We propose an analytical model to estimate the interface temperature $\varTheta _{\varGamma }$ and the Nusselt number $Nu$ for an evaporating two-layer Rayleigh–Bénard configuration in statistically ...stationary conditions. The model is based on three assumptions: (i) the Oberbeck–Boussinesq approximation can be applied to the liquid phase, while the gas thermophysical properties are generic functions of thermodynamic pressure, local temperature and vapour composition, (ii) the Grossmann–Lohse theory for thermal convection can be applied to the liquid and gas layers separately and (iii) the vapour content in the gas can be taken as the mean value at the gas–liquid interface. We validate this setting using direct numerical simulations in a parameter space composed of the Rayleigh number ($10^6\leq Ra\leq 10^8$) and the temperature differential ($0.05\leq \varepsilon \leq 0.20$), which modulates the variation of state variables in the gas layer. To better disentangle the variable property effects on $\varTheta _\varGamma$ and $Nu$, simulations are performed in two conditions. First, we consider the case of uniform gas properties except for the gas density and gas–liquid diffusion coefficient. Second, we include the variation of specific heat capacity, dynamic viscosity and thermal conductivity using realistic equations of state. Irrespective of the employed setting, the proposed model agrees very well with the numerical simulations over the entire range of $Ra$–$\varepsilon$ investigated.
Free Convective Heat Transfer is a thorough survey of various kinds of free-convective flows and heat transfer. Reference data are accompanied by a large number of photographs originating from ...different optical visualization methods illustrating the different types of flow. The formulas derived from numerical and analytical investigations are valuable tools for engineering calculations. They are written in their most compact and general form in order to allow for an extensive range of different variants of boundary and initial conditions, which, in turn, leads to a wide applicability to different flow types. Some specific engineering problems are solved in the book as exemplary applications of these formulas.
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FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NUK, OBVAL, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
We study the evolution of a melting front between the solid and liquid phases of a pure incompressible material where fluid motions are driven by unstable temperature gradients. In a plane-layer ...geometry, this can be seen as classical Rayleigh–Bénard convection where the upper solid boundary is allowed to melt due to the heat flux brought by the fluid underneath. This free-boundary problem is studied numerically in two dimensions using a phase-field approach, classically used to study the melting and solidification of alloys, which we dynamically couple with the Navier–Stokes equations in the Boussinesq approximation. The advantage of this approach is that it requires only moderate modifications of classical numerical methods. We focus on the case where the solid is initially nearly isothermal, so that the evolution of the topography is related to the inhomogeneous heat flux from thermal convection, and does not depend on the conduction problem in the solid. From a very thin stable layer of fluid, convection cells appear as the depth – and therefore the effective Rayleigh number – of the layer increases. The continuous melting of the solid leads to dynamical transitions between different convection cell sizes and topography amplitudes. The Nusselt number can be larger than its value for a planar upper boundary, due to the feedback of the topography on the flow, which can stabilize large-scale laminar convection cells.
We present an experimental and numerical study of turbulent thermal convection in the presence of an effective horizontal buoyancy that generates extra shear at the boundary. Geometrical confinements ...are also applied by varying the streamwise and spanwise aspect ratios of the convection cell to condense the plumes. With these, we systematically explore the effects of plume and shear on heat transfer. It is found that a streamwise confinement results in increased plume coverage but decreased shear compared with spanwise confinement. The fact that streamwise confinement leads to a higher vertical heat transfer efficiency than the spanwise confined case suggests that the increase of plume coverage is the dominant effect responsible for the enhanced heat transfer. Our results highlight the potential applications of coherent structure manipulation in efficient passive heat transfer control and thermal engineering. We also analyse the energetics of the present system and derive the expression of mixing efficiency accordingly. The mixing efficiency is found to increase with both the buoyancy ratio and streamwise dimension.
We present an experimental study of the large-scale vortex (or large-scale circulation, LSC) in turbulent Rayleigh–Bénard convection in a $\varGamma =\text {diameter}/\text {height}=2$ cylindrical ...cell. The working fluid is deionized water with Prandtl number ($Pr$) around 5.7, and the Rayleigh number ($Ra$) ranges from $7.64\times 10^7$ to $6.06\times 10^8$. We measured the velocity field in various vertical cross-sectional planes by using the planar particle image velocimetry technique. The velocity measurement in the LSC central plane shows that the flow is in the single-roll form, and the centre of the single-roll (vortex) does not always stay at the centre of the cell; instead, it orbits periodically in the direction opposite to the flow direction of the LSC, with its trajectory in the shape of an ellipse. The velocity measurements in the three vertical planes in parallel to the LSC central plane indicate that the flow is in the vortex tube form horizontally filling almost the whole cell, and the centre line of the vortex tube is consistent with the so-called ‘jump rope’ form proposed by a previous study that combined numerical simulation and local velocity measurements in the low $Pr$ case (Vogt et al., Proc. Natl Acad. Sci. USA, vol. 115, 2018, pp. 12674–12679). In addition, we found that the oscillation of the local velocity in $\varGamma =2$ cells originates from the periodical orbiting of the vortex centre. Our velocity measurements further indicate that the vortex centre orbiting is absent in $\varGamma =1$ cells, at least in the $Ra$ range of our experiments.
A novel experiment was performed in rotating Rayleigh–Bénard convection (RRBC), wherein the convection cell with radius $R$ was shifted away from the rotation axis by a distance $d$. In this case, ...the centrifugal force felt by a fluid parcel (characterized by the Froude number $Fr$) can be decomposed into an axisymmetrical part $Fr_R$ and a directed one $Fr_d$. It has been reported that the global heat transport enhances at $Fr_{d,c}$ and then reaches an optimal state at $Fr_{d,max}$ (Hu et al., Phys. Rev. Lett., vol. 127, 2021, 244501). In this paper, the local properties after the offset effects set in are investigated further, which show different features before and after $Fr_{d,max}$. The local temperature measurements at the cell centre reveal that the bulk flow turns from a turbulent state into a laminar state at $Fr_{d,max}$, which is consistent with the particle image velocimetry results. This transition can be qualitatively understood by an equivalent tilted RRBC system. As for the hot and cold coherent structures near the sidewall, their vertical temperature variations reach a minimum at $Fr_{d,max}$, implying that these structures are mostly uniform in the vertical direction at $Fr_{d,max}$. Their temperature contrasts show a linear dependence on $Fr_d$ and start to deviate from this linear behaviour when $Fr_d>Fr_{d,max}$. Besides the dominant effects of $Fr_d$, the secondary effects of $Fr_R$ are also investigated. Due to the positive effect of $+Fr_R$ on the cold structure and the negative effect of $-Fr_R$ on the hot one, the cold structure is more coherent than the hot one, but its size is smaller. The shift of the cold cluster centre from the farthest point is also larger than the shift of the hot one from the nearest point.
The influence of tilt on flow reversals in two-dimensional thermal convection in rectangular cells with two typical aspect ratios,
$\unicodeSTIX{x1D6E4}=\text{width/height}=1$
and 2, are investigated ...by means of direct numerical simulations. For
$\unicodeSTIX{x1D6E4}=1$
, tilt tends to suppress flow reversals. However, it is found that flow reversals characterized by two main rolls are promoted by tilt for
$\unicodeSTIX{x1D6E4}=2$
, which are even observed for some cases of small Prandtl numbers (
$Pr$
) and large tilt angles (
$\unicodeSTIX{x1D6FD}$
). Different from level cases where the four corner rolls all have opportunities to grow and trigger a flow reversal, the reversals in an anticlockwise tilted cell with
$\unicodeSTIX{x1D6E4}=2$
are always led by the growth of the bottom-right or the top-left corner roll. Tilt is favourable for the growth of the bottom-right or the top-left corner roll and thus for breaking the balance between the two main rolls and triggering a flow reversal. The mode decomposition analysis shows that the appearance of the intermediate single-roll mode is crucial for reversals, and flow reversals in a tilted cell with
$\unicodeSTIX{x1D6E4}=2$
can be viewed as a mode competition process between single-roll mode and horizontally adjacent double-roll mode. They can only occur in a limited range of
$\unicodeSTIX{x1D6FD}$
where the two modes have comparable strength. Furthermore, the Nusselt numbers at the hot plate
$Nu_{h}$
and at the cold plate
$Nu_{c}$
behave differently during a flow reversal for
$\unicodeSTIX{x1D6E4}=2$
due to the preference of single corner roll growth.
We report on the transition between two regimes of heat transport in a radiatively driven convection experiment, where a fluid gets heated up within a tunable heating length
$\ell$
in the vicinity of ...the bottom of the tank. The first regime is similar to that observed in standard Rayleigh–Bénard experiments, the Nusselt number
$Nu$
being related to the Rayleigh number
$Ra$
through the power law
$Nu\sim Ra^{1/3}$
. The second regime corresponds to the ‘ultimate’ or mixing-length scaling regime of thermal convection, where
$Nu$
varies as the square root of
$Ra$
. Evidence for these two scaling regimes has been reported in Lepot et al. (Proc. Natl Acad. Sci. USA, vol. 115, 2018, pp. 8937–8941), and we now study in detail how the system transitions from one to the other. We propose a simple model describing radiatively driven convection in the mixing-length regime. It leads to the scaling relation
$Nu\sim (\ell /H)Pr^{1/2}Ra^{1/2}$
, where
$H$
is the height of the cell and
$Pr$
is the Prandtl number, thereby allowing us to deduce the values of
$Ra$
and
$Nu$
at which the system transitions from one regime to the other. These predictions are confirmed by the experimental data gathered at various
$Ra$
and
$\ell$
. We conclude by showing that boundary layer corrections can persistently modify the Prandtl number dependence of
$Nu$
at large
$Ra$
, for
$Pr\gtrsim 1$
.
For over 16 years, the Precipitation Radar of the Tropical Rainfall Measuring Mission (TRMM) satellite detected the three‐dimensional structure of significantly precipitating clouds in the tropics ...and subtropics. This paper reviews and synthesizes studies using the TRMM radar data to present a global picture of the variation of convection throughout low latitudes. The multiyear data set shows convection varying not only in amount but also in its very nature across the oceans, continents, islands, and mountain ranges of the tropics and subtropics. Shallow isolated raining clouds are overwhelmingly an oceanic phenomenon. Extremely deep and intense convective elements occur almost exclusively over land. Upscale growth of convection into mesoscale systems takes a variety of forms. Oceanic cloud systems generally have less intense embedded convection but can form very wide stratiform regions. Continental mesoscale systems often have more intense embedded convection. Some of the most intense convective cells and mesoscale systems occur near the great mountain ranges of low latitudes. The Maritime Continent and Amazonia exhibit convective clouds with maritime characteristics although they are partially or wholly land. Convective systems containing broad stratiform areas manifest most strongly over oceans. The stratiform precipitation occurs in various forms. Often it occurs as quasi‐uniform precipitation with strong melting layers connected with intense convection. In monsoons and the Intertropical Convergence Zone, it takes the form of closely packed weak convective elements. Where fronts extend into the subtropics, broad stratiform regions are larger and have lower and sloping melting layers related to the baroclinic origin of the precipitation.
Key Points
Deep convection takes different forms over land, ocean, and mountainous terrain
Location of deep convective precipitation on Earth depends on life cycle stage
Stratiform precipitation seen by TRMM varies in type and structure
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FZAB, GIS, IJS, KILJ, NLZOH, NUK, OILJ, SAZU, SBCE, SBMB, UL, UM, UPUK
We perform direct numerical simulations of wall sheared Rayleigh–Bénard convection for Rayleigh numbers up to $Ra=10^{8}$, Prandtl number unity and wall shear Reynolds numbers up to $Re_{w}=10\,000$. ...Using the Monin–Obukhov length $L_{MO}$ we observe the presence of three different flow states, a buoyancy dominated regime ($L_{MO}\lesssim \unicodeSTIX{x1D706}_{\unicodeSTIX{x1D703}}$; with $\unicodeSTIX{x1D706}_{\unicodeSTIX{x1D703}}$ the thermal boundary layer thickness), a transitional regime ($0.5H\gtrsim L_{MO}\gtrsim \unicodeSTIX{x1D706}_{\unicodeSTIX{x1D703}}$; with $H$ the height of the domain) and a shear dominated regime ($L_{MO}\gtrsim 0.5H$). In the buoyancy dominated regime, the flow dynamics is similar to that of turbulent thermal convection. The transitional regime is characterized by rolls that are increasingly elongated with increasing shear. The flow in the shear dominated regime consists of very large-scale meandering rolls, similar to the ones found in conventional Couette flow. As a consequence of these different flow regimes, for fixed $Ra$ and with increasing shear, the heat transfer first decreases, due to the breakup of the thermal rolls, and then increases at the beginning of the shear dominated regime. In the shear dominated regime the Nusselt number $Nu$ effectively scales as $Nu\sim Ra^{\unicodeSTIX{x1D6FC}}$ with $\unicodeSTIX{x1D6FC}\ll 1/3$, while we find $\unicodeSTIX{x1D6FC}\simeq 0.30$ in the buoyancy dominated regime. In the transitional regime, the effective scaling exponent is $\unicodeSTIX{x1D6FC}>1/3$, but the temperature and velocity profiles in this regime are not logarithmic yet, thus indicating transient dynamics and not the ultimate regime of thermal convection.
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