Convection due to an unstable density difference across a permeable membrane
TL;DR: In this article, the authors studied the near-membrane flow in all the regimes consisting of sheet plumes formed from the unstable layers of fluid near the membrane, and proposed a phenomenology which predicts the observed $Nu_b\sim Ra^2/Sc$ scaling of Nusselt number.
Abstract: We study natural convection driven by unstable concentration differences of NaCl across a horizontal permeable membrane at Rayleigh numbers (Ra) of $10^{10}$ to $10^{11}$; the 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, viz. 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 bserved at higher concentration differences $(\Delta C)$ 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 $Nu_b\sim Ra^2/Sc$ scaling of 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. Increase in Ra results in more number of closely and regularly spaced sheet plumes. The mean plume spacing in the advection regime $\overline{\lambda}_b$, is larger than the mean plume spacing in Rayleigh - $B\'{e}nard$ Convection $(\overline{\lambda})$, and shows a different Ra dependence. The plume spacings in the advection regime $(\lambda_b)$ show a common log-normal probability density function at all Ra. We propose a phenomenology which predicts $\overline{\lambda}_b\sim \sqrt{Z_wZ_{V_i}}$, where $Z_w$ and $Z_{V_i}$ are respectively the near-wall length scales in Rayleigh - $B\'{e}nard$ convection (RBC) and due to the advection velocity. In the combined regime, which occur at intermediate values of $\Delta C$, the flux scales as $(\Delta C/2)^{4/3}$. At lower driving potentials, in the diffusion regime, the flux scaling is similar to that in turbulent RBC.
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TL;DR: In this article, the Nusselt number and the Reynolds number depend on the Rayleigh number Ra and the Prandtl number Pr, and the thicknesses of the thermal and the kinetic boundary layers scale with Ra and Pr.
Abstract: 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.
1,372 citations
Cites background from "Convection due to an unstable densi..."
...…recent papers were devoted to them (Breuer et al. (2004), Funfschilling and Ahlers (2004), Haramina and Tilgner (2004), Puthenveettil et al. (2005), Puthenveettil and Arakeri (2005, 2008), Shishkina and Wagner (2006, 2008), Theerthan and Arakeri (1998, 2000), Xi et al. (2004), Zhou et al. (2007b),…...
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Journal Article•
TL;DR: In this article, the Nusselt number in turbulent thermal convection in a cylindrical container of aspect ratio 4 was measured and the data showed that the log Nu-log Ra slope saturates at a value close to 1/3, as observed previously by us in experiments with smaller aspect ratios.
Abstract: We report measurements of the Nusselt number, Nu, in turbulent thermal convection in a cylindrical container of aspect ratio 4. The highest Rayleigh number achieved was Ra=2×10 13 . Except for the last half a decade or so of Ra, experimental conditions obey the Boussinesq approximation accurately. For these conditions, the data show that the log Nu-log Ra slope saturates at a value close to 1/3, as observed previously by us in experiments with smaller aspect ratios. The increasing slope over the last half a decade of Ra is inconclusive because the corresponding conditions are non-Boussinesq. Finally, we report a modified scaling relation between the plume advection frequency and Ra that collapses data for different aspect ratios.
102 citations
TL;DR: In this article, a systematic experimental study of geometric and physical properties of thermal plumes in turbulent Rayleigh-Benard convection using the thermochromic-liquid-crystal (TLC) technique is presented.
Abstract: We present a systematic experimental study of geometric and physical properties of thermal plumes in turbulent Rayleigh–Benard convection using the thermochromic-liquid-crystal (TLC) technique. The experiments were performed in three water-filled cylindrical convection cells with aspect ratios 2, 1 and 0.5 and over the Rayleigh number range 5×107≤Ra≤1011. TLC thermal images of horizontal plane cuts at various depths below the top plate were acquired. Three-dimensional (3D) images of thermal plumes were then reconstructed from the 2D slices of the temperature field. The results show that the often-called sheetlike plumes are really 1D structures and may be called rodlike plumes. We find that the number densities for both sheetlike/rodlike and mushroomlike plumes have power-law dependence on Ra with scaling exponents of ~0.3, which is close to that between the Nusselt number Nu and Ra. This result suggests that it is the plume number that primarily determines the scaling exponent of the Nu–Ra scaling relation. The evolution of the aspect ratio of sheetlike/rodlike plumes reveals that, as Ra increases, the plume geometry changes from more-elongated to less-elongated. Our study of the plume area fraction (fraction of coverage over the surface of the plate) further reveals that the increase in plume numbers with Ra mainly comes from an increase in plume emission, rather than fragmentation of plumes. In addition, the area, perimeter and shape complexity of the 2D horizontal cuts of sheetlike/rodlike plumes were studied and all are found to obey log-normal distributions.
69 citations
TL;DR: In this article, the authors present planforms of line plumes formed on horizontal surfaces in turbulent convection, along with the length of plumes measured from these planforms, in a six decade range of Rayleigh numbers (10(5) < Ra < 10(11)) and at three Prandtl numbers (Pr = 0.7, 5.2, 602).
Abstract: 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 (10(5) < Ra < 10(11)) and at three Prandtl numbers (Pr = 0.7, 5.2, 602). 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 (L(p)/A), made dimensionless by the near-wall length scale in turbulent convection (Z(w)), remains constant for a given fluid. The Nusselt number is shown to be directly proportional to L(p)H/A for a given fluid layer of height H. The increase in Pr has a weak influence in decreasing L(p)/A. 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.
39 citations
Cites background or methods from "Convection due to an unstable densi..."
...The wide separation of time scales made the system quasi-steady and hence comparable with steady convection systems such as RBC; the reader is referred to Puthenveettil & Arakeri (2005, 2008) and Ramareddy & Puthenveettil (2011) for details....
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...The experiments with the membrane were unsteady, but the time scales of variation of the bulk concentration and flux were much larger than the time scale of variation of the large-scale flow (PA; Puthenveettil & Arakeri 2008)....
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...…the planforms of the plume structure at any instant had a lognormal distribution of spacings, with the variance of λ being of the same order as λ (PA; Puthenveettil & Arakeri 2008; Ramareddy & Puthenveettil 2011), a single mean spacing may not be a complete measure to characterize the structure;…...
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...Further details of the experimental setup, visualization procedure and the planforms could be found in PA and Puthenveettil et al. (2005) and Puthenveettil & Arakeri (2008)....
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TL;DR: In this paper, a systematic experimental study of geometric and statistical properties of thermal plumes in turbulent Rayleigh-B\'{e}nard convection using the thermochromic-liquid-crystal (TLC) technique is presented.
Abstract: We present a systematic experimental study of geometric and statistical properties of thermal plumes in turbulent Rayleigh-B\'{e}nard convection using the thermochromic-liquid-crystal (TLC) technique. The experiments were performed in three water-filled cylindrical convection cells with aspect ratios 2, 1, and 0.5 and over the Rayleigh-number range $5\times10^7 \leq Ra \leq 10^{11}$. TLC thermal images of horizontal plane cuts at various depths below the top plate were acquired. Three-dimensional images of thermal plumes were then reconstructed from the two-dimensional slices of the temperature field. The results show that the often-called sheetlike plumes are really one-dimensional structures and may be called rodlike plumes. We find that the number densities for both sheetlike/rodlike and mushroomlike plumes have power-law dependence on $Ra$ with scaling exponents of $\sim 0.3$, which is close to that between the Nusselt number $Nu$ and $Ra$. This result suggests that it is the plume number that primarily d ermines the scaling exponent of the $Nu$-$Ra$ scaling relation. The evolution of the aspect ratio of sheetlike/rodlike plumes reveals that as $Ra$ increases the plume geometry changes from more-elongated to less-elongated. Our study of the plume area fraction (fraction of coverage over the surface of the plate) further reveals that the increased plume numbers with $Ra$ mainly comes from increased plume emission, rather than fragmentation of plumes. In addition, the area, perimeter, and the shape complexity of the two-dimensional horizontal cuts of sheetlike/rodlike plumes were studied and all are found to obey log-normal distributions.
37 citations
References
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Book•
01 Jan 1973TL;DR: CRC handbook of chemistry and physics, CRC Handbook of Chemistry and Physics, CRC handbook as discussed by the authors, CRC Handbook for Chemistry and Physiology, CRC Handbook for Physics,
Abstract: CRC handbook of chemistry and physics , CRC handbook of chemistry and physics , کتابخانه مرکزی دانشگاه علوم پزشکی تهران
52,268 citations
TL;DR: Turbulent convection exemplifies many of the startling aspects of turbulent flows that have been uncovered in the past two decades, but frequently exhibits a novel twist as discussed by the authors, as in the case of free shear flows, convection can organize into large-scale vortical structures, but these then react back in subtle ways on the boundary layers which ultimately sustain them.
Abstract: Turbulent convection exemplifies many of the startling aspects of turbulent flows that have been uncovered in the past two decades, but frequently exhibits a novel twist. Thus, as in the case of free shear flows, convection can organize into large-scale vortical structures, but these then react back in subtle ways on the boundary layers which ultimately sustain them. Thermal plumes are a coherent mode of heat transport, analogous to boundary layer bursts, yet their overall effect can be surprisingly close to the structureless predictions of mixing length theory. Convection cells are closed, which facilitates their experimental control, but fluctuations never exit and there is a dynamically determined bulk forcing. While the single pass mode characteristic of wind tunnel experiments seems simpler, the convection cell is, in ways to be discussed, more constrained. This review aims to familiarize the turbulence researcher with con vergent lines of investigation in convection and also to remind those working in convection that turbulence is not a new subject. To situate convection within the gamut of other turbulent flows, let us by way of introduction contrast the directions in which convection has developed with research on the turbulent boundary layer. From the onset of convection up to Rayleigh numbers Ra � 1 0 times critical, there is a great wealth of information about flow structures (which can be visualized from above), and their relative stabilities (Busse 198 1 ) . Turbulence, in the sense of many coupled modes, and not just sensitive dependence on initial conditions, can arise for low Ra in large aspect ratio
633 citations
611 citations
"Convection due to an unstable densi..." refers background in this paper
...is the characteristic velocity scale in the bulk known as the Deardorff velocity scale (Deardorff 1970) which gives an appropriate estimate of the large-scale flow velocity in high-Rayleigh-number convection (see PA)....
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...Here, W∗ = (gβqH ) 1/3 (2.6) is the characteristic velocity scale in the bulk known as the Deardorff velocity scale (Deardorff 1970) which gives an appropriate estimate of the large-scale flow velocity in high-Rayleigh-number convection (see PA)....
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TL;DR: Investigating thermal transport over eleven orders of magnitude of the Rayleigh number, using cryogenic helium gas as the working fluid, finds no evidence for a transition to the Ra1/2 regime, and studies the variation of internal temperature fluctuations with Ra, and probe velocity statistics indirectly.
Abstract: Turbulent convection occurs when the Rayleigh number (Ra)--which quantifies the relative magnitude of thermal driving to dissipative forces in the fluid motion--becomes sufficiently high. Although many theoretical and experimental studies of turbulent convection exist, the basic properties of heat transport remain unclear. One important question concerns the existence of an asymptotic regime that is supposed to occur at very high Ra. Theory predicts that in such a state the Nusselt number (Nu), representing the global heat transport, should scale as Nu proportional to Ra(beta) with beta = 1/2. Here we investigate thermal transport over eleven orders of magnitude of the Rayleigh number (10(6) < or = Ra < or = 10(7)), using cryogenic helium gas as the working fluid. Our data, over the entire range of Ra, can be described to the lowest order by a single power-law with scaling exponent beta close to 0.31. In particular, we find no evidence for a transition to the Ra(1/2) regime. We also study the variation of internal temperature fluctuations with Ra, and probe velocity statistics indirectly.
562 citations
"Convection due to an unstable densi..." refers background in this paper
...The experimentally observed scaling law is Nu ∼ Ran where n shows a large variation between 0.20 to 0.382, with the majority of the exponents being slightly less than 0.3 (Siggia 1994; Niemela et al. 2000; Chavanne et al. 2001; Xia, Lam & Zhou 2002; Sun et al. 2005; Niemela & Sreenivasan 2006)....
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01 Nov 1999
TL;DR: In this paper, the authors investigate thermal transport over eleven orders of magnitude of the Rayleigh number (106,≤ Ra ≤10 17), using cryogenic helium gas as the working fluid, and find no evidence for a transition to the Ra1/2 regime.
Abstract: Turbulent convection occurs when the Rayleigh number (Ra)—which quantifies the relative magnitude of thermal driving to dissipative forces in the fluid motion—becomes sufficiently high. Although many theoretical and experimental studies of turbulent convection exist, the basic properties of heat transport remain unclear. One important question concerns the existence of an asymptotic regime that is supposed to occur at very high Ra. Theory predicts that in such a state the Nusselt number (Nu), representing the global heat transport, should scale as Nu ∝ Raβ with β = 1/2. Here we investigate thermal transport over eleven orders of magnitude of the Rayleigh number (106 ≤ Ra ≤ 10 17), using cryogenic helium gas as the working fluid. Our data, over the entire range of Ra, can be described to the lowest order by a single power-law with scaling exponent β close to 0.31. In particular, we find no evidence for a transition to the Ra1/2 regime. We also study the variation of internal temperature fluctuations with Ra, and probe velocity statistics indirectly.
460 citations