Session H11: Rayleigh-Benard Convection II

Effects of Pr on Optimal Heat Transport in Rayleigh-B\'{e}nard Convection

Steady flows that optimize heat transport are obtained for two-dimensional Rayleigh-B\'{e}nard convection with no-slip horizontal walls for a variety of Prandtl numbers $Pr$ and Rayleigh number up to $Ra\sim 10^{9}$. The presence of two local maxima of $Nu$ with different horizontal wavenumbers at the same $Ra$ leads to the emergence of two different flow structures as candidates for optimizing the heat transport where the Nusselt number $Nu$ is a non-dimensional measure of the vertical heat transport. For $Pr \leq 7$, optimal transport is achieved at the smaller maximal wavenumber whereas for $Pr > 7$ at high-enough $Ra$ the optimal structure occurs at the larger maximal wavenumber. Three regions are observed in the optimal mean temperature profiles, $\overline{T}\left(y\right)$: 1.) $d\overline{T}/dy < 0$ in the boundary layers, 2.) $d\overline{T}/dy > 0$ ($Pr\leq 7$) or $d\overline{T}/dy < 0$ ($Pr>7$) in the central region, and 3.) $d\overline{T}/dy > 0$ between the boundary layers and central region. We also search for a signature of these optimal structures in a fully-developed turbulent flow by employing modal decompositions such as the proper orthogonal decomposition and the Koopman mode decomposition.   

Optimal heat transport

The transport of heat by buoyancy driven flows, i.e., thermal convection plays a central role in many natural phenomena and an understanding of how to control its mechanisms is relevant to many engineering applications. In this talk we will consider a variational formulation of optimal heat transport in simple geometries. Numerical results, limits on heat transport, and a comparison to Rayleigh-B\'enard convection will be presented.   

A theory for optimal heat transfer in a partitioned convection cell

We report a theory explaining recent observation of significant enhancement of heat transfer in a partitioned Rayleigh-B\'{e}nard convection (RBC), where vertical adiabatic boards are inserted into the enclosure with narrow channel left open between partition boards and the cooling/heating plates. An enhancement of heat transfer of up to 2.7 times is observed compared to normal RBC cell without partitions. It is found that laminar wall jet is formed in the narrow horizontal channel, which makes the thermal boundary layer thinner. Two asymptotic trends, a channel flow and a boundary layer, describe the motions of the jets in the horizontal channel, and the competition between them gives rise to an optimized state for the global heat transfer, with an optimal width of the sub-cell W/H $=$0.038-0.083 for $\Gamma =$1, and an optimal spacing of the horizontal channel b/H$=$0.011 for $\Gamma =$5. The former (channel) yields a heat flux linearly proportional to b for small b, whereas the latter (boundary layer) follows -2/3-law for large b. We suggest that the partitioned RBC provides a vehicle for heat enhancement with a wide range of industrial applications.   

Tailoring boundary geometry to optimize heat transport in turbulent convection

Turbulent Rayleigh-B\'enard convection between planar horizontal boundaries is a classical example of the challenge posed by multiple interacting scales in fluid dynamics. The detailed description by which hot fluid rises and cold fluid descends focuses on the nature of the interaction between the boundary layers and the turbulent interior of the flow. Here, by tailoring the geometry of the upper boundary we manipulate this boundary layer -- interior flow interaction, and study the turbulent transport of heat in two-dimensional Rayleigh-B\'enard convection with numerical simulations using the Lattice Boltzmann method. By fixing the roughness amplitude of the upper boundary and varying the wavelength $\lambda$, we find that the exponent $\beta$ in the Nusselt-Rayleigh scaling relation, $Nu-1 \propto Ra^\beta$, is maximized at $\lambda \equiv \lambda_{\mathrm{max}} \approx (2 \pi)^{-1}$, but decays to the planar value in both the large ($\lambda \gg \lambda_{\mathrm{max}}$) and small ($\lambda \ll \lambda_{\mathrm{max}}$) wavelength limits. The changes in the exponent originate in the nature of the coupling between the boundary layer and the interior flow. We present a simple scaling argument embodying this coupling, which describes the maximal convective heat flux.   

The effects of Prandtl number on flow over a vertical heated cylinder

Flow over a two dimensional heated cylinder is analyzed numerically using a hybrid finite element-finite volume method. We assume the flow direction to be opposite to the direction of gravity. It is fundamental in fluid dynamics that the von Karman vortex street appears in the wake of the cylinder above the Reynolds number of approximately 47. On heating the cylinder surface, the Strouhal number (St), which is the non dimensional representation of the vortex shedding frequency, increases. The gradual increase in St is followed by a sudden drop at a particular value of Richardson number (Ri), defined as the relative dominance of the buoyancy force to the inertia force reported as a sudden breakdown of the Karman vortex. Our simulations show that upon further increase in Ri, recirculation bubble reappears. The present numerical work discusses the physical reasons behind this phenomenon and the effects of Prandtl number (defined as the ratio of viscous diffusion to the moment um diffusion) on Richardson number at which break down occurs.   

Heat-flux enhancement by vapour-bubble nucleation in Rayleigh-B\'enard turbulence

We report on turbulent convective heat transport enhancement and local temperature modifications in the bulk due to vapour-bubble nucleation at the bottom plate of a Rayleigh-B\'enard cylindrical sample (aspect ratio 1.0, diameter of 8.8 cm) filled with liquid. Etched microcavities acted as nucleation sites. Only the central area of the bottom plate (diameter of 2.5 cm) with an array of microcavities was heated. The Nusselt-number $Nu$ was investigated as a function of the bottom plate superheat $T_h$ by varying the temperature of the bottom plate $T_b$ and keeping a fixed difference between $T_b$ and the top plate temperature $T_t$, $T_b - T_t \simeq 16$ K. Nusselt-number of both 1- and 2-phase flow for the same $T_h$ value was obtained; 2-phase-$Nu$ was increasingly enhanced relative to the 1-phase $Nu$ for increasing $T_h$. Varying the cavity density between 69 and 0.3 per mm$^2$ had only a small effect on the global $Nu$ enhancement; however $Nu$ per active site decreased as the cavity density increased. $Nu$ of an isolated nucleating site was found to be limited by the rate at which it could host a phase change.   

Azimuthal Decomposition of Wide Aspect-Ratio, Turbulent Rayleigh-Benard Convection in a Cylindrical Cell

Turbulent Rayleigh-Benard convection (RBC) is considered an ideal problem for studying the thermal convection that occurs in nature, and it is typically studied in finite cylindrical or rectangular domains. Cylindrical domains have an advantage because they prevent geometric effects from defining preferential horizontal directions in the flow. This allows the large scale patterns to drift azimuthally and mimic the dynamics of convection in applications where geometric constraints are minimal. The large scale pattern for RBC in small aspect-ratio ($\Gamma$) domains is a single roll-cell that spans the entire domain, and the azimuthal drift for this pattern can be fairly energetic. As $\Gamma$ is increased the single-roll cell breaks into a multi-roll cell pattern, and the time scale for azimuthal motion increases substantially. In this presentation we investigate azimuthal properties of the velocity and temperature fields in a 6.3 $\Gamma$ cell with a Rayleigh number of $1 \times 10^8$ and a Prandtl number of 6.7. Statistical independence in the azimuthal direction is investigated for each field, and a detailed decomposition of the multi-roll cell pattern is presented. These analysis' are performed through temporal and spatial averaging techniques and Fourier decomposition.   

Variation of effective roll number on MHD Rayleigh-Benard convection confined in a small-aspect ratio box

MHD Rayleigh-Benard convection was studied experimentally using a box filled with liquid metal with five in aspect ratio and square horizontal cross section. Applying horizontal magnetic field organizes the convection motion into quasi-two dimensional rolls arranged parallel to the magnetic field. The number of rolls has tendency, decreases with increasing Rayleigh number $Ra$ and increases with increasing Chandrasekhar number $Q$. To fit the box with relatively smaller aspect ratio, the convection rolls take regime transition accompanying variation of the roll number against variations of $Ra$ and $Q$. We explored convection regimes in a ranges, $2 \times 10^3 < Q < 10^4$ and $5 \times 10^3 < Ra < 3 \times 10^5$ using ultrasonic velocity profiling that can capture time variations of instantaneous velocity profile. In a range $Ra/Q \sim 10$, we found periodic flow reversals in which five rolls periodically change the direction of their circulation with gradual skew of rolls. We performed POD analysis on the spatio-temporal velocity distribution obtained by UVP and indicated that that the periodic flow reversals consist of periodic emergence of 4-rolls mode in dominant 5-rolls mode. POD analysis also provided evaluation of effective number of rolls as a more objective approach.   

Roughness-triggered turbulent boundary layers in Rayleigh-Bénard convection

We present an analysis of the velocity fields in a Rayleigh-Bénard cell with a rough bottom plate. Beyond a critical Rayleigh number, the cell undergoes a transition towards a regime of enhanced heat transfer. The threshold is reached when the boundary layer thickness is smaller than the roughness size. We have obtained velocity fields using PIV near the obstacles, as well as the local heat-flux on the bottom plate. This has allowed us to test and improve our previous interpretation of the roughness-induced heat transfer enhancement mechanisms as a roughness-trigerred transition to turbulent boundary layers, see Salort, \emph{et al.}, Phys. Fluids \textbf{26}, 015112 (2014). The velocity profiles on the top of the obstacle are indeed quite different above and below the transition. Below the transition, the profile is fairly compatible with profiles obtained in the smooth case. Above the transition, for $z^+ > 30$, the velocity profile is closer to the logarithmic profile that one would expect in the case of a turbulent boundary layer, and the slope is close to the classical value of 2.40. The offset however is slightly lower than the classical 5.84, as can be expected on a rough surface.   

Inertial effects on heat transfer in superhydrophobic microchannels

This work numerically studies the effects of inertia on thermal transport in superhydrophbic microchannels. An infinite parallel plate channel comprised of structured superhydrophbic walls is considered. The structure of the superhydrophobic surfaces consists of square pillars organized in a square array aligned with the flow direction. Laminar, fully developed flow is explored. The flow is assumed to be non-wetting and have an idealized flat meniscus. A shear-free, adiabatic boundary condition is used at the liquid/gas interface, while a no-slip, constant heat flux condition is used at the liquid/solid interface. A wide range of Peclet numbers, relative channel spacing distances, and relative pillar sizes are considered. Results are presented in terms of Poiseuille number, Nusselt number, hydrodynamic slip length, and temperature jump length. Interestingly, the thermal transport is varied only slightly by inertial effects for a wide range of parameters explored and compares well with other analytical and numerical work that assumed Stokes flow. It is only for very small relative channel spacing and large Peclet number that inertial effects exert significant influence. Overall, the heat transfer is reduced for the superhydrophbic channels in comparison to classic smooth walled channels.