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.

For millimetre to micron sized bubbles, floating at the free surface of different low viscosity fluids with different surface tensions, and then collapsing, we study the ensuing expansion of the outer radius of the hole (ro) at the free surface, as well as its velocity of expansion (uo). Since the thin film cap of the bubble disintegrates before the hole in it reaches the static rim, the hole expansion at intermediate times occurs as if it initiates at the bubble’s static rim of radius Rr; the evolution of ro then results to be a strong function of gravity, since Rr depends strongly on the bubble radius R. A scaling analysis, which includes the increase in the tip radius due to mass accumulation and the resulting change in the retraction force, along with the gravity effects by considering the hole radius in excess of its initial static radius, re = ro − Rr, results in a novel scaling law re/R∼(t/tc)4/7, where tc=ρR3/σ is the capillary time scale; this scaling law is shown to capture the evolution of the hole radii in the present study. The dimensionless velocities of hole expansion, namely, the Weber numbers of hole expansion, Weo=ρuo2R/σ, scale as Weo∼(t/tc)−6/7, independent of gravity effects, matching the observations. We also show that these Weber numbers, which reduce with time, begin with a constant initial Weber number of 64, while the viscous limit of the present phenomena occurs when the bubble Ohnesorge number Oh=μ/σρR≃0.24.

Hole expansion from a bubble at a liquid surface

Dynamics of Collapse of Free Surface Bubbles

Abstract We present a simple, novel kinematic criterion – that uses only the horizontal velocity fields and is free of arbitrary thresholds – to separate line plumes from local boundary layers in a plane close to the hot plate in turbulent convection. We first show that the horizontal divergence of the horizontal velocity field ($\boldsymbol {\nabla _H} \boldsymbol {\cdot } \boldsymbol {u}$) has negative and positive values in two-dimensional (2D), laminar similarity solutions of plumes and boundary layers, respectively. Following this observation, based on the understanding that fluid elements predominantly undergo horizontal shear in the boundary layers and vertical shear in the plumes, we propose that the dominant eigenvalue ($\lambda _D$) of the 2D strain rate tensor is negative inside the plumes and positive inside the boundary layers. Using velocity fields from our experiments, we then show that plumes can indeed be extracted as regions of negative $\lambda _D$, which are identical to the regions with negative $\boldsymbol {\nabla _H} \boldsymbol {\cdot } \boldsymbol {u}$. Exploring the connection of these plume structures to Lagrangian coherent structures (LCS) in the instantaneous limit, we show that the centrelines of such plume regions are captured by attracting LCS that do not have dominant repelling LCS in their vicinity. Classifying the flow near the hot plate based on the distribution of eigenvalues of the 2D strain rate tensor, we then show that the effect of shear due to the large-scale flow is felt more in regions close to where the local boundary layers turn into plumes. The lengths and areas of the plume regions, detected by the $\boldsymbol {\nabla _H}\boldsymbol {\cdot }\boldsymbol {u}$ criterion applied to our experimental and computational velocity fields, are then shown to agree with our theoretical estimates from scaling arguments. Using velocity fields from numerical simulations, we then show that the $\boldsymbol {\nabla _H}\boldsymbol {\cdot }\boldsymbol {u}$ criterion detects all the upwellings, while the available criteria based on temperature and flux thresholds miss some of these upwellings. The plumes detected by the $\boldsymbol {\nabla _H}\boldsymbol {\cdot }\boldsymbol {u}$ criterion are also shown to be thicker at Prandtl numbers ($Pr$) greater than one, expectedly so, due to the thicker velocity boundary layers of the plumes at $Pr>1$.

/pdf/on-separating-plumes-from-boundary-layers-in-turbulent-297hd41w.pdf

On separating plumes from boundary layers in turbulent convection

The wetting or dewetting of a solid substrate by a liquid involves the motion of the contact line between the two phases. One of the parameters that govern the dynamics of the flow near a moving contact line is the local Reynolds number, ). Furthermore, we quantify the effect of inertia by analysing its contribution to the velocity at the liquid interface. We also provide the leading-order inertial correction to the dynamic contact angles predicted by the classical Cox–Voinov model; while inertia has considerable effect on the hydrodynamic flow fields, we find that it has little to no influence on the dynamic contact angles.

Baburaj A. Puthenveettil

Papers

Hole expansion from a bubble at a liquid surface

Dynamics of Collapse of Free Surface Bubbles

On separating plumes from boundary layers in turbulent convection

Inertial effects on the flow near a moving contact line

Integral Analysis of Laminar Indirect Free Convection Boundary Layers with Weak Blowing for Schmidt No. ∼ 1