Effect of Prandtl number on heat transport enhancement in Rayleigh-B\'enard convection under geometrical confinement

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.Comment: 25 pages, 11 figures, and 1 table in main tex

A bouncing oil droplet in a stratified liquid and its sudden death

Droplets can self-propel when immersed in another liquid in which a concentration gradient is present. Here we report the experimental and numerical study of a self-propelling oil droplet in a vertically stratified ethanol/water mixture: At first, the droplet sinks slowly due to gravity, but then, before having reached its density matched position, jumps up suddenly. More remarkably, the droplet bounces repeatedly with an ever increasing jumping distance, until all of a sudden it stops after about 30 min. We identify the Marangoni stress at the droplet/liquid interface as responsible for the jumping: its strength grows exponentially because it pulls down ethanol-rich liquid, which in turn increases its strength even more. The jumping process can repeat because gravity restores the system. Finally, the sudden death of the jumping droplet is also explained. Our findings have demonstrated a type of prominent droplet bouncing inside a continuous medium with no wall or sharp interface.Comment: 6 pages, 4 figure

From zonal flow to convection rolls in Rayleigh-B\'enard convection with free-slip plates

Rayleigh-B\'enard (RB) convection with free-slip plates and horizontally periodic boundary conditions is investigated using direct numerical simulations. Two configurations are considered, one is two-dimension (2D) RB convection and the other one three-dimension (3D) RB convection with a rotating axis parallel to the plate. We explore the parameter range of Rayleigh numbers Ra from $10^7 to$10^9$and Prandtl numbers$Pr$from$1$to$100. We show that zonal flow, which was observed, for example, by Goluskin \emph{et al}. \emph{J. Fluid. Mech.} 759, 360-385 (2014) for \Gamma=2$, is only stable when$\Gamma$is smaller than a critical value, which depends on$Ra$and$Pr$. With increasing$\Gamma$, we find a second regime in which both zonal flow and different convection roll states can be statistically stable. For even larger$\Gamma$, in a third regime, only convection roll states are statistically stable and zonal flow is not sustained. For the 3D simulations, we fix$Ra=10^7$and$Pr=0.71$, and compare the flow for$\Gamma=8$and$\Gamma = 16$. We demonstrate that with increasing aspect ratio$\Gamma$, zonal flow, which was observed for small$\Gamma=2\pi by von Hardenberg \emph{et al}. \emph{Phys. Rev. Lett.} 15, 134501 (2015), completely disappears for \Gamma=16$. For such large$\Gamma$only convection roll states are statistically stable. In between, here for medium aspect ratio$\Gamma = 8$, the convection roll state and the zonal flow state are both statistically stable. What state is taken depends on the initial conditions, similarly as we found for the 2D case.Comment: 26 pages, 12 figure

Two-layer Thermally Driven Turbulence: Mechanisms for Interface Breakup

It is commonly accepted that the breakup criteria of drops or bubbles in turbulence is governed by surface tension and inertia. However, also {\it{buoyancy}} can play an important role at breakup. In order to better understand this role, here we numerically study Rayleigh-B\'enard convection for two immiscible fluid layers, in order to identify the effects of buoyancy on interface breakup. We explore the parameter space spanned by the Weber number $5\leq We \leq 5000$ (the ratio of inertia to surface tension) and the density ratio between the two fluids $0.001 \leq \Lambda \leq 1$, at fixed Rayleigh number $Ra=10^8$ and Prandtl number $Pr=1$. At low $We$, the interface undulates due to plumes. When $We$ is larger than a critical value, the interface eventually breaks up. Depending on $\Lambda$, two breakup types are observed: The first type occurs at small $\Lambda \ll 1$ (e.g. air-water systems) when local filament thicknesses exceed the Hinze length scale. The second, strikingly different, type occurs at large $\Lambda$ with roughly $0.5 < \Lambda \le 1$ (e.g. oil-water systems): The layers undergo a periodic overturning caused by buoyancy overwhelming surface tension. For both types the breakup criteria can be derived from force balance arguments and show good agreement with the numerical results.Comment: 13 pages, 7 figure

Buoyancy-driven attraction of active droplets

For dissolving active oil droplets in an ambient liquid, it is generally assumed that the Marangoni effect results in repulsive interactions, while the buoyancy effects caused by the density difference between the droplets, diffusing product and the ambient fluid are usually neglected. However, it has been observed in recent experiments that active droplets can form clusters due to buoyancy-driven convection (KrÜger et al., Eur. Phys. J. E, vol. 39, 2016, pp. 1-9). In this study we numerically analyse the buoyancy effect, in addition to the propulsion caused by Marangoni flow (with its strength characterized by the Péclet number). The buoyancy effects have their origin in (i) the density difference between the droplet and the ambient liquid, which is characterized by the Galileo number; and (ii) the density difference between the diffusing product (i.e. filled micelles) and the ambient liquid, which can be quantified by a solutal Rayleigh number. We analyse how the attracting and repulsing behaviour of neighbouring droplets depends on the control parameters, and. We find that while the Marangoni effect leads to the well-known repulsion between the interacting droplets, the buoyancy effect of the reaction product leads to buoyancy-driven attraction. At sufficiently large, even collisions between the droplets can take place. Our study on the effect of further shows that with increasing, the collision becomes delayed. Moreover, we derive that the attracting velocity of the droplets, which is characterized by a Reynolds number, is proportional to, where is the distance between the neighbouring droplets normalized by the droplet radius. Finally, we numerically obtain the repulsive velocity of the droplets, characterized by a Reynolds number, which is proportional to. The balance of attractive and repulsive effect leads to, which agrees well with the transition curve between the regimes with and without collision.</p

Flow structure transition in thermal vibrational convection

This study investigates the effect of vibration on the flow structure transitions in thermal vibrational convection (TVC) systems, which occur when a fluid layer with a temperature gradient is excited by vibration. Direct numerical simulations of TVC in a two-dimensional enclosed square box were performed over a range of dimensionless vibration amplitudes $0.001 \le a \le 0.3$ and angular frequencies $10^{2} \le \omega \le 10^{7}$, with a fixed Prandtl number of 4.38. The flow visualisation shows the transition behaviour of flow structure upon the varying frequency, characterising three distinct regimes, which are the periodic-circulation regime, columnar regime and columnar-broken regime. Different statistical properties are distinguished from the temperature and velocity fluctuations at the boundary layer and mid-height. Upon transition into the columnar regime, columnar thermal coherent structures are formed, in contrast to the periodic oscillating circulation. These columns are contributed by merging of thermal plumes near the boundary layer, and the resultant thermal updrafts remain at almost fixed lateral position, leading to a decrease in fluctuations. We further find that the critical point of this transition can be described nicely by the vibrational Rayleigh number $Ra_\mathrm{vib}$. As the frequency continues to increase, entering the so-called columnar-broken regime, the columnar structures are broken, and eventually the flow state becomes a large-scale circulation, characterised by a sudden increase in fluctuations. Finally, a phase diagram is constructed to summarise the flow structure transition over a wide range of vibration amplitude and frequency parameters.Comment: 14 pages, 9 figure