Moderate Rayleigh Numbers Research Articles

Interactive free convection from a pair of vertical tube-arrays at moderate Rayleigh numbers

Steady laminar free convection from a pair of vertical arrays of equally-spaced, horizontal isothermal cylinders set in free air, is studied numerically. A specifically developed computer-code based on the SIMPLE-C algorithm is used for the solution of the mass, momentum and energy transfer governing equations. Simulations are performed for pairs of tube-arrays consisting of 1–4 circular cylinders, for center-to-center horizontal and vertical spacings from 1.4 to 24 cylinder-diameters, and from 2 to 12 cylinder-diameters, respectively, and for values of the Rayleigh number based on the cylinder-diameter in the range between 10 2 and 10 4. It is found that any cylinder may exhibit either enhanced or reduced Nusselt numbers with respect to the case of single tube-array, depending on its location in the array, on the geometry of the array, as well as on the Rayleigh number. Heat transfer dimensionless correlating equations are also proposed.

NUMERICAL ANALYSIS OF THE NATURAL CONVECTION IN HORIZONTAL ANNULI AT LOW AND MODERATE Ra

The natural convection at low and moderate Rayleigh numbers (Ra) incylindrical horizontal annuli with imposed temperatures in both surfaces isnumerically studied. This flow inside concentric cylinders classic configuration has a wide range of practical and technological applications, which justifies its growing studies efforts. In this work, the governing equations are discretized by the volume finite technique over a staggered grid, with second-order accuracy in space and time. The flow pattern is presented by several Rayleigh numbers, with an analysis of the heat transfer coefficient and flow properties. Furthermore, a three-dimensional field is shown at a moderate Ra number. The results showed a good agreement with the experimental data.

Mass transfer from inclined cylinders at moderate Rayleigh number including the effects of end face boundary conditions

Natural convection from inclined cylinders is studied using an electrochemical mass transfer technique. The effect of end face boundary conditions (active vs inactive) has been studied for a range of Rayleigh numbers 4 × 10 9 < Ra L ∗ < 1 × 10 12 , angles of inclination 0°, 30°, 60° and 90°, cylinder aspect ratios (length-to-diameter) of 0.63, 1.48 and 2.34 and a nominal Schmidt number of 2280. The data indicate that at these aspect ratios, the boundary condition on the bottom end face significantly affects the mass transfer on the curved part of the cylinder, due to depleted solution from the bottom rising along the cylinder. Inactive base data can be correlated by a single length scale that includes the effect of aspect ratio as well as inclination angle. Active base data do not correlate with this length scale, but match well with the sum of correlations for the mass transfer from downward-facing discs and the inactive base data in the current study.

Cell pattern of thermal convection induced by internal heating

A numerical simulation of thermal convection induced by internal heating in a three-dimensional thin flat box is performed. The upper boundary is held at a constant temperature and the lower boundary is thermally insulated with a uniform internal heating in a fluid layer. We observe hexagonal cells with downwelling at their center in moderate Rayleigh number (RaH≲30RaHc). As the Rayleigh number increases, the cell size increases. We also observe a pattern transition: sheet-like downwellings gradually develop around the columnar downwelling with increasing Rayleigh number. At ∼30RaHc, hexagonal cells with sheet downwelling at their center (spoke pattern) is observed. These sheets can be seen only near the upper boundary. At 30RaHc∼70RaHc, the cells experience successive transitions to the pattern which are dominated by sinking sheet.

Numerical upper bounds on convective heat transport in a layer of fluid of finite Prandtl number: Confirmation of Howard’s analytical asymptotic single-wave-number bound

By means of the Howard-Busse method of the optimum theory of turbulence we investigate numerically the upper bounds on the Nusselt number in a heated-from-below horizontal layer of fluid of finite Prandtl number for the case of rigid boundaries. The bounds are obtained by the solutions of the Euler-Lagrange equations of a variational problem possessing up to three wave numbers. The obtained results are compared to the numerical results for the case of fluid layer with stress-free boundaries [N. K. Vitanov and F. H. Busse, “Upper bounds on heat transport in a horizontal fluid layer with stress-free boundaries,” ZAMP 48, 310 (1997)] as well as to the numerical and analytical asymptotic results obtained by Howard [“Heat transport by turbulent convection,” J. Fluid Mech. 17, 405 (1963)], Busse [“On Howard’s upper bound for heat transport by turbulent convection,” J. Fluid Mech. 37, 457 (1969)], and Strauss [“On the upper bounding approach to thermal convection at moderate Rayleigh numbers, II. Rigid boundaries,” Dyn. Atm. Oceans 1, 77 (1976)]. We show that for low and intermediate Rayleigh numbers the numerical bounds are positioned below the analytical asymptotic bounds obtained by Howard and Busse. For large Rayleigh numbers the numerical bounds tend to approach the analytical asymptotic bounds. We confirm numerically the bound obtained by Howard for the case of one-wave-number solution of the Euler-Lagrange equations. As the region of validity of the results of the analytical asymptotic theory for solutions of the Euler-Lagrange equations with two and three wave numbers lies in the area of very high Rayleigh numbers the values of the second and third wave numbers are different from their analytical asymptotic values for the values of the Rayleigh number reached by the numerical computation.

Gain-scheduled controller for the suppression of convection at high Rayleigh number

Recent studies in the feedback control of Rayleigh-Bénard convection indicate that one can sustain the no-motion state at a moderate supercritical Rayleigh number (Ra) using only proportional compensation. However, stabilization occurs at a much higher Rayleigh number using linear-quadratic-Gaussian (LQG) control synthesis. The restriction is that the convection model is linear. In this paper, we show that a comparable degree of stabilization is achievable for a fully nonlinear convection state. The process is demonstrated in two stages using a fully nonlinear, 3D Boussinseq model, compensated by a reduced-order LOQ controller and a gain-schedule table. In the first stage a fully-developed convective state is suppressed through the control action at a moderate supercritical Ra. After the residual convection decays to a sufficiently small amplitude, in the second stage, we increase the Ra by a large step and switch the compensator gains using the gain-schedule table. During this change the control action is in place. Our nonlinear simulation results suggest that the nonlinear system can be stabilized to the limit predicted by the linear analysis. The simulation shows that the large Ra jump induces a large transient temperature in the conductive component, which appears to have very small impact on the stabilization.

Distribution of heat sources in vertical open channels with natural convection

In this paper we use the constructal method to determine the optimal distribution and sizes of discrete heat sources in a vertical open channel cooled by natural convection. Two classes of geometries are considered: (i) heat sources with fixed size and fixed heat flux, and (ii) single heat source with variable size and fixed total heat current. In both classes, the objective is the maximization of the global thermal conductance between the discretely heated wall and the cold fluid. This objective is equivalent to minimizing temperature of the hot spot that occurs at a point on the wall. The numerical results show that for low Rayleigh numbers (∼10 2), the heat sources select as optimal location the inlet plane of the channel. For configuration (i), the optimal location changes as the Rayleigh number increases, and the last (downstream) heat source tends to migrate toward the exit plane, which results in a non-uniform distribution of heat sources on the wall. For configuration (ii) we also show that at low and moderate Rayleigh numbers ( Ra M ∼ 10 2 and 10 3) the thermal performance is maximized when the heat source does not cover the entire wall. As the flow intensity increases, the optimal heat source size approaches the height of the wall. The importance to free the flow geometry to morph toward the configuration of minimal global resistance (maximal flow access) is also discussed.

Emergent Anisotropy and Flow Alignment in Viscous Rock

A novel class of nonlinear, visco-elastic rheologies has recently been developed by MUHLHAUS et al. (2002a, b). The theory was originally developed for the simulation of large deformation processes including folding and kinking in multi-layered visco-elastic rock. The orientation of the layer surfaces or slip planes in the context of crystallographic slip is determined by the normal vector the so-called director of these surfaces. Here the model (MUHLHAUS et al., 2002a, b) is generalized to include thermal effects; it is shown that in 2-D steady states the director is given by the gradient of the flow potential. The model is applied to anisotropic simple shear where the directors are initially parallel to the shear direction. The relative effects of textural hardening and thermal softening are demonstrated. We then turn to natural convection and compare the time evolution and approximately steady states of isotropic and anisotropic convection for a Rayleigh number Ra=5.64x10(5) for aspect ratios of the experimental domain of 1 and 2, respectively. The isotropic case has a simple steady-state solution, whereas in the orthotropic convection model patterns evolve continuously in the core of the convection cell, which makes only a near-steady condition possible. This near-steady state condition shows well aligned boundary layers, and the number of convection cells which develop appears to be reduced in the orthotropic case. At the moderate Rayleigh numbers explored here we found only minor influences in the change from aspect ratio one to two in the model domain.

Zonal jets in rotating convection with mixed mechanical boundary conditions

Large-scale zonal flows, as observed on the giant planets, can be driven by thermal convection in a rapidly rotating spherical shell. Most previous models of convectively-driven zonal flow generation have utilized stress-free mechanical boundary conditions ( FBC) for both the inner and the outer surfaces of the convecting layer. Here, using 3D numerical models, we compare the FBC case to the case with a stress free outer boundary and a non-slip inner boundary, which we call the mixed case ( MBC). We find significant differences in surface zonal flow profiles produced by the two cases. In low to moderate Rayleigh number FBC cases, the main equatorial jet is flanked by a strong, high-latitude retrograde jets in the northern and southern hemispheres. For the highest Rayleigh number FBC case, the equatorial jet is flanked by strong reversed jets as well as two additional large-scale alternating jets at higher latitudes. The MBC cases feature stronger equatorial jets but, much weaker, small-scale alternating zonal flows are found at higher latitudes. Our high Rayleigh number FBC results best compare with the zonal flow pattern observed on Jupiter, where the equatorial jet is flanked by strong retrograde jets as well as small-scale alternating jets at high latitude. In contrast, the MBC results compare better with the observed flow pattern on Saturn, which is characterized by a dominant prograde equatorial jet and a lack of strong high latitude retrograde flow. This may suggest that the mechanical coupling at the base of the jovian convection zone differs from that on Saturn.

Exploring super-critical properties of secondary flows of natural convection in inclined channels

The super-critical secondary flow properties of natural convection in an inclined channel (NCIC) have been explored numerically by virtue of Boussinesq type Navier–Stokes equations. The primary flow in the inclined channel heated from below was assumed as fully developed so that the secondary flow in the cross-section can be studied as a Bénard type problem without regarding the primary speed profiles but allowing the buoyancy force depending on the channel inclination. The ratios of channel spacing to width for the inclined channel have been considered to recognize the sensitivity to the secondary flow patterns from the ratio, with the Rayleigh number changing from 10 4 to 10 6. For moderate Rayleigh numbers ranging from 10 4 to 10 5, it was revealed that the numerically evaluated spatial wavelength of the secondary flow appears a relatively good consistence with measured data, and the pattern of the secondary flow is sensitive to the inclination angle.

Direct numerical simulations of free convection beneath an air–water interface at low Rayleigh numbers

Direct numerical simulations of a cooling air–water interface were employed to determine the structure of the temperature, velocity, and vorticity fields in the thin thermal boundary layer formed at the free surface. The simulations were performed at low to moderate Rayleigh numbers. In this flow, the turbulence is initiated by the Rayleigh instability at the interface and is maintained by buoyant production. Visualizations of the flow reveal that the temperature field at the interface is composed of large warm patches surrounded by cooler dense fluid which accumulates in thin bands. The cool fluid associated with the bands initially falls in sheets, but rapidly forms descending tubes and plumes. The turbulence statistics were scaled both with outer and inner variables. The latter scaling is based on the so-called surface strain model which is essentially consistent with Townsend’s inner scaling. It is found that the temperature statistics collapse well using inner variables. On the other hand, the vertical velocity scales well with inner variables within the thermal boundary layer, but at greater depths it becomes more appropriate to use outer scaling. The anisotropic nature of the velocity statistics in the core of the flow is ascribed to the relatively low Rayleigh numbers used in the simulations. An explanation for this anisotropy is offered based on a detailed examination of the turbulence kinetic energy balances.

From stationary to complex time-dependent flows at moderate Rayleigh numbers in two-dimensional annular thermal convection

Two-dimensional nonlinear thermal convection in a cylindrical annulus is numerically analyzed for a Boussinesq fluid of low Prandtl number σ=0.025. For a fixed value of the radius ratio, η=0.3, different types of steady columnar patterns are found. The stability of these convection patterns and the spatial interaction between them, which result in the formation of mixed modes, are investigated by considering the full nonlinear set of Navier–Stokes equations. Special attention is paid to the strong spatial interaction of the initially unstable modes with wavenumbers n=2 and n=4, which leads, through global bifurcations, to multiple stable quasi-periodic states of the system. A detailed picture of the nonlinear dynamics until temporal chaotic patterns set in is presented and understood in terms of local and global symmetry-breaking bifurcations of the O(2)-symmetric system.

Numerical Investigation of Natural Convection inside Complex Enclosures

In this article, the analyses of heat transfer and free convective motion have been carried out numerically for various structures. The solution is based on a finite element method with the frontal solver to examine the flow parameters and heat transfer characteristics. Several dome configurations--such as flat, inclined, and dome shapes--are considered for the top of the enclosure. A general conic equation is considered to represent the dome as circular, elliptical, parabolic, or hyperbolic shape. The findings from this study indicate that the convective phenomenon is greatly influenced by the shape of the top cover dome and tends to form a secondary core even at a moderate Rayleigh number when compared with an equivalent rectangular enclosure. In addition, the circular and elliptical shapes of the dome give higher heat transfer rate. The effect of various "offset" of the dome and inclined roof on convective heat transfer is also found to be quite significant. However, beyond 0.3 of offset of the top cover for the dome and inclined roof, the change in overall heat transfer rate is minimal. The heat transfer coefficients of dome shaped and inclined roof enclosures are given and discussed.

Natural convection from narrow horizontal plates at moderate Rayleigh numbers

The present work deals with the natural convection flow and heat transfer from a horizontal plate cooled from above. Experiments are carried out for rectangular plates having aspect ratios between φ=0.036 and 0.43 and Rayleigh numbers in the range 290⩽ Ra w⩽3.3×10 5. These values of Ra w and φ have been selected below those commonly considered in previous research in view of a future application to the design of printed circuit boards. The plates are made of two different metals, copper and steel. The choice of a metal is relevant to the present problem because the plates are heated by means of an electric current. Important variations of the surface temperature are observed along the transverse direction for the steel plates. The surface of the copper plates is almost isothermal because of the high thermal conductivity of the metal. Calculations for a semi-infinite plate are carried out to predict the transverse profiles of the surface temperature and heat flux and to visualize the structure of the flow. Three-dimensional calculations are also used at a qualitative level to observe the changes in the flow structure due to the finite length of the plate. Present results are compared with both previous experimental work and analyses that are based on boundary layer theory. It is shown that analyses for an infinite boundary layer are not completely applicable to the present problem because of its different physics. The most relevant feature of the natural convection flow, which is not predicted by boundary layer analyses, is a thermal plume rising near the center of the plate. Present heat transfer results differ from previous experimental work because of the lower Rayleigh numbers and aspect ratios investigated here. The Nusselt number is found to depend on Ra w n , with the exponent n=0.17 being lower than most of the values reported in the literature. This comparatively low value is related to the transverse conduction of heat through the air, which becomes increasingly significant as Ra w approaches zero. It is shown that such a low- Ra w effect can be accounted for in a physically consistent manner by adding a constant term to the heat transfer correlation. On the other hand, it is found that the Nusselt number does not significantly depend on the aspect ratio in the range of φ investigated contrary to what has been previously reported for wider plates.

Convective heat transport in compressible fluids.

We present hydrodynamic equations of compressible fluids in gravity as a generalization of those in the Boussinesq approximation used for nearly incompressible fluids. They account for adiabatic processes taking place throughout the cell (the piston effect) and those taking place within plumes (the adiabatic temperature gradient effect). Performing two-dimensional numerical analysis, we reveal some unique features of plume generation and convection in transient and steady states of compressible fluids. As the critical point is approached, the overall temperature changes induced by plume arrivals at the boundary walls are amplified, giving rise to overshoot behavior in transient states and significant noise in the temperature in steady states. The velocity field is suggested to assume a logarithmic profile within boundary layers. Random reversal of macroscopic shear flow is examined in a cell with unit aspect ratio. We also present a simple scaling theory for moderate Rayleigh numbers.

Natural convection inside dome shaped enclosures

In the present paper the analysis of heat transfer and free convective motion have been carried out numerically for dome shaped enclosures. The solution method is based on the finite element technique with the frontal solver and is used to examine the flow parameters and the heat transfer characteristics inside dome shaped enclosures of various offsets. In formulating the solution a general conic equation is considered to represent the dome of circular, elliptical, parabolic and hyperbolic shapes. The numerical results indicate that the circular and elliptical shapes of dome give higher heat transfer rate and offset of the dome effects convective heat transfer quite significantly. However, beyond 0.3 top dome offset, the change in overall heat transfer rate is not significant. In addition, the convective phenomenon influenced by a dome shaped cover results in establishing a secondary core region even at a moderate Rayleigh number when compared with an equivalent rectangular enclosure. A good comparison between the present numerical predictions and the previous published data is achieved.

An adaptive remeshing technique for laminar natural convection problems

An adaptive remeshing procedure based on interpolation error has been developed for solving laminar natural convection problems. A simple relation between the Rayleigh number and the equilibration constant is introduced to increase the efficiency of the method. Unstructured meshes have been regenerated based on an initial solution and used in the calculations. This avoids the expensive mesh sensitivity study. Two typical natural convection problems are solved to demonstrate the present technique. Excellent performance has been observed at moderate and high Rayleigh numbers although present method gives good results over the whole range of Rayleigh numbers considered.

Experimental laminar Rayleigh-Bénard convection in a cubical cavity at moderate Rayleigh and Prandtl numbers

Rayleigh-Benard convection in a cubical cavity with adiabatic or conductive sidewalls is experimentally analyzed at moderate Rayleigh numbers (Ra ≤ 8 × 104) using silicone oil (Pr=130) as the convecting fluid. Under these conditions the flow is steady and laminar. Three single-roll-type structures and an unstable toroidal roll have been observed inside the cavity with nearly adiabatic sidewalls. The sequence from the conductive state consists of a toroidal roll that evolves to a diagonally oriented single roll with increasing Rayleigh number. This diagonal roll, which is stabilized by the effect of the small but finite conductivity of the walls, shifts its axis of rotation towards to two opposite walls, and back to the diagonal orientation to allow for the increase in circulation that occurs as the Rayleigh number is further increased. Conduction at the sidewalls modifies this sequence in the sense that the two initial single rolls finally evolve into a four-roll structure. Once formed, this four-roll structure remains stable when decreasing the Rayleigh number until the initial single diagonally oriented roll is again recovered. The topology and the velocity fields of all structures, characterized with visualization and particle image velocimetry, respectively, are in good agreement with numerical results reported previously for the cavity with adiabatic walls, as well as with the numerical predictions obtained in the present study for perfectly conducting lateral walls.

Electrochemical Mass Transfer at Moderate Rayleigh Numbers

Electrochemical Mass Transfer at Moderate Rayleigh Numbers

Computation of turbulent buoyant flows in enclosures with low-Reynolds-number k-ω models

This work deals with the computation of turbulent buoyant convection flows with thermal stratification using the low-Reynoldsnumber (LRN) k-x model. When applying the k-e model to buoyancy-driven cavity flows induced by diAerentially heated side walls, a problem commonly encountered at moderate Rayleigh numbers (Raa 10 10 ‐1 0 12 ) is that the model is not capable of giving gridindependent predictions owing to the transition regime along the vertical walls. It was found that the buoyancy source term for the turbulence kinetic energy, Gk, exhibits strong grid sensitivity, as this term is modelled with the Standard Gradient DiAusion Hypothesis (SGDH). By introducing a damping function into this term, the above grid-dependence problem is eliminated and, additionally, the modelled Gk renders correct asymptotic behavior near the vertical wall. The mechanism held in the k-x model for describing the onset of transition is analyzed. The present approach is simple for practical use and gives reasonable predictions. ” 1999 Elsevier Science Inc. All rights reserved.