B. Gebhart

Buoyancy-induced flow arising from a line thermal source on an adiabatic vertical surface

Abstract A study of the laminar naturai convection flow arising from a steady line thermal source positioned at the leading edge of a vertical adiabatic surface is carried out. The resulting two-dimensional boundary-layer flow is analyzed and the governing equations solved numerically for a Prandtl number range of 0.01–100. The dependence of the surface temperature, the velocity level, the boundary region thickness and other physical aspects of the flow on the Prandtl number is determined. The numerical results obtained allow an evaluation of the velocity and temperature fields in the generated flow. The results are also compared with those obtained for a freely rising plane plume and for a vertical isothermal surface. Several interesting features concerning this and similar flows are brought out.

An experimental investigation of natural convection flow on an inclined surface

Abstract A thermocouple and two hot-film anemometer probes in the form of an “inverted V” were used to investigate the boundary region flow formed over an inclined surface dissipating a uniform heat flux. Temperature and the longitudinal and transverse components of velocity were measured. A single longitudinal vortex system arises. It causes a spanwise variation of the temperature and velocity fields, which results in a spanwise variation of heat transfer. The spanwise mean-flow variation, or vortex system, arises first downstream for angles greater than 11°. Our local measurements permit detailed comparison with well-established disturbance mechanisms in vertical flows. Frequency filtering of periodic disturbances is again found. The beginning of transition is postulated in terms of disturbance magnitude. The location is found to depend on the distance from the leading edge, in addition to the Grashof number, in the same way as in vertical flows.

An experimental determination of transition limits in a vertical natural convection flow adjacent to a surface

This paper reports the results of an experimental investigation to determine transition mechanisms and limits in gases at high pressure levels. We sought also to refine further the parameters for transition, in particular the role of kinematic viscosity. In flow adjacent to a vertical uniform-flux surface in nitrogen, pressures to 16 atm were used. Both mean and disturbance quantities for the temperature and velocity fields were measured for various values of the heat flux, downstream location and ambient pressure level. Hot-wire and fine thermocouple probes were used. We found that the velocity and thermal fields remain closely coupled. Velocity, or fluid-dynamic, transition is immediately followed by thermal transition. Each was detected as a decrease in the rate of increase of both the maximum velocity and the overall temperature difference, respectively, from the laminar downstream trends. Also, the ends of transition for the velocity and the thermal fields, respectively, signalled by no further appreciable change in the intermittency distributions, were simultaneous. These results re-affirm the finding that the events of transition are not correlated by the Grashof number alone. An additional dependence on both downstream location and pressure level arises. A fixed value of the parameter

Transition and transport in a buoyancy driven flow in water adjacent to a vertical uniform flux surface

Q_{BT} = qB^{frac{2}{15}}= 290

Buoyancy force reversals in vertical natural convection flows in cold water

characterizes the beginning of transition, where q is the fifth root of the local non-dimensional wall heat flux and B is the unit Grashof number. The end of transition, on the other hand, is best correlated by

Higher order approximations to the natural convection flow over a uniform flux vertical surface

Q_{ET} = QB^{frac{1}{30}} = 11.4

Heat transfer and ice-melting in ambient water near its density extremum

, where Q is the fifth root of the local non-dimensional total heat convected in the boundary region. A re-examination of other transition studies, in both gases and liquids, supports these correlations, although many such data were not determined with fast response to local sensors. There remains a small level of uncertainty in establishing exact limits for transition, since the apparently proper standards for determining them are very difficult to apply precisely in experiments. However, such limits are very important in separating regimes of different transport mechanisms.

Visualization of the flow adjacent to a vertical ice surface melting in cold pure water

Abstract This experiment, in water at room temperature, accurately maps changes in the mean temperature distribution from laminar flow, through changes during transition, to full turbulent flow. Temperature disturbance levels are also determined. Simultaneously, very accurate measurements of local heat-transfer parameters were made to high Rayleigh numbers. Our objective was to interrelate different past criteria for the beginning of thermal transition and to find any characteristic patterns during transition. It was found that such criteria are internally consistent but some systematically disagree with others because of differences in methodology. Mean profiles and transport are relatively unambiguous. Further, some events during transition may also be correlated, from our measurements, both in terms of mean profiles and heat transfer. Transport in laminar, in transition and in turbulent flow are mapped. In the first regime, it agrees with theory and in the last, with the correlation of Vliet and Liu [1]. Mean temperature distributions in full turbulence are in very close agreement with the new transport theory of George and Capp [2].

Buoyancy induced flows adjacent to horizontal surfaces in water near its density extremum

Calculated numerical results are presented for laminar buoyancy-induced flows driven by thermal transport to or from a vertical isothermal surface in cold pure and saline water wherein a density extremum arises. The present calculations specifically explore the consequences of temperature conditions wherein the buoyancy force reverses across the thermal region owing to the presence of a density extremum within the region. Such conditions commonly occur in terrestrial waters and in technological processes utilizing cold water. The linear approximation of density dependence on temperature, used in conventional analysis, is here replaced by a very accurate non-linear density equation of state for both pure and saline water. This permits an accurate treatment of such flows for bounding temperatures up to 20 °C at ambient salinity and pressure levels from 0 to 40 p.p.t. and 1 to 1000 bars, respectively. The results may be applied to the melting or slow freezing of a vertical ice surface in pure water as well as to a heated or cooled vertical isothermal surface in pure or saline water. For example, buoyancy force reversals arise for a vertical ice surface at 0 °C melting in fresh water between 4 °C and 8 °C at atmospheric pressure. Temperature conditions for which buoyancy force reversals occur are of special interest because of the resulting anomalous flow behaviour and low surface heat-transfer rates. The transition from conditions with no buoyancy-force reversal to those resulting in a large buoyancy-force reversal is accompanied by as much as 50% decrease in surface heat transfer. This produces a corresponding trend in the melt rate of a vertical ice surface in pure water. Sufficiently strong buoyancy force reversals are found to cause local flow reversal either at the edge of the flow layer or near the surface. Conditions are determined for which flow reversals occur at each of these locations. These local flow reversals are the precursors of convective inversion, that is, of the reversal of the net flow direction with changing ambient medium temperature. Limits on conditions for convective inversion are determined. Calculated transport is compared with previous experimental results, with good agreement throughout the several regions of such complicated flows. The calculations indicate that such flows are relatively very weak. However, their form may lead to early laminar instability.

The interaction of unequal laminar plane plumes

Abstract Perturbation analysis of higher order boundary-layer effects for convection flow over a semi-infinite vertical uniform flux surface is presented. Using asymptotic matching technique, three term inner and outer expansions have been obtained. Eigenvalues and their eigenfunctions associated with the inner expansions have also been investigated and it has been shown that their contribution to these three term expansions is identically zero. The numerical results for Pr = 0.733 and 6.7 show that the higher order corrections to the local temperature difference and the local skin friction are negative but are positive to the local Nusselt number. Considerations of global momentum and buoyancy indicate an indeterminacy of O(1) in the expression for total drag.