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Confined inclined thermal convection in low-Prandtl-number fluids

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Abstract

Any tilt of a Rayleigh–Bénard convection cell against gravity changes the global flow structure inside the cell, which leads to a change of the heat and momentum transport. Especially sensitive to the inclination angle is the heat transport in low-Prandtl-number fluids and confined geometries. The purpose of the present work is to investigate the global flow structure and its influence on the global heat transport in inclined convection in a cylindrical container of diameter-to-height aspect ratio $\unicode[STIX]{x1D6E4}=1/5$ . The study is based on direct numerical simulations where two different Prandtl numbers $Pr=0.1$ and 1.0 are considered, while the Rayleigh number, $Ra$ , ranges from $10^{6}$ to $10^{9}$ . For each combination of $Ra$ and $Pr$ , the inclination angle is varied between 0 and $\unicode[STIX]{x03C0}/2$ . An optimal inclination angle of the convection cell, which provides the maximal global heat transport, is determined. For inclined convection we observe the formation of two system-sized plume columns, a hot and a cold one, that impinge on the opposite boundary layers. These are related to a strong increase in the heat transport.

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... More recently an extension of the RBC and VC sys-tem has been studied which takes into account the effect of an arbitrary inclination angle (β) of the bottom wall of the system with respect to the horizontal, so that the limiting cases β = 0 o corresponds to RBC and β = 90 o to VC. Such a system, denoted as confined inclined convection (CIC) [19,20] [24]. While the first studies (water and silicon-oil) revealed a weak monotonous decrease of the heat flux, with a maximal reduction of 20% with respect to the RBC value, the second group of studies displayed always an overall increase of Nu with the existence of an optimal inclination around β ≃ 65 o − 70 o for which the increment was 20% in the Γ = 1, and remarkably as high as 1100% in the Γ = 1/20 system. ...
... Furthermore, Nu(β) appears to be a non-monotonic function characterized either by one or two local maxima [19]. A second series of numerical studies have been conducted at Pr = 0.1 and Ra ranging from 10 6 to 10 9 in a narrower cylindrical system (Γ = 1/5) which showed that Nu increases up to 120% with the inclination angle [20]. It is worth mentioning also the only detailed study that reports in parallel experimental and numerical results [25]. ...
... Their intensity if we compare the two Ra varies of about a factor √ 5 in a agreement with the approximate scaling law Re ∼ Ra 1/2 [49], which is larger than the variation detected in the amplitude of Nu (which scales roughly as ∼ Ra 1/3 or ∼ Ra 1/4 ). Also note that the magnitude of Re for β = 0 • is much smaller than one observed in RBC at small Pr [20]. For what concerns the β dependency, we observe that in the turbulence dominated regime all the curves increase to a maximum corresponding to a weak tilting at β ≃ 10 • and then they drop rapidly till a value Re ∼ 2 for angles which correspond roughly to the occurrence of a local minimum in the Nu(β) (see again of Fig. 5). ...
Preprint
We investigate the dependency of the magnitude of heat transfer in a convection cell as a function of its inclination by means of experiments and simulations. The study is performed with a working fluid of large Prandtl number, $Pr \simeq 480$, and at Rayleigh numbers $Ra \simeq 10^{8}$ and $Ra \simeq 5 \times 10^{8}$ in a quasi-two-dimensional rectangular cell with unit aspect ratio. By changing the inclination angle ($\beta$) of the convection cell, the character of the flow can be changed from moderately turbulent, for $\beta = 0^o$, to laminar and steady at $\beta = 90^o$. The global heat transfer is found to be insensitive to the drastic reduction of turbulent intensity, with maximal relative variations of the order of $20\%$ at $Ra \simeq 10^{8}$ and $10\%$ at $Ra \simeq 5 \times 10^{8}$, while the Reynolds number, based on the global root-mean- square velocity, is strongly affected with a decay of more than $85\%$ occurring in the laminar regime. We show that the intensity of the heat flux in the turbulent regime can be only weakly enhanced by establishing a large scale circulation flow by means of small inclinations. On the other hand, in the laminar regime the heat is transported solely by a slow large scale circulation flow which exhibits large correlations between the velocity and temperature fields. For inclination angles close to the transition regime in-between the turbulent-like and laminar state, a quasi-periodic heat-flow bursting phenomenon is observed.
... (104) and (103) for a cylinder, we conclude that in the limit of the cylinder aspect ratio → 0, the critical Rayleigh number for the onset of convection in a cylinder is significantly smaller than that for the optimal box with the equal area of the plates. More precisely, Ra c, (cylinder)/Ra c, (parallelepiped) ≈ 0.63, → 0 (for equal plate area); (107) see the pink solid line in Fig. 11, for small . In Fig. 11 we compared the cylindrical and the optimal parallelepiped containers, i.e., those with x = y . ...
... As soon as y → x , the difference between Ra c, ( y < x ) and Ra c, ( y = x ) certainly vanishes. However, for any fixed x , this difference grows dramatically when y → 0, as one can see in Fig This result supports the known fact of laminarization of the confined Rayleigh-Bénard flow even for very large Ra, as soon as the width of the three-dimensional domain becomes too small [104][105][106][107]. ...
... Based on the results of (almost) Oberbeck-Boussinesq experiments and direct numerical simulations of Rayleigh-Bénard convection in cylindrical containers [59,87,89,107,, let us now illustrate that the derived relevant length scale in Rayleigh-Bénard convection is , Eq. (125), and the corresponding Rayleigh number is Ra , Eq. (134). For cylindrical Rayleigh-Bénard convection cells, c u is defined by Eq. (103), which in combination with Eq. (135) gives the relevant scaling quantity ...
Article
Full-text available
To study turbulent thermal convection, one often chooses a Rayleigh-Bénard flow configuration, where a fluid is confined between a heated bottom plate, a cooled top plate of the same shape, and insulated vertical sidewalls. When designing a Rayleigh-Bénard setup, for specified fluid properties under Oberbeck-Boussinesq conditions, the maximal size of the plates (diameter or area), and maximal temperature difference between the plates, Δmax, one ponders: Which shape of the plates and aspect ratio Γ of the container (ratio between its horizontal and vertical extensions) would be optimal? In this article, we aim to answer this question, where under the optimal container shape, we understand such a shape, which maximizes the range between the maximal accessible Rayleigh number and the critical Rayleigh number for the onset of convection in the considered setup, Rac,Γ. First we prove that Rac,Γ∝(1+cuΓ−2)(1+cθΓ−2), for some cu>0 and cθ>0. This holds for all containers with no-slip boundaries, which have a shape of a right cylinder, whose bounding plates are convex domains, not necessarily circular. Furthermore, we derive accurate estimates of Rac,Γ, under the assumption that in the expansions (in terms of the Laplace eigenfunctions) of the velocity and reduced temperature at the onset of convection, the contributions of the constant-sign eigenfunctions vanish, both in the vertical and at least in one horizontal direction. With that we derive Rac,Γ≈(2π)4(1+cuΓ−2)(1+cθΓ−2), where cu and cθ are determined by the container shape and boundary conditions for the velocity and temperature, respectively. In particular, for circular cylindrical containers with no-slip and insulated sidewalls, we have cu=j112/π2≈1.49 and cθ=(j̃11)2/π2≈0.34, where j11 and j̃11 are the first positive roots of the Bessel function J1 of the first kind or its derivative, respectively. For parallelepiped containers with the ratios Γx and Γy, Γy≤Γx≡Γ, of the side lengths of the rectangular plates to the cell height, for no-slip and insulated sidewalls we obtain Rac,Γ≈(2π)4(1+Γx−2)(1+Γx−2/4+Γy−2/4). Our approach is essentially different to the linear stability analysis, however, both methods lead to similar results. For Γ≲4.4, the derived Rac,Γ is larger than Jeffreys' result Rac,∞J≈1708 for an unbounded layer, which was obtained with linear stability analysis of the normal modes restricted to the consideration of a single perturbation wave in the horizontal direction. In the limit Γ→∞, the difference between Rac,Γ→∞=(2π)4 for laterally confined containers and Jeffreys' Rac,∞J for an unbounded layer is about 8.8%. We further show that in Rayleigh-Bénard experiments, the optimal rectangular plates are squares, while among all convex plane domains, circles seem to match the optimal shape of the plates. The optimal Γ is independent of Δmax and of the fluid properties. For the adiabatic sidewalls, the optimal Γ is slightly smaller than 1/2 (for cylinder, about 0.46), which means that the intuitive choice of Γ=1/2 in most Rayleigh-Bénard experiments is right and justified. For the given plate diameter D and maximal temperature difference Δmax, the maximal attainable Rayleigh number range is about 3.5 orders of magnitudes smaller than the order of the Rayleigh number based on D and Δmax. Deviations from the optimal Γ lead to a reduction of the attainable range, namely, as log10(Γ) for Γ→0 and as log10(Γ−3) for Γ→∞. Our theory shows that the relevant length scale in Rayleigh-Bénard convection in containers with no-slip boundaries is ℓ∼D/Γ2+cu=H/1+cu/Γ2. This means that in the limit Γ→∞, ℓ equals the cell height H, while for Γ→0, it is rather the plate diameter D.
... slender cylindrical cells of small diameter-to-height aspect ratio Γ = D/H. Not only a single-roll mode (SRM) of the LSC, but also a double-roll mode (DRM) -composed of two rolls on top of each other -was found for cylindrical cells with Γ = 1, 1/2, 1/3 and 1/5 [15][16][17]. Experimental studies with water (Pr ≈ 5) found that the SRM is characterized by a slightly enhanced heat transport (≈ 0.5 %) compared to the DRM [15,16]. It was also found that small-Γ systems spend more time in the DRM than in the SRM. ...
... It was also found that small-Γ systems spend more time in the DRM than in the SRM. Direct Numerical Simulations (DNS) [17] for Ra = 10 6 , Pr = 0.1 and Γ = 1/5 showed that the heat transport of the DRM is only 80 % compared to the SRM. ...
... The DNS were conducted at Ra = 5 × 10 6 , Pr = 0.1 and Γ = 1/5, using a mesh of 256×128×22 nodes in z, ϕ and r-directions, which is of sufficient resolution [17,26]. We consider the volume-averaged instantaneous heat transport Nu(t) ≡ ( u z (t) θ(t) − κ ∂ z θ(t) )/(κ∆/H) (Nusselt number) and the Reynolds number, Re(t) ≡ H u 2 /ν, which is based on the kinetic energy. ...
Preprint
Turbulent Rayleigh-B\'enard convection in slender cylindrical cells exhibits rich dynamics of the large-scale circulation (LSC), with several rolls stacked on top of each other. We propose that the elliptical instability is the mechanism which causes the twisting and breaking of the LSC into multiple rolls and that the volume-averaged heat and momentum transport, represented by the Nusselt number and Reynolds number, is generally weaker for larger number $n$ of the LSC rolls. This is supported by direct numerical simulations for $Ra=5\times10^5$, $Pr=0.1$, $H=5D$ and $1\leq~n\leq4$.
... For pure RB convection, in particular, the heat transfer and flow structure have been studied extensively ). There also has been rapid progress in the modelling of RB convection with additional effects, such as multiphase RB convection (Lakkaraju et al. 2013;Wang, Mathai & Sun 2019), convection with rough walls (Shishkina & Wagner 2011;Wagner & Shishkina 2015;Zhu et al. 2017Zhu et al. , 2019Jiang et al. 2018), tilted convection (Shishkina & Horn 2016;Wang et al. 2018;Zwirner & Shishkina 2018;Jiang, Sun & Calzavarini 2019) and partitioned RB convection (Bao et al. 2015). In the numerical study of confined inclined RB convection in low-Prandtl-number fluids, Zwirner & Shishkina (2018) identified significant heat transfer enhancement, which is closely related to the organization of the plumes in inclined convection, namely the formation of system-sized plume columns impinging on the opposite boundary layers. ...
... There also has been rapid progress in the modelling of RB convection with additional effects, such as multiphase RB convection (Lakkaraju et al. 2013;Wang, Mathai & Sun 2019), convection with rough walls (Shishkina & Wagner 2011;Wagner & Shishkina 2015;Zhu et al. 2017Zhu et al. , 2019Jiang et al. 2018), tilted convection (Shishkina & Horn 2016;Wang et al. 2018;Zwirner & Shishkina 2018;Jiang, Sun & Calzavarini 2019) and partitioned RB convection (Bao et al. 2015). In the numerical study of confined inclined RB convection in low-Prandtl-number fluids, Zwirner & Shishkina (2018) identified significant heat transfer enhancement, which is closely related to the organization of the plumes in inclined convection, namely the formation of system-sized plume columns impinging on the opposite boundary layers. Also porous-media convection can be understood as geometrically modified RB convection, with correspondingly modified heat transfer and flow structure. ...
... The coexistence of the two competing effects leads to the non-monotonic behaviour of Nu(φ) as φ is decreased from 1, as shown in figure 2. The heat transfer enhancement is consistent with the counterintuitive observation that an appropriate strength of a stabilizing force can enhance heat transfer by increasing flow coherence (Chong et al. 2017). Significant enhancement of the heat transfer due to the increased flow coherence was also observed in the confined inclined convection in low-Pr fluids (Zwirner & Shishkina 2018). Due to the emergence of system-sized plume columns and the interaction of these plume columns with the opposed boundary layers in inclined convection, an increase of the heat transfer by a factor of approximately 2.3 can be realized. ...
Article
Full-text available
We perform a numerical study of the heat transfer and flow structure of Rayleigh-Bénard (RB) convection in (in most cases regular) porous media, which are comprised of circular, solid obstacles located on a square lattice. This study is focused on the role of porosity φ in the flow properties during the transition process from the traditional RB convection with φ = 1 (so no obstacles included) to Darcy-type porous-media convection with φ approaching 0. Simulations are carried out in a cell with unity aspect ratio, for Rayleigh number Ra from 10 5 to 10 10 and varying porosities φ, at a fixed Prandtl number Pr = 4.3, and we restrict ourselves to the two-dimensional case. For fixed Ra, the Nusselt number Nu is found to vary non-monotonically as a function of φ; namely, with decreasing φ, it first increases, before it decreases for φ approaching 0. The non-monotonic behaviour of Nu(φ) originates from two competing effects of the porous structure on the heat transfer. On the one hand, the flow coherence is enhanced in the porous media, which is beneficial for the heat transfer. On the other hand, the convection is slowed down by the enhanced resistance due to the porous structure, leading to heat transfer reduction. For fixed φ, depending on Ra, two different heat transfer regimes are identified, with different effective power-law behaviours of Nu versus Ra, namely a steep one for low Ra when viscosity dominates, and the standard classical one for large Ra. The scaling crossover occurs when the thermal boundary layer thickness and the pore scale are comparable. The influences of the porous structure on the temperature and velocity fluctuations, convective heat flux and energy dissipation rates are analysed, further demonstrating the competing effects of the porous structure to enhance or reduce the heat transfer.
... For pure RB convection, in particular, the heat transfer and flow structure have been studied extensively ). There also has been rapid progress in the modelling of RB convection with additional effects, such as multiphase RB convection (Lakkaraju et al. 2013;Wang et al. 2019), convection with rough walls (Shishkina & Wagner 2011;Wagner & Shishkina 2015;Zhu et al. 2017;Jiang et al. 2018;Zhu et al. 2019), tilted convection (Shishkina & Horn 2016;Zwirner & Shishkina 2018;Wang et al. 2018;Jiang et al. 2019), and partitioned RB convection (Bao et al. 2015). In the numerical study of confined inclined RB convection in low-Prandtl-number fluids, Zwirner & Shishkina (2018) identified significant heat transfer enhancement, which is closely related to the organization of the plumes in inclined convection, namely, the formation of system-sized plume columns impinging on the opposite boundary layers. ...
... There also has been rapid progress in the modelling of RB convection with additional effects, such as multiphase RB convection (Lakkaraju et al. 2013;Wang et al. 2019), convection with rough walls (Shishkina & Wagner 2011;Wagner & Shishkina 2015;Zhu et al. 2017;Jiang et al. 2018;Zhu et al. 2019), tilted convection (Shishkina & Horn 2016;Zwirner & Shishkina 2018;Wang et al. 2018;Jiang et al. 2019), and partitioned RB convection (Bao et al. 2015). In the numerical study of confined inclined RB convection in low-Prandtl-number fluids, Zwirner & Shishkina (2018) identified significant heat transfer enhancement, which is closely related to the organization of the plumes in inclined convection, namely, the formation of system-sized plume columns impinging on the opposite boundary layers. Also porousmedia convection can be understood as geometrically modified RB convection, with correspondingly modified heat transfer and flow structure. ...
... The coexistence of the two competing effects leads to the non-monotonic behaviour of N u(φ) as φ is decreased from 1, as shown in figure 2. The heat transfer enhancement is consistent with the counterintuitive observation that an appropriate strength of a stabilizing force can enhance heat transfer by increasing flow coherence (Chong et al. 2017). Significant enhancement of the heat transfer due to the increased flow coherence was also observed in the confined inclined convection in low-P r fluids (Zwirner & Shishkina 2018). Due to the emergence of system-sized plume columns and the interaction of these plume columns with the opposed boundary layers in inclined convection, an increase of the heat transfer by a factor of approximately 2.3 can be realized. ...
Preprint
Full-text available
We perform a numerical study of the heat transfer and flow structure of Rayleigh-B\'enard (RB) convection in (in most cases regular) porous media, which are comprised of circular, solid obstacles located on a square lattice. This study is focused on the role of porosity $\phi$ in the flow properties during the transition process from the traditional RB convection with $\phi=1$ (so no obstacles included) to Darcy-type porous-media convection with $\phi$ approaching 0. Simulations are carried out in a cell with unity aspect ratio, for the Rayleigh number $Ra$ from $10^5$ to $10^{10}$ and varying porosities $\phi$, at a fixed Prandtl number $Pr=4.3$, and we restrict ourselves to the two dimensional case. For fixed $Ra$, the Nusselt number $Nu$ is found to vary non-monotonously as a function of $\phi$; namely, with decreasing $\phi$, it first increases, before it decreases for $\phi$ approaching 0. The non-monotonous behaviour of $Nu(\phi)$ originates from two competing effects of the porous structure on the heat transfer. On the one hand, the flow coherence is enhanced in the porous media, which is beneficial for the heat transfer. On the other hand, the convection is slowed down by the enhanced resistance due to the porous structure, leading to heat transfer reduction. For fixed $\phi$, depending on $Ra$, two different heat transfer regimes are identified, with different effective power-law behaviours of $Nu$ vs $Ra$, namely, a steep one for low $Ra$ when viscosity dominates, and the standard classical one for large $Ra$. The scaling crossover occurs when the thermal boundary layer thickness and the pore scale are comparable. The influences of the porous structure on the temperature and velocity fluctuations, convective heat flux, and energy dissipation rates are analysed, further demonstrating the competing effects of the porous structure to enhance or reduce the heat transfer.
... These results are all performed with relatively small β. In recent years, thermal convection in tilted cells with a wide range of β (0 • ≤ β ≤ 90 • ) has also been investigated [59,104,106,107,270,271,282,283,[289][290][291]. Ref. [270] experimentally studied thermal convection in a tilted cavity for Ra ≈ 4.3 × 10 9 and P r ≈ 6.7, they found that the LSC is sensitive to the symmetry of the system and the LSC changes gradually from oblique ellipse-like to square-like, they also found that the N u monotonically decreases with increasing β for 0 • ≤ β ≤ 90 • . ...
... This system is well suited for the study of buoyancy-and shear flow-driven instabilities in the Earth's atmosphere and hydrosphere [66]. Due to its importance, thermal convection in tilted containers has been studied extensively in recent years both with small tilt angles [155,160,261,[286][287][288] and with relatively wide range of tilt angles (0 • < β < 90 • ) [59,104,107,270,271,[279][280][281][282][283][289][290][291]. For a convection cell with aspect ratio Γ = 1, Ref. [270] experimentally studied convection in a tilted cavity for the Rayleigh number Ra ≈ 4.3 × 10 9 and the Prandtl number P r ≈ 6.7, they found that the large-scale circulation (LSC) is sensitive to the symmetry of the system and the LSC changes gradually from oblique ellipse-like to square-like, they also found that the detailed direct numerical simulations (DNS) for 10 6 ≤ Ra ≤ 10 8 and 0.1 ≤ P r ≤ 100 in a cylindrical cell of unit aspect ratio, they showed that N u(β)/N u(0) dependence is not universal and is strongly influenced by a combination of Ra and P r, and the N u(β)/N u(0) dependence is a complicated, non-monotonic function of β. ...
... Several studies performed in convection cells with Γ < 1 also reported that tilt can induce heat transport enhancement [279][280][281]283]. Recently, Ref. [290] studied thermal convection in a tilted cylindrical cell with Γ = 1/5 for P r = 0.1 and 1.0 using DNS. They determined the optimal tilt angle, which provides the maximal global heat transport. ...
Thesis
Full-text available
Turbulent thermal convection driven by a temperature difference is omnipresent in nature and it plays an important role in numerous industrial applications. Rayleigh-Bénard convection (RBC) where a fluid layer in a cavity heated from below and cooled from above, is a classical model problem for the study of thermal convection. Another model problem is vertical convection (VC) where the fluid layer is heated/cooled at the sides. Both RBC and VC can be viewed as extreme cases of tilted convection where the tilt angle is 0° for RBC and 90° for VC. This thesis studies the above three model problems for thermal convection, i.e. Rayleigh-Bénard convection (Part I), tilted convection (Part II), and vertical convection (Part III). In Part I, we study non-Oberbeck-Boussinesq (NOB) effects either due to density maximum of cold water near 4℃ (Chap. 2) or due to large temperature differences (Chap. 3) in turbulent RBC. We also investigate multistability of convection roll states (Chap. 4) and metastability of the zonal flow (Chap. 5) in turbulent RBC. In Part II, we focus on tilted convection. We report that tilting can promote flow reversals in a two-dimensional convection cell with aspect ratio 2 (Chap. 6). The global flow organization and heat transport for two-dimensional tilted convection with small aspect ratio 0.5 (Chap. 7) and large aspect ratios (Chap. 8) are also discussed. We further study NOB effects in three-dimensional tilted convection in Chap. 9. In Part III, we investigate vertical convection, and we particularly focus on NOB effects in two-dimensional vertical convection in Chap. 10.
... Furthermore, for low Pr Շ 1 this angle provides a single maximum for NuðhÞ, and for greater inclinations, NuðhÞ exhibits a monotonous decay. [14][15][16][17][18] At intermediate and high Pr, the influence of the tilt on the heat transfer seems to be more involved. Increasing the tilt pushes the turbulent threshold to higher Ra at high Prandtl numbers. ...
... There are much fewer studies than on Nusselt number, and we are not aware of research on the effect of the inclination on the Reynolds number in melting PCM. DNS in cylinders with C ¼ 1=5 filled with fluids at Pr ¼ 0:1; 1 and Ra 2 ½10 6 -10 9 15 finds a tilt that maximizes the Reynolds number. While the behavior is the same as for Nu, the maximum value of Re occurs at lower inclinations. ...
... The laminar regime h > h c exhibits a smooth downward trend with increasing the inclination, as a consequence of the reduced strength of the convective motions at higher inclinations within the laminar regime. Similar to the exponent a, the angles slightly above h c exhibit a strong increase in the exponent b with respect to the turbulent regime.While at high Rayleigh convection frequently occurs a maximum of Re at a non-zero inclination in a wide range of Prandtl numbers,15,19,21,23 we find that the high dispersion of b and downtrend at h < h c does not favor an extreme value of Re. We notice, however, that other results in RB convection are similar to this work. ...
Article
Full-text available
We report two-dimensional simulations and analytic results on the effect of the inclination on the transient heat transfer, flow, and melting dynamics of a phase change material within a square domain heated from one side. The liquid phase has Prandtl number Pr = 60.8, Stefan number Ste = 0.49, and Rayleigh numbers extend over eight orders of magnitude 0 ≤ R a ≤ 6.6 · 10 8 for the largest geometry studied. The tilt determines the stability threshold of the base state. Above a critical inclination, there exists only a laminar flow at the melted phase, irrespective of the Rayleigh number. Below that inclination, the base state destabilizes following two paths according to the inclination: either leading to a turbulent state for angles near the critical inclination or passing through a regime of plume coarsening before reaching the turbulent state for smaller angles. We find that the Nusselt and Reynolds numbers follow a power law as N u ∼ R a α , R e ∼ R a β in the turbulent regime. Small inclinations reduce very slightly α and strongly β. The inclination leads to subduction of the kinematic boundary layer into the thermal boundary layer. The scaling laws of the Nusselt and Reynolds numbers and boundary layers are in agreement with different results at high Rayleigh convection. However, some striking differences appear as the stabilization of turbulent states with further increasing of the Rayleigh number. We find as well that the turbulent regime exhibits a higher dispersion in quantities related to heat transfer and flow dynamics on smaller domains.
... Several studies focused on how lateral confinement in one direction influences the heat transport and flow structures [15][16][17][18][19], though only a few studies focused on lateral confinement in two directions, e.g., slender cylindrical cells of small diameter-to-height aspect ratio Γ ¼ D=H. Not only a single-roll mode (SRM) of the LSC, but also a double-roll mode (DRM)composed of two rolls on top of each other-was found for cylindrical cells with Γ ¼ 1, 1=2, 1=3, and 1=5 [20][21][22]. Experimental studies with water (Pr ≈5) found that the SRM is characterized by a slightly enhanced heat transport (≈0.5%) compared to the DRM [20,21]. It was also found that small-Γ systems spend more time in the DRM than in the SRM. ...
... It was also found that small-Γ systems spend more time in the DRM than in the SRM. Direct numerical simulations (DNS) [22] for Ra ¼ 10 6 , Pr ¼ 0.1, and Γ ¼ 1=5 showed that the heat transport of the DRM is only 80% compared to the SRM. ...
... The DNS were conducted at Ra ¼ 5 × 10 6 , Pr ¼ 0.1, and Γ ¼ 1=5, using a mesh of 256 × 128 × 22 nodes in z, φ, and r directions, which is of sufficient resolution [22,32]. We consider the volume-averaged instantaneous heat transport NuðtÞ ≡ ½hu z ðtÞθðtÞi − κh∂ z θðtÞi=ðκΔ=HÞ (Nusselt number) and the Reynolds number, ReðtÞ≡ H ffiffiffiffiffiffiffiffiffi hu 2 i p =ν, which is based on the kinetic energy. ...
Article
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The large-scale circulation (LSC) of fluid is one of the main concepts in turbulent thermal convection as it is known to be important in global heat and mass transport in the system. In turbulent Rayleigh-Bénard convection (RBC) in slender containers, the LSC is formed of several dynamically changing convective rolls that are stacked on top of each other. The present study reveals the following two important facts: (i) the mechanism which causes the twisting and breaking of a single-roll LSC into multiple rolls is the elliptical instability and (ii) the heat and momentum transport in RBC, represented by the Nusselt (Nu) and Reynolds (Re) numbers, is always stronger (weaker) for smaller (larger) number n of the rolls in the LSC structure. Direct numerical simulations support the findings for n=1,…,4 and the diameter-to-height aspect ratio of the cylindrical container Γ=1/5, the Prandtl number Pr=0.1 and Rayleigh number Ra=5×105. Thus, Nu and Re are, respectively, 2.5 and 1.5 times larger for a single-roll LSC (n=1) than for a LSC with n=4 rolls.
... and Scheel and Schumacher [15] performed simulations for flows in a vertical cylinder to obtain empirical correlations for Nusselt number (Nu) as a function of Ra and Pr, and evaluated critical Ra when the boundary layer transitions to turbulence. Zwirner and Shiskina [16] performed simulations in an inclined cylinder (at different angles from horizontal to vertical) to study the effect of cylinder inclination on global flow structure and heat transport. The study identified that coherent large-scale circulation plays an important role in the enhancement of heat transfer, and the enhancement magnitude depends on the inclination angle, Ra, and Pr. ...
... Inclined cylinder(θ = 0 to 90°) Zwirner and Shiskina [16] 0.1, 1 10 8 , 10 9 (10 7 to 10 9 ) ...
Article
A direct numerical simulation (DNS) database is presented for turbulent flow and heat transfer in a vertical channel for Reynolds number Reτ = 150 and 640, Prandtl number Pr = 0.004, 0.025 and 0.71, with and without buoyancy forcing (Ri = 0 and 0.15 or 0.21). The effects of Pr on mean and turbulent flow and thermal transport in mixed convective conditions are discussed. Aiding/opposing buoyant conditions result in acceleration/deceleration of mean flow and reduction/enhancement of turbulence. Flow turbulence is highly anisotropic and dominated by the streamwise component on the aiding side, and becomes two-dimensional on the opposing side with a decrease in Pr. Temperature distributions depend on the relative role of molecular and turbulent thermal transport. The former increases with decreasing Pr and the latter with increasing Re. Buoyancy affects thermal transport through augmentation of the wall-normal turbulent heat flux, v′θ′¯, which is more pronounced for higher Pr. A priori analysis of the DNS datasets shows that a variable formulation for turbulent Prandtl number in Reynolds-averaged Navier-Stokes simulations performs well for both high- and low-Pr flows without buoyancy and in stable convective regimes, but yields relatively high error in unstable convective regimes.
... This system is well suited for the study of buoyancy-and shear flow-driven instabilities in the earth's atmosphere and hydrosphere [21]. Due to its importance, thermal convection in tilted containers has been studied extensively both with small tilt angles [22][23][24][25][26][27] and with a relatively wide range of tilt angles (0 • < β < 90 • ) [28][29][30][31][32][33][34][35][36][37][38][39][40]. For a convection cell with aspect ratio = 1, Guo et al. [31] experimentally studied convection in a tilted cavity for the Rayleigh number Ra ≈ 4.3 × 10 9 and the Prandtl number Pr ≈ 6.7. ...
... Several studies performed in convection cells with < 1 also reported that tilt can induce heat transport enhancement [28][29][30]35]. Recently, Zwirner and Shishkina [39] studied thermal convection in a tilted cylindrical cell with = 1 5 for Pr = 0.1 and 1.0 using DNS. They determined the optimal tilt angle, which provides the maximal global heat transport. ...
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The coexistence of multiple turbulent states was reported in several recent studies in different flows. We present in this work that multiple turbulent states also exist for thermal convection in two-dimensional tilted cells with large aspect ratios (\Gamma= width/height) through direct numerical simulations for the Rayleigh number Ra = 10^7 and the Prandtl number Pr = 0.71. The considered \Gamma ranges from 1 to 16. The tilt angle β varies from 0◦ to 180◦. Multiple states are identified for small β with \Gamma>= 2, where the different flow states are reflected in different numbers of convection rolls. The corresponding Nu is generally higher for the flow state with more convection rolls. Moreover, flow mode transitions between different roll states are observed for large \Gamma>= 8 when β is larger than a critical value. The effect of cell tilting on Nu and Re is also investigated. It is found that for \Gamma<= 4, Nu first increases with increasing β and then declines after reaching its local maximum. However, Nu generally decreases monotonically with increasing β for \Gamma = 8, 12, and 16. This indicates that the idea to enhance heat transfer by tilting the cell can be realized only for relatively small \Gamma for the present system. It is also found that the previous finding that Re decreases monotonically with increasing β for large β with \Gamma = 1 does not hold for large-\Gamma cases.
... and Scheel and Schumacher [15] performed simulations for flows in a vertical cylinder to obtain empirical correlations for Nusselt number (Nu) as a function of Ra and Pr, and evaluated critical Ra when the boundary layer transitions to turbulence. Zwirner and Shiskina [16] performed simulations in an inclined cylinder (at different angles from horizontal to vertical) to study the effect of cylinder inclination on global flow structure and heat transport. The study identified that coherent large-scale circulation plays an important role in the enhancement of heat transfer, and the enhancement magnitude depends on the inclination angle, Ra, and Pr. ...
... Inclined cylinder(θ = 0 to 90°) Zwirner and Shiskina [16] 0.1, 1 10 8 , 10 9 (10 7 to 10 9 ) ...
Article
Linear eddy viscosity-based Reynolds-averaged Navier-Stokes (RANS), hybrid RANS/Large Eddy Simulation (LES), and LES models are applied for simulations of plane channel flow at Reynolds number Reτ = 150 and 640. Results are obtained for fluids with Prandtl number (Pr) of 0.025 and 0.71, for cases without buoyancy (Gr = 0) and for a vertical channel flow with buoyancy (Ri = 0.1–0.13). The objective is to evaluate the predictive capability of each class of model for low-Pr flow typical of liquid metal reactor cooling systems. For this purpose, RANS, partially-averaged Navier-Stokes (PANS) and LES computations are performed on grids which contain 0.67%, 2%, and 5% of the grid points used in the Direct Numerical Simulation (DNS), respectively, and the predictions of the mean and turbulent flow and thermal quantities are validated against DNS results. RANS results obtained using Kays variable turbulent Prandtl number (PrT) formulation were more accurate than those using constant PrT for low-Pr forced convective flows, but did not play a significant role for mixed convective flows. The variable PrT formulation likewise did not have significant effect for hybrid RANS/LES or LES computations, wherein > 90% turbulence was resolved. The RANS model performed reasonably well for flows without buoyancy, but for mixed convective flows, significantly under predicted the sharp pressure gradient around the mid-channel, and the limitations were identified due to over prediction of heat transfer by wall-normal fluctuations on the aiding side. Both PANS and dynamic Smagorinsky (DSM) showed similar predictions, and under predicted the mid-channel temperature gradient for the low-Pr vertical channel case and over predicted flow turbulence on the opposing flow side and thermal fluctuations on the aiding flow side. The Wall-Adapting Local Eddy-viscosity (WALE) model performed best among the LES subgrid stress models. The improved predictions by WALE over PANS and DSM were identified because of higher modeled dissipation predictions away from the wall. Future work will focus on investigation of non-linear RANS models and extend the validation for higher Re flows.
... There, the fluid layer, heated on one surface and cooled from the opposite surface, is tilted with respect to the gravity direction, so that both, buoyancy and shear drive the flow in this case. This type of convection was studied previously, in particular, by [20,11,76,2,59,87,47] and more recently by [24,50,80,43,70,78,51,40,98]. ...
... A tiny local increase of Nu with a small inclination of the RBC cell filled with a fluid of Pr > 1 is possible only when a two-roll form of the global Large Scale Circulation (LSC) is present in RBC, which usually almost immediately transforms into a single-roll form of the LSC with any inclination [87]. The single-roll LSC is known to be more efficient in the heat transport than its double-roll form, as it was proved in the measurements [91,87] and DNS [98]. Thus, all available experimental and numerical results on IC show that the Nu(β)/Nu(0) dependence is a complex function of Ra, Pr and Γ, which cannot be represented as a simple combination of their power functions. ...
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Inclined turbulent thermal convection by large Rayleigh numbers in extremely small-Prandtl-number fluids is studied based on results of both, measurements and high-resolution numerical simulations. The Prandtl number $Pr\approx0.0093$ considered in the experiments and the Large-Eddy Simulations (LES) and $Pr=0.0094$ considered in the Direct Numerical Simulations (DNS) correspond to liquid sodium, which is used in the experiments. Also similar are the studied Rayleigh numbers, which are, respectively, $Ra=1.67\times10^7$ in the DNS, $Ra=1.5\times10^7$ in the LES and $Ra=1.42\times10^7$ in the measurements. The working convection cell is a cylinder with equal height and diameter, where one circular surface is heated and another one is cooled. The cylinder axis is inclined with respect to the vertical and the inclination angle varies from $\beta=0^\circ$, which corresponds to a Rayleigh-B\'enard configuration (RBC), to $\beta=90^\circ$, as in a vertical convection (VC) setup. The turbulent heat and momentum transport as well as time-averaged and instantaneous flow structures and their evolution in time are studied in detail, for different inclination angles, and are illustrated also by supplementary videos, obtained from the DNS and experimental data. To investigate the scaling relations of the mean heat and momentum transport in the limiting cases of RBC and VC configurations, additional measurements are conducted for about one decade of the Rayleigh numbers around $Ra=10^7$ and $Pr\approx0.009$. With respect to the turbulent heat transport in inclined thermal convection by low $Pr$, a similarity of the global flow characteristics for the same value of $RaPr$ is proposed and analysed, based on the above simulations and measurements and on complementary DNS for $Ra=1.67\times10^6$, $Pr=0.094$ and $Ra=10^9$, $Pr=1$.
... The concept of inclined convection (IC) is a generalisation of RBC and VC, i.e. the fluid layer between the parallel plates is tilted with respect to the direction of gravity, and both buoyancy and shear act on the flow. This type of convection was studied previously by Daniels, Wiener & Bodenschatz (2003), Chillà et al. (2004), Sun, Xi & Xia (2005), Ahlers, Brown & Nikolaenko (2006b), Riedinger et al. (2013), Weiss & Ahlers (2013) and Langebach & Haberstroh (2014), and more recently by Frick et al. (2015), Mamykin et al. (2015), Vasil'ev et al. (2015), Kolesnichenko et al. (2015), Shishkina & Horn (2016), , Mandrykin & Teimurazov (2019), Khalilov et al. (2018) and Zwirner & Shishkina (2018). ...
... A tiny local increase of Nu with a small inclination of the RBC cell filled with a fluid of Pr > 1 is possible only when a two-roll form of the global large-scale circulation (LSC) is present in RBC, which usually almost immediately transforms into a singleroll form of the LSC with any inclination (Weiss & Ahlers 2013). The single-roll LSC is known to be more efficient in the heat transport than its double-roll form, as was proved in the measurements (Xi & Xia 2008;Weiss & Ahlers 2013) and DNS (Zwirner & Shishkina 2018). ...
Article
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The influence of the cell inclination on the heat transport and large-scale circulation in liquid metal convection - Volume 884 - Lukas Zwirner, Ruslan Khalilov, Ilya Kolesnichenko, Andrey Mamykin, Sergei Mandrykin, Alexander Pavlinov, Alexander Shestakov, Andrei Teimurazov, Peter Frick, Olga Shishkina
... With a further increase in , an increase in the Nusselt number is observed, although the dependence of the heat transfer intensity on the inclination angle can be rather complicated. Experimental [4,7,11] and numerical [6,10,12] studies show that the maximum heat transfer between hot and cold surfaces is realized in an inclined position of the cavity. ...
... The flow structure and the heat transfer intensity also depend on the aspect ratio of the cavity. It was established experimentally and numerically that convection in cylinders with different aspect ratios ( ) is significantly different in the case of liquid metals [4,[6][7][8] and in the case of liquids with large [12]. In this paper, convection of liquid sodium ( ) in an inclined cylinder with a unit aspect ratio at a Rayleigh number is studied numerically. ...
Article
Turbulent convection of liquid sodium (Prandtl number Pr = 0.0093) in a cylinder of unit aspect ratio, heated at one end face and cooled at the other, is studied numerically. The flow regimes with inclination angles β = 0°, 20°, 40°, 70° with respect to the vertical are considered. The Rayleigh number is 1.5 × 107 . Three-dimensional nonstationary simulations allow one to get instant and average characteristics of the process and to study temperature pulsation fields. A mathematical model is based on the Boussinesq equations for thermogravitational convection with use of the LES (large-eddy simulations) approach for small-scale turbulence modeling. Simulations were carried out with a nonuniform numerical grid consisting of 2.9 × 106 nodes. It is shown that the flow structure strongly depends on β. The large-scale circulation (LSC) exists in the cylinder at any β. Under moderate inclination (β = 20°), the strong oscillations of the LSC orientation angle with dominant frequency are observed. Increasing the inclination up to 40° leads to stabilization of the large-scale flow and there is no dominant frequency of oscillations in this case. It is shown that more intensive temperature pulsations occur at small cylinder inclinations. At any β the regions with intensive pulsations are concentrated in the areas along low and upper cylinder faces. The maximum values of pulsations occur in the area close to lateral walls, where hot and cold fluid flows collide. The intensity of temperature pulsations decreases with increasing distance from the lateral walls. The Reynolds number which characterizes the total energy of the flow reaches its maximum value at β = 20° and then decreases with increasing β. The mean flow has maximum intensity at β = 40°. Turbulent velocity pulsation energy decreases monotonically with increasing inclination angle. It is shown that the inclination leads to an increase in heat transfer along the cylinder axis. The Nusselt number at β = 40° is 26% higher than that in the vertical cylinder.
... Hartmann et al. (2021) have shown that heat transport enhancement in confined RBC generally occurs in cylindrical and cuboid domains, but flow organization, optimal confinement Γ −1 and amplitude of heat transport enhancement are strongly influenced by the cell geometry. Further, Zwirner & Shishkina (2018) have shown that an inclined gravity in addition to confinement is able to significantly enhance the heat transport. Chong et al. (2017) have nicely revealed more striking similarities in heat transport enhancement for individually examined types of stabilization: confinement, rotation and an additional stabilizing buoyant scalar field as in double-diffusive convection. ...
... When Γ −1 is increased beyond the optimum, the growing impact of the sidewalls reduces the heat transport (Chong et al. 2015;Chong & Xia 2016). Similarly, coherent flow structures in wall-normal direction help to maximize the heat transport in inclined-confined RBC (Zwirner & Shishkina 2018). ...
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Moderate rotation and moderate horizontal confinement similarly enhance the heat transport in Rayleigh–Bénard convection (RBC). Here, we systematically investigate how these two types of flow stabilization together affect the heat transport. We conduct direct numerical simulations of confined-rotating RBC in a cylindrical set-up at Prandtl number $\textit {Pr}=4.38$ , and various Rayleigh numbers $2\times 10^{8}\leqslant {\textit {Ra}}\leqslant 7\times 10^{9}$ . Within the parameter space of rotation (given as inverse Rossby number $0\leqslant {\textit {Ro}}^{-1}\leqslant 40$ ) and confinement (given as height-to-diameter aspect ratio $2\leqslant \varGamma ^{-1}\leqslant 32$ ), we observe three heat transport maxima. At lower ${\textit {Ra}}$ , the combination of rotation and confinement can achieve larger heat transport than either rotation or confinement individually, whereas at higher ${\textit {Ra}}$ , confinement alone is most effective in enhancing the heat transport. Further, we identify two effects enhancing the heat transport: (i) the ratio of kinetic and thermal boundary layer thicknesses controlling the efficiency of Ekman pumping, and (ii) the formation of a stable domain-spanning flow for an efficient vertical transport of the heat through the bulk. Their interfering efficiencies generate the multiple heat transport maxima.
... [2,3,[15][16][17][18]. For Γ < 1, the LSC forms multiple rolls arranged on top of each other [4,[19][20][21]. The particular LSC configuration determines the magnitude of transferred heat which is quantified by the Nusselt number N u [17,21,22]. ...
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The large-scale flow structure and the turbulent transfer of heat and momentum are directly measured in highly turbulent liquid metal convection experiments for Rayleigh numbers varied between $4 \times 10^5$ and $\leq 5 \times 10^9$ and Prandtl numbers of $0.025~\leq~Pr~\leq ~0.033$. Our measurements are performed in two cylindrical samples of aspect ratios $\Gamma =$ diameter/height $= 0.5$ and 1 filled with the eutectic alloy GaInSn. The reconstruction of the three-dimensional flow pattern by 17 ultrasound Doppler velocimetry sensors detecting the velocity profiles along their beamlines in different planes reveals a clear breakdown of the large-scale circulation for $\Gamma = 0.5$. As a consequence, the scaling laws for heat and momentum transfer inherit a dependence on the aspect ratio. We show that this breakdown of coherence is accompanied with a reduction of the Reynolds number $Re$. The scaling exponent $\beta$ of the power law $Nu\propto Ra^{\beta}$ crosses over from $\beta=0.221$ to 0.124 when the liquid metal flow at $\Gamma=0.5$ reaches $Ra\gtrsim 2\times 10^8$.
... In recent years, thermal convection in tilted cells with a wide range of β (0 • ≤ β ≤ 90 • ) has also been investigated. 14,15,[24][25][26][27][28][29][30][31][32] Guo et al. 24 experimentally studied thermal convection in a tilted cavity for Ra ≈ 4.3 × 10 9 and Pr ≈ 6.7; they found that the LSC is sensitive to the symmetry of the system and the LSC changes gradually from oblique ellipse-like to square-like, and they also found that the Nu monotonically decreases with increasing β for 0 • ≤ β ≤ 90 • . Shishkina and Horn 25 performed detailed direct numerical simulations (DNS) for 10 6 ≤ Ra ≤ 10 8 and 0.1 ≤ Pr ≤ 100 in a cylindrical cell with Γ = 1, and they showed that Nu(β)/Nu(0) dependence is not universal and is strongly influenced by a combination of Ra and Pr, and the Nu(β)/Nu(0) dependence is a complicated, non-monotonic function of β. ...
Article
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Thermal convection in a two-dimensional tilted cell with aspect ratio (Γ = width/height) 0.5 is studied using direct numerical simulations. The considered tilt angle β ranges from 0 deg. to 90 deg. The Prandtl number Pr dependence is first studied in the range of 0.01 ≤ Pr ≤ 100 for a fixed Rayleigh number Ra = 10^7. The Ra dependence is also investigated in the range of 10^6 ≤ Ra ≤ 10^9 for a fixed Pr = 0.71. Different flow states are identified over the β − Pr parameter space. It is found that the flow tends to organize in stable vertically-stacked double-roll state (DRS) for small Pr and small β, while this DRS becomes unstable and flow reversals happen with the increase of β. This finding complements our previous study of flow reversals in tilted cells with Γ = 1 and 2 [Wang et al., J. Fluid Mech. 849, 355–372 (2018)]. For relatively larger Pr, the flow gives way to a stable triple-roll state or an unstable triple-roll state for small β. Moreover, multiple states in the turbulent regime are found for Ra ≥ 10^8, between which the flow can or cannot switch. In the latter case, the Nu are different for the two states with the same number of convection rolls, but different orientations. It is found that the Nu(β)/Nu(0) and Re(β)/Re(0) dependence is strongly influenced by a combination of Ra and Pr. In the present system, we interestingly find that the earlier conclusion that Nu decreases with increasing β close to β=90◦ for Γ=1 does not hold for the present Γ=0.5 case with small Pr.
... This phenomenon in fact has two aspects. The first one is enhancement of scalar transport by a stabilising force, be it drag force due to geometrical confinement in the case of CRB (Huang, Kaczorowski, Ni & Xia 2013;Chong, Huang, Kaczorowski & Xia 2015;Zwirner & Shishkina 2018;Chong et al. 2018b), or the Coriolis force in the case of RRB Stevens et al. 2009;Weiss, Wei & Ahlers 2016), or the stabilising temperature field in the case of DDC (Yang, Verzicco & Lohse 2016). The second aspect is the existence of an optimal enhancement corresponding to certain value of the control parameter that characterises the stabilizing force, i.e. the aspect ratio for CRB, the Rossby number for RRB and the density ratio for DDC, respectively. ...
Preprint
We present a numerical study of quasistatic magnetoconvection in a cubic Rayleigh-B\'enard (RB) convection cell subjected to a vertical external magnetic field. For moderate values of the Hartmann number Ha, we find an enhancement of heat transport. Furthermore, a maximum heat transport enhancement is observed at certain optimal $Ha_{opt}$. The enhanced heat transport may be understood as a result of the increased coherency of the thermal plumes, which are elementary heat carriers of the system. To our knowledge this is the first time that a heat transfer enhancement by the stabilising Lorentz force in quasistatic magnetoconvection has been observed. We further found that the optimal enhancement may be understood in terms of the crossing between the thermal and the momentum boundary layers (BL) and the fact that temperature fluctuations are maximum near the position where the BLs cross. These findings demonstrate that the heat transport enhancement phenomenon in the quasistatic magnetoconvection system belongs to the same universality class of stabilising$-$destabilising ($S$-$D$) turbulent flows as the systems of confined Rayleigh-B\'enard (CRB), rotating Rayleigh-B\'enard (RRB) and double-diffusive convection (DDC). This is further supported by the findings that the heat transport, boundary layer ratio and the temperature fluctuations in magnetoconvection at the boundary layer crossing point are similar to the other three cases.
... This phenomenon in fact has two aspects. The first one is enhancement of scalar transport by a stabilising force, be it drag force due to geometrical confinement in the case of CRB (Huang et al. 2013;Chong et al. 2015Zwirner & Shishkina 2018), or the Coriolis force in the case of RRB Stevens et al. 2009;Weiss, Wei & Ahlers 2016), or the stabilising temperature field in the case of DDC (Yang, Verzicco & Lohse 2016). The second aspect is the existence of an optimal enhancement corresponding to certain value of the control parameter that characterises the stabilising force, i.e. the aspect ratio for CRB, the Rossby number for RRB and the density ratio for DDC, respectively. ...
Article
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We present a numerical study of quasistatic magnetoconvection in a cubic Rayleigh–Bénard (RB) convection cell subjected to a vertical external magnetic field. For moderate values of the Hartmann number $Ha$ (characterising the strength of the stabilising Lorentz force), we find an enhancement of heat transport (as characterised by the Nusselt number $Nu$ ). Furthermore, a maximum heat transport enhancement is observed at certain optimal $Ha_{opt}$ . The enhanced heat transport may be understood as a result of the increased coherence of the thermal plumes, which are elementary heat carriers of the system. To our knowledge this is the first time that a heat transfer enhancement by the stabilising Lorentz force in quasistatic magnetoconvection has been observed. We further found that the optimal enhancement may be understood in terms of the crossing of the thermal and the momentum boundary layers (BL) and the fact that temperature fluctuations are maximum near the position where the BLs cross. These findings demonstrate that the heat transport enhancement phenomenon in the quasistatic magnetoconvection system belongs to the same universality class of stabilising–destabilising (S–D) turbulent flows as the systems of confined Rayleigh–Bénard (CRB), rotating Rayleigh–Bénard (RRB) and double-diffusive convection (DDC). This is further supported by the findings that the heat transport, boundary layer ratio and temperature fluctuations in magnetoconvection at the boundary layer crossing point are similar to the other three cases. A second type of boundary layer crossing is also observed in this work. In the limit of $Re\gg Ha$ , the (traditionally defined) viscous boundary $\unicode[STIX]{x1D6FF}_{v}$ is found to follow a Prandtl–Blasius-type scaling with the Reynolds number $Re$ and is independent of $Ha$ . In the other limit of $Re\ll Ha$ , $\unicode[STIX]{x1D6FF}_{v}$ exhibits an approximate ${\sim}Ha^{-1}$ dependence, which has been predicted for a Hartmann boundary layer. Assuming the inertial term in the momentum equation is balanced by both the viscous and Lorentz terms, we derived an expression $\unicode[STIX]{x1D6FF}_{v}=H/\sqrt{c_{1}Re^{0.72}+c_{2}Ha^{2}}$ (where $H$ is the height of the cell) for all values of $Re$ and $Ha$ , which fits the obtained viscous boundary layer well.
... The aspect ratio can also be an additional control parameter for confined thermal convection problems (van der Poel et al. 2011;Zwirner & Shishkina 2018), but at present, we restrict our analyses to a fixed value. Our simulations cover the values of Ra = 1.3 × 10 8 -1.3 × 10 9 and for Pr = 7, corresponding to water. ...
Preprint
We present results on the effect of dispersed droplets in vertical natural convection (VC) using direct numerical simulations based on a two-way fully coupled Euler-Lagrange approach with a liquid phase and a dispersed droplets phase. For increasing thermal driving, characterised by the Rayleigh number, $Ra$, of the two analysed droplet volume fractions, $\alpha = 5\times10^{-3}$ and $\alpha = 2\times 10^{-2}$, we find non-monotonic responses to the overall heat fluxes, characterised by the Nusselt number, $Nu$. The $Nu$ number is larger when the droplets are thermally coupled to the liquid. However, $Nu$ retains the effective scaling exponents that are close to the ${1/4}$-laminar VC scaling, suggesting that the heat transport is still modulated by thermal boundary layers. Local analyses reveal the non-monotonic trends of local heat fluxes and wall-shear stresses: Whilst regions of high heat fluxes are correlated to increased wall-shear stresses, the spatio-temporal distribution and magnitude of the increase is non-universal, implying that the overall heat transport is obscured by competing mechanisms. Most crucially, we find that the transport mechanisms inherently depend on the dominance of droplet driving to thermal driving that can quantified by (i) the bubblance parameter $b$, which measures the ratio of energy produced by the dispersed phase and the energy of the background turbulence, and (ii) $Ra_d/Ra$, where $Ra_d$ is the droplet Rayleigh number, which we introduce in this paper. When $b \lesssim O(10^{-1})$ and $Ra_d/Ra \lesssim O(100)$, the $Nu$ scaling is expected to recover to the VC scaling without droplets, and comparison with $b$ and $Ra_d/Ra$ from our data supports this notion.
... Scheel and Schumacher 2 identified a transition between the rotationally constrained and the weakly rotating turbulent states in rotating Rayleigh-Bénard convection with liquid gallium that differs substantially from moderate-Prandtl number convection. The main differences are due to the more diffuse temperature field, more vigorous velocity field, coarser yet fewer production of thermal plumes in low-Prandtl number convection [11][12][13] . ...
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We report the statistical properties of temperature and thermal energy dissipation rate in low-Prandtl number turbulent Rayleigh-Bénard convection. High resolution two-dimensional direct numerical simulations were carried out for the Rayleigh number (Ra) of 10⁶ ≤ Ra ≤ 10⁷ and the Prandtl number (Pr) of 0.025. Our results show that the global heat transport and momentum scaling in terms of Nusselt number (Nu) and Reynolds number (Re) are Nu = 0.21Ra0.25 and Re = 6.11Ra0.50, respectively, indicating that scaling exponents are smaller than those for moderate-Prandtl number fluids (such as water or air) in the same convection cell. In the central region of the cell, probability density functions (PDFs) of temperature profiles show stretched exponential peak and the Gaussian tail; in the sidewall region, PDFs of temperature profiles show a multimodal distribution at relatively lower Ra, while they approach the Gaussian profile at relatively higher Ra. We split the energy dissipation rate into contributions from bulk and boundary layers and found the locally averaged thermal energy dissipation rate from the boundary layer region is an order of magnitude larger than that from the bulk region. Even if the much smaller volume occupied by the boundary layer region is considered, the globally averaged thermal energy dissipation rate from the boundary layer region is still larger than that from the bulk region. We further numerically determined the scaling exponents of globally averaged thermal energy dissipation rates as functions of Ra and Re.
... These three typical configurations of thermal convection are characterized by the type of the coupling of gravity with buoyancy. The inclined layer convection is particularly suitable for the study of buoyancy and shear flow driven instabilities (For example, convection in atmosphere and hydrosphere, meteorology, and oceanography etc.), which makes ILC an important stability problem and the literature is vast, where a wide variety of experimental [13,14,23,32,38,40] and theoretical [6,7,12,41] studies have been conducted for decades. The control parameter for the onset of the instability is found to increase rapidly on increasing the inclination of the layer with respect to the horizontal. ...
Article
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A nonlinear stability analysis of convection in an inclined layer of a viscous, incompressible fluid is carried out using energy method. The nonlinear stability boundary determined by the Rayleigh number $$\text {Ra}$$ for the underlying dynamical system is found to depend upon the inclination of the layer with respect to the horizontal and Prandtl number of the fluid. A comparison with the linear instability boundary for the considered hydrodynamic system indicates a parametric region where the nonlinear stability of the basic flow is uncertain.
... We use water as the working fluid and the Prandtl number is kept at Pr = 4.3 throughout this work. For small Prandtl number working fluids (gases or liquid metals), the system may behave quite differently due to its relatively large thermal diffusivity (Zwirner & Shishkina 2018;Zürner et al. 2019;Zwirner et al. 2020). Combining experiments with DNS, we apply streamwise and spanwise confinements to turbulent thermal convection in the presence of an effective horizontal buoyancy using the platform of tilted RBC reported in Zhang et al. (2021). ...
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We present an experimental and numerical study of turbulent thermal convection in the presence of an effective horizontal buoyancy that generates extra shear at the boundary. Geometrical confinements are also applied by varying the streamwise and spanwise aspect ratios of the convection cell to condense the plumes. With these, we systematically explore the effects of plume and shear on heat transfer. It is found that a streamwise confinement results in increased plume coverage but decreased shear compared with spanwise confinement. The fact that streamwise confinement leads to a higher vertical heat transfer efficiency than the spanwise confined case suggests that the increase of plume coverage is the dominant effect responsible for the enhanced heat transfer. Our results highlight the potential applications of coherent structure manipulation in efficient passive heat transfer control and thermal engineering. We also analyse the energetics of the present system and derive the expression of mixing efficiency accordingly. The mixing efficiency is found to increase with both the buoyancy ratio and streamwise dimension.
... Geometrical confinement provides an additional way to enhance heat transfer via plume condensation [14,15], despite for an intermediate range of the cell aspect ratio only. Very recently, the combination of inclination of the convection cell and confined geometries was found to be able to lead to an increase of heat transport for several times, especially for low Prandtl numbers and relatively low Rayleigh numbers [16,17]. Up to now, how to severely improve the efficiency of the global heat transport through the RB convection is still highly desirable, which is the main goal of the present investigation. ...
... Geometrical confinement provides an additional way to enhance heat transfer via plume condensation (14,15), despite an intermediate range of the cell aspect ratio only. Very recently, the combination of inclination of the convection cell and confined geometries was found to be able to lead to an increase of heat transport for several times, especially for low Prandtl numbers and relatively low Rayleigh numbers (16,17). Up to now, how to severely improve the efficiency of the global heat transport through the RB convection is still highly desirable, which is the main goal of the present investigation. ...
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Thermal turbulence is well known as a potent means to convey heat across space by a moving fluid. The existence of the boundary layers near the plates, however, bottlenecks its heat-exchange capability. Here, we conceptualize a mechanism of thermal vibrational turbulence that breaks through the boundary-layer limitation and achieves massive heat-transport enhancement. When horizontal vibration is applied to the convection cell, a strong shear is induced to the body of fluid near the conducting plates, which destabilizes thermal boundary layers, vigorously triggers the eruptions of thermal plumes, and leads to a heat-transport enhancement by up to 600%. We further reveal that such a vibration-induced shear can very efficiently disrupt the boundary layers. The present findings open a new avenue for research into heat transport and will also bring profound changes in many industrial applications where thermal flux through a fluid is involved and the mechanical vibration is usually inevitable.
... (where is the kinematic viscosity and is the thermal diffusivity), e.g., liquid metals, is actively studied both experimentally [1,2] and numerically [3][4][5]. Liquids with are characterized by a large value of the thickness ratio of thermal and dynamic boundary layers. Because of this, in the case of turbulent flow regimes (for large Grashof numbers ) the dynamic boundary layer is very thin (see, for example, review [6]). ...
... [2,3,[15][16][17][18]. For Γ < 1, the LSC forms multiple rolls arranged on top of each other [4,[19][20][21]. The particular LSC configuration determines the magnitude of transferred heat which is quantified by the Nusselt number N u [17,21,22]. ...
Article
The large-scale flow structure and the turbulent transfer of heat and momentum are directly measured in highly turbulent liquid metal convection experiments for Rayleigh numbers varied between 4×10^{5} and ≤5×10^{9} and Prandtl numbers of 0.025≤Pr≤0.033. Our measurements are performed in two cylindrical samples of aspect ratios Γ=diameter/height=0.5 and 1 filled with the eutectic alloy GaInSn. The reconstruction of the three-dimensional flow pattern by 17 ultrasound Doppler velocimetry sensors detecting the velocity profiles along their beam lines in different planes reveals a clear breakdown of coherence of the large-scale circulation for Γ=0.5. As a consequence, the scaling laws for heat and momentum transfer inherit a dependence on the aspect ratio. We show that this breakdown of coherence is accompanied with a reduction of the Reynolds number Re. The scaling exponent β of the power law Nu∝Ra^{β} crosses eventually over from β=0.221 to 0.124 when the liquid metal flow at Γ=0.5 reaches Ra≳2×10^{8} and the coherent large-scale flow is completely collapsed.
... where ≡ θ h − θ c . The aspect ratio (L x /L z ) can also be an additional control parameter for confined thermal convection problems (van der Poel, Stevens & Lohse 2011 ;Zwirner & Shishkina 2018), but at present, we restrict our analyses to a fixed value. Our simulations cover the values of Ra = 1.3 × 10 8 -1.3 × 10 9 and for Pr = 7, corresponding to water. ...
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Non-monotonic transport mechanisms in vertical natural convection with dispersed light droplets - Volume 900 - Chong Shen Ng, Vamsi Spandan, Roberto Verzicco, Detlef Lohse
... In many circumstances, adding asymmetry to the system can orient the LSC. This has been done before by inclining the apparatus [54][55][56][57], and is also commonly done in experiments (though with much smaller inclinations) to fix the LSC in place for visualization and measurement purposes. ...
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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 $Ra=10^8$.
... This is important for accurate calculations of the Reynolds number from experimental data and also because the scaling theories (Grossmann & Lohse 2000;Shishkina 2016) are built upon the notion of wind. One challenge for further theoretical studies is that, in VC, unlike RBC, there is a non-closed term in the exact relation between the global kinetic energy dissipation rate and the vertical convective heat transport (Ng et al. 2015;Zwirner & Shishkina 2018). Also, for the future, a deeper investigation of the local flow organization is necessary, because it, unlike the global quantities, strongly depends on the aspect ratio. ...
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Using complementary experiments and direct numerical simulations, we study turbulent thermal convection of a liquid metal (Prandtl number $\textit {Pr}\approx 0.03$ ) in a box-shaped container, where two opposite square sidewalls are heated/cooled. The global response characteristics like the Nusselt number ${\textit {Nu}}$ and the Reynolds number $\textit {Re}$ collapse if the side height $L$ is used as the length scale rather than the distance $H$ between heated and cooled vertical plates. These results are obtained for various Rayleigh numbers $5\times 10^3\leq {\textit {Ra}}_H\leq 10^8$ (based on $H$ ) and the aspect ratios $L/H=1, 2, 3$ and $5$ . Furthermore, we present a novel method to extract the wind-based Reynolds number, which works particularly well with the experimental Doppler-velocimetry measurements along vertical lines, regardless of their horizontal positions. The extraction method is based on the two-dimensional autocorrelation of the time–space data of the vertical velocity.
... VC was also called convection in a differentially heated vertical box in many early papers (Paolucci & Chenoweth 1989;Le Quéré & Behnia 1998). Both RBC and VC can be viewed as extreme cases of the more general so called tilted convection (Guo et al. 2015;Shishkina & Horn 2016;Wang et al. 2018b,a;Zwirner & Shishkina 2018;Zwirner et al. 2020), with the tilt angle of 0 • for RBC and 90 • for VC. We focus on VC in this study. ...
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Vertical convection is investigated using direct numerical simulations over a wide range of Rayleigh numbers $10^7\le Ra\le10^{14}$ with fixed Prandtl number $Pr=10$, in a two-dimensional convection cell with unit aspect ratio. It is found that the dependence of the mean vertical centre temperature gradient $S$ on $Ra$ shows three different regimes: In regime I ($Ra \lesssim 5\times10^{10}$), $S$ is almost independent of $Ra$; In the newly identified regime II ($5\times10^{10} \lesssim Ra \lesssim 10^{13}$), $S$ first increases with increasing $Ra$ (regime ${\rm{II}}_a$), reaches its maximum and then decreases again (regime ${\rm{II}}_b$); In regime III ($Ra\gtrsim10^{13}$), $S$ again becomes only weakly dependent on $Ra$, being slightly smaller than in regime I. The transitions between diffeereent regimes are discussd. In the three different regimes, significantly different flow organizations are identified: In regime I and regime ${\rm{II}}_a$, the location of the maximal horizontal velocity is close to the top and bottom walls; However, in regime ${\rm{II}}_b$ and regime III, banded zonal flow structures develop and the maximal horizontal velocity now is in the bulk region. The different flow organizations in the three regimes are also reflected in the scaling exponents in the effective power law scalings $Nu\sim Ra^\beta$ and $Re\sim Ra^\gamma$. In regime I, the fitted scaling exponents ($\beta\approx0.26$ and $\gamma\approx0.51$) are in excellent agreement with the theoretical predication of $\beta=1/4$ and $\gamma=1/2$ for laminar VC (Shishkina, {\it{Phys. Rev. E.}} 2016, 93, 051102). However, in regimes II and III, $\beta$ increases to a value close to 1/3 and $\gamma$ decreases to a value close to 4/9. The stronger $Ra$ dependence of $Nu$ is related to the ejection of plumes and larger local heat flux at the walls.
... For low Prandtl number, on the other hand, Frick et al. (2015), Teimurazov & Frick (2017) and Khalilov et al. (2018), using liquid sodium as working fluid, found that there exists an optimal tilting angle for heat transport. Shishkina & Horn (2016) and Zwirner & Shishkina (2018) conducted direct numerical simulations (DNS) for wide ranges of Rayleigh number, Prandtl number and different aspect ratios. They found that the normalized Nusselt number Nu(β)/Nu(0) has a complicated, non-monotonic dependence on β as well as Ra and Pr. ...
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We report direct numerical simulations (DNS) of the Nusselt number $Nu$ , the vertical profiles of mean temperature $\varTheta (z)$ and temperature variance $\varOmega (z)$ across the thermal boundary layer (BL) in closed turbulent Rayleigh–Bénard convection (RBC) with slippery conducting surfaces ( $z$ is the vertical distance from the bottom surface). The DNS study was conducted in three RBC samples: a three-dimensional cuboid with length $L = H$ and width $W = H/4$ ( $H$ is the sample height), and two-dimensional rectangles with aspect ratios $\varGamma \equiv L/H = 1$ and $10$ . The slip length $b$ for top and bottom plates varied from $0$ to $\infty$ . The Rayleigh numbers $Ra$ were in the range $10^{6} \leqslant Ra \leqslant 10^{10}$ and the Prandtl number $Pr$ was fixed at $4.3$ . As $b$ increases, the normalised $Nu/Nu_0$ ( $Nu_0$ is the global heat transport for $b = 0$ ) from the three samples for different $Ra$ and $\varGamma$ can be well described by the same function $Nu/Nu_0 = N_0 \tanh (b/\lambda _0) + 1$ , with $N_0 = 0.8 \pm 0.03$ . Here $\lambda _0 \equiv L/(2Nu_0)$ is the thermal boundary layer thickness for $b = 0$ . Considering the BL fluctuations for $Pr>1$ , one can derive solutions of temperature profiles $\varTheta (z)$ and $\varOmega (z)$ near the thermal BL for $b \geqslant 0$ . When $b=0$ , the solutions are equivalent to those reported by Shishkina et al. ( Phys. Rev. Lett. , vol. 114, 2015, 114302) and Wang et al. ( Phys. Rev. Fluids , vol. 1, 2016, 082301(R)), respectively, for no-slip plates. For $b > 0$ , the derived solutions are in excellent agreement with our DNS data for slippery plates.
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Computational codes for direct numerical simulations of Rayleigh–Bénard (RB) convection are compared in terms of computational cost and quality of the solution. As a benchmark case, RB convection at Ra=10⁸ and Pr=1 in a periodic domain, in cubic and cylindrical containers is considered. A dedicated second-order finite-difference code (AFID/RBFLOW) and a specialized fourth-order finite-volume code (GOLDFISH) are compared with a general purpose finite-volume approach (OPENFOAM) and a general purpose spectral-element code (NEK5000). Reassuringly, all codes provide predictions of the average heat transfer that converge to the same values. The computational costs, however, are found to differ considerably. The specialized codes AFID/RBFLOW and GOLDFISH are found to excel in efficiency, outperforming the general purpose flow solvers NEK5000 and OPENFOAM by an order of magnitude with an error on the Nusselt number Nu below 5%. However, we find that Nu alone is not sufficient to assess the quality of the numerical results: in fact, instantaneous snapshots of the temperature field from a near wall region obtained for deliberately under-resolved simulations using NEK5000 clearly indicate inadequate flow resolution even when Nu is converged. Overall, dedicated special purpose codes for RB convection are found to be more efficient than general purpose codes.
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We study, using direct numerical simulations, the effect of geometrical confinement on heat transport and flow structure in Rayleigh-B\'enard convection in fluids with different Prandtl numbers. Our simulations span over two decades of Prandtl number $Pr$, $0.1 \leq Pr \leq 40$, with the Rayleigh number $Ra$ fixed at $10^8$. The width-to-height aspect ratio $\Gamma$ spans between $0.025$ and $0.25$ while the length-to-height aspect ratio is fixed at one. We first find that for $Pr \geq 0.5$, geometrical confinement can lead to a significant enhancement in heat transport as characterized by the Nusselt number $Nu$. For those cases, $Nu$ is maximal at a certain $\Gamma = \Gamma_{opt}$. It is found that $\Gamma_{opt}$ exhibits a power-law relation with $Pr$ as $\Gamma_{opt}=0.11Pr^{-0.06}$, and the maximal relative enhancement generally increases with $Pr$ over the explored parameter range. As opposed to the situation of $Pr \geq 0.5$, confinement-induced enhancement in $Nu$ is not realized for smaller values of $Pr$, such as $0.1$ and $0.2$. The $Pr$ dependence of the heat transport enhancement can be understood in its relation to the coverage area of the thermal plumes over the thermal boundary layer (BL) where larger coverage is observed for larger $Pr$ due to a smaller thermal diffusivity. We further show that $\Gamma_{opt}$ is closely related to the crossing of thermal and momentum BLs, and find that $Nu$ declines sharply when the thickness ratio of the thermal and momentum BLs exceeds a certain value of about one. In addition, through examining the temporally averaged flow fields and 2D mode decomposition, it is found that for smaller $Pr$ the large-scale circulation is robust against the geometrical confinement of the convection cell.
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In thermal convection for very large Rayleigh numbers , the thermal and viscous boundary layers are expected to undergo a transition from a classical state to an ultimate state. In the former state, the boundary-layer thicknesses follow a laminar-like Prandtl-Blasius-Polhausen scaling, whereas in the latter, the boundary layers are turbulent with logarithmic corrections in the sense of Prandtl and von Kármán. Here, we report evidence of this transition via changes in the boundary-layer structure of vertical natural convection (VC), which is a buoyancy-driven flow between differentially heated vertical walls. The numerical dataset spans values from to and a constant Prandtl number value of . For this range, the VC flow has been previously found to exhibit classical state behaviour in a global sense. Yet, with increasing , we observe that near-wall higher-shear patches occupy increasingly larger fractions of the wall areas, which suggest that the boundary layers are undergoing a transition from the classical state to the ultimate shear-dominated state. The presence of streaky structures - reminiscent of the near-wall streaks in canonical wall-bounded turbulence - further supports the notion of this transition. Within the higher-shear patches, conditionally averaged statistics yield a logarithmic variation in the local mean temperature profiles, in agreement with the log law of the wall for mean temperature, and an effective power-law scaling of the local Nusselt number. The scaling of the latter is consistent with the logarithmically corrected power-law scaling predicted for ultimate thermal convection for very large. Collectively, the results from this study indicate that turbulent and laminar-like boundary layer coexist in VC at moderate to high and this transition from the classical state to the ultimate state manifests as increasingly larger shear-dominated patches, consistent with the findings reported for Rayleigh-Bénard convection and Taylor-Couette flows.
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We study the effect of severe geometrical confinement in Rayleigh–Bénard convection with a wide range of width-to-height aspect ratio Γ , 1/128 Γ 1, and Rayleigh number Ra, 3 × 10 4 Ra 1 × 10 11 , at a fixed Prandtl number of Pr = 4.38 by means of direct numerical simulations in Cartesian geometry with no-slip walls. For convection under geometrical confinement (decreasing Γ from 1), three regimes can be recognized (Chong et al., Phys. Rev. Lett., vol. 115, 2015, 264503) based on the global and local properties in terms of heat transport, plume morphology and flow structures. These are Regime I: classical boundary-layer-controlled regime; Regime II: plume-controlled regime; and Regime III: severely confined regime. The study reveals that the transition into Regime III leads to totally different heat and momentum transport scalings and flow topology from the classical regime. The convective heat transfer scaling, in terms of the Nusselt number Nu, exhibits the scaling Nu − 1 ∼ Ra 0.61 over three decades of Ra at Γ = 1/128, which contrasts sharply with the classical scaling Nu − 1 ∼ Ra 0.31 found at Γ = 1. The flow in Regime III is found to be dominated by finger-like, long-lived plume columns, again in sharp contrast with the mushroom-like, fragmented thermal plumes typically observed in the classical regime. Moreover, we identify a Rayleigh number for regime transition, Ra * = (29.37/Γ) 3.23 , such that the scaling transition in Nu and Re can be clearly demonstrated when plotted against Ra/Ra * .
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We report a new thermal boundary layer equation for turbulent Rayleigh-Benard convection for Prandtl number Pr>1 that takes into account the effect of turbulent fluctuations. These fluctuations are neglected in existing equations, which are based on steady-state and laminar assumptions. Using this new equation, we derive analytically the mean temperature profiles in two limits: (a) Pr>1, Pr~1 and (b) Pr>>1. These two theoretical predictions are in excellent agreement with the results of our direct numerical simulations for Pr=4.38 (water) and Pr=2547.9 (glycerol) respectively.
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We report an experimental study of structures and dynamics of the large-scale mean flow in Rayleigh–Bénard convection cells with aspect ratio (Γ)1, 1/2, and 1/3. It is found that both a single circulating roll flow structure and one with two vertically stacked counter-rotating rolls exist in the three aspect ratio cells. The average percentages of time that the large-scale mean flow spends in the single-roll mode (SRM) and the double-roll mode (DRM) are 87.1% and 0.8% for Γ = 1, 69.5% and 7.9% for Γ = 1/2, and 26.7% and 34.1% for Γ = 1/3. Several routes of transitions among the different flow modes are identified. In addition, different structures for the DRM are found and their relative weights are determined. We also show direct evidence that the SRM is more efficient for heat transfer than the DRM. Although the difference is very small, it shows how changes in internal flow state can manifest in the global transport properties of the system. It is also found that the time interval between successive flow mode transitions has an exponential distribution, suggesting a Poisson process for the underlying dynamics. The duration of the flow mode transition is found to be log-normally distributed.
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We report an experimental and numerical study of the effect of spatial confinement in turbulent thermal convection. It is found that when the width of the convection cell is narrowed, the heat-transfer efficiency increases significantly despite the fact that the overall flow is slowed down by the increased drag force from the sidewalls. Detailed experimental and numerical studies show that this enhancement is brought about by the changes in the dynamics and morphology of the thermal plumes in the boundary layers and in the large-scale flow structures in the bulk. It is found that the confined geometry produces more coherent and energetic hot and cold plume clusters that go up and down in random locations, resulting in more uniform and thinner thermal boundary layers. The study demonstrates how changes in turbulent bulk flow can influence the boundary layer dynamics and shows that the prevalent mode of heat transfer existing in larger aspect ratio convection cells, in which hot and cold thermal plumes are carried by the large-scale circulation along opposite sides of the sidewall, is not the most efficient way for heat transport.
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A systematic theory for the scaling of the Nusselt number Nu and of the Reynolds number Re in strong Rayleigh-Benard convection is suggested and shown to be compatible with recent experiments. It assumes a coherent large-scale convection roll ('wind of turbulence') and is based on the dynamical equations both in the bulk and in the boundary layers. Several regimes are identified in the Rayleigh number Ra versus Prandtl number Pr phase space, defined by whether the boundary layer or the bulk dominates the global kinetic and thermal dissipation, respectively, and by whether the thermal or the kinetic boundary layer is thicker. The crossover between the regimes is calculated. In the regime which has most frequently been studied in experiment (Ra ≤ 1011) the leading terms are Nu ~ Ra(1/4) Pr(1/8), Re ~ Ra(1/2) Pr(-3/4) for Pr ≤ 1 and Nu ~ Ra(1/4) Pr(-1/12), Re ~ Ra(1/2) Pr(-5/6) for Pr ≥ 1. In most measurements these laws are modified by additive corrections from the neighbouring regimes so that the impression of a slightly larger (effective) Nu vs. Ra scaling exponent can arise. The most important of the neighbouring regimes towards large Ra are a regime with scaling Nu ~ Ra(1/2) Pr(1/2), Re ~ Ra(1/2) Pr(-1/2) for medium Pr ('Kraichnan regime'), a regime with scaling Nu ~ Ra(1/5) Pr(1/5), Re ~ Ra(2/5) Pr(-3/5) for small Pr, a regime with Nu ~ Ra(1/3), Re ~ Ra(4/9) Pr(-2/3) for larger Pr, and a regime with scaling Nu ~ Ra(3/7) Pr(-1/7), Re ~ Ra(4/7) Pr(-6/7) for even larger Pr. In particular, a linear combination of the 1/4 and the (1/3) power laws for Nu with Ra, Nu = 0.27Ra(1/4) + 0.038Ra(1/3) (the prefactors follow from experiment), mimics a (2/7) power-law exponent in a regime as large as ten decades. For very large Ra the laminar shear boundary layer is speculated to break down through the non-normal-nonlinear transition to turbulence and another regime emerges. The theory presented is best summarized in the phase diagram figure 2 and in table 2.
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The progress in our understanding of several aspects of turbulent Rayleigh-Benard convection is reviewed. The focus is on the question of how the Nusselt number and the Reynolds number depend on the Rayleigh number Ra and the Prandtl number Pr, and on how the thicknesses of the thermal and the kinetic boundary layers scale with Ra and Pr. Non-Oberbeck-Boussinesq effects and the dynamics of the large-scale convection-roll are addressed as well. The review ends with a list of challenges for future research on the turbulent Rayleigh-Benard system.
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The unifying theory of scaling in thermal convection (Grossmann & Lohse (2000)) (henceforth the GL theory) suggests that there are no pure power laws for the Nusselt and Reynolds numbers as function of the Rayleigh and Prandtl numbers in the experimentally accessible parameter regime. In Grossmann & Lohse (2001) the dimensionless parameters of the theory were fitted to 155 experimental data points by Ahlers & Xu (2001) in the regime $3\times 10^7 \le Ra \le 3 \times 10^{9}$ and $4\le Pr \le 34$ and Grossmann & Lohse (2002) used the experimental data point from Qiu & Tong (2001) and the fact that Nu(Ra,Pr) is independent of the parameter a, which relates the dimensionless kinetic boundary thickness with the square root of the wind Reynolds number, to fix the Reynolds number dependence. Meanwhile the theory is on one hand well confirmed through various new experiments and numerical simulations. On the other hand these new data points provide the basis for an updated fit in a much larger parameter space. Here we pick four well established (and sufficiently distant) Nu(Ra,Pr) data points and show that the resulting Nu(Ra,Pr) function is in agreement with almost all established experimental and numerical data up to the ultimate regime of thermal convection, whose onset also follows from the theory. One extra Re(Ra,Pr) data point is used to fix Re(Ra,Pr). As Re can depend on the definition and the aspect ratio the transformation properties of the GL equations are discussed in order to show how the GL coefficients can easily be adapted to new Reynolds number data while keeping Nu(Ra,Pr) unchanged.
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The stability of the buoyancy-driven parallel shear flow of a variable-viscosity Newtonian fluid between vertical or inclined plates maintained at different temperatures is studied theoretically. The analysis is capable of dealing with arbitrary viscosity-temperature relations. Depending on the Prandtl number, angle of inclination, and form of the viscosity-temperature variation, the flow may become unstable with respect to two-dimensional longitudinal or transverse disturbances. Outstanding questions arising in previous investigations of the stability of parallel free-convection flows of constant-viscosity fluids in inclined slots and of variable-viscosity fluids in vertical slots are discussed. We find that, in a variable-viscosity fluid, non-monotonic dependence of the critical Rayleigh number on the inclination angle can occur at significantly higher Prandtl numbers than is possible in the constant-viscosity case. Results are also presented for the stability of the free-convection flow of several glycerol-water solutions in an inclined slot.
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Rayleigh–Bénard cells are one of the simplest systems for exploring the laws of natural convection in the highly turbulent limit. However, at very high Rayleigh numbers (Ra1012) and for Prandtl numbers of the order of one, experiments fall into two categories: some evidence a steep enhancement of the heat transfer while others do not. The origin of this apparent disagreement is at present still unexplained. This puzzling situation motivated a systematic study of the triggering of the regime with an enhanced heat transfer, originally named the 'Ultimate Regime' of convection. High-accuracy heat transfer measurements have been conducted in convection cells with various aspect ratios and different specificities, such as altered boundary conditions or obstacles inserted in the flow. The two control parameters, the Rayleigh and Prandtl numbers, have been varied independently to disentangle their relative influence. Among other results, it is found that (i) most experiments reaching very high Ra are not in disagreement if small differences in Prandtl number are taken into account, (ii) the transition is not directly triggered by the large-scale circulation present in the cell and (iii) the sidewalls of the cell have a significant influence on the transition. The characteristics of this Ultimate Regime are summarized and compared with the R Kraichnan prediction for the asymptotic regime of convection.
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Recent experimental, numerical and theoretical advances in turbulent Rayleigh-Bénard convection are presented. Particular emphasis is given to the physics and structure of the thermal and velocity boundary layers which play a key role for the better understanding of the turbulent transport of heat and momentum in convection at high and very high Rayleigh numbers. We also discuss important extensions of Rayleigh-Bénard convection such as non-Oberbeck-Boussinesq effects and convection with phase changes.
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We report measurements of properties of turbulent thermal convection of a fluid with a Prandtl number $\Pra=4.38$ in a cylindrical cell with an aspect ratio $\Gamma=0.50$. The rotational symmetry was broken by a small tilt of the sample axis relative to gravity. Measurements of the heat transport (as expressed by the Nusselt number \Nu), as well as of large-scale-circulation (LSC) properties by means of temperature measurements along the sidewall, are presented. In contradistinction to similar experiments using containers of aspect ratio $\Gamma=1.00$ \cite[]{ABN06} and $\Gamma=0.50$ \cite[]{CRCC04,SXX05,RGKS10}, we see a very small increase of the heat transport for tilt angles up to about 0.1 rad. Based on measurements of properties of the LSC we explain this increase by a stabilization of the single-roll state (SRS) of the LSC and a de-stabilization of the double-roll state (DRS) (it is known from previous work that the SRS has a slightly larger heat transport than the DRS). Further, we present quantitative measurements of the strength of the LSC, its orientation, and its torsional oscillation as a function of the tilt angle.
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We present an experimental study of the convective motion of gas contained in a cubic cell heated from the side. The Rayleigh number (Ra) is varied from 4×105 to 1×1011 by changing the pressure of the gas. Using local temperature probes and shadowgraph visualization, we observe two distinct types of motion coexistent in the cell: turbulent flow and waves. A turbulent large scale circulation around the periphery of the cell, with side eddies along each plate, is observed for Ra>3×107. The turbulent fluctuations are confined to the regions near the hot and cold plates, while the bulk of the cell is stably stratified. We measure the thermal boundary layer thickness; its scaling with Ra has an exponent close to 2/7, as measured in Rayleigh-Bénard convection. In the central part of the cell, we observe internal waves, with a frequency corresponding to the Brunt-Väisälä frequency of the mean vertical temperature gradient. This system provides a laboratory environment for the study of fluctuation-generated gravitational waves in stratified gases.
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We derive the dependence of the Reynolds number Re and the Nusselt number Nu on the Rayleigh number Ra and the Prandtl number Pr in laminar vertical convection (VC), where a fluid is confined between two differently heated isothermal vertical walls. The boundary layer equations in laminar VC yield two limiting scaling regimes: Nu∼Pr1/4Ra1/4, Re∼Pr−1/2Ra1/2 for Pr≪1 and Nu∼Pr0Ra1/4, Re∼Pr−1Ra1/2 for Pr≫1. These theoretical results are in excellent agreement with direct numerical simulations for Ra from 105 to 1010 and Pr from 10−2 to 30. The transition between the regimes takes place for Pr around 10−1.
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By means of direct numerical simulations (DNS) we investigate the effect of a tilt angle ${\it\beta}$ , $0\leqslant {\it\beta}\leqslant {\rm\pi}/2$ , of a Rayleigh–Bénard convection (RBC) cell of aspect ratio 1, on the Nusselt number $\mathit{Nu}$ and Reynolds number $\mathit{Re}$ . The considered Rayleigh numbers $\mathit{Ra}$ range from $10^{6}$ to $10^{8}$ , the Prandtl numbers range from 0.1 to 100 and the total number of the studied cases is 108. We show that the $\mathit{Nu}\,({\it\beta})/\mathit{Nu}(0)$ dependence is not universal and is strongly influenced by a combination of $\mathit{Ra}$ and $\mathit{Pr}$ . Thus, with a small inclination ${\it\beta}$ of the RBC cell, the Nusselt number can decrease or increase, compared to that in the RBC case, for large and small $\mathit{Pr}$ , respectively. A slight cell tilt may not only stabilize the plane of the large-scale circulation (LSC) but can also enforce an LSC for cases when the preferred state in the perfect RBC case is not an LSC but a more complicated multiple-roll state. Close to ${\it\beta}={\rm\pi}/2$ , $\mathit{Nu}$ and $\mathit{Re}$ decrease with increasing ${\it\beta}$ in all considered cases. Generally, the $\mathit{Nu}({\it\beta})/\mathit{Nu}(0)$ dependence is a complicated, non-monotonic function of ${\it\beta}$ .
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We report the Prandtl-number (Pr) and Rayleigh-number (Ra) dependencies of the Reynolds number (Re) and mean convective heat transport, measured by the Nusselt number (Nu), in horizontal convection (HC) systems, where the heat supply and removal are provided exclusively through a lower horizontal surface of a fluid layer. For laminar HC, we find that Re∼Ra2/5Pr−4/5, Nu∼Ra1/5Pr1/10 with a transition to Re∼Ra1/2Pr−1, Nu∼Ra1/4Pr0 for large Pr. The results are based on direct numerical simulations for Ra from 3×108 to 5×1010 and Pr from 0.05 to 50 and are explained by applying the Grossmann-Lohse approach [J. Fluid Mech. 407, 27 (2000)] transferred from the case of Rayleigh-Bénard convection to the case of laminar HC.
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The equations of charged particle motion in the earth's magnetic field are analyzed. Analytical data allow us to estimate important parameters of the charge dynamics, such as the charge capture conditions, oscillation amplitude and frequency, and longitudinal drift, and also relate these parameters to the parameters of the charged particles and the height of their trajectory. The results may be used in designing space vehicles intended for investigation of the near-earth space, specifically, earth's magnetosphere, with charged particle beams.
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We report a new thermal boundary layer equation for turbulent Rayleigh–Bénard convection for Prandtl number Pr>1 that takes into account the effect of turbulent fluctuations. These fluctuations are neglected in existing equations, which are based on steady-state and laminar assumptions. Using this new equation, we derive analytically the mean temperature profiles in two limits: (a) Pr≳1 and (b) Pr≫1. These two theoretical predictions are in excellent agreement with the results of our direct numerical simulations for Pr=4.38 (water) and Pr=2547.9 (glycerol), respectively.
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Turbulent convective heat transfer in a closed cylinder with aspect ratio L = 5D (D is the diameter and L is the cylinder length) filled with liquid sodium, heated at one end face and cooled at the other, is studied experimentally for three different positions: vertical, inclined at 45 degrees to the vertical and horizontal. The Rayleigh number, which is determined by the superimposed temperature difference and the cylinder diameter, varies within the range (2-10) . 10(6). It is shown that the convective heat transfer along the cylinder is most effective in the inclined cylinder, where an intense large-scale circulation exists on a background of developed small-scale turbulence. In the horizontal cylinder, the turbulence is weak, but the large-scale circulation provides moderate heat transfer. In the vertical cylinder, the large-scale circulation is absent, the turbulent fluctuations are most active, but the heat transfer is the weakest. The dependence of the Nusselt number on the Rayleigh and the Prandtl numbers, and the dependence of the Reynolds number on the Grashof number are shown and discussed. Copyright (C) EPLA, 2015
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This paper presents results of an experimental investigation of convective heat transfer in liquids confined by two parallel plates and inclined at various angles with respect to the horizontal. The experiments covered a range of Rayleigh numbers between 5(10)4 and 7.17(10)8 , and Prandtl numbers between 0.02 and 11,560. Tests were made in rectangular and circular containers having copper plates and insulating walls. The liquids used were water, silicone oils, and mercury. The test results indicate that the heat transfer coefficients for all liquids investigated at the various angles, from horizontal to vertical, may be determined from the relationship Nu = C(Ra) 1/3(Pr) 0.074 The constant, C, is a function of the angle of inclination. It varies from C = 0.069 for the horizontal case when heated from below to C = 0.049 for the vertical case. For the test cells used, no effect on the Nusselt number had been detected for the vertical case by the change of the ratio of height of cell to distance between plates. The ratio for these tests was varied from 4.41 to 16.56.
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In this paper we report measurements of the heat flux in a slightly tilted channel (angle less than 45°), filled with water, that connects two chambers: the hot in the lower part and the cold on the top. We show that different regimes develop depending on the angle and the applied power. We put in evidence a hard turbulent regime, a soft turbulent regime, a laminar regime, and an intermittent one. In the last regime, the flow oscillates between laminar and turbulent, which locks the temperature gradient to a constant value. We characterize those regimes thanks to the measurement of the axial gradient of temperature and to the measurement of the power. We model them giving descriptions in term of Nusselt and Rayleigh numbers. The soft turbulence to hard turbulence transition is interpreted as the birth of the inertial range of developed turbulence. This transition, which appears in several systems, is particularly clear here, thanks to its consequences on heat transport properties.
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We report on a numerical study of the aspect-ratio dependency of Rayleigh-Bénard convection, using direct numerical simulations. The investigated domains have equal height and width while the aspect ratio Γ of depth per height is varied between 1/10 and 1. The Rayleigh numbers Ra for this study variate between 105 and 109, while the Prandtl number is Pr = 0.786. The main focus of the study concerns the dependency of the Nusselt number Nu and the Reynolds number Re on Ra and Γ. It turns out that due to Γ, differences to the cubic case (i.e., Γ = 1) in Nu of up to 55% and in Re of up to 97% occur, which decrease for increasing Ra. In particular for small Γ sudden drops in the Ra-scaling of Nu and Re appear for Ra ≈ 106. Further analysis reveals that these correspond to the onset of unsteady motion accompanied by changes in the global flow structure. The latter is investigated by statistical analysis of the heat flux distribution on the bottom and top plates and a decomposition of the instantaneous flow fields into two-dimensional modes. For Ra slightly above the onset of unsteady motion (i.e., Ra ≈ 106) for all considered Γ ⩽ 1/3 a four-roll structure is present, which corresponds to thermal plumes moving vertically through the domain's center. For Ra ≥ 107, also for small Γ, a single-roll structure is dominant, in agreement with two-dimensional simulations and experiments at larger Ra and Pr.
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Heat transfer between a uniformly heated flat plate and an impinging circular air jet was investigated experimentally to determine the values of the local and average Nusselt numbers, particularly for small values of Reynolds number and jet spacing. A heat transfer correlation was developed, which extends the existing database to Reynolds number and jet spacing values as low as 6000 and one jet diameter, respectively. Experimental results provided useful information of interest to potential industrial applications regarding the radius of the heat transfer area and jet spacing for maximizing the average Nusselt number.
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Different coverages of Cs on the Si(100)2×1 surface were studied by high-resolution core-level spectroscopy and angle-resolved valence-band spectroscopy. The observation of two Cs 4d components, separated by~0.4 eV, with approximately the same intensity at saturation coverage is consistent with the double-layer model for alkali-metal adsorption proposed earlier in the literature. Si 2p spectra from the saturated surface exhibit two surface shifted components that are interpreted as the up and down atom components of asymmetric Si dimers. The existence of asymmetric Si dimers on the Si(100)2×1-Cs surface is in contrast to the symmetric Si dimer structure reported earlier for the Si(100)2×1-K surface. The Cs saturated surface exhibits a metallic character as evidenced by a finite emission at the Fermi level in the valence-band spectra and by the large singularity index (Doniach-S&breve;unjic line shape) needed in order to fit the Si 2p and Cs 4d core-level spectra.
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This review summarizes results for Rayleigh-Bénard convection that have been obtained over the past decade or so. It concentrates on convection in compressed gases and gas mixtures with Prandtl numbers near one and smaller. In addition to the classical problem of a horizontal stationary fluid layer heated from below, it also briefly covers convection in such a layer with rotation about a vertical axis, with inclination, and with modulation of the vertical acceleration.
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The transitions from the onset of convection to fully developed turbulence of a Rayleigh–Be´nard flow, in a low-aspect-ratio cell and in mercury, are studied through three-dimensional numerical simulation of the Navier–Stokes equations. The calculation of the growth rate of the azimuthal energy modes permitted the accurate determination of the critical Rayleigh number for the establishment of the convective regime (Rac=3750) which is in good agreement with analytical and other numerical results. Increasing the Rayleigh number, the flow remained steady up to Ra≃2.11×104 when an oscillatory instability was observed. Further increases in the Rayleigh produced a chaotic state through the period doubling mechanism and finally the turbulent state was achieved. It is shown that for Ra&ges;Rac the mean flow consists of a large-scale convective cell which persists in the whole range of studied Rayleigh numbers (Ra⩽106). The dependence of the Nusselt number over the Rayleigh number is also analyzed and, for Ra&ges;3.75×104, when the turbulent state is reached, a power law in quantitative agreement with previous results at higher Ra is observed.
Article
Nonlinear interaction between transverse disturbances and longitudinal rolls has been investigated for flow in an inclined slot with a heated lower wall when both modes of instability occur at nearly the same value of the control parameter. This condition is shown to be possible for a fluid with Prandtl number greater than 0.263 897, For slightly supercritical values of the Rayleigh number (R) when the critical Rayleigh number for longitudinal rolls RLC is somewhat less than that for transverse stationary rolls, RSC, and for transverse travelling waves, RTC, longitudinal rolls occur first and then remain stable as R is increased beyond RSC or RTC; no mixed mode state occurs. In contrast, if RSc or RTc is slightly below RLC, pure transverse modes exist for only a relatively small range of R beyond RSC or RTC. Thereafter, a three-dimensional mixed mode state occurs well before RLC is reached, i.e. three-dimensionality sets in on a subcritical basis. As R approaches RLC the contribution of the transverse mode decreases continuously until a pure longitudinal roll state emerges for R slightly greater than RLC. Mixed mode convection is also investigated for a special choice of parameters when three modes, namely transverse stationary rolls, transverse travelling waves and longitudinal rolls, become unstable simultaneously. Longitudinal rolls again emerge as the stable supercritical state.