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.

We experimentall y study the collapse of cavities formed by bubbles of different radius R at the free surface of five different fluids. The tangential and normal velocities of the disturbance front on the bubble surface were measured and shown that the tangential velocities scale as the capillary velocity (σ/ρR) 1/2 . The total time for the completion of cavity collapse was measured and shown to scale as capillary time scale R 3/2 /(σ/ρ) 1/2 . We hence infer that the cavity collapse is dominated by surface tension and inertia with viscosity not playing a role in the regime of our investigation. Such a behavior of cavity dynamics is different from the behavior of jet velocity discussed by Sangeeth et.al.[9] wherein the jet velocity showed a clear viscous cut off. The rate of change of cavity volume was found to be constant with time, the implications of which needs to be investigated further.

IUTAM Symposium on Multiphase flows with phase change: challenges and opportunities, Hyderabad, India (December 08 - December 11, 2014) Dynamics of collapse of free surface bubbles

Baburaj A. Puthenveettil

Papers

IUTAM Symposium on Multiphase flows with phase change: challenges and opportunities, Hyderabad, India (December 08 - December 11, 2014) Dynamics of collapse of free surface bubbles