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 rupture of the thin film at the top of a bubble floating at a liquid-gas interface leads to the axisymmetric collapse of the bubble cavity. We present scaling laws for such a cavity collapse, established from experiments conducted with bubbles spanning a wide range of Bond (${10^{-3}<Bo\leq1}$) and Ohnesorge numbers (${10^{-3}<Oh<10^{-1}}$), defined with the bubble radius $R$. The cavity collapse is a capillary-driven process, with a dependency on viscosity and gravity affecting, respectively, precursory capillary waves on the cavity boundary, and the static bubble shape. The collapse is characterised by tangential and normal velocities of the kink, formed by the intersection of the concave cavity opening after the top thin film rupture, with the convex bubble cavity boundary. The tangential velocity $U_t$ is constant during the collapse and is shown to be $U_t=4.5~U_c{\mathcal{W}}_R$, where $U_c$ is the capillary velocity and ${\mathcal{W}}_R(Oh,Bo)={(1-\sqrt{Oh {\mathscr{L}}} )^{-1/2}}$ is the wave resistance factor due to the precursory capillary waves, with $\mathscr{L}(Bo)$ being the path correction of the kink motion. The movement of the kink in the normal direction is part of the inward shrinkage of the whole cavity due to the sudden reduction of gas pressure inside the bubble cavity after the thin film rupture. This normal velocity is shown to scale as $U_c$ in the equatorial plane, while at the bottom of the cavity $\overline{U}_{nb}=U_c(Z_c/R)({\mathcal{W}_R}/ {\mathscr{L}})$, where $Z_c(Bo)$ is the static cavity depth. The total volume flux of cavity-filling, which is entirely contributed by this shrinking, scales as ${Q_T\simeq 2\pi R Z_c U_c}$; remains a constant throughout the collapse.

Dynamics of collapse of free-surface bubbles: effects of gravity and viscosity

Abstract In Rayleigh Bénard convection, for a range of Prandtl numbers $4.69 \leqslant Pr \leqslant 5.88$ and Rayleigh numbers $5.52\times 10^5 \leqslant Ra \leqslant 1.21\times 10^9$, we study the effect of shear by the inherent large-scale flow (LSF) on the local boundary layers on the hot plate. The velocity distribution in a horizontal plane within the boundary layers at each $Ra$, at any instant, is (A) unimodal with a peak at approximately the natural convection boundary layer velocities $V_{bl}$; (B) bimodal with the first peak between $V_{bl}$ and $V_{L}$, the shear velocities created by the LSF close to the plate; or (C) unimodal with the peak at approximately $V_{L}$. Type A distributions occur more at lower $Ra$, while type C occur more at higher $Ra$, with type B occurring more at intermediate $Ra$. We show that the second peak of the bimodal type B distributions, and the peak of the unimodal type C distributions, scale as $V_{L}$ scales with $Ra$. We then show that the areas of such regions that have velocities of the order of $V_{L}$ increase exponentially with increase in $Ra$ and then saturate. The velocities in the remaining regions, which contribute to the first peak of the bimodal type B distributions and the single peak of type A distributions, are also affected by the shear. We show that the Reynolds number based on these velocities scale as $Re_{bs}$, the Reynolds number based on the boundary layer velocities forced externally by the shear due to the LSF, which we obtained as a perturbation solution of the scaling relations derived from integral boundary layer equations. For $Pr=1$ and aspect ratio $\varGamma =1$, $Re_{bs} \sim Ra^{0.375}$ for small shear, similar to the observed flux scaling in a possible ultimate regime. The velocity at the edge of the natural convection boundary layers was found to increase with $Ra$ as $Ra^{0.35}$; since $V_{bl}\sim Ra^{1/3}$, this suggests a possible shear domination of the boundary layers at high $Ra$. The effect of shear, however, decreases with increase in $Pr$ and with increase in $\varGamma$, and becomes negligible for $Pr\geqslant 100$ at $\varGamma =1$ or for $\varGamma \geqslant 20$ at $Pr=1$, causing $Re_{bs}\sim Ra_w^{1/3}$.

Effect of shear on local boundary layers in turbulent convection

We measure the velocity field in a horizontal field near the hot plate in turbulent convection using stereo PIV for ${10^6<Ra_w<10^9}$ and ${5.2<Pr<4}$. We then extract the line plumes from this velocity field using a divergence criterion using the PIV technique on the obtained plume structures, which gives us the velocity field of the plume motion. The statistical analysis of this velocity field of the plume motion shows the coexistence of two different kinds of motion of the plumes, lateral merging and motion along the plumes.

Velocity of Line Plumes on the Hot Plate in Turbulent Natural Convection

We present a simple, new criterion to extract line plumes from the velocity fields, without using the temperature field, in a horizontal plane close to the plate in turbulent convection. The existing coherent structure detection criteria from velocity fields, proposed for shear driven wall turbulence, are first shown to be inadequate for turbulent convection. Based on physical arguments, we then propose that the negative values of $\overline{\nabla_H}\cdot\overline{u}$, the horizontal divergence of the horizontal velocity field extracts the plume regions from the velocity field. The $\overline{\nabla_H}\cdot\overline{u}$ criterion is then shown to predict the total length of plumes correctly over a Rayleigh number range of $7.7\times 10^5\le Ra_w\le 1.43\times 10^9$ and a Prandtl number range of $3.96\le Pr\le 5.3$. The thickness of the plume region predicted by the present criterion is approximately same as that obtained by a threshold of RMS temperature fluctuations, $\sqrt{T'^2}=0.2 \Delta T$, where $\Delta T$ is the temperature drop across the fluid layer.

A criterion to detect line plumes from velocity fields in turbulent convection

We present plume structures at high Rayleigh numbers (Ra&lt;sub&gt;H&lt;/sub&gt; &amp;#126; 10&lt;sup&gt;11&lt;/sup&gt;) and Schmidt numbers (Sc &amp;#126; 675), in convection driven by density differences across a horizontal membrane, when vertical transpiration velocities in the range of 0.002 cm/s &amp;le; V&lt;sub&gt;o&lt;/sub&gt; &amp;le; 0.065 cm/s are imposed. The density differences across the membrane are created by having brine with higher concentrations above the membrane and ammonium chloride with lower concentrations below it, with the bottom fluid being forced upward across the membrane by gravitational heads to create transpiration. Planar laser induced fluorescence (PLIF) visualizations in a horizontal plane grazing the top of the membrane, with refractive index matching, brings out the plume structures and their evolution with Ra&lt;sub&gt;H&lt;/sub&gt; and V&lt;sub&gt;o&lt;/sub&gt;. The plume structures show complex dendritic patterns of line plumes with plume-free patches; the plume-free patches decrease in area and number, with decrease in Ra&lt;sub&gt;H&lt;/sub&gt; and V&lt;sub&gt;o&lt;/sub&gt;. At low V&lt;sub&gt;o&lt;/sub&gt;, these plume-free patches have aligned plumes around them, while at larger V&lt;sub&gt;o&lt;/sub&gt;, such aligned plumes are not seen. We expect these patches to be due to the impingement of large-scale flow at low V&lt;sub&gt;o&lt;/sub&gt;, while they could be due to the modification of boundary layer instability at large V&lt;sub&gt;o&lt;/sub&gt;. These plumes become closer with increase in Ra&lt;sub&gt;H&lt;/sub&gt;, while they become more separated with increase in V&lt;sub&gt;o&lt;/sub&gt;. We quantify this variation by measuring the total plume length (L&lt;sub&gt;p&lt;/sub&gt;) by skeletonizing the plume structures and calculate the mean plume spacing (&amp;lambda;) from L&lt;sub&gt;p&lt;/sub&gt;; &amp;lambda; show nontrivial, non-power law dependence on Ra&lt;sub&gt;H&lt;/sub&gt; and V&lt;sub&gt;o&lt;/sub&gt;, different from those observed for the case of no throughflow and low throughflows. 

Baburaj A. Puthenveettil

Papers

Dynamics of collapse of free-surface bubbles: effects of gravity and viscosity

Effect of shear on local boundary layers in turbulent convection

Velocity of Line Plumes on the Hot Plate in Turbulent Natural Convection

A criterion to detect line plumes from velocity fields in turbulent convection

Plume structures in natural convection with transpiration