The boundary layer equations for plane, incompressible, and steady flow are 

$$\matrix{ {u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = - {1 \over \varrho }{{\partial p} \over {\partial x}} + v{{{\partial ^2}u} \over {\partial {y^2}}},} \cr {0 = {{\partial p} \over {\partial y}},} \cr {{{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}} = 0.} \cr }$$

Boundary Layer Theory

The progress in our understanding of several aspects of turbulent Rayleigh-Benard convection is reviewed. The focus is on the question of how the Nusselt number and the Reynolds number depend on the Rayleigh number Ra and the Prandtl number Pr, and on how the thicknesses of the thermal and the kinetic boundary layers scale with Ra and Pr. Non-Oberbeck-Boussinesq effects and the dynamics of the large scale convection roll are addressed as well. The review ends with a list of challenges for future research on the turbulent Rayleigh-Benard system.

/pdf/heat-transfer-and-large-scale-dynamics-in-turbulent-rayleigh-39oz2czpiy.pdf

Heat transfer and large scale dynamics in turbulent Rayleigh-Bénard convection

Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves

The properties of the structure functions and other small-scale quantities in turbulent Rayleigh-Benard convection are reviewed, from an experimental, theoretical, and numerical point of view. In particular, we address the question of whether, and if so where in the flow, the so-called Bolgiano-Obukhov scaling exists, i.e., Sθ(r) ∼ r2/5 for the second-order temperature structure function and Su(r) ∼ r6/5 for the second-order velocity structure function. Apart from the anisotropy and inhomogeneity of the flow, insufficiently high Rayleigh numbers, and intermittency corrections (which all hinder the identification of such a potential regime), there are also reasons, as a matter of principle, why such a scaling regime may be limited to at most a decade, namely the lack of clear scale separation between the Bolgiano length scale LB and the height of the cell.

Small-Scale Properties of Turbulent Rayleigh-Bénard Convection

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 geometrically different plan forms of near wall plume structure in turbulent natural convection, visualised by driving the convection using concentration differences across a membrane, are shown to have a common multifractal spectrum of singularities for Rayleigh numbers in the range $10^{10}- 10^{11}$ at Schmidt number of 602. The scaling is seen for a length scale range of $2^5$ and is independent of the Rayleigh number, the flux, the strength and nature of the large scale flow, and the aspect ratio. Similar scaling is observed for the plume structures obtained in the presence of a weak flow across the membrane. This common non trivial spatial scaling is proposed to be due to the same underlying generating process of the near wall plume structures.

/pdf/multifractal-nature-of-plume-structure-in-high-rayleigh-q6fr8az233.pdf

Multifractal nature of plume structure in high Rayleigh number convection

In natural convection, driven by an unstable density difference due to a heavier fluid (brine) above a lighter fluid (water) across a horizontal permeable membrane, we discover a new regime of convection, where the Sherwood number (Sh) scales approximately as the Rayleigh number (Ra). Inferring from the planforms of plume structure on the membrane and the estimates of velocity through the membrane, we show that such a regime occurs when advection balances diffusion in the membrane, i.e. the Peclet number based on the membrane thickness (Pe) is of order one. The advection is inferred to be caused by the impingement of the large-scale flow on the membrane. Utilizing mass balance and symmetry assumptions in the top and the bottom fluids, we derive an expression for the concentration profile in the membrane pore in the new regime by solving the convection–diffusion equation in the membrane pore; this helps us to obtain the concentration drops above and below the membrane that drive the convection. We find that the net flux, normalised by the diffusive flux corresponding to the concentration drop on the side opposite to the impingement of the large-scale flow remains constant throughout the new regime. On the basis of this finding, we then obtain an expression for the flux scaling in the new regime which matches with the experiments; the expression has the correct asymptotes of flux scaling in the advection and the diffusion regimes. The plume spacings in the new regime are distributed lognormally, and their mean follows the trend in the advection regime.

The Pe ~ 1 regime of convection across a horizontal permeable membrane

We study the spreading of a film from ethanol–water droplets of radii on the surface of a deep water layer for various concentrations of ethanol in the drop. Since the drop is lighter ( ), it stays at the surface of the water layer during the spreading of the film from the drop; the film is more viscous than the underlying water layer since . Inertial forces are not dominant in the spreading since the Reynolds numbers based on the film thickness are in the range . The spreading is surface-tension-driven since the film capillary numbers are in the range and the drop Bond numbers are in the range . We observe that, when the drop is brought in contact with the water surface, capillary waves propagate from the point of contact, followed by a radially expanding, thin circular film of ethanol–water mixture. The film develops instabilities at some radius to form outward-moving fingers at its periphery while it is still expanding, till the expansion stops at a larger radius. The film then retracts, during which time the remaining major part of the drop, which stays at the centre of the expanding film, thins and develops holes and eventually mixes completely with water. The radius of the expanding front of the film scales as and shows a dependence on the concentration of ethanol in the drop as well as on , and is independent of the layer height . Using a balance of surface tension and viscous forces within the film, along with a model for the fraction of the drop that forms the thin film, we obtain an expression for the dimensionless film radius , in the form , where , with the time scale and is a function of . Similarly, we show that the dimensionless velocity of film spreading, , scales as .

Film spreading from a miscible drop on a deep liquid layer

In the presence of shear, the mean spacing between coherent structures on horizontal surfaces in turbulent Rayleigh-B\'enard convection, in excess of the no-shear spacing, is shown to be proportional to $R\phantom{\rule{0}{0ex}}{e}^{3}\phantom{\rule{0}{0ex}}{Z}_{s}\phantom{\rule{0}{0ex}}h/(D\phantom{\rule{0}{0ex}}R\phantom{\rule{0}{0ex}}a)$, where ${Z}_{s}\phantom{\rule{0}{0ex}}h$ is the viscous-shear length and $D$ a nonmonotonic function of $P\phantom{\rule{0}{0ex}}r$.

Effect of shear on coherent structures in turbulent convection

The multiplier plots in page 130 are erroneous. The corrected plots can be seen in JFM, 526,pp245-256, 2005.

Baburaj A. Puthenveettil

Papers

Multifractal nature of plume structure in high Rayleigh number convection

The Pe ~ 1 regime of convection across a horizontal permeable membrane

Film spreading from a miscible drop on a deep liquid layer

Effect of shear on coherent structures in turbulent convection

Investigations On High Rayleigh Number Turbulent Free Convection