Olga Shishkina’s research while affiliated with Max Planck Institute for Dynamics and Self-Organization and other places

Radial unstable stratification is a potential source of turbulence in the cold regions of accretion disks. To investigate this thermal effect, here we focus on two-dimensional Rayleigh–Bénard convection in an annulus subject to radially dependent gravitational acceleration $g \propto 1/r$ . Next to the Rayleigh number Ra and Prandtl number Pr , the radius ratio $\eta$ , defined as the ratio of inner and outer cylinder radii, is a crucial parameter governing the flow dynamics. Using direct numerical simulations for Pr=1 and Ra in the range from $10^7$ to $10^{10}$ , we explore how variations in $\eta$ influence the asymmetry in the flow field, particularly in the boundary layers. Our results show that in the studied parameter range, the flow is dominated by convective rolls and that the thermal boundary-layer (TBL) thickness ratio between the inner and outer boundaries varies as $\eta ^{1/2}$ . This scaling is attributed to the equality of velocity scales in the inner ( $u_i$ ) and outer ( $u_o$ ) regions. We further derive that the temperature drops in the inner and outer TBLs scale as $1/(1+\eta ^{1/2})$ and $\eta ^{1/2}/(1+\eta ^{1/2})$ , respectively. The scalings and the temperature drops are in perfect agreement with the numerical data.

We offer a new model for the heat transfer and the turbulence intensity in strongly driven Rayleigh-Bénard turbulence (the so-called ultimate regime), which in contrast to hitherto models is consistent with the new mathematically exact heat transfer upper bound of Choffrut [Upper bounds on Nusselt number at finite Prandtl number, .] and thus enables extrapolations of the heat transfer to geo- and astrophysical flows. The model distinguishes between four subregimes of the ultimate regime and well describes the measured heat transfer in various large-Rayleigh experiments. In this new representation, which properly accounts for the Prandtl number dependence, the onset to the ultimate regime is seen in all available large-Rayleigh datasets, though at different Rayleigh numbers, as to be expected for a non-normal–nonlinear instability. Published by the American Physical Society 2024

Thermally driven turbulent flows are omnipresent in nature and technology. A good understanding of the physical principles governing such flows is key for numerous problems in oceanography, climatology, geophysics, and astrophysics for problems involving energy conversion, heating and cooling of buildings and rooms, and process technology. In the physics community, the most popular system to study wall-bounded thermally driven turbulence has been Rayleigh-Bénard flow, i.e., the flow in a box heated from below and cooled from above. The dimensionless control parameters are the Rayleigh number Ra (the dimensionless heating strength), the Prandtl number Pr (the ratio of kinematic viscosity to thermal diffusivity), and the aspect ratio Γ of the container. The key response parameters are the Nusselt number Nu (the dimensionless heat flux from the bottom to the top) and the Reynolds number Re (the dimensionless strength of the turbulent flow). While there is good agreement and understanding of the dependences Nu(Ra,Pr,Γ) up to Ra∼1011 (the “classical regime”), for even larger Ra in the so-called ultimate regime of Rayleigh-Bénard convection the experimental results and their interpretations are more diverse. The transition of the flow to this ultimate regime, which is characterized by strongly enhanced heat transfer, is due to the transition of laminar-type flow in the boundary layers to turbulent-type flow. Understanding this transition is of the utmost importance for extrapolating the heat transfer to large or strongly thermally driven systems. Here the theoretical results on this transition to the ultimate regime are reviewed and an attempt is made to reconcile the various experimental and numerical results. The transition toward the ultimate regime is interpreted as a non-normal–nonlinear and thus subcritical transition. Experimental and numerical strategies are suggested that can help to further illuminate the transition to the ultimate regime and the ultimate regime itself, for which a modified model for the scaling laws in its various subregimes is proposed. Similar transitions in related wall-bounded turbulent flows such as turbulent convection with centrifugal buoyancy and Taylor-Couette turbulence are also discussed.

Three-dimensional direct numerical simulations of rotating Rayleigh–Bénard convection in the planar geometry with no-slip top and bottom and periodic lateral boundary conditions are performed for a broad parameter range with the Rayleigh number spanning in $5\times 10^{6}\leq Ra \leq 5\times 10^{13}$ , Ekman number within $5\times 10^{-9}\leq Ek \leq 5\times 10^{-5}$ and Prandtl number Pr=1 . The thermal and Ekman boundary layer (BL) statistics, temperature drop within the thermal BL, interior temperature gradient and scaling behaviours of the heat and momentum transports (reflected in the Nusselt Nu and Reynolds numbers Re ) as well as the convective length scale are investigated across various flow regimes. The global and local momentum transports are examined via the Re scaling derived from the classical theoretical balances of viscous–Archimedean–Coriolis (VAC) and Coriolis–inertial–Archimedean (CIA) forces. The VAC-based Re scaling is shown to agree well with the data in the cellular and columnar regimes, where the characteristic convective length scales as the onset length scale ${\sim } Ek^{1/3}$ , while the CIA-based Re scaling and the inertia length scale $\sim (ReEk)^{1/2}$ work well in the geostrophic turbulence regime for $Ek\leq 1.5\times 10^{-8}$ . The examinations of Nu , global and local Re , and convective length scale as well as the temperature drop within the thermal BL and its thickness scaling behaviours, indicate that for extreme parameters of $Ek\leq 1.5\times 10^{-8}$ and $80\lesssim RaEk^{4/3}\lesssim 200$ , we have reached the diffusion-free geostrophic turbulence regime.

We offer a new model for the heat transfer and the turbulence intensity in strongly driven Rayleigh-Benard turbulence (the so-called ultimate regime), which in contrast to hitherto models is consistent with the new mathematically exact heat transfer upper bound of Choffrut et al. [J. Differential Equations 260, 3860 (2016)] and thus enables extrapolations of the heat transfer to geo- and astrophysical flows. The model distinguishes between four subregimes of the ultimate regime and well describes the measured heat transfer in various large-Ra experiments. In this new representation, which properly accounts for the Prandtl number dependence, the onset to the ultimate regime is seen in all available large-Ra data sets, though at different Rayleigh numbers, as to be expected for a non-normal-nonlinear instability.

We investigate states of rapidly rotating Rayleigh-Bénard convection in a cylindrical cell over a range of Rayleigh numbers 3 × 10 5 ≤ Ra ≤ 5 × 10 9 and Ekman numbers 10 − 6 ≤ Ek ≤ 10 − 4 for Prandtl number Pr = 0.8 and aspect ratios 1 / 5 ≤ Γ ≤ 5 using direct numerical simulations. We characterize, for perfectly insulating sidewall boundary conditions, the first transition to convection via wall mode instability and the nonlinear growth and instability of the resulting wall mode states, including a secondary transition to time dependence. We show how the radial structure of the vertical velocity u z and the temperature T is captured well by the linear eigenfunctions of the wall mode instability where the radial width of u z is δ u z ∼ Ek 1 / 3 r / H whereas δ T ∼ e − k r ( k is the wave number of a laterally infinite wall mode state). The disparity in spatial scales for Ek = 10 − 6 means that the heat transport is dominated by the radial structure of u z since T varies slowly over the radial scale δ u z . We further describe how the transition to a state of bulk convection is influenced by the presence of the wall mode states. We use temporal and spatial scales as measures of the local state of convection and the Nusselt number Nu as representative of global transport. Our results elucidate the evolution of the wall state of rotating convection and confirm that wall modes are strongly linked with the boundary zonal flow being the robust remnant of nonlinear wall mode states. We also show how the heat transport ( Nu ) contributions of wall modes and bulk modes are related and discuss approaches to disentangling their relative contributions. Published by the American Physical Society 2024

The validity of the Oberbeck–Boussinesq (OB) approximation in Rayleigh–Bénard (RB) convection is studied using the Gray & Giorgini ( Intl J. Heat Mass Transfer , vol. 19, 1976, pp. 545–551) criterion that requires that the residuals, i.e. the terms that distinguish the full governing equations from their OB approximations, are kept below a certain small threshold $\hat {\sigma }$ . This gives constraints on the temperature and pressure variations of the fluid properties (density, absolute viscosity, specific heat at constant pressure $c_p$ , thermal expansion coefficient and thermal conductivity) and on the magnitudes of the pressure work and viscous dissipation terms in the heat equation, which all can be formulated as bounds regarding the maximum temperature difference in the system, $\varDelta$ , and the container height, L . Thus for any given fluid and $\hat {\sigma }$ , one can calculate the OB-validity region (in terms of $\varDelta$ and L ) and also the maximum achievable Rayleigh number ${{Ra}}_{max,\hat {\sigma }}$ , and we did so for fluids water, air, helium and pressurized SF $_6$ at room temperature, and cryogenic helium, for $\hat {\sigma }=5\,\%$ , $10\,\%$ and $20\,\%$ . For the most popular fluids in high- ${{Ra}}$ RB measurements, which are cryogenic helium and pressurized SF $_6$ , we have identified the most critical residual, which is associated with the temperature dependence of $c_p$ . Our direct numerical simulations (DNS) showed, however, that even when the values of $c_p$ can differ almost twice within the convection cell, this feature alone cannot explain a sudden and strong enhancement in the heat transport in the system, compared with its OB analogue.

The geostrophic turbulence in rapidly rotating thermal convection exhibits characteristics shared by many highly turbulent geophysical and astrophysical flows. In this regime, the convective length and velocity scales and heat flux are all diffusion-free, i.e. independent of the viscosity and thermal diffusivity. Our direct numerical simulations (DNS) of rotating Rayleigh–Bénard convection in domains with no-slip top and bottom and periodic lateral boundary conditions for a fluid with the Prandtl number Pr=1 and extreme buoyancy and rotation parameters (the Rayleigh number up to $Ra=3\times 10^{13}$ and the Ekman number down to $Ek=5\times 10^{-9}$ ) indeed demonstrate all these diffusion-free scaling relations, in particular, that the dimensionless convective heat transport scales with the supercriticality parameter $\widetilde {Ra}\equiv Ra\, Ek^{4/3}$ as $Nu-1\propto \widetilde {Ra}^{3/2}$ , where Nu is the Nusselt number. We further derive and verify in the DNS that with the decreasing $\widetilde {Ra}$ , the geostrophic turbulence regime undergoes a transition into another geostrophic regime, the convective heat transport in this regime is characterized by a very steep $\widetilde {Ra}$ -dependence, $Nu-1\propto \widetilde {Ra}^{3}$ .