Heat transfer enhancement in Rayleigh-B\'enard convection using a single passive barrier

In this numerical study on Rayleigh-B\'enard convection we seek to improve the heat transfer by passive means. To this end we introduce a single tilted conductive barrier in an aspect ratio one cell, breaking the symmetry of the geometry and to channel the ascending hot and descending cold plumes. We study the global and local heat transfer and the flow organization for Rayleigh numbers $10^5 \leq Ra \leq 10^9$ for a fixed Prandtl number of $Pr=4.3$. We find that the global heat transfer can be enhanced up to $18\%$, and locally around $800\%$. The averaged Reynolds number is decreased when a barrier is introduced, even for those cases where the global heat transfer is increased. We map the entire parameter space spanned by the orientation and the size of a single barrier for $Ra=10^8$.

FIG. 1: Rayleigh-Bénard geometry with a single infinitesimally thin barrier (gray). The center of the cell is indicated by a black disk and denoted o. The barrier is centered around o, has a length , and is oriented with an angle φ with respect to the horizontal (dashed line). The convection cell has unit width W and therefore the aspect ratio Γ = W/L = 1.
moving upwards and downwards, respectively, driving a large scale circulation (LSC). Sometimes, however, countergradient motions of thermal plumes have been found, with ascending cold plumes and descending hot plumes. These misoriented motions of plumes are associated with the bulk dynamics and the competition between the corner rolls and LSC, and they can lead to strongly negative local heat flux [25,26]. One way to control the plume dynamics is by adding barriers or obstacles, for which one might-naively-guess that adding them to the geometry probably decreases the flow and therefore the thermal transport, though the opposite can be true [27,28].
Another aspect of the misoriented dynamics of convective flow is the reversal of the LSC. Flow reversal in RB convection has received persistent interest, since it may provide important insights to many natural phenomena, such as the reversal processes of geomagnetic field and atmosphere [29,30]. With the complex three-dimensional (3D) flow dynamics suppressed [31][32][33][34], flow reversal in two-dimensional (2D) and quasi-2D systems has received much attention recently, where the complex interaction between the LSC and corner rolls plays an important role to reversals [35][36][37][38][39][40][41][42][43][44][45][46]. While most of previous studies considered idealized symmetric systems, it is desirable to examine how symmetry-breaking effects influence flow reversals, since asymmetry is omnipresent in real world [39,44]. In many circumstances, adding asymmetry to the system can orient the LSC. This has been done before by inclining the apparatus [47][48][49][50], and is also commonly done in experiments (though with much smaller inclinations) to fix the LSC in place for visualization and measurement purposes.
In the current work we explore the possibility of increasing the thermal transport and controlling flow dynamics via passive means. We study RB convection in a standard geometry of aspect ratio Γ = W/L = 1. A single passive angled flow barrier is included inside the convection cell to stabilize the LSC, to guide the plume motions, and to enhance the heat transfer, see Fig. 1.
The remainder of the paper is organized as follows. The numerical details are given in §II. The main results are presented and discussed in §III, focusing on the effects of barrier on the heat transfer, flow organization, and flow reversals. In §IV the main findings are summarized. Finally, two appendices are provided to give more details on the grid resolution and the influence of the barrier on the flow structure.

II. NUMERICAL DETAILS
As shown in Fig. 1, we consider the RB convection in a cell of unit aspect ratio, where a straight, solid conductive barrier with infinitesimal thickness and length is added, with the barrier center collapsing on the cell center o. The fluid flow and heat transport are described by the dimensionless incompressible Navier-Stokes equations, the convection-diffusion equation, and the continuity equation within the Boussinesq approximation: ∂ u ∂t + u · ∇ u = − ∇p + Pr Ra ∇ 2 u + T e z , ∂T ∂t + u · ∇T = 1 √ Pr Ra ∇ 2 T, where u = (u, w) is the velocity vector in the (x, z)-plane, p is the fluid pressure, and T is the temperature. e z is the unit vector along the vertical z-direction. The two dimensionless numbers in the governing equations (2) are the Rayleigh number Ra and Prandtl number Pr, which are given in §I. For the non-dimensionalization, the cell height L, the temperature difference ∆ between the top and bottom plates, and the free-fall velocity √ gβ∆L are used as the characteristic quantities for length, temperature, and velocity, respectively.
The cell is assumed to have unit width (W = L = 1) and the barrier configuration is quantified by two geometric parameters, namely, the barrier length and the angle φ the barrier makes with respect to the horizontal axis. There is a maximum barrier length m , beyond which the convection cell is divided into two sub-regions without mass transfer between each other. The maximum length m for complete blockage is dependent on the barrier angle φ. For our square unit cell we have m = min sec(φ) , csc(φ) .
No-slip and no-penetration boundary conditions are imposed at all solid boundaries, including the barrier surfaces. The horizontal top and bottom plates are isothermal and the sidewalls are thermally insulated. The fluid and barrier are assumed to have the same thermal properties. Practically, no thermal boundary conditions are explicitly enforced for the thin barrier, and the temperature equation is solved in the whole domain.
The governing equations (2) were solved using a second-order finite-difference method implemented in the opensource AFiD code [51,52]. Time integration is performed using the fractional-step third-order scheme for the explicit terms and the Crank-Nicolson scheme for the implicit terms. The grid is stretched along the vertical direction so that finer resolution is used near the horizontal plates to resolve the boundary layers. The barrier is taken into account using the immersed boundary method [16,18,53], which is versatile for the treatment of complex geometries, while the fluid motion can be conveniently solved based on a Cartesian grid.
Adequate grid resolutions are used to resolve the boundary layers and the bulk flows, satisfying the resolution requirement for the direct numerical simulations of thermal convection [54,55]. A finer resolution is employed when is close to m such that the barrier end is close to the wall with a small gap. The flow in the gap is resolved with at least 10 grid points. We note that for certain parameters ( , φ) the thermal boundary layers on the horizontal plates can be highly non-uniform with the local boundary layer thickness near the barrier end much smaller than the averaged thickness, as shown in Fig. 8(a). The grid is chosen such that the whole thermal boundary layer is well resolved, which agrees with the recommendations in Ref. [54]. For the highest Ra (Ra = 10 9 ) 2250 grid points are used in the vertical direction for = 1.35 and φ = π/4, and the thermal boundary layers are everywhere resolved with at least 11 grid points. Simulation parameters including the grid resolutions are presented in Table I. More details of grid resolution are given in Appendix A. To guarantee that the resolution used is sufficient, grid independence studies were performed at selected parameters, and the numerical details are also provided in Appendix A. The statistics of the global response parameters are based on data over sufficiently long times and the difference of Nusselt numbers based on the first and second halves of the simulations after the initial transients is within 1%.  In this study the effects of a passive barrier on the heat transfer and flow organization are investigated for varying Rayleigh numbers Ra and barrier geometric properties ( , φ), at a fixed Prandtl number Pr = 4.3. The barrier angle φ takes values from 0 to π/2. For given φ, the barrier length is varied from 0 to m , as shown in Table I.
We mainly consider the 2D problem, since the 2D simulations are much less computationaly demanding, allowing us to explore a larger parameter space. According to Ref. [56], for a large range of Ra and large Pr > 1, the Nu versus Ra scalings in 2D and 3D convection are comparable up to constant prefactors, which justifies our 2D simulations at Pr = 4.3 for the traditional RB convection without barriers. A set of 3D simulations were also performed for comparison.

A. Heat transfer
We first examine the influence of the barrier on the global heat transfer. Fig. 2 shows the dependence of Nu on Ra for various values of at a fixed barrier angle φ = π/4. For φ = π/4, the maximum barrier length is m = √ 2. In the traditional RB convection with = 0, the increase of Nu with Ra follows an effective power law Nu ∝ Ra 0.3 , consistent with previous results in the literature [26,56,57]. When a barrier is introduced, the modification of Nu is dependent on both Ra and . Interestingly, the addition of a flow barrier (a restriction) can lead to heat transfer enhancement. For = 1.0 the absolute values of Nu are uniformly larger than those of = 0 for the parameters examined, while the scaling behavior of Nu with Ra is similar to that of the traditional RB convection, suggesting that the heat transfer is still governed mainly by the thermal boundary layers [1]. In the completely blocking configuration with = m , the convection cell is divided into two sub-regions of a triangular geometry. Between the two sub-regions heat is transferred via thermal conduction through the barrier. On the horizontal plates of the top and bottom sub-regions the temperature is fixed at 0 and 1, respectively. Based on symmetry, the temperature on the inclined boundary (the barrier surface) is assumed to be 0.5 for simplification. Each triangular sub-system may be comparable to a traditional RB system with an effective temperature drop equal to 0.5 and an effective cell height less than L (≈ 0.6L), resulting in a smaller effective Ra (roughly 10 times smaller). Thus, it is expected that for φ = π/4 the heat transfer is significantly reduced in the completely blocking configuration with = m = √ 2, as observed in Fig. 2. When is close to m , the top and bottom sub-regions are connected via two small gaps between the barrier ends and the boundaries of convection cell. It is found that the dependence of Nu on Ra exhibits two different regimes, see the curves for = 1.3 and = 1.35 in Fig. 2. When Ra is low, the heat transfer is reduced compared with that in the traditional RB convection. The reduced heat transfer is attributed to the small gap and large coherence length scale of the flow at small Ra. The convection between the two sub-regions is restricted by the small gap, and the system is close to the completely blocking configuration when is close to m . When Ra is low enough, the small gap is deeply immersed in the thermal boundary layer, and has negligible influence on the heat transfer, as shown in Fig. 2 that the Nusselt numbers for the three largest collapse for Ra = 10 5 . When Ra is high enough, the heat transfer is mildly affected by the barrier, and Nu follows a similar effective power law as that in the traditional RB convection. Due to the smaller coherence length scale of the flow at higher Ra, a small gap becomes less restrictive. The critical Ra separating the two regimes is dependent on . When is closer to m , the gap becomes smaller, and correspondingly the critical Ra becomes larger. Note that in the presence of a conductive barrier, the thermal conduction state is still a solution of the system. The convection onset will be delayed to higher Ra in the presence of a barrier due to the enhanced drag on the convection. We now further examine the influence of barrier length on the heat transfer. We show in Figs. 3(a, b) the dependence of the absolute and normalized Nusselt numbers on for different Ra at a fixed barrier angle φ = π/4. When is small, the heat transfer is only mildly influenced by the barrier, as expected. As increases, the influences of the barrier become more significant, and heat transfer enhancement is observed for a broad range of . The global heat transfer can be enhanced up to 18% in the parameter space explored. When is further increased to reach m , the top and bottom sub-regions are effectively blocked, and Nu decreases rapidly. Similar behavior holds for different Ra, showing the robustness for the barrier to enhance/reduce the heat transfer at different . The anomalous, non-smooth variation of Nu with Ra is associated with the change of flow structure, such as the size of the plumes in the cell corner. Further discussions on the change of corner flow with the increase of are presented in Appendix B, including visualizations of the streamlines. Figs. 3(a, b) also include the results of 3D simulations with spanwise depth 1/4. It is found that the 3D cases show similar behavior as their 2D counterparts, providing a stronger evidence that the heat transfer enhancement by adding a barrier is also relevant to physical systems. Fig. 3(c) shows the dependence of Nu with the barrier length for different φ at Ra = 10 8 . Heat transfer enhancement is observed for a large range of φ. However, no evident heat transfer enhancement is observed for φ = 0, which is attributed to the fact that when the barrier is aligned horizontally (and thus not symmetry-breaking), the interaction between the colder (hotter) fluid with the bottom (top) boundary layer is weakened by the diffusion effects during the travel over half the cell height. In the completely blocking configuration with φ = 0, the system consists of two vertically stacked RB cells with height L b = L/2, aspect ratio Γ b = 2 and temperature drop ∆ b = ∆/2, where the subscript b indicates quantities of the completely blocking sub-systems. Thus the effective Rayleigh number of the sub-systems Ra b is Ra/16. Using the effective scaling law Nu ∝ Ra 0.3 with the effects of aspect ratio neglected for simplicity, it is expected that Nu b ≈ 0.44Nu, which is consistent with our simulation results. When the barrier aligns vertically with φ = π/2 and = m , Nu is only mildly influenced as compared to that in the traditional RB convection. In this configuration, the RB cell consists of two horizontally adjacent RB cells with aspect ratio Γ b = 0.5 and Rayleigh number Ra b = Ra which share a common wall and thus can also exchange heat between them. In Fig. 3(d) we display the normalized Nusselt numbers in the parameter space ( x , z ) at Ra = 10 8 , where It is found that the heat transfer is only mildly affected for small ; for barriers with ≤ 0.5 the Nusselt numbers are all within 2.5% (green and yellow data points in Fig. 3(d)) the nominal case of no barrier. Considerable heat transfer enhancement is observed for intermediate , and heat transfer is reduced in the completely blocking configurations for different φ (points on the top and on the right in the phase space). Fig. 3(d) shows that the influence of barrier on the heat transfer is dependent on φ. With the increase of φ above π/4, the heat transfer reduction in the completely blocking configuration is alleviated, as now the two compartments both directly connect with the top and bottom boundaries, without going through the conducting barrier. When φ is relatively small, hardly any heat transfer enhancement is observed which is consistent with the observation in Fig. 3(c).

B. Flow organization
We now focus on the effects of the barrier on the flow organization. Fig. 4 shows the instantaneous temperature fields superimposed with velocity vectors for various and φ at Ra = 10 8 . When a short barrier is introduced in the cell, the flow organization is close to that of the traditional RB convection. The flow consists of a well-organized LSC and two corner rolls, see e.g. Figs. 4(a, e, i). The temperature in the bulk is well mixed, and the temperature drop is dominated by the thermal boundary layers near the horizontal plates. The barrier is located in the center of the LSC, where the flow velocity is low and temperature is uniform. It is expected that a short barrier has only a mild influence on the heat transfer, as shown in §III A.
Although the flow structure is qualitatively unaffected by a short barrier, we can see that a short barrier can obviously distort the LSC and influence the competition between the LSC and the corner rolls. Besides, an inclined barrier will break the parity symmetry of the system with respect to the horizontal mid-plane. Thus, flow reversal process is expected to be influenced by the barrier, as shown in §III C.
When the barrier becomes longer, the flow organization is significantly modified. It is appropriate to regard the convection cell to be the combination of two sub-regions with mass and heat exchange between each other. Complex vortex structures develop in each sub-region, as depicted in Figs. 4(b, f, j). When the barrier becomes even larger, the gaps connecting the two sub-regions become smaller. Due to the suppressed mass transfer between the two subregions, the fluid mixing in the cell becomes less efficient. For φ < π/2, the fluid in the top (bottom) sub-region is colder (hotter) due to the imbalance between the cold and hot boundaries in each sub-region, suggesting that a temperature boundary layer gradually develops on the barrier surface, see Figs. 4(c, g). Note that strong jets develop across the small gaps, which are attributed to the convergence of fluid flow and the pressure imbalance near the gaps, which originates from the distinct temperature values in the two sub-regions.
The interaction of the jets with the boundary layers can promote plume generation, which is beneficial for the heat transfer [58]. For φ = 0, the jets travel half the cell height before interacting with the boundary layers, in contrast to the cases for larger φ. The phenomena of jet formation and the resulting heat transfer enhancement are similar to those found in the partitioned RB convection [27]. When vertical partitions are inserted in the convection system, coherent convective flow is established spontaneously, with the hotter and cooler fluid moving upwards and downwards, respectively. Due to the horizontal pressure drops across the thin gaps between the partitions and horizontal plates, strong horizontal jets form and sweep the thermal boundary layers, realizing efficient heat transfer. When = m , the two sub-regions are completely separated without mass transfer between each other. Heat is transferred between the two sub-regions via thermal conduction through the barrier. For φ < π/2, the averaged temperature values in the two sub-regions are different, particularly for small φ. When φ is relatively small, the temperature difference on the two sides of the barrier is significant and a thermal boundary layer appears on the barrier, which can also generate thermal plumes, as shown in Figs. 4(d, h). See also movie S4 in the Supplemental Material for an illustration of the flow dynamics in the completely blocking configuration with φ = π/4. For φ > π/4, each of the two sub-regions contains part of the top and bottom plates, and heat can be directly transferred between the two plates through thermal convection besides of conduction. When φ = π/2, the two sub-regions are similar to the traditional RB convection in a cavity with aspect ratio Γ b = 1/2. For φ ≤ π/4, heat is transferred across the barrier through thermal conduction.
When the barrier is put in the cell, it is expected that the flow strength will be reduced due to the additional drag exerted by the barrier on the flow. We now quantitatively examine how the flow strength is affected by the barrier. The flow strength is measured by the volume-averaged Reynolds number Re, which is defined as where . . . V,t denotes averaging the bracketed value over the entire volume and time. Fig. 5 shows the dependence of  Re on Ra for different for the case of φ = π/4. In the traditional RB convection with = 0, Re follows an effective scaling law, Re ∝ Ra 0.6 . When the barrier is introduced, Re is uniformly decreased for all Ra under consideration, as expected. The scaling behavior of Re with Ra is found to be insensitive to the presence of barrier despite of the remarkable changes of flow organization and strength. It is interesting to compare Fig. 5 with the behavior of Nu in Fig. 2. While in the completely blocking configuration with = m , both Nu and Re are significantly reduced compared to those in the traditional RB convection, Nu is very robust compared to Re when the barrier has a length < m , particularly for high Ra. The slight decrease of from m results in a mild change of flow strength and a huge heat transfer enhancement, demonstrating the effect of jets to enhance the heat transfer. To further investigate the jet structure, we show in Fig. 6(a) the profile of the time-averaged horizontal velocity u t (z) along the vertical slice crossing the barrier end at Ra = 10 8 , with a barrier angle φ = π/4 and a barrier length = 1.38, where . . . t indicates averaging over time. The existence of jet structure is clearly demonstrated and the jet is found to approximately take a parabolic profile. Fig. 6(b) shows the maximum value u jet of the jet velocity | u t | as a function of 1 − / m . For the parameters explored, u jet increases for increasing (except for the case = m ), showing the formation process of the jet. When is large enough with Ra fixed, it is expected that u jet will decrease with and vanishes at = m . We quantify the width of the jet using the distance δ jet of the extreme point of velocity from the horizontal plate, which is found to decrease with decreasing gap width. The capability of mass transfer by the jet is measured using the jet Reynolds number Re jet = u jet δ jet Ra/P r, which is shown in Fig. 6(b) as a function of 1 − / m . It is found that Re jet decreases with increasing , indicating that the mass transfer between the two sub-regions is suppressed as the gap is narrowed, even though a strong jet develops.
We now focus on how the barrier influences the global Reynolds number. We display in Figs. 7(a, b) the absolute and normalized Reynolds numbers as functions of for different Ra at a fixed barrier angle φ = π/4, showing that the flow strength is reduced in the presence of a barrier. With larger the flow strength generally becomes smaller, which is attributed to the fact that, for larger the drag force becomes stronger and the buoyancy force acting on the plumes is reduced when the temperature mixing is less efficient. Non-monotonic variation of Re with is also observed, which is associated with the change of flow organization, such as the modification of corner flow demonstrated in Appendix B. It is interesting that, in the presence of a barrier, the heat transfer can be enhanced for a broad range of , even though the flow strength is significantly reduced. The heat transfer enhancement can therefore be attributed to the well-organized, direction-oriented motion (funneling) of the plumes and the direct interaction of the jets with the boundary layers. In Figs. 7(a, b) the results of a set of 3D simulations with spanwise depth 1/4 are also included, showing a similar behavior. In the 3D case, thanks to the strong friction of the cell walls, a relatively short barrier has only a mild influence on the flow strength. Thus, Re is insensitive to the increase of when is relatively small; while when becomes large, the influence of the barrier becomes significant, and Re is found to drop rapidly with the further increase of . the reduced flow strength in the presence of a barrier. The reduction of the flow strength is found to be less significant at larger φ. It is attributed to the larger buoyancy force acting on the plumes compared to the cases with smaller φ, where the mixing of the hot and cold fluid is less efficient. Considering that the present flow is in the 'classical' regime of turbulent RB convection and the heat transfer is limited by the thermal boundary layers, where heat is mainly transferred via thermal conduction, it is interesting to study how the barrier influences the properties of the thermal boundary layers. To show that, we plot in Fig. 8(a) the time-averaged local heat flux Nu(x) = −∂ z T t at the top plate for different at Ra = 10 8 and φ = π/4. The horizontal coordinates of the barrier ends with 0 < < m are indicated by dashed lines. It is found that as increases, Nu(x) increases rapidly above the barrier end, which is attributed to the direct impact of hot fluid to the top boundary layer. Similar phenomenon appears at the bottom plate due to the rotational symmetry of the system. For the parameters considered, the maximum Nu max of Nu(x) is enhanced up to around 800%. The results imply that the thermal boundary layers at the horizontal plates are highly nonuniform in the presence of a long barrier at φ = π/4. Fig. 8(b) shows the dependence of Nu max with the barrier length for different φ. It is found that Nu max increases as 1 − / m approaches 0 ( approaches m ), except for the case of φ = 0, where the interaction between the jet and thermal boundary layer is less significant than for the other cases. The growth rate of Nu max with decreasing 1 − / m is dependent on φ and is always slower than (1 − / m ) −1 . Thus, for φ ≥ π/4, the decreasing rate of the local thermal boundary layer thickness is slower than the gap width as increases. For large enough , the thermal boundary layer thickness will become larger than the gap width, and the system gradually approaches the completely blocking configuration with increasing . We plot in Fig. 9 the profiles of time-averaged temperature T t and horizontal velocity u t at the horizontal mid-plane for different barrier lengths at Ra = 10 8 and φ = π/4, to show the effects of the barrier on the mean flow. Note that for the case of Ra = 10 8 and = 0 the flow exhibits flow reversals, and u t vanishes at the horizontal mid-plane. The temperature and velocity profiles are anti-symmetric with respect to the center point for all values of due to the symmetry of the system under rotation by π. Without barrier, the time-averaged temperature in the bulk is uniform and equal to the arithmetic mean temperature T m = 0.5, as shown in Fig. 9(a). For > 0 the temperature values in the interiors of the two sub-regions stay uniform and gradually deviate from T m for increasing , indicating the emergence of a thermal boundary layer on the barrier surface. The overshoots of the temperature profile near the horizontal plates are associated with the formation of the horizontal jets near the barrier ends. Meanwhile, velocity boundary layers also emerge on the barrier surface with the increase of , as shown in Fig. 9(b). The gradients of temperature and velocity on the barrier increase with .
We now quantitatively investigate the influence of the barrier on the temperature distribution. Fig. 10 at Ra = 10 8 and φ = π/4. For different values of , PDF(δT ) takes a symmetric form, which is consistent with the rotational symmetry of the system. In the traditional RB convection with = 0, PDF(δT ) peaks at δT = 0, which is associated with the well-mixed center core where T ≈ T m . When a short barrier is present, the temperature distribution is basically unchanged as in the case of = 0.4. When the barrier becomes longer, the value of PDF(δT ) at δT = 0 decreases and PDF(δT ) now has two peaks located symmetrically with respect to δT = 0, which is attributed to the reduced temperature mixing in the cell. For increasing the two peaks become more and more separated, consistent with the time-averaged temperature profiles in Fig. 9(a). We observe that rather than having a single bulk temperature for small , we move towards a configuration for which we have an upper and lower bulk temperature for increasing .
How are the local heat transfer properties influenced when symmetry-breaking effects are introduced by a barrier? To address this question, we plot in Fig. 10(b) the PDFs of the convective heat flux wδT along the vertical direction in the whole cell for Ra = 10 8 and φ = π/4. The variation of corresponding skewness S(wδT ) with is shown in the inset. PDF(wδT ) shows a long tail for strong upward convective heat transfer with wδT > 0, resulting from the coupling between the temperature and vertical velocity through the buoyancy term in the momentum equation. The left branch of PDF(wδT ) shows that there is a finite probability for the occurrence of counter-gradient convective heat transfer, as observed in Refs. [26]. The asymmetric form of PDF(wδT ) about wδT = 0 implies that, on average, heat is transferred from the hot bottom plate to the cold top plate. With the increase of , the strength of the countergradient convective heat transfer is reduced, as revealed by the decreased probability of the left branch of PDF(wδT ). It indicates that due to the symmetry-breaking effects introduced by the barrier, the flow motion is guided-and funneled-in such a way that hotter (colder) fluid tends to flow upwards (downwards), which is beneficial for the heat transfer. This effect is confirmed by the increase of the skewness S(wδT ) with as depicted in the insect of Fig. 10(b), showing that PDF(wδT ) becomes more asymmetric for larger . When is further increased to reach m , the probability of the left branch of PDF(wδT ) increases due to the enhanced counter-gradient convective heat transfer along the barrier surface, and concomitantly, S(wδT ) decreases rapidly.

C. Flow reversals
In this section we focus on how the barrier influences the flow reversals. To this end we fix the Rayleigh number to Ra = 10 8 . At this Ra, frequent flow reversals are observed in the traditional RB convection. An example of the reversal process is shown in movie S1 of the Supplemental Material. In a square cell, the occurrence of flow reversals can be precisely inferred from the evolution of the global angular momentum of the flow field [35,44]. The angular momentum with respect to the cell center o (see Fig. 1) is defined as where . . . V represents averaging over the entire cell. Positive (negative) values of L o indicate that the flow field rotates counter-clockwise (clockwise) in the average sense. We first examine how the barrier length affects the reversals. Fig. 11(a) shows the time series of L o for different at φ = π/4. Corresponding PDFs of L o are given in Fig. 11(b). The statistical samples are large enough to have good statistical convergence. The longest simulation was run for around 8 × 10 4 free-fall times. In the absence of a barrier, frequent reversal events are observed, and PDF(L o ) has a symmetric, bi-modal shape, demonstrating that the flow states with clockwise and anti-clockwise LSCs are equally probable. These two flow states are equivalent due to the parity symmetry of the system with respect to the horizontal mid-plane. Transition between the two flow states occurs randomly. When a short barrier is included in the cell, the parity symmetry of the original system is partially broken. As shown in Fig. 4, the flow field still consists of a well-developed LSC and two corner rolls. However, the states with counter-rotating LSCs are no longer equivalent. Despite that, frequent reversals between the two states are observed for = 0.2, and PDF(L o ) still takes a symmetric, bi-modal profile in Fig. 11(b). Thus, a very short barrier does not qualitatively influence the reversal process. Quantitatively, the reversal frequency is slightly increased since a short barrier can distort and dampen the LSC and affect the competition between the LSC and corner rolls.
With the increase of , the symmetry-breaking effects of the barrier become more significant, strongly influencing the reversal events. The time series of L o for = 0.3 and 0.4 in Fig. 11(a) show that reversal events still exist. However, it is found that the negative values of L o become less probable as compared to the positive values, which is also demonstrated by the asymmetric profiles of PDF(L o ) in Fig. 11(b). The asymmetry of PDF(L o ) originates from the fact that there is no complete reversal following some of the cessations of the anti-clockwise state. When a cessation occurs, L o decreases and vanishes. Then L o becomes negative due to the inertia. There is a finite probability for L o to become again positive after being negative transiently, indicating the re-establishment of the anti-clockwise state. See movie S2 in the Supplemental Material for an illustration of the unsuccessful reversal process. Due to the strong symmetry-breaking effects, the flow prefers the anti-clockwise state statistically. PDF(L o ) displays three bumps at = 0.3 and 0.4 in Fig. 11(b). The right and left bumps correspond to the anti-clockwise and clockwise states, respectively, and the middle one is associated with the frequent cessation events without complete reversal. For even larger , the cessation events are also inhibited, since the traditional LSC is broken, and the rotation of the flow is biased in one direction, as illustrated by movie S3 in the Supplemental Material.
We show in Figs. 11(c, d) the PDFs of L o for various φ at = 0.2 and = 0.5, respectively. When the barrier is very short, the profiles of PDF(L o ) for different barrier angles collapse, indicating that the flow dynamics are qualitatively unaffected by the barrier. PDF(L o ) takes a symmetric, bi-modal form, demonstrating that the LSC can take the clockwise and anti-clockwise directions with similar probabilities, and flow reversal occurs, even though the parity symmetry of the system is slightly broken with 0 < φ < π/2. When the barrier becomes longer, the effects of the barrier on the flow dynamics become stronger, as revealed by the profiles of PDF(L o ) for various φ at = 0.5 in Fig. 11(d). When the barrier aligns horizontally with φ = 0, PDF(L o ) takes a symmetric, bi-modal form. The symmetry of PDF(L o ) is a manifestation of the intrinsic symmetry of the system. The bi-modal form indicates the occurrence of reversal events. When 0 < φ < π/2, PDF(L o ) becomes asymmetric due to the strong symmetrybreaking effects introduced by the barrier. When φ = π/2, PDF(L o ) recovers to a symmetric form, consistent with the symmetry of the system. In contrast with the φ = 0 case, PDF(L o ) at φ = π/2 has only one peak located at L o = 0. The discrepancy is associated with the different flow organizations. For φ = π/2 both the left and right sub-regions prefer a vertically-stacked two-roll state, rather than having a single large global roll. Thus, L o has a high probability to vanish. While for φ = 0 the simulation shows that both the top and bottom sub-regions prefer a single-roll state. Thus, L o generally takes a negative or positive value depending on the orientation of the rolls.

IV. CONCLUSIONS
In summary, we performed a numerical study of turbulent RB convection with a single, centered, passive, conductive barrier. The barrier is added aiming to control the LSC, the thermal transport, and to funnel the hot and cold plumes. The influences of the barrier on the heat transfer, flow organization, and reversal events are studied for a myriad of barrier lengths and angles φ. In the presence of a short barrier, the heat transfer and flow structure are only mildly affected. With the increase of , the influences of the barrier become stronger. We find that the global heat transfer can be considerably enhanced (up to 18%) even though the average flow velocity is significantly reduced, which is due to the guiding and focusing of the cold and hot plumes. The local flux on the horizontal plates is also greatly enhanced (up to 800% for the parameters considered). The flow organization can be significantly changed due to the constraints of a long barrier. With the increase of , thermal and kinetic boundary layers gradually develop on the barrier surface, and the gradients of temperature and velocity on the barrier are dependent on the length and angle of the barrier. When the gap between the barrier end and the cell boundaries is small, strong jets form across the gap due to the convergence of fluid flow and the pressure imbalance. The interaction between the jets and boundary layers is beneficial for the heat transfer due to enhanced plume generation. When the barrier is long enough, the convection cell is divided into two sub-regions without mass transfer between each other. In this case, the heat transfer and flow dynamics are highly dependent on the barrier angle φ. For large φ, each of the two sub-regions contains part of the top and bottom plates, and the global heat transfer is mildly influenced by the barrier; while when φ is small, the system can be regarded as a combination of two vertically-stacked RB cells with reduced effective Ra, and the heat transfer is significantly reduced.
To further examine the effects of the barrier on the LSC, the influence of the barrier on the flow reversal is studied. When the barrier is very short, the reversal process is qualitatively unchanged for different values of φ, even though the parity symmetry of the system is slightly broken for 0 < φ < π/2. When the barrier becomes longer, flow reversals are gradually suppressed due to the stronger symmetry-breaking effect of the barrier, which demonstrates the stabilizing effects of the barrier on the LSC.
The results highlight that the turbulent thermal transport can be enhanced by a passive barrier even though the flow strength is reduced, demonstrating a simple and promising way to enhance the heat transfer in technological applications. In the future, it will be of interest to extend this study to larger parameter space, including Pr, Γ, and multiple barriers to achieve even better flow control.
Numerical details of the typical grid resolutions in the 2D simulations with φ = π/4 are provided in Table II. A finer resolution is used when is close to m as a 'jet' is created between the horizontal plate and the barrier end and the thermal boundary layer near the barrier end becomes thinner.  To guarantee the sufficiency of grid resolution, grid independence studies were performed at selected parameters, particularly for large , where the barrier end is close to the wall, and thermal and velocity boundary layers develop on the barrier surfaces, as shown in Fig. 9. The numerical details of the grid independence studies are shown in Table  III. Overall, we find that the relative errors of Nu and Re based on two grids of distinct resolutions are all within 1% or less.
showing the appearance of a peak near the barrier end, as already depicted in Fig. 8(a). For both values of Ra, at relatively small , plumes arise from the thermal boundary layers along the sidewalls. In the top-right and bottom-left corners, small circulations are observed, which are connected to separation vortices attached on the sidewalls. When is slightly increased, the size of the plumes in the top-right and bottom-left corners is decreased, and the separation vortices are suppressed. Furthermore, a marked drop of the global heat transfer is observed with even a slight increase of . From Fig. 13 it is found that with the increase of , the peak value Nu max of Nu(x) is decreased, demonstrating the influence of the change of corner flow structure on the heat transfer. As is further increased, the circulations in the top-right and bottom-left corners are also suppressed, while Nu max is significantly increased, due to the impact of strong flow of hot and cold fluids on the opposite boundary layers. The global Nu is also significantly increased. Thus, the change of the local flow structure in the corner and near the boundary layer has a significant influence on the heat transfer, and a gradual variation of may result in an evident and non-monotonic change of Nu, due to the disappearance of the corner flow for increasing .