15 results
Dynamics of collapse of free-surface bubbles: effects of gravity and viscosity
- Sangeeth Krishnan, Baburaj A. Puthenveettil, E.J. Hopfinger
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- Journal:
- Journal of Fluid Mechanics / Volume 980 / 10 February 2024
- Published online by Cambridge University Press:
- 06 February 2024, A36
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The rupture of the thin film at the top of a bubble at a liquid–gas interface leads to an 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 (${10^{-3}< Bo\leq 1}$) 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 the normal interface velocity ($U_n$) and by the tangential wave propagation velocity of the kink ($U_t$), defined by the intersection of the concave cavity boundary formed after the rupture of the thin film with the convex boundary of the bubble cavity. During the collapse, $U_t$ remains constant and is shown to be $U_t=4.5U_c{\mathcal {W}}_R$, 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 $\bar {U}_{nb}=U_c(Z_c/R)({\mathcal {W}_R}/ {\mathscr {L}})$, where $Z_c(Bo)$ is the static cavity depth. The filling rate of the cavity, which remains a constant throughout the collapse, is shown to be entirely determined by the shrinking velocity and scales as ${Q_T\simeq 2{\rm \pi} R Z_c U_c}$. From $Q_T$ we recover the jet velocity scaling, thereby relating the cavity collapse with the jet velocity scaling.
Effect of shear on local boundary layers in turbulent convection
- Prafulla P. Shevkar, Sanal K. Mohanan, Baburaj A. Puthenveettil
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- Journal:
- Journal of Fluid Mechanics / Volume 962 / 10 May 2023
- Published online by Cambridge University Press:
- 08 May 2023, A41
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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 $Re_{bs}\sim Ra_w^{1/3}$.
On separating plumes from boundary layers in turbulent convection
- Prafulla P. Shevkar, R. Vishnu, Sanal K. Mohanan, Vipin Koothur, Manikandan Mathur, Baburaj A. Puthenveettil
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- Journal:
- Journal of Fluid Mechanics / Volume 941 / 25 June 2022
- Published online by Cambridge University Press:
- 25 April 2022, A5
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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 $Pr>1$.
Inertial effects on the flow near a moving contact line
- Akhil Varma, Anubhab Roy, Baburaj A. Puthenveettil
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- Journal:
- Journal of Fluid Mechanics / Volume 924 / 10 October 2021
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- 16 August 2021, A36
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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, $\rho$. At sufficient proximity to the moving contact line, where $\rho \ll 1$, the flow is dominated by viscous forces over inertia. However, further away from the contact line, or at higher speeds of motion, inertia is also expected to be influential. In such cases, the current contact line models, which assume Stokes flow and neglect inertia entirely, would be inaccurate in describing the hydrodynamic flow fields. Hence, to account for inertia, here we perform a regular perturbation expansion in $\rho$, of the streamfunction near the Stokes solution. We, however, find that the leading-order inertial correction thus obtained is singular at a critical contact angle of $0.715 {\rm \pi}$. We resolve this spurious mathematical singularity by incorporating the eigenfunction terms, which physically represent flows due to disturbances originating far from the contact line. In particular, we propose a stick slip on the solid boundary – arising from local surface heterogeneities – as the mechanism that generates these disturbance flows. The resulting singularity-free, inertia-corrected streamfunction shows significant deviation from the Stokes solution in the visco-inertial regime ($\rho \sim 1$). 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.
Scaling in concentration-driven convection boundary layers with transpiration
- G. V. Ramareddy, P. J. Joshy, Gayathri Nair, Baburaj A. Puthenveettil
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- Journal:
- Journal of Fluid Mechanics / Volume 903 / 25 November 2020
- Published online by Cambridge University Press:
- 17 September 2020, A3
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We study concentration-driven natural convection boundary layers on horizontal surfaces, subjected to a weak, surface normal, uniform blowing velocity $V_i$ for three orders of range of the dimensionless blowing parameter $10^{-8}\le J=Re_x^3/Gr_x\le 10^{-5}$, where $Re_x$ and $Gr_x$ are the local Reynolds and Grashof numbers at the horizontal location $x$, based respectively on $V_i$ and ${\rm \Delta} C$, the concentration difference across the boundary layer. We formulate the integral boundary layer equations, with the assumption of no concentration drop within the species boundary layer, which is valid for weak blowing into the thin species boundary layers that occur at the high Schmidt number ($Sc \simeq 600$) of concentration-driven convection. The equations are then numerically solved to show that the species boundary layer thickness $\delta _d = 1.6\,x(Re_x/Gr_x)^{1/4}$, the velocity boundary layer thickness $\delta _v=\delta _d Sc^{1/5}$, the horizontal velocity $u = V_i(Gr_x/Re_x)^{1/4}f(\eta )$, where $\eta =y/\delta _v$, and the drag coefficient based on $V_i$, $C_D = 2.32/\sqrt {J}$. We find that the vertical profile of the horizontally averaged dimensionless concentration across the boundary layer becomes, surprisingly, independent of the blowing and the species diffusion effects to follow a $Gr_y^{2/3}$ scaling, where $Gr_y$ is the Grashof number based on the vertical location $y$ within the boundary layer. We then show that the above profile matches the experimentally observed mean concentration profile within the boundary layers that form on the top surface of a membrane, when a weak flow is forced gravitationally from below the horizontal membrane that has brine above it and water below it. A similar match between the theoretical scaling of the species boundary layer thickness and its experimentally observed variation is also shown to occur.
Film spreading from a miscible drop on a deep liquid layer
- Raj Dandekar, Anurag Pant, Baburaj A. Puthenveettil
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- Journal:
- Journal of Fluid Mechanics / Volume 829 / 25 October 2017
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- 14 September 2017, pp. 304-327
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We study the spreading of a film from ethanol–water droplets of radii $0.9~\text{mm}<r_{d}<1.1~\text{mm}$ on the surface of a deep water layer for various concentrations of ethanol in the drop. Since the drop is lighter ($\unicode[STIX]{x1D709}=\unicode[STIX]{x1D70C}_{l}/\unicode[STIX]{x1D70C}_{d}>1.03$), 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 $\unicode[STIX]{x1D712}=\unicode[STIX]{x1D707}_{l}/\unicode[STIX]{x1D707}_{d}>0.38$. Inertial forces are not dominant in the spreading since the Reynolds numbers based on the film thickness $h_{f}$ are in the range $0.02<Re_{f}<1.4$. The spreading is surface-tension-driven since the film capillary numbers are in the range $0.0005<Ca_{f}<0.0069$ and the drop Bond numbers are in the range $0.19<Bo_{d}<0.56$. 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 $r_{f}\sim t^{1/4}$ and shows a dependence on the concentration of ethanol in the drop as well as on $r_{d}$, and is independent of the layer height $h_{l}$. 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 $r_{f}^{\ast }=r_{f}/r_{d}$, in the form $fr_{f}^{\ast }={t_{\unicode[STIX]{x1D707}d}^{\ast }}^{1/4}$, where $t_{\unicode[STIX]{x1D707}d}^{\ast }=t/t_{\unicode[STIX]{x1D707}d}$, with the time scale $t_{\unicode[STIX]{x1D707}d}=\unicode[STIX]{x1D707}_{d}r_{d}/\unicode[STIX]{x0394}\unicode[STIX]{x1D70E}$ and $f$ is a function of $Bo_{d}$. Similarly, we show that the dimensionless velocity of film spreading, $Ca_{d}=u_{f}\unicode[STIX]{x1D707}_{d}/\unicode[STIX]{x0394}\unicode[STIX]{x1D70E}$, scales as $4f^{4}Ca_{d}={r_{f}^{\ast }}^{-3}$.
On the scaling of jetting from bubble collapse at a liquid surface
- Sangeeth Krishnan, E. J. Hopfinger, Baburaj A. Puthenveettil
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- Journal:
- Journal of Fluid Mechanics / Volume 822 / 10 July 2017
- Published online by Cambridge University Press:
- 08 June 2017, pp. 791-812
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We present scaling laws for the jet velocity resulting from bubble collapse at a liquid surface which bring out the effects of gravity and viscosity. The present experiments conducted in the range of Bond numbers $0.004<Bo<2.5$ and Ohnesorge numbers $0.001<Oh<0.1$ were motivated by the discrepancy between previous experimental results and numerical simulations. We show here that the actual dependence of $We$ on $Bo$ is determined by the gravity dependency of the bubble immersion (cavity) depth which has no power-law variation. The power-law variation of the jet Weber number, $We\sim 1/\sqrt{Bo}$, suggested by Ghabache et al. (Phys. Fluids, vol. 26 (12), 2014, 121701) is only a good approximation in a limited range of $Bo$ values ($0.1<Bo<1$). Viscosity enters the jet velocity scaling in two ways: (i) through damping of precursor capillary waves which merge at the bubble base and weaken the pressure impulse, and (ii) through direct viscous damping of the jet formation and dynamics. These damping processes are expressed by a dependence of the jet velocity on Ohnesorge number from which critical values of $Oh$ are obtained for capillary wave damping, the onset of jet weakening, the absence of jetting and the absence of jet breakup into droplets.
Dynamics of line plumes on horizontal surfaces in turbulent convection
- G. S. Gunasegarane, Baburaj A. Puthenveettil
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- Journal of Fluid Mechanics / Volume 749 / 25 June 2014
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- 14 May 2014, pp. 37-78
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We study the dynamics of line plumes on the bottom horizontal plate in turbulent convection over six decades of Rayleigh number $(10^5<\mathit{Ra}_w<10^{11})$ and two decades of Prandtl number or Schmidt number ($0.7<\mathit{Pr}<5.3$, $\mathit{Sc}=602$). 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 ($\overline{V_{sh}}$) scales as the large-scale flow velocity ($V_{LS}$), with the fraction of the length of the plumes affected by shear increasing with $\mathit{Ra}_w$ as $L_{ps}/L_p\sim \mathit{Ra}_w^{0.054} \mathit{Pr}^{-0.12}$. The mean time of initiation of the plumes $\overline{t^{*}}$, scales as the diffusive time scale near the plate, $Z_w^2/\alpha $, where $Z_w$ is the appropriate length scale near the plate, 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 ${\sim }70\, \%$ of the length of the plumes undergoing merging at $\mathit{Ra}_w\approx 10^{11}$ and $\mathit{Sc}= 602$. The fraction of the length of the plumes that undergoes merging decreases with increase in $\mathit{Ra}_w$ as, $L_{pm}/L_p \sim \mathit{Ra}_w^{-0.054} \mathit{Pr}^{0.12}$; the exponents of $\mathit{Ra}_w$ and $\mathit{Pr}$ being of the same magnitude but of opposite sign as that in the relation for $L_{ps}/L_p$. Measurements of the locational means of the velocities of merging of the plumes $(V_m)$ show that $V_m$ 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 $\mathit{Ra}_w-\mathit{Pr}$ combination. The merging velocities at all $\mathit{Ra}_w$ and $\mathit{Pr}$ have a common lognormal distribution, but their mean and variance increased with increasing $\mathit{Ra}_w$ and decreasing $\mathit{Pr}$. Using mass and momentum balance of the region between two merging plumes, we show that the spatiotemporal mean merging velocities ($\overline{V_m}$), which are an order lower than $\overline{V_{sh}}$, 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 $\overline{V_m}$ scales as the diffusive velocity scale near the plate $\nu /Z_w$. The Reynolds number in terms of $\overline{V_m}$ and the layer height $H(\mathit{Re}_H)$ scales as $\mathit{Ra}_w^{1/3}$, in the same way as the Nusselt number ($\mathit{Nu}$) scales approximately; therefore $\mathit{Re}_{H}\sim \mathit{Nu}$. These relations imply that $\mathit{Re}_w= \overline{V_m}Z_w/\nu $ a Reynolds number near the plate, is an invariant for a given fluid in turbulent convection.
Motion of drops on inclined surfaces in the inertial regime
- Baburaj A. Puthenveettil, Vijaya K. Senthilkumar, E. J. Hopfinger
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- Journal:
- Journal of Fluid Mechanics / Volume 726 / 10 July 2013
- Published online by Cambridge University Press:
- 30 May 2013, pp. 26-61
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We present experimental results on high-Reynolds-number motion of partially non-wetting liquid drops on inclined plane surfaces using: (i) water on fluoro-alkyl silane (FAS)-coated glass; and (ii) mercury on glass. The former is a high-hysteresis ($3{5}^{\circ } $) surface while the latter is a low-hysteresis one (${6}^{\circ } $). The water drop experiments have been conducted for capillary numbers $0. 0003\lt Ca\lt 0. 0075$ and for Reynolds numbers based on drop diameter $137\lt Re\lt 3142$. The ranges for mercury on glass experiments are $0. 0002\lt Ca\lt 0. 0023$ and $3037\lt Re\lt 20\hspace{0.167em} 069$. It is shown that when $Re\gg 1{0}^{3} $ for water and $Re\gg 10$ for mercury, a boundary layer flow model accounts for the observed velocities. A general expression for the dimensionless velocity of the drop, covering the whole $Re$ range, is derived, which scales with the modified Bond number ($B{o}_{m} $). This expression shows that at low $Re$, $Ca\sim B{o}_{m} $ and at large $Re$, $Ca \sqrt{Re} \sim B{o}_{m} $. The dynamic contact angle (${\theta }_{d} $) variation scales, at least to first-order, with $Ca$; the contact angle variation in water, corrected for the hysteresis, collapses onto the low-$Re$ data of LeGrand, Daerr & Limat (J. Fluid Mech., vol. 541, 2005, pp. 293–315). The receding contact angle variation of mercury has a slope very different from that in water, but the variation is practically linear with $Ca$. We compare our dynamic contact angle data to several models available in the literature. Most models can describe the data of LeGrand et al. (2005) for high-viscosity silicon oil, but often need unexpected values of parameters to describe our water and mercury data. In particular, a purely hydrodynamic description requires unphysically small values of slip length, while the molecular-kinetic model shows asymmetry between the wetting and dewetting, which is quite strong for mercury. The model by Shikhmurzaev (Intl J. Multiphase Flow, vol. 19, 1993, pp. 589–610) is able to group the data for the three fluids around a single curve, thereby restoring a certain symmetry, by using two adjustable parameters that have reasonable values. At larger velocities, the mercury drops undergo a change at the rear from an oval to a corner shape when viewed from above; the corner transition occurs at a finite receding contact angle. Water drops do not show such a clear transition from oval to corner shape. Instead, a direct transition from an oval shape to a rivulet appears to occur.
Length of near-wall plumes in turbulent convection
- Baburaj A. Puthenveettil, G. S. Gunasegarane, Yogesh K. Agrawal, Daniel Schmeling, Johannes Bosbach, Jaywant H. Arakeri
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- Journal:
- Journal of Fluid Mechanics / Volume 685 / 25 October 2011
- Published online by Cambridge University Press:
- 20 September 2011, pp. 335-364
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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 () and at three Prandtl numbers (). 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 (), made dimensionless by the near-wall length scale in turbulent convection (), remains constant for a given fluid. The Nusselt number is shown to be directly proportional to for a given fluid layer of height . The increase in has a weak influence in decreasing . 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.
The Pe ~ 1 regime of convection across a horizontal permeable membrane
- G. V. RAMA REDDY, BABURAJ A. PUTHENVEETTIL
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- Journal of Fluid Mechanics / Volume 679 / 25 July 2011
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- 17 May 2011, pp. 476-504
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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 Péclet 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.
Evolution and breaking of parametrically forced capillary waves in a circular cylinder
- BABURAJ A. PUTHENVEETTIL, E. J. HOPFINGER
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- Journal of Fluid Mechanics / Volume 633 / 25 August 2009
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- 25 August 2009, pp. 355-379
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We present results on parametrically forced capillary waves in a circular cylinder, obtained in the limit of large fluid depth, using two low-viscosity liquids whose surface tensions differ by an order of magnitude. The evolution of the wave patterns from the instability to the wave-breaking threshold is investigated in a forcing frequency range (f = ω/2π = 25–100 Hz) that is around the crossover frequency (ωot) from gravity to capillary waves (ωot/2≤ω/2≤4ωot). As expected, near the instability threshold the wave pattern depends on the container geometry, but as the forcing amplitude is increased the wave pattern becomes random, and the wall effects are insignificant. Near breaking, the distribution of random wavelengths can be fitted by a Gaussian. A new gravity–capillary scaling is introduced that is more appropriate, than the usual viscous scaling, for low-viscosity fluids and forcing frequencies <103 Hz. In terms of these scales, a criterion is derived to predict the crossover from capillary- to gravity-dominated breaking. A wave-breaking model is developed that gives the relation between the container and the wave accelerations in agreement with experiments. The measured drop size distribution of the ejected drops above the breaking threshold is well approximated by a gamma distribution. The mean drop diameter is proportional to the wavelength determined from the dispersion relation; this wavelength is also close to the most likely wavelength of the random waves at drop ejection. The dimensionless drop ejection rate is shown to have a cubic power law dependence on the dimensionless excess acceleration ε′d an inertial–gravitational ligament formation model is consistent with such a power law.
Convection due to an unstable density difference across a permeable membrane
- BABURAJ A. PUTHENVEETTIL, JAYWANT H. ARAKERI
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- Journal of Fluid Mechanics / Volume 609 / 25 August 2008
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- 31 July 2008, pp. 139-170
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We study natural convection driven by unstable concentration differences of sodium chloride (NaCl) across a horizontal permeable membrane at Rayleigh numbers (Ra) of 1010 to 1011 and Schmidt number (Sc)=600. A layer of brine lies over a layer of distilled water, separated by the membrane, in square-cross-section tanks. The membrane is permeable enough to allow a small flow across it at higher driving potentials. Based on the predominant mode of transport across the membrane, three regimes of convection, namely an advection regime, a diffusion regime and a combined regime, are identified. The near-membrane flow in all the regimes consists of sheet plumes formed from the unstable layers of fluid near the membrane. In the advection regime observed at higher concentration differences (ΔC) across the membrane, there is a slow overturning through-flow across the membrane; the transport across the membrane occurs mostly by advection. This phenomenology explains the observed Nub~Ra2/Sc scaling of the Nusselt number. The planforms of sheet plumes near the membrane show a dendritic structure due to the combined influence of the mean shear due to the large-scale flow and the entrainment flow of the adjacent plumes. The near-membrane dynamics show initiation, elongation and merger of plumes; a movie is available with the online version of the paper. Increase in Ra results in a larger number of closely and regularly spaced sheet plumes. The mean plume spacing in the advection regime , is larger than the mean plume spacing in Rayleigh–Bénard convection (), and shows a different Ra-dependence. The plume spacings in the advection regime (λb) show a common log-normal probability density function at all Ra. We propose a phenomenology which predicts ~ , where Zw and are, respectively, the near-wall length scales in Rayleigh–Bénard convection (RBC) and due to the advection velocity. In the combined regime, which occurs at intermediate values of ΔC, the flux scales as (ΔC/2)4/3. At lower driving potentials, in the diffusion regime, the flux scaling is similar to that in turbulent RBC.
Plume structure in high-Rayleigh-number convection
- BABURAJ A. PUTHENVEETTIL, JAYWANT H. ARAKERI
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- Journal:
- Journal of Fluid Mechanics / Volume 542 / 10 November 2005
- Published online by Cambridge University Press:
- 25 October 2005, pp. 217-249
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Near-wall structures in turbulent natural convection at Rayleigh numbers of $10^{10}$ to $10^{11}$ at A Schmidt number of 602 are visualized by a new method of driving the convection across a fine membrane using concentration differences of sodium chloride. The visualizations show the near-wall flow to consist of sheet plumes. A wide variety of large-scale flow cells, scaling with the cross-section dimension, are observed. Multiple large-scale flow cells are seen at aspect ratio (AR)= 0.65, while only a single circulation cell is detected at AR= 0.435. The cells (or the mean wind) are driven by plumes coming together to form columns of rising lighter fluid. The wind in turn aligns the sheet plumes along the direction of shear. the mean wind direction is seen to change with time. The near-wall dynamics show plumes initiated at points, which elongate to form sheets and then merge. Increase in rayleigh number results in a larger number of closely and regularly spaced plumes. The plume spacings show a common log–normal probability distribution function, independent of the rayleigh number and the aspect ratio. We propose that the near-wall structure is made of laminar natural-convection boundary layers, which become unstable to give rise to sheet plumes, and show that the predictions of a model constructed on this hypothesis match the experiments. Based on these findings, we conclude that in the presence of a mean wind, the local near-wall boundary layers associated with each sheet plume in high-rayleigh-number turbulent natural convection are likely to be laminar mixed convection type.
The multifractal nature of plume structure in high-Rayleigh-number convection
- BABURAJ A. PUTHENVEETTIL, G. ANANTHAKRISHNA, JAYWANT H. ARAKERI
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- Journal:
- Journal of Fluid Mechanics / Volume 526 / 10 March 2005
- Published online by Cambridge University Press:
- 25 February 2005, pp. 245-256
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The geometrically different planforms of near-wall plume structure in turbulent natural convection, visualized 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 for the near-wall plume structures.