Session H14: Convection and Buoyancy-driven Flows: Turbulent Rayleigh-Benard

Elusive transition to the ultimate regime of turbulent RBC: Dynamics of LSC in high-Ra cryogenic helium experiments

Non Oberbeck-Boussinesq (NOB) effects may increase the heat transfer efficiency of turbulent Rayleigh-B\'enard convection (RBC), when the top plate temperature approaches the saturation vapor curve (SVC) even far away from the critical point of the working fluid. Our recent experimental study [1] using cryogenic $^4$He under conditions as close as possible to the Goettingen study using SF$_6$ [2] argues that the claim of having observed the transition to Kraichnan’s ultimate $Nu(Ra)$ scaling is likely due to NOB effects, and the important issue of transition to the ultimate state of RBC remains open. I will present here a detailed analysis of large-scale circulation (LSC) dynamics in the experiment [1]. I will discuss dependences of the Reynolds numbers associated with LSC circulation and sloshing and of the LSC reversal frequency on the position in the p-T diagram of $^4$He, in particular on the boundary layer asymmetry due to NOB conditions near the SVC. [1] P. Urban, P. Hanzelka, T. Králík, M. Macek, V. Musilová and L. Skrbek, Phys. Rev. E 99, 011101(R) (2019). [2] X. He, D. Funfschilling, H. Nobach, E. Bodenschatz, and G. Ahlers, Phys. Rev. Lett. 108, 024502 (2012).   

Velocity and Thermal Boundary Layer Equations for Turbulent Rayleigh-B\'enard Convection

In turbulent Rayleigh-B\'enard convection, the boundary layers (BLs) are non-steady with fluctuations, the time-averaged large-scale circulating velocity vanishes far away from the top and bottom plates, and the motion arises from buoyancy. There is no existing BL theory that successfully incorporates all these physical effects. We have derived a full set of BL equations for both the temperature and velocity fields from the Boussinesq equations for a quasi-two-dimensional flow above a heated plate, accounting for all these physical effects. We use the commonly employed concepts of eddy viscosity and eddy thermal diffusivity to study fluctuations and propose a closure model to relate them to the stream function. This full set of BL equations enables us to obtain the time-averaged velocity and temperature BL profiles, in the form of similarity solutions, for general Prandtl number (Pr) in terms of two parameters $k_1$ and $k_2$ that measure the size of fluctuations. We have demonstrated that with a suitable choice of $k_1$ and $k_2$, our theoretical results are good approximations of both the time-averaged velocity and temperature profiles obtained in direct numerical simulations especially at low Pr, for which theoretical results are challenging to obtain.   

Effects of Prandtl number in quasi-two-dimensional turbulent Rayleigh-Bénard convection

We report an experimental study of the Prandtl (\textit{Pr}) number effects on flow pattern and local temperature fluctuation in quasi-two-dimensional turbulent Rayleigh-Bénard convection. The experiments were conducted in four rectangular cells with same aspect ratio but different heights, the Rayleigh number \textit{Ra} range (1e9 -- 2e10) remains unchanged while \textit{Pr} is varied from 11.6 to 157.4. The flow patterns visualized by the shadowgraph show that thermal plumes become more slender as \textit{Pr} increases, and their organized-motion is more concentrated towards the sidewall. The mean flow strength, characterized by the Reynolds number \textit{Re}, becomes weaker with the increase of \textit{Pr}, i.e. \textit{Ra}$^{\mathrm{0.57}}$\textit{Pr}$^{\mathrm{-0.81}}$. It is further found that the temperature fluctuations in the center ($\sigma_{\mathrm{c}}$/$\Delta $T) and near sidewall ($\sigma_{\mathrm{s}}$/$\Delta $T) behave different, i.e. Pr$^{\mathrm{-0.19}}$Ra$^{\mathrm{-0.28}}$ and Pr$^{\mathrm{0.10}}$Ra$^{\mathrm{-0.20}}$, respectively. This result quantitatively demonstrates that, as \textit{Pr} increases, thermal plumes prefer to move along the sidewall rather than traveling through the center of the cell.   

Effect of slip boundary conditions on the heat flux and near-wall temperature equations in turbulent Rayleigh-B\'{e}nard convection

We present direct numerical simulations (DNS) of the heat transport and near-wall temperature profiles in turbulent Rayleigh-B\'{e}nard convection (RBC) with slip boundary conditions (BCs) on horizontal walls. The mean horizontal velocity on the wall is assumed as $\boldsymbol{u}_w=(b/L)(\partial{\boldsymbol{u}}/\partial{\boldsymbol{n}})|_w$. Here $L$ is the height of RBC sample, $b$ is the slip length with $b = 0$ for no-slip BC and $b \rightarrow \infty$ for free-slip BC. The simulations were for $0 \leq b/L \rightarrow \infty$ and the Prandtl numbers $Pr = 4.3$ in the Rayleigh-number range $10^{8} \lesssim Ra \lesssim \times 10^{10}$. As $b/L$ increases, we found that the ratio of dimensionless heat flux, as expressed by the Nusselt number follows $Nu/Nu_0 = 0.8\times tanh(100\times b/L) + 1$, where $Nu_0$ is the Nusselt number for $b = 0$. Considering the boundary layer fluctuations, we derived the equation $\Theta(\xi)=\int_0^{\xi}(1+p^x\eta^x)^{-n}d\eta$ for the mean temperature profile $\Theta(\xi)$ near the horizontal surface, where $p = \Gamma(1+1/x)\Gamma(n-1/x)/\Gamma(n)$ with $2 \leq x \leq 3$ depending on $b/L$ and $n>1$ for varying geometries of the convection sample.   

A systematic investigation of the inverse cascade and flow speed scaling behavior in rapidly rotating Rayleigh-Benard convection

Rotating convection is a robust driving mechanism for the inverse kinetic energy cascade, which is characterized by the transfer of energy from small-scale convective motions to large-scale motions. One of the consequences of this process is the formation of so-called large-scale vortices (LSVs) that fill the entire flow domain. However, the domain of existence for these vortices and the behavior of their saturated amplitude is not well-constrained. A systematic set of simulations of the asymptotically-reduced governing equations show that: (1) the presence of LSVs is characterized by a critical small-scale Reynolds number over a wide range of Prandtl numbers; (2) the amplitude of the LSVs scales predictively with the horizontal dimensions of the flow domain; and (3) the amplitude of the LSVs saturates with increasing Rayleigh number. Furthermore, we find that LSVs are present in flow regimes not previously known to harbor them. A scaling law is developed for the convective Reynolds number as a function of Rayleigh number and Prandtl number, which can be used to predict the presence of LSVs.   

High Rayleigh Number Convection in a Slender Cylinder for Prandtl Number of 1

We report results from direct numerical simulations of turbulent Rayleigh-B\'enard convection for a Prandtl number of 1 and aspect ratio of 0.1, with Rayleigh numbers varying from $1 \times 10^8$ to $1 \times 10^{14}$, and possibly higher. We present the dependence of the heat transport (the Nusselt number) and momentum transport (the Reynolds number) on the Rayleigh number for this parameter regime. We will also discuss the global structure of the convection flow and details of the increasingly intermittent boundary layer dynamics. Various strategies for improving the efficiency of statistical convergence will also be presented.   

Resolved Energy Budget of Superstructures in Rayleigh-B\'{e}nard Convection

Turbulent Rayleigh-B\'{e}nard convection shows a complex interaction between coherent large-scale flow patterns, so-called turbulent superstructures, and small-scale fluctuations. Here, we use direct numerical simulations to study the impact of turbulent fluctuations on large-scale patterns. To separate the superstructures and small-scale fluctuations, we employ a filtering approach, which retains the physical space information and therefore complements spectral analysis techniques. Focussing on the resolved energy budget of the superstructures, we characterize the different contributions, such as the resolved power input, direct dissipation and the energy transfer rate between scales. The results show that, as expected, the energy transfer differs significantly between the bulk and close to the wall. At large Rayleigh numbers, the energy input into the superstructures is primarily balanced by a direct energy transfer to smaller scales in the bulk. Here, the energy transfer acts as an effective dissipation for the superstructures. However, close to the wall the energy transfer is more complex and may even drive the superstructures. Besides a characterization, these results can help to develop an effective description of turbulent superstructures in convection.   

Inhomogeneous Nusselt Number Distribution in Turbulent Rayleigh-B\'{e}nard Convection

The Nusselt number (Nu) scaling with Rayleigh number (Ra) has been the focus in the study of Rayleigh-B\'{e}nard (RB) convection, inspiring the Grossman-Lohse proposal. However, different apparatus seem to display different scalings. We investigate this issue by considering inhomogenous Nu distribution along horizontal plate, with four flow regimes, i.e. corner-roll, jet-impingement, wind-shearing and plume-ejecting. In corner-roll, Nu scaling follows a modified mixing zone model, with a scale correction $r\sim Ra^{-0.085}$. In jet-impingement region, a streamwise momentum similarity yields an Nu distribution: $Nu_{ji}=0.2Ra^{0.3}\exp(-1.2(x-x_{rea})/L_{ji})$. In wind-shearing region, side wall does not affect the similarity of the momentum and thermal boundary layer. In plume-ejecting region, after choosing the proper scale of plume scale as the buffer layer thickness and the ejecting velocity as large scale circulation velocity, the balance of inertia and buoyancy yields $Nu_{pe}\sim Ra^{0.369}$. In summary, four local Nu-scalings are obtained: $1/3$, $0.30$, $0.262$, and $0.369$, and the global Nu scaling is then obtained by adding all four with multiplying them with their spatial domain sizes. This model provides an alternative interpretation for Nu-scaling transition.   

Temperature fluctuations in turbulent Rayleigh-B{\'e}nard convection

Non-Gaussian fluctuations with an exponential tail in their probability density function (PDF) are often observed in nonequilibrium steady states (NESSs) and one does not understand why they appear so often. Turbulent Rayleigh-B{\'e}nard convection (RBC) is an example of such a NESS, in which the measured PDF $P(\delta T)$ of temperature fluctuations $\delta T$ in the central region of the flow has a long exponential tail. Here we show that because of the dynamic heterogeneity in RBC, the exponential PDF is generated by a convolution of a set of dynamics modes conditioned on a constant local thermal dissipation rate $\epsilon$. The conditional PDF $G(\delta T \vert \epsilon)$ of $\delta T$ under a constant $\epsilon$ is found to be of Gaussian form and its variance $\sigma^2_T$ for different values of $\epsilon$ follows an exponential distribution. The convolution of the two distribution functions gives rise to the exponential PDF $P(\delta T)$. This work thus provides a physical mechanism of the observed exponential distribution of $\delta T$ in RBC and also sheds new light on the origin of non-Gaussian fluctuations in other NESSs.   

Thermal boundary layer properties in turbulent thermal convection: Effects of Prandtl number

We report experimental measurements of thermal boundary layer properties in turbulent Rayleigh-Bénard convection with the Prandtl (Pr) number being varied from 11.6 to 157.4. The experiments were conducted in rectangular convection cells over the Rayleigh number \textit{Ra} range of 1e9 \textasciitilde 2e10 Both the mean temperature and its root mean square profiles were measured by a thermistor that is movable along the central vertical axis of the cell. These results are compared with the recently derived boundary layer equations by Shishkina et al. (Phys. Rev. Lett., vol. 114, 2015, 114302) and by Wang et al. (Phys. Rev. Fluids, vol. 1, 2016, 082301), with a focus on the effects of Pr number.   

Near free-fall oscillatory velocities in liquid metal rotating convection

The geomagnetic field is induced by liquid metal flow inside Earth´s outer core as a self-excited dynamo. Buoyancy drives the liquid metal flow because the iron rich core is cooling from its primordial state through heat loss to the mantle. The rotation of the Earth and Lorentz forces alter the resulting convective flow. However, the detailed flow topology is largely unknown. Here we will investigate the effect of rotation on a low Prandtl number thermal convection by means of laboratory experiments and DNS. Therefore, we consider a rotating Rayleigh-B\'enard convection setup in an upright cylindrical vessel of aspect ratio $\Gamma = D / H = 2$. We investigate supercriticalities in the range of $1 \leq \widetilde{Ra} < 20$ and Ekman numbers $4 \times 10^{-5} \leq Ek \leq 5 \times 10^{-6}$ in liquid gallium at $Pr = 0.03$. By means of ultrasound-Doppler velocity measurements, we find that oscillatory convection generates velocities approaching the freefall velocity. Multi-modal bulk oscillations dominate the vertical velocity field over the whole range of supercriticalities investigated. Additionally, coherent mean zonal flows and time-mean helicity suggesting that these oscillatory flows can be relevant for dynamo action.