⇦Chinese Physics Letters, 2017, Vol. 34, No. 10, Article code 104401⇨ Thermal Convection in a Tilted Rectangular Cell with Aspect Ratio 0.5 * Qi Wang(王启)1, Bo-Lun Xu(徐博伦)1, Shu-Ning Xia(夏树宁)2, Zhen-Hua Wan(万振华)1**, De-Jun Sun(孙德军)1** Affiliations 1Department of Modern Mechanics, University of Science and Technology of China, Hefei 230027 2Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072 Received 28 June 2017 ^*Supported by the National Natural Science Foundation of China under Grant Nos 11572314, 11232011 and 11621202, and the Fundamental Research Funds for the Central Universities.
^**Corresponding author. Email: wanzh@ustc.edu.cn; dsun@ustc.edu.cn Citation Text: Wang Q, Xu B L, Xia S N, Wan Z H and Sun D J 2017 Chin. Phys. Lett. 34 104401 Abstract Thermal convection in a three-dimensional tilted rectangular cell with aspect ratio 0.5 is studied using direct numerical simulations within both Oberbeck–Boussinesq (OB) approximation and strong non-Oberbeck–Boussinesq (NOB) effects. The considered Rayleigh numbers $Ra$ range from $10^5$ to $10^7$, the working fluid is air at 300 K, and the corresponding Prandtl number $Pr$ is 0.71. Within the OB approximation, it is found that there exist multiple states for $Ra=10^5$ and hysteresis for $Ra=10^6$. For a relatively small tilt angle $\beta$, the large-scale circulation can either orient along one of the vertical diagonal planes (denoted by $M_{\rm d}$ mode) or orient parallel to the front wall (denoted by $M_{\rm p}$ mode). Which of the two modes transports heat more efficiently is not definitive, and it depends on the Rayleigh number $Ra$. For $Ra=10^7$ and $\beta=0^\circ$, the time-averaged flow field contains four rolls in the upper half and lower half of the cell, respectively, $M_{\rm d}$ and $M_{\rm p}$ modes only developing in tilted cells. By investigating NOB effects in tilted convection for fixed $Ra=10^6$, it is found that the NOB effects on the Nusselt number $Nu$, the Reynolds number $Re$ and the central temperature $T_{\rm c}$ for different $\beta$ ranges are different. NOB effects can either increase or decrease $Nu$, $Re$ and $T_{\rm c}$ when $\beta$ is varied. DOI:10.1088/0256-307X/34/10/104401 PACS:44.25.+f, 47.20.Bp, 47.27.te, 47.55.pb © 2017 Chinese Physics Society Article Text Thermal convection is ubiquitous in nature and is important in many industrial process. An important model to study thermal convection is the Rayleigh–Bénard convection (RBC) where a fluid layer in a box is heated from below and cooled from above.[1-4] In an industrial process, the heat transfer in heat exchangers can be enhanced by tilting the cell. The relationship between the tilt angle and the Nusselt number $Nu$ has a direct interest for reducing the loss of energy in honeycomb solar collector plates. In recent years, the effect of cell tilting on RBC has been studied numerically[5] and experimentally.[6,7] These results all are performed within the Oberbeck–Boussinesq (OB) approximation, and they are all restricted to a cell of unit aspect ratio. Another typical aspect ratio in the studies of RBC is 0.5.[8,9] However, there are less works devoted to tilted convection in a cell with aspect ratio 0.5. In this work, we try to fill this gap by investigating thermal convection in a tilted cell with aspect ratio 0.5 within the OB approximation. The OB approximation, being more mathematical than physical, assumes the thermal properties of the working fluid to be constant and the density is constant everywhere except within the buoyancy term. Obviously, this approximation is valid only for small temperature differences. In many practical applications, such as thermal insulation systems in nuclear reactors, the temperature differences can be as large as several hundreds of Kelvin degrees. Thus non-Oberbeck–Boussinesq (NOB) effects should be taken into consideration and are worthy of in-depth study. The influence of NOB effects on RBC has been studied extensively in recent years.[10-13] However, there are less works devoted to NOB effects in tilted convection, as tilted convection in industrial process may be subjected to large temperature differences. In this work, we also study NOB effects in tilted convection. The problem is described in Fig. 1. The tilt angle $\beta$ is such that $\beta=0$ corresponds to RBC. When the cell rotates counterclockwise about $x$ axis, the tilt angle increases. The considered Rayleigh numbers $Ra$ range from $10^5$ to $10^7$. The working fluid is air with reference temperature 300 K, and the corresponding $Pr$ is 0.71. Both the length-to-height aspect ratio and the width-to-height aspect ratio are fixed at 0.5. Two important response parameters of the system are the Nusselt number $Nu$ and the Reynolds number $Re$ which are defined by $$ Nu=\frac{Q}{k\Delta T/H},~~Re=\frac{\sqrt{\langle {\boldsymbol u}\cdot{\boldsymbol u}\rangle}H}{\nu},~~ \tag {1} $$ where $Q$ is the heat flux across any horizontal plane, $k$ is the thermal conductivity of the fluid, $\Delta T=T_{\rm H}-T_{\rm C}$ is the temperature difference between the hot and cold plate, $H$ is the height of the cell, $\nu$ is the kinematic viscosity of the fluid, and ${\boldsymbol u}$ is the velocity vector. The low-Mach-number equations[14] are adopted to study NOB effects. The governing equations are solved by our in-house code $lMn3d$. All spatial terms are discretized using the second-order central difference scheme. Time integration is accomplished using an Adams–Bashforth scheme for the nonlinear terms and a Crank–Nicolson scheme for the viscous and diffusion terms. The two-dimensional version of the code $lMn2d$ which is directly developed from $lMn3d$ has already been validated and described in our previous work.[13] Computation grids of $48\times48\times96$, $64\times64\times128$, $96\times96\times192$ are used for $Ra=10^5$, $Ra=10^6$ and $Ra=10^7$, respectively. The according time steps for these three grids are 0.01, 0.008 and 0.003.

Fig. 1. Sketch of a tilted rectangular cell with length $L$, width $W$ and height $H$.

Fig. 2. Time-averaged velocity streamlines for $Ra=10^5$ ((a), (d)), $Ra=10^6$ ((b), (e)), and $Ra=10^7$ ((c), (f)). Here (a) $M_{\rm d}$ mode for $\beta=0^\circ$, (b) $M_{\rm d}$ mode for $\beta=3^\circ$, (c) $\beta=0^\circ$, (d) $M_{\rm p}$ mode for $\beta=0^\circ$, (e) $M_{\rm p}$ mode for $\beta=3^\circ$, and (f) $M_{\rm d}$ mode for $\beta=5^\circ$. The color coding denotes the magnitude of time-averaged vertical velocity normalized by free-fall velocity $w/U_{\rm f}$.

We first discuss the influence of cell tilting on the flow organization. Figure 2 shows time-averaged velocity streamlines for different combinations of $Ra$ and $\beta$, and the color contour denotes the magnitude of the time-averaged vertical velocity normalized by free-fall velocity $w/U_{\rm f}$. Note that the free-fall velocity is defined by $U_{\rm f}=\sqrt{g\alpha \Delta TH}$, where $\alpha$ is the thermal expansion coefficient, and $g$ is acceleration due to gravity. For $Ra=10^5$, we observe two different flow states under different initial conditions. One contains a large-scale-circulation (LSC) oriented along one of the diagonal planes which is denoted by $M_{\rm d}$ mode as shown in Fig. 2(a). This kind of flow structure also prevails in a cubic cell even at much larger $Ra$. The other one contains an LSC oriented parallel to the front wall denoted by $M_{\rm p}$ mode as shown in Fig. 2(d). For small $\beta$, both the modes can be stable, while only the $M_{\rm p}$ mode exists for $\beta>8^\circ$. The $Nu$ value of the $M_{\rm d}$ mode is larger than that of the $M_{\rm p}$ mode, which may be related to larger flow intensity denoted by the Reynolds number $Re$ as shown in the following. For $Ra=10^6$, the two modes can also be stable at small $\beta$ as shown for $\beta=3^\circ$ in Figs. 2(b) and 2(e). However, different from $Ra=10^5$, $Nu$ of the $M_{\rm p}$ mode is larger than that of the $M_{\rm d}$ mode for $Ra=10^6$. Therefore, which of the two modes transports heat more efficiently is not definitive, and it depends on $Ra$. For $Ra=10^7$, the flow organization is qualitatively different for $\beta=0^\circ$ as shown in Fig. 2(c). Four rolls develop in the upper half and lower half of the cell, respectively, and there is no apparent LSC. The $M_{\rm d}$ mode only exists when the cell is tilted as shown in Fig. 2(f) for $\beta=5^\circ$. For even larger $\beta$, the flow will organize in the $M_{\rm p}$ mode.

Fig. 3. Absolute and normalized Nusselt and Reynolds numbers in tilted convection for $Ra=10^5$ (squares), $Ra=10^6$ (circles) and $Ra=10^7$ (diamonds), as a function of tilt angle $\beta$. (a) Absolute Nusselt numbers $Nu$. (b) Normalized Nusselt numbers $Nu(\beta)/Nu(0)$. (c) Absolute Reynolds numbers $Re$. (d) Normalized Reynolds numbers $Re(\beta)/Re(0)$. The corresponding values for RBC ($\beta=0^\circ$) are used for normalization. For $Ra=10^5$, $Nu$ and $Re$ of $M_{\rm p}$ mode for RBC are used for normalization, while for $Ra=10^6$, we use $Nu$ and $Re$ of $M_{\rm d}$ mode.

We then investigate the influence of cell tilting on $Nu$ and $Re$. Figures 3(a) and 3(b) show the absolute and normalized $Nu$ as a function of $\beta$ for different $Ra$. It should be mentioned that for $Ra=10^5$, $Nu$ of the $M_{\rm p}$ mode for RBC is used for normalization, while for $Ra=10^6$ only the $M_{\rm d}$ mode exists for RBC, and we use the corresponding $Nu$ for normalization. Multiple states are found for $Ra=10^5$ and hysteresis is found for $Ra=10^6$ for small $\beta$ when $\beta$ increases or decreases. The hysteresis of $Ra=10^6$ is more clearly shown in Figs. 5(a) and 5(b). From Fig. 3(b), it is seen at first glance that $Nu(\beta)/Nu(0)$ is strongly influenced by $Ra$ while no apparent regularity-dependence on $Ra$ can be found. For $Ra=10^5$, $Nu$ of $M_{\rm p}$ mode first increases as $\beta$ increases until it reaches the maximum value at about $50^\circ$, then it decreases with increasing $\beta$. However, for the $M_{\rm d}$ mode, $Nu$ decreases monotonically with increasing $\beta$. For $Ra=10^6$, the behavior of $Nu(\beta)/Nu(0)$ is much more complicated, and there exist at least three maxima and two minima. For $Ra=10^7$, $Nu$ experiences an increase when the cell is minor tilted, which is different from that in a cylindrical cell with aspect ratio 0.5 where $Nu$ decreases in a tilted cell at higher $Ra$.[15] This demonstrates that the Rayleigh number plays an important role in the influence of tilting on $Nu$, which is also found in an aspect ratio 1 cell.[5,6] The value of $Nu$ has two maxima and one minimum, and the first and second maxima locate near $\beta=30^\circ$ and $\beta=80^\circ$, respectively. Compared with the Nusselt number $Nu$, the Reynolds number $Re$ shows a much more regular dependence on $\beta$ as shown in Figs. 3(c) and 3(d). The normalization in Fig. 3(d) is the same as that of $Nu$. Normally, $Re$ first increases with $\beta$, after reaching its maximum, it decreases monotonously with increasing $\beta$, which is similar to the cases in a cylindrical cell of unity aspect ratio.[3] It is also seen that $Re$ of the $M_{\rm d}$ mode is higher than that of $M_{\rm p}$ mode for $Ra=10^5$, which is in contrast to that of $Ra=10^6$. The same conclusion is also drawn for $Nu$ as shown above, which implies that the higher $Nu$ may be related to faster fluid motion.

Fig. 4. Time evolution of the Nusselt number $Nu$ for $Ra=10^6$ with NOB effects where temperature difference $\Delta T=360$ K for $\beta=1^\circ$ (a) and $\beta=2^\circ$ (b). Instantaneous streamlines in (a) at different times: $t=960t_{\rm f}$ (c), $t=5760t_{\rm f}$ (d), and $t=10560t_{\rm f}$ (e).

The above results are performed within the strict OB approximation. We now investigate strong NOB effects on the flow organization, the Nusselt number $Nu$, the Reynolds number $Re$, and the central temperature $T_{\rm c}$ in tilted convection for fixed $Ra=10^6$. The temperature difference $\Delta T$ is $360$ K for the considered NOB effects, corresponding to $T_{\rm H}=480$ K and $T_{\rm C}=120$ K. For $\beta=0^\circ$ and $\beta=1^\circ$, there exists flow mode transition between single-roll mode (SRM) and double-roll mode (DRM) for the NOB cases. Figure 4(a) shows the time evolution of $Nu$ for $\beta=1^\circ$ and Figs. 4(c)–4(e) show the instantaneous streamlines at the time $t=960t_{\rm f}$, $t=5760t_{\rm f}$ and $t=10560t_{\rm f}$, respectively. It is seen that the flow organizes in SRM at time $t=960t_{\rm f}$ and $t=5760t_{\rm f}$ with relatively larger $Nu$, while the orientation of the LSC is different. At time $t=10560t_{\rm f}$, the flow organizes in DRM with smaller $Nu$. It is already experimentally found that $Nu$ of SRM is larger than that of DRM at higher $Ra$ within the OB approximation.[16] The flow mode transition is also observed in experiment within the OB approximation.[17] Thus our numerical results confirm the phenomenon and show that it also exists when strong NOB effects exist. As $\beta$ increases to $2^\circ$, the flow organizes in stable $M_{\rm d}$ mode similar to Fig. 4(d), and the reorientation of LSC and flow mode transition is suppressed. Figure 4(b) shows the time evolution of $Nu$ for $\beta=2^\circ$, and it is evident that $Nu$ of $2^\circ$ is larger than that of $1^\circ$.

Fig. 5. (a) The Nusselt number $Nu$ as a function of tilt angle $\beta$ within the OB approximation (green circles) and with NOB effects (red squares). (b) The Reynolds number $Re$ as a function of tilt angle $\beta$ within the OB approximation (green circles) and with NOB effects (red squares). (c) Central temperature deviation $T_{\rm c}-T_{\rm m}$ as a function of tilt angle for the NOB cases.

Figures 5(a) and 5(b) show $Nu$ and $Re$ as a function of $\beta$ for the OB and NOB cases, respectively. It is evident that the influence of NOB effects is different for different $\beta$ ranges. For $\beta=0^\circ$ and $1^\circ$, flow mode transition exists for the NOB cases and both $Nu$ and $Re$ are decreased compared with the OB cases. For $2^\circ \le\beta\le 23^\circ$, $Nu$ and $Re$ of both the $M_{\rm d}$ and $M_{\rm p}$ modes are increased when considering NOB effects. However, for larger $\beta>23^\circ$, $Nu$ and $Re$ decrease compared with the OB cases, and the influence of NOB effects on $Re$ at $\beta\ge60^\circ$ is very small. Quantitatively, the maximum reduction of $Nu$ is 11% at $\beta=0^\circ$, which is due to changed flow modes, and the maximum enhancement of $Nu$ is 7% at $\beta=5^\circ$ in the $M_{\rm p}$ mode. For $Re$, the maximum reduction is 4% at $\beta=0^\circ$ and $30^\circ$ while the maximum enhancement is 7% at $\beta=3^\circ$ in the $M_{\rm d}$ mode. The present finding demonstrates complicated influence of NOB effects on tilted convection at relatively low $Ra=10^6$. For higher $Ra$ where the flow is in the fully turbulent state, we expect that this influence will be small as already found in turbulent RBC.[6] The influence of NOB effects on tilted convection at higher $Ra$ will be the goal of our future research. A major influence of NOB effects on RBC is the breaking of top-down symmetry, and this will cause the deviation of central temperature $T_{\rm c}$ from the arithmetic mean temperature of hot and cold plates $T_{\rm m}=(T_{\rm H}+T_{\rm C})/2$. We show in Fig. 5(c) this temperature deviation $T_{\rm c}-T_{\rm m}$ as a function of $\beta$. It is seen that at small $\beta$ the central temperature decreases compared with $T_{\rm m}$, while $T_{\rm c}$ at larger $\beta$ is normally increased except for $\beta=40^\circ$ where $T_{\rm c}$ is slightly decreased. This demonstrates that for different $\beta$, NOB effects have a varying influence on $T_{\rm c}$. In summary, the influence of cell tilting on thermal convection in a three-dimensional rectangular cell with aspect ratio 0.5 has been numerically studied within both the OB approximation and the strong NOB effects caused by large temperature differences. It is found that there exist multiple states for $Ra=10^5$ and hysteresis for $Ra=10^6$. Both $M_{\rm d}$ and $M_{\rm p}$ modes can be stable in a specific $\beta$ range. It is not definitive which of the two modes transports heat more efficiently, and it depends on $Ra$. For $Ra=10^7$ and $\beta=0^\circ$, the time-averaged flow field contains four rolls in the upper half and lower half of the cell, respectively. The $M_{\rm d}$ and $M_{\rm p}$ modes only develop in tilted cells. By investigating the influence of NOB effects on tilted convection for fixed $Ra=10^6$, it is found that there is flow mode transition between SRM and DRM for $\beta=0^\circ$ and $1^\circ$. The SRM is shown to transport heat more efficiently. In different $\beta$ ranges, the influence of NOB effects on $Nu$, $Re$ and $T_{\rm c}$ is different for considered $Ra=10^6$. NOB effects can either increase or decrease $Nu$, $Re$ and $T_{\rm c}$ which depends on $\beta$. References Heat transfer and large scale dynamics in turbulent Rayleigh-BÃ©nard convectionSmall-Scale Properties of Turbulent Rayleigh-BÃ©nard ConvectionCurrent trends and future directions in turbulent thermal convectionPhenomenology of buoyancy-driven turbulence: recent resultsThermal convection in inclined cylindrical containersThe effect of cell tilting on turbulent thermal convection in a rectangular cellEvolution and statistics of thermal plumes in tilted turbulent convectionRadial boundary layer structure and Nusselt number in RayleighâBÃ©nard convectionThermal boundary layer profiles in turbulent Rayleigh-BÃ©nard convection in a cylindrical sampleNon-OberbeckâBoussinesq effects in strongly turbulent RayleighâBÃ©nard convectionNon-Oberbeck-Boussinesq Effects in Gaseous Rayleigh-BÃ©nard ConvectionOn non-OberbeckâBoussinesq effects in three-dimensional RayleighâBÃ©nard convection in glycerolFlow reversals in RayleighâBÃ©nard convection with non-OberbeckâBoussinesq effectsAzimuthal Symmetry, Flow Dynamics, and Heat Transport in Turbulent Thermal Convection in a Cylinder with an Aspect Ratio of 0.5How heat transfer efficiencies in turbulent thermal convection depend on internal flow modesFlow mode transitions in turbulent thermal convection

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