate the suppression process. Overall, Figure 12 provides a comprehensive overview of the combined effects of buoyancy, partition height, and partition number on the global thermal performance of the cavity.

While Figure 12 provides the absolute trends of $\overline{Nu}$ with respect to the number of partitions and partition height, it is often more instructive to examine the relative loss of heat-transfer efficiency. To this end, the reduction percentage, defined as the normalized decrease in $\overline{Nu}$ relative to the baseline case ($N=0$), is introduced. This representation emphasizes the comparative impact of partitions across different Rayleigh numbers and geometrical configurations, and the corresponding results are presented in Figure 13.

The reduction percentage is defined as the relative decrease in the surface-averaged Nusselt number compared to the baseline case without partitions ($N=0$):

$$Reduction \left(\%\right)={\displaystyle \frac{\overline{{Nu}_0}-\overline{{Nu}_{N,H}}}{\overline{{Nu}_0}}}\times 100$$

(9)

where $\overline{{Nu}_0}$ is the surface-averaged Nusselt number for the open cavity ($N=0$), and $\overline{{Nu}_{N,H}}$ denotes the value for a cavity with partition number $N$ and height $H$. This metric quantifies the relative loss of convective heat-transfer efficiency induced by the partitions.

As shown in Figure 13a at $Ra={10}^3$, the reduction percentage varies almost linearly with partition height regardless of the number of partitions, reflecting the dominance of conduction when buoyancy is weak. However, at higher Rayleigh numbers—$Ra={10}^4$ in Figure 13b, $Ra={10}^5$ in Figure 13c, and $Ra={10}^6$ in Figure 13d—the trends progressively deviate from linearity. For small $N$, the reduction rises sharply with increasing $H$, indicating that strong buoyancy amplifies the blocking effect of tall partitions. In contrast, at larger $H$ and $N$, the growth rate diminishes and the curves flatten, showing that the system approaches a conduction-dominated limit where further partitioning yields little additional suppression. Thus, Figure 13 highlights a clear progression: linear dependence at low $Ra$ transitions into nonlinear behavior at higher $Ra$, with sharp sensitivity at small $N$ and saturation at tall partitions.

Quantitatively, the suppression of global heat-transfer efficiency exhibits a strong dependence on both partition height and number. For $Ra={10}^3$, the reduction increases gradually, exceeding 50% at $H=0.5 \mathrm{and} N\ge 4$, and reaching about 89% at $H=0.9$ and $N=7$. As $Ra$ increases to $Ra={10}^4$, the reduction becomes more distinct, surpassing 60% for $H\ge 0.5$ and $N\ge 2$, and rising to approximately 94% at $H=0.9$ and $N=7$. As $Ra$ increases to $Ra={10}^5$, the reduction exceeds 50% at $H=0.5 \mathrm{a}\mathrm{n}\mathrm{d} N\ge 4$, and attains a maximum of 97% at $H=0.9$ and $N=7$, indicating near-complete flow blockage. At $Ra={10}^6$, suppression begins at much smaller heights: under $N=2$, the reduction increases from about 8% to 22% as $H$ rises from 0.3 to 0.5, while for $N\ge 4$ it steeply increases from approximately 19% to 45%, and attains a maximum of 98.5% at $H=0.9$ and $N=7$, representing almost total inhibition of convective heat transfer.

Across all $Ra$, a reduction of 50% is consistently observed for $H=0.7$ and $N \ge 3$, confirming that partition height dominates global suppression while partition number modulates its rate.

4. Conclusions

To support the design of efficient passive thermal management systems, this study has presented the first systematic investigation of natural convection in partitioned cavities by simultaneously varying the Rayleigh number ($\mathit{R}\mathit{a}={\mathbf{10}}^{\mathbf{3}}$–${\mathbf{10}}^{\mathbf{6}}$), partition height ($\mathit{H}=\mathbf{0.1}–\mathbf{0.9}$), and partition number ($\mathit{N}=\mathbf{0}–\mathbf{7}$). The numerical model was validated against established benchmark results for the open cavity ($\mathit{N}=\mathbf{0}$), showing excellent agreement in both flow fields and surface-averaged Nusselt numbers. Building on this validated framework, the analysis was then extended across a broad parametric space, enabling a consistent classification of flow modes and a quantitative assessment of how geometric confinement and buoyancy strength interact to suppress convective transport.

The convection fields were classified into four fundamental flow modes—single circulation (SC), corner vortices (CV), secondary vortices (SV), and stagnant flow (SF)—and their combinations. A unified flow-mode map was constructed to provide a consistent framework for interpreting complex partitioned convection patterns. Within this framework, the study identified two distinct types of transitions: slot-scale transitions, characterized by nonlinear back-and-forth changes among localized vortical structures, and partition-dominated transitions, which reorganize the backbone circulation in a more unidirectional manner. This distinction provides a mechanistic basis for interpreting how geometric confinement drives the observed hierarchy of transitions, in which SC evolves into dual-mode states such as SC–CV, SC–SV, and SC–SF, and eventually into multi-mode composites such as SC–CV–SV, SC–SV–SF, and SC–CV–SV–SF.

Thermal field analysis demonstrated that partitions systematically reorganize the temperature distribution: short partitions induce local distortions, intermediate partitions promote multi-vortex recirculation, and tall partitions suppress lower-cavity convection to produce conduction-dominated stagnant zones. These structural changes were directly reflected in local and global Nusselt number distributions. While SC maintained monotonic boundary-layer profiles, CV and SV shifted or distorted heat transfer toward the upper domain, and SF nearly eliminated convective transport. Composite modes, in contrast, generated irregular multi-peaked Nusselt profiles that captured the simultaneous action of multiple vortical mechanisms. Importantly, these observations established a clear relationship between flow modes and heat transfer, linking qualitative flow regimes with quantitative thermal performance.

The surface-averaged Nusselt number ($\overline{\mathit{N}\mathit{u}}$) decreased monotonically with the number of partitions, but partition height emerged as the dominant factor in determining the degree of suppression. Reduction-percentage analysis revealed a sharp initial drop followed by saturation, demonstrating that tall partitions alone are sufficient to enforce conduction-dominated behavior. At low $\mathit{R}\mathit{a}={\mathbf{10}}^{\mathbf{3}}$, the reduction varied almost linearly with partition height, independent of the number of partitions. At higher $\mathit{R}\mathit{a}={\mathbf{10}}^{\mathbf{4}}$ to ${\mathbf{10}}^{\mathbf{6}}$, however, the dependence became nonlinear: sharp sensitivity appeared at small $\mathit{N}$ and large $\mathit{H}$, while further partitioning yielded only marginal additional reduction. Taken together, these results establish a systematic basis for understanding and predicting how slot-scale and partition-dominated mechanisms jointly modulate convective efficiency.

Quantitatively, the global heat-transfer efficiency decreases by up to 75–85% under the most confined configurations ($\mathit{H}=\mathbf{0.9},\mathit{N}\ge \mathbf{6}$) compared with the open cavity. Moderate partitions ($\mathit{H}\approx \mathbf{0.5},\mathit{N}=\mathbf{3}–\mathbf{4}$) yield an intermediate reduction of 40–60%, while shallow partitions ($\mathit{H}\le \mathbf{0.3}$) cause less than 20% loss even at high $\mathit{R}\mathit{a}$. These numerical trends demonstrate that partition height acts as the dominant control parameter, whereas the number of partitions primarily accelerates the suppression rate. Accordingly, the present results provide a quantitative guideline for balancing insulation and convective cooling in partitioned enclosures. From a design perspective, for low-Rayleigh-number regimes where thermal insulation is prioritized, tall partitions ($\mathit{H}\ge \mathbf{0.8}$) are most effective, whereas for higher Rayleigh numbers requiring controlled convective cooling, moderate partitions ($\mathit{H}\approx \mathbf{0.5}$ with $\mathit{N}=\mathbf{3}–\mathbf{4}$) offer a practical balance between heat-transfer suppression and circulation.

Collectively, these findings consolidate a unified understanding of how geometric confinement governs buoyancy-driven transport across a broad parameter space, while indicating that the two-dimensional steady laminar framework captures the dominant buoyancy-driven flow and heat-transfer mechanisms. In addition, the present analysis is restricted to steady-state conditions and transient thermal loading may introduce additional time-dependent effects associated with fluid thermal inertia and baffle heat capacity, leading to quantitative variations in the system response. Nevertheless, the geometric mechanisms governing flow-mode transitions and convective suppression identified in this study are expected to remain qualitatively relevant. They also provide practical design guidelines for thermal-management systems, including passive cooling of electronic enclosures, enhancement of building energy efficiency, and suppression of natural convection in storage tanks. Future extensions may include three-dimensional configurations, transient responses under variable heating, or optimization studies targeting hybrid partition geometries and materials, thereby enabling a more effective balance between convective transport, structural constraints, and insulation requirements.