Recent experimental, numerical and theoretical advances in turbulent Rayleigh-Benard convection are presented. Particular emphasis is given to the physics and structure of the thermal and velocity boundary layers which play a key role for the better understanding of the turbulent transport of heat and momentum in convection at high and very high Rayleigh numbers. We also discuss important extensions of Rayleigh-Benard convection such as non-Oberbeck-Boussinesq effects and convection with phase changes.

https://link.springer.com/content/pdf/10.1140%2Fepje%2Fi2012-12058-1.pdf

New perspectives in turbulent Rayleigh-Bénard convection.

The structure of the boundary layers in turbulent Rayleigh‐B´ enard convection is studied by means of three-dimensional direct numerical simulations. We consider convection in a cylindrical cell at aspect ratio one for Rayleigh numbers of RaD 3 10 9 and 3 10 10 at fixed Prandtl number PrD 0:7. Similar to the experimental results in the same setup and for the same Prandtl number, the structure of the laminar boundary layers of the velocity and temperature fields is found to deviate from the prediction of Prandtl‐Blasius‐Pohlhausen theory. Deviations decrease when a dynamical rescaling of the data with an instantaneously defined boundary layer thickness is performed and the analysis plane is aligned with the instantaneous direction of the large-scale circulation in the closed cell. Our numerical results demonstrate that important assumptions of existing classical laminar boundary layer theories for forced and natural convection are violated, such as the strict twodimensionality of the dynamics or the steadiness of the fluid motion. The boundary layer dynamics consists of two essential local dynamical building blocks, a plume detachment and a post-plume phase. The former is associated with larger variations of the instantaneous thickness of velocity and temperature boundary layer and a fully three-dimensional local flow. The post-plume dynamics is connected with the large-scale circulation in the cell that penetrates the boundary region from above. The mean turbulence profiles taken in localized sections of the boundary layer for each dynamical phase are also compared with solutions of perturbation expansions of the boundary layer equations of forced or natural convection towards mixed convection. Our analysis of both boundary layers shows that the near-wall dynamics combines elements of forced Blasius-type and natural convection.

Boundary layer structure in turbulent Rayleigh-Bénard convection

We simulate numerically Boussinesq convection in non-rotating spherical shells for a fluid with a unity Prandtl number and Rayleigh numbers up to $10^9$. In this geometry, curvature and radial variations of the gravitationnal acceleration yield asymmetric boundary layers. A systematic parameter study for various radius ratios (from $\eta=r_i/r_o=0.2$ to $\eta=0.95$) and gravity profiles allows us to explore the dependence of the asymmetry on these parameters. We find that the average plume spacing is comparable between the spherical inner and outer bounding surfaces. An estimate of the average plume separation allows us to accurately predict the boundary layer asymmetry for the various spherical shell configurations explored here. The mean temperature and horizontal velocity profiles are in good agreement with classical Prandtl-Blasius laminar boundary layer profiles, provided the boundary layers are analysed in a dynamical frame, that fluctuates with the local and instantaneous boundary layer thicknesses. The scaling properties of the Nusselt and Reynolds numbers are investigated by separating the bulk and boundary layer contributions to the thermal and viscous dissipation rates using numerical models with $\eta=0.6$ and a gravity proportional to $1/r^2$. We show that our spherical models are consistent with the predictions of Grossmann \& Lohse's (2000) theory and that $Nu(Ra)$ and $Re(Ra)$ scalings are in good agreement with plane layer results.

/pdf/turbulent-rayleigh-b-enard-convection-in-spherical-shells-2phvwga699.pdf

Turbulent Rayleigh-B\'enard convection in spherical shells

We simulate numerically Boussinesq convection in non-rotating spherical shells for a fluid with a Prandtl number of unity and for Rayleigh numbers up to . In this geometry, curvature and radial variations of the gravitational acceleration yield asymmetric boundary layers. A systematic parameter study for various radius ratios (from to ) and gravity profiles allows us to explore the dependence of the asymmetry on these parameters. We find that the average plume spacing is comparable between the spherical inner and outer bounding surfaces. An estimate of the average plume separation allows us to accurately predict the boundary layer asymmetry for the various spherical shell configurations explored here. The mean temperature and horizontal velocity profiles are in good agreement with classical Prandtl–Blasius laminar boundary layer profiles, provided the boundary layers are analysed in a dynamical frame that fluctuates with the local and instantaneous boundary layer thicknesses. The scaling properties of the Nusselt and Reynolds numbers are investigated by separating the bulk and boundary layer contributions to the thermal and viscous dissipation rates using numerical models with and with gravity proportional to . We show that our spherical models are consistent with the predictions of Grossmann & Lohse’s (J. Fluid Mech., vol. 407, 2000, pp. 27–56) theory and that and scalings are in good agreement with plane layer results.

/pdf/turbulent-rayleigh-benard-convection-in-spherical-shells-2x2059ks7w.pdf

Turbulent Rayleigh-Bénard convection in spherical shells

Selection of efficient solvent in solvent-aided thermal recovery of bitumen

We present planforms of line plumes formed on horizontal surfaces in turbulent convection, along with the length of line plumes measured from these planforms, in a six decade range of Rayleigh numbers (10(5) < Ra < 10(11)) and at three Prandtl numbers (Pr = 0.7, 5.2, 602). Using geometric constraints on the relations for the mean plume spacings, we obtain expressions for the total length of near-wall plumes on horizontal surfaces in turbulent convection. The plume length per unit area (L(p)/A), made dimensionless by the near-wall length scale in turbulent convection (Z(w)), remains constant for a given fluid. The Nusselt number is shown to be directly proportional to L(p)H/A for a given fluid layer of height H. The increase in Pr has a weak influence in decreasing L(p)/A. These expressions match the measurements, thereby showing that the assumption of laminar natural convection boundary layers in turbulent convection is consistent with the observed total length of line plumes. We then show that similar relationships are obtained based on the assumption that the line plumes are the outcome of the instability of laminar natural convection boundary layers on the horizontal surfaces.

/pdf/length-of-near-wall-plumes-in-turbulent-convection-2e5x3xw0ka.pdf

Length of near-wall plumes in turbulent convection

We study the dynamics of line plumes on the bottom horizontal plate in turbulent convection over six decades of Rayleigh number and two decades of Prandtl number or Schmidt number ( , ). From the visualisations of these plumes in a horizontal plane close to the plate, we identify the main dynamics as (i) motion along the plumes, (ii) lateral merging of the plumes and (iii) initiation of the plumes; various other minor types of motion also occur along with these main dynamics. In quantifying the three main motions, we first find that the spatiotemporal mean velocity along the length of the plumes ( ) scales as the large-scale flow velocity ( ), with the fraction of the length of the plumes affected by shear increasing with as . The mean time of initiation of the plumes , scales as the diffusive time scale near the plate, , where is the appropriate length scale near the plate, in agreement with Howard (Proc. 11th Int. Congress Applied Mechanics, Munich, 1964, pp. 1109–1115). Merging occurs in a large fraction of the area of the plate, with of the length of the plumes undergoing merging at and . The fraction of the length of the plumes that undergoes merging decreases with increase in as, ; the exponents of and being of the same magnitude but of opposite sign as that in the relation for . Measurements of the locational means of the velocities of merging of the plumes show that is a constant during each merging cycle at any location. However, the values of these constant velocities depend on the location and the time of measurement, since the merging velocities are affected by the local shear, which is a function of space and time at any combination. The merging velocities at all and have a common lognormal distribution, but their mean and variance increased with increasing and decreasing . Using mass and momentum balance of the region between two merging plumes, we show that the spatiotemporal mean merging velocities ( ), which are an order lower than , scale as the entrainment velocity at the sides of the plumes, averaged over the height of the diffusive layer near the plate. This implies that scales as the diffusive velocity scale near the plate . The Reynolds number in terms of and the layer height scales as , in the same way as the Nusselt number ( ) scales approximately; therefore . These relations imply that a Reynolds number near the plate, is an invariant for a given fluid in turbulent convection.

/pdf/dynamics-of-line-plumes-on-horizontal-surfaces-in-turbulent-3tt99mdvnr.pdf

Dynamics of line plumes on horizontal surfaces in turbulent convection

We study the dynamics of line plumes on the bottom plate in turbulent convection over six decades of Rayleigh number $(10^5<Ra_w<10^{11})$ and two decades of Prandtl number $(0.7<Pr<600)$. We qualitatively identify the main dynamics as motion along the plumes, lateral merging of the plumes and initiation of the plumes; various other minor types of motion also occur along with these main dynamics. The mean velocity along the length of the plumes scales as the large scale flow velocity, with the fraction of the length of the plumes affected by shear increasing with $Ra_w$ as $L_{ps}/L_p\sim Ra_w^{0.04} Pr^{-0.1}$. In agreement with \cite{how}, the mean time of initiation of the plumes $\bar{t^*}$, scales as the diffusive time scale near the plate, $Z_w^2/\alpha$, where $Z_w$ is the appropriate length scale near the plate. The fraction of the length of the plumes that undergoes merging decreases with increase in $Ra_w$ as, $L_{pm}/L_p \sim Ra_w^{-0.04} Pr^{0.1}$. The plumes merge with a constant velocity during their merging cycle, however, the values of these constant velocities depend on the location and the time of measurement. The merging velocities at all $Ra_w$ and $Pr$ have a common lognormal distribution, but their mean and variance increased with increasing $Ra_w$ and decreasing $Pr$. $\bar{V_m}$, which are an order lower than the large scale velocities, scale as the average entrainment velocity at the sides of the plumes. This implies that $\bar{V_m}$ scales as the velocity scale near the plate $\nu/Z_w$. $Re_H$, the Reynolds number interms of $\bar{V_m}$ and the layer height $H$ scales as $Ra^{1/3}$, in the same way as the Nusselt number $Nu$; therefore $Re_{H}\sim Nu$. These relations imply that $Re_w=\bar{V_m}Z_w/\nu$ the Reynolds number near the plate is an invariant for a given fluid in turbulent convection.

G. S. Gunasegarane

Papers

Length of near-wall plumes in turbulent convection

Dynamics of line plumes on horizontal surfaces in turbulent convection

Dynamics of line plumes on horizontal surfaces in turbulent convection