Vertical End Walls Research Articles

A laboratory model of a convectively driven ocean

Many large-scale flows in the ocean are driven by an imposed horizontal density gradient and the resulting circulation is strongly influenced by the Earth's rotation. Some of the essential features of such flows have been incorporated into a laboratory model by differentially heating and cooling the vertical end walls of a shallow rectangular cavity rotating about a vertical axis. Buoyancy driven boundary currents produced by the heating and cooling were unstable and the resulting eddy structures eventually dominated the system. A broad mean flow, perpendicular to the side walls, developed in the central region of the cavity. The resultant steady-state flow had a barotropic component consisting of two large-scale gyres of cyclonic and anticyclonic senses. In this paper, the evolution of the flow towards its final steady state is described. Measurements of the time-scales for the establishment of stratification in the cavity, point to the existence of two dynamical density adjustment modes.

Convection Near the Temperature of Maximum Density for High Rayleigh Number, Low Aspect Ratio, Rectangular Cavities

Natural convection in rectangular cavities with the vertical endwalls differentially heated about the temperature of maximum density of water was studied in the laboratory. In order to control heat losses, the experiments were conducted in a large cold room with the air temperature maintained at 4°C. The Rayleigh number was varied over the range 105 to 108 and the aspect ratio over the range 0.1 to 0.5. The flow field consisted of a double cellular circulation, and the observed temperature field, flow field, and heat transfer rates were compared with analytical predictions in the diffusive regime. The experiments demonstrated that the interior sinking flow was unstable for Ra > 107.

Blockage of natural convection boundary layer flow in a multizone enclosure

Two separate mechanisms can be responsible for natural convection flow between the hot and cold zones of a multizone enclosure: (1) bulk density differences created by temperature differences between the fluid in the hot and cold zones (bulk-density-driven flow) and, (2) thermosyphon “pumping” generated by boundary layers or plumes (motion-pressure-driven flow). This paper reports the results of an experimental study that examines the transition between flow regimes, as a function of aperture size, in a two-zone enclosure with heated and cooled end walls. A constant heat flux boundary condition was maintained on one vertical end wall, and an isothermal cold temperature sink was maintained on the opposite vertical end wall. All of the remaining surfaces were highly insulated. The transition between the boundary-layer-driven regime and the bulk-density-driven regime was established as a function of the geometry of the aperture in a partition that separated the hot and cold zones. The results of the study demonstrate that transition from the boundary-layer-driven regime to the bulk-density-driven regime is caused by blockage of the boundary layer flow when the area of the flow aperture is reduced below a critical value. A preliminary model has been developed which predicts that the critical aperture area for the onset of flow blockage is directly proportional to the number of active heat transfer surfaces and inversely proportional to the Rayleigh number which characterizes the level of heating and cooling provided to the active heat transfer surfaces.

Transient three‐dimensional flow in an enclosure with a hot spot on a vertical wall

AbstractA numerical method for the solution of the vector potential/vorticity vector formulation of the transient, fully three‐dimensional Navier‐Stokes energy and continuity equations has been applied to simulate the development of natural convective flow within a ‘box’ after a sudden temperature change on a vertical portion of the wall. Only one cavity size has been considered, this having a vertical height of three times its width and a horizontal length of six times its width. A single heated rectangular hot spot or ‘element’ on an otherwise adiabatic wall is centred between the vertical end walls. The opposite vertical wall is held at the intial fluid temperature, and all other walls are assumed to be adiabatic. Fluid properties have been assumed constant except for the density change with temperature that gives rise to the buoyancy force. The numerical method is an underrelaxation Gauss‐Seidel method using finite differencing at each time step. Solutions have been obtained for a Prandtl number of 0.71, for Rayleigh numbers, based on the width, of between 0 and 100000 and for a number of heated element locations and sizes.

The effect of slab width on the stability of natural convection in confined saturated porous media

Stability boundaries separating different states of bimodal convection in a box of saturated porous material with impermeable bounding faces are determined. Two opposing vertical end walls are always insulated, while the thermal conditions on the other set of opposing vertical side walls range from insulated to perfectly conducting, as measured by the Biot number based on box height. The temperature difference between the hot lower plate and the cooler upper plate provides the mechanism for instability. The eigenvalue problem for the critical Rayleigh number is solved numerically over a range of box sizes and side-wall heat transfer conditions. For small values of the distance H2 between conducting side walls, agreement with previous asymptotic analysis is obtained. New results for boxes with planform dimensions both comparable to the box height show the existence of isolated regions rich in bimodal structure. These islands are separated by broad regions having modal properties identical to those found for conducting side walls in the limit H2→0. The numerical results for O(1) box dimensions continue to exhibit the stabilizing effect of side-wall heat transfer as previously observed for thin slabs.

The influence of side wall heat transfer on convection in a confined saturated porous medium

The effect of side wall heat transfer on the stability of natural convection in a vertically oriented finite slab of saturated porous material is considered. All six bounding faces are impermeable. A temperature contrast between the top and bottom horizontal surfaces provides the mechanism for destabilization. The narrow vertical end walls are perfectly insulated. The thermal conditions on the broad vertical side walls range from perfectly insulating to fully conducting as determined by the value of B, the Biot number based on slab height. An asymptotic analysis of the general solution is made in the narrow gap limit ε≪1, where ε is the cross-slab width-to-height ratio. In this case the relevant heat transfer variable is B̄=εB, the Biot number based on the narrow slab width. In the limit ε → 0 when B̄=O(1), including the case B̄→∞ that corresponds to a linear side wall temperature profile, tall, vertical, three-dimensional, finger-like cells are found at the critical Rayleigh number Rc=π2/ε2. In the limit B̄ → 0 corresponding to perfect insulation, one obtains two-dimensional, O(1) aspect ratio rolls with axes normal to the side walls at the critical value Rc=4π2. These two-dimensional rolls predominate only for B̄=O(ε2), and transition to tall narrow three-dimensional cells occurs when O(ε2)≪ B ≪ O(1).

Three‐dimensional flow beneath a slow spreading ridge axis: A dynamic contribution to the deepening of the Median Valley toward fracture zones

The axial valley along a slow spreading mid‐ocean ridge may be explained by a vertical pressure gradient due to viscous flow of asthenosphere upwelling in a relatively narrow conduit beneath the ridge axis. Along a ridge segment, the axial valley floor deepens by as much as 2–3 km over distances of several tens of kilometers approaching a ridge‐transform intersection. This deepening may be explained, in part, by a pressure gradient associated with horizontal flow in the ridge axis conduit. Vanishing vertical velocity on a conduit end wall at a ridge‐transform intersection results in reduced vertical flow within about one conduit width of the intersection. Horizontal flow along the conduit, toward the intersection, must occur to form lithosphere at a uniform rate along the ridge axis. A simple model with a conduit of uniform width that terminates with vertical planar endwalls at ridge‐transform intersections is considered. The model is based on an analytical solution for flow in a narrow conduit and boundary layer approximations for the flow structure near the vertical end walls. This model indicates that the induced horizontal flow and pressure gradient extend for a distance along the ridge axis of several lithosphere thicknesses from an intersection. For reasonable values of the conduit width and asthenosphere viscosity the model predicts that induced horizontal flow can contribute significantly to the deepening of an axial valley approaching an intersection.

Experiments on transient natural convection in a cavity

A laboratory experiment is used to study the transient flow in an initially isothermal cavity at temperature T0 following the rapid change of the two vertical endwalls to temperatures T0 ± ΔT respectively. Individual temperature records are taken and the transient flow in the entire cavity is visualized with the aid of a tracer technique. It is shown that an oscillatory approach to final steady-state conditions exists for certain flow regimes, although the form of the oscillatory response is different to that suggested by previous work. It is argued that this oscillatory behaviour is due to the inertia of the flow entering the interior of the cavity from the sidewall boundary layers, which may lead to a form of internal hydraulic jump if the Rayleigh number is sufficiently large.

Natural Convection in a Rectangular Porous Cavity With One Permeable Endwall

This paper describes a theoretical and experimental study of two-dimensional, buoyancy-driven flow in a rectangular porous cavity with one permeable endwall. Connected to a constant temperature tank, the permeable end allows for natural recharge and discharge of the saturating fluid. The other vertical endwall is impermeable and maintained at a constant but higher temperature, thus inducing a buoyancy-driven flow. The theoretical study includes an asymptotic analysis developed for a shallow cavity with one permeable endwall and the numerical solutions of the power-law difference representation of the full governing equations. The experimental system consists of water-saturated glass beads packed in a rectangular cavity with a length-to-height aspect ratio of 3.17, for which the Rayleigh number can vary up to 120. Measurements were made of the steady-state temperature distribution in the cavity and the heat transfer rate from the impermeable endwall. It is shown that the constant pressure and temperature assumptions at the permeable wall, as employed in the theoretical analysis, satisfactorily predict the experimental data. Results are also compared with those existing in the literature.

Low Prandtl number convection between differentially heated end walls

Two-dimensional convection in a horizontal cavity of height D and length L, driven by differential heating of the two vertical end walls, is described. Of particular interest is the range of external parameters, at low Prandtl numbers, for which the circulation contains a parallel flow core. The simple core solution, asymptotically valid at small aspect ratio D/L, breaks down either as end effects extend into the center of the cavity, or as a secondary shear flow instability develops in the core itself.

A note on the stability of low-Prandtl-number Hadley circulations

This paper discusses the stability of the flow of a low-Prandtl-number liquid contained in a shallow slot with differentially heated vertical endwalls. The effect of thermally insulating boundaries at the top and bottom of the container on wavelength selection is emphasized. Stability calculations indicate that, for Prandtl number Pr in the range 0.015 < Pr < 0.27, the first perturbations to grow are overstable (oscillatory) longitudinal rolls with axes perpendicular to the endwalls, and with very large cross-stream wavelengths of about 9 to 15 layer depths. Previous studies using thermally conducting boundaries predict critical wavelengths of about three layer depths. The new results are in substantial agreement with an experiment using a differentially heated layer of mercury with aspect ratio (depth/length) 0.047 in both horizontal directions. The implications of the long-wavelength instability for the interpretation of thermal oscillations observed in other smaller-aspect-ratio configurations is discussed.

LECLERCQIA COMPLEXA (LYCOPSIDA, MIDDLE DEVONIAN): ITS ANATOMY, AND THE INTERPRETATION OF PYRITE PETRIFACTIONS

The development and/or modification of special acid etching and macerating techniques permits pyritized fossils of Leclercqia complexa to be separated into their carbonaceous wall remains and pyrite lumen—pit casts. These carbonized and pyritic portions can then be studied separately by both light and SEM microscopy and information from both modes compared with that obtainable from carbonaceous compression fossils of L. complexa. This combination of techniques, preservation modes and methods of analysis, allows a synthesis of information obtainable from each, as well as providing a check on errors of interpretation due to preservation mode and method of observation. The developmental sequence of the secondary wall thickenings of the protoxylem elements is shown to be annular to connected annular and spiral, with both exarch and mesarch maturation. The metaxylem shows a transition through reticulate to multiseriate round, oval, and elongate bordered‐pitted tracheids. Different wall facets of separated pyrite tracheid casts can be examined by the SEM and variations in wall structure and pitting characterized. Cells of the outer cortex are 6 to 10 times longer than wide, have thickened vertical walls and end walls that vary from nearly horizontal to angles of overlap of 50 degrees.

Leclercqia complexa (Lycopsida, Middle Devonian): Its Anatomy, and the Interpretation of Pyrite Petrifactions

The development and/or modification of special acid etching and macerating techniques permits pyritized fossils of Leclercqia complexa to be separated into their carbonaceous wall remains and pyrite lumen—pit casts. These carbonized and pyritic portions can then be studied separately by both light and SEM microscopy and information from both modes compared with that obtainable from carbonaceous compression fossils of L. complexa. This combination of techniques, preservation modes and methods of analysis, allows a synthesis of information obtainable from each, as well as providing a check on errors of interpretation due to preservation mode and method of observation. The developmental sequence of the secondary wall thickenings of the protoxylem elements is shown to be annular to connected annular and spiral, with both exarch and mesarch maturation. The metaxylem shows a transition through reticulate to multiseriate round, oval, and elongate bordered-pitted tracheids. Different wall facets of separated pyrite tracheid casts can be examined by the SEM and variations in wall structure and pitting characterized. Cells of the outer cortex are 6 to 10 times longer than wide, have thickened vertical walls and end walls that vary from nearly horizontal to angles of overlap of 50 degrees.

Natural convection in a shallow cavity with differentially heated end walls. Part 3. Experimental results

The steady motion of water in an enclosed rectangular cavity with differentially heated vertical end walls was studied experimentally, and the results are compared with the findings of parts 1 and 2. The depth-to-length ratios of the cavities were 102 and 1·9 × 102, and the Rayleigh number was allowed to vary sufficiently to enable a study to be made of the transition from a flow driven by the vertical wall boundary layers to one sustained by a longitudinal temperature gradient in the central sections of the cavity.