Showing papers on "Natural convection published in 1977"

A Correlating Equation for Forced Convection From Gases and Liquids to a Circular Cylinder in Crossflow

Abstract: A single comprehensive equation is developed for the rate of heat and mass transfer from a circular cylinder in crossflow, covering a complete range of Pr (or Sc) and the entire range of Re for which data are available. This expression is a lower bound (except possibly for RePr < 0.2); free-stream turbulence, end effects, channel blockage, free convection, etc., may increase the rate. In the complete absence of free convection, the theoretical expression of Nakai and Okazaki may be more accurate for RePr < 0.2. The correlating equation is based on theoretical results for the effect of Pr in the laminar boundary layer, and on both theoretical and experimental results for the effect of Re. The process of correlation reveals the need for theoretical results for the effect of Pr in the region of the wake. Additional experimental data for the effect of Pr at small Pe and for the effect of Re during the transition in the point of separation are also needed.

Free convection about a vertical flat plate embedded in a porous medium with application to heat transfer from a dike

Abstract: An analysis is made for steady free convection about a vertical flat plate embedded in a saturated porous medium at high Rayleigh numbers. Within the framework of boundary layer approximations, similarity solutions are obtained for a class of problems where wall temperature varies as xλ, i.e., a power function of distance from the origin where wall temperature begins to deviate from that of the surrounding fluids. Analytical expressions are obtained for boundary layer thickness, local and overall surface heat flux, and local and average heat transfer coefficients. Application to convective heat transfer about an isothermal dike intruded in an aquifer is discussed.

Convective instability: A physicist's approach

Abstract: A number of apparently disparate problems from engineering, meteorology, geophysics, fluid mechanics and applied mathematics are considered under the unifying heading of natural convection. After a review of the mathematical framework that serves to delineate these problems, the heuristic approach to Benard and Rayleigh convection is discussed with special attention to buoyancy and surface tension. Then consideration is given to some aspects of scaling, and the nondimensionalization of equations to a given problem. The thermohydrodynamic description of a Newtonian fluid is presented, and the Boussinesq--Oberbeck model. This is followed by a treatment of the linear stability problem, and a description of the basic ideas of Landau and Hopf concerning the bifurcation of secondary solutions. Quantitative, though approximate, estimations are given for quantities belonging to the nonlinear steady convective regime: flow velocity and temperature distribution. Higher-order, though steady, bifurcations are discussed, as well as the transition to turbulence, along with such time-dependent phenomena as relaxation oscillations. The paper concludes with an Appendix showing a simple application of the Leray--Schauder topological degree of a mapping.

Three-dimensional natural convection in a box: a numerical study

Abstract: The solution of the steady-state Navier–Stokes equations in three dimensions has been obtained by a numerical method for the problem of natural convection in a rectangular cavity as a result of differential side heating. In the past, this problem has generally been treated as though it were two-dimensional. The solutions explore the three-dimensional motion generated by the presence of no-slip adiabatic end walls. For Ra = 10 4 , the three-dimensional motion is shown to be the result of the inertial interaction of the rotating flow with the stationary walls together with a contribution arising from buoyancy forces generated by longitudinal temperature gradients. The inertial effect is inversely dependent on the Prandtl number, whereas the thermal effect is nearly constant. For higher values of Ra , multiple longitudinal flows develop which are a delicate function of Ra, Pr and the cavity aspect ratios.

A comprehensive correlating equation for laminar, assisting, forced and free convection

Abstract: A correlating equation for assisting convection was developed by combining correlating equations for pure free and pure forced convection. These component equations are based on laminar boundary-layer theory for an isothermal, vertical plate. Theoretical values for assisting convection indicate that the third root of the sum of the third powers gives the best representation, as contrasted with the choice and rationalization of the second or fourth power by prior investigators. This expression was modified by the addition of a limiting value Nuo to obtain a better representation below the domain of boundary-layer theory and was generalized for uniform heating and for spheres and horizontal cylinders by the appropriate choice of the characteristic length.

Analysis of melting in the presence of natural convection in the melt region

Abstract: An analysis of multidimensional melting is performed which takes account of natural convection induced by temperature differences in the liquid melt. Consideration is given to the melt region created by a heated vertical tube embedded in a solid which is at its fusion temperature. Solutions were obtained by an implicit finite-difference scheme tailored to take account of the movement of the liquid-solid interface as melting progresses. The results differed decisively from those corresponding to a conventional pure-conduction model of the melting problem. The calculated heat transfer rate at the tube wall decreased at early times and attained a minimum, then increased and achieved a maximum, and subsequently decreased. This is in contrast to the pure conduction solution whereby the heat transfer rate decreases monotonically with time. The thickness of the melt region was found to vary along the length of the tube, with the greatest thickness near the top. This contrasts with the uniform thickness predicted by the conduction solution. These findings indicate that natural convection effects, although unaccounted for in most phase change analyses, are of importance and have to be considered.

Heat thermal structure in the interfacial boundary layer measured in an open tank of water in turbulent free convection

Abstract: The thermal structure in the boundary layer and its relation to the heat flux from the cooling and evaporating surface of a deep tank of water are investigated. When a deep layer of water in contact with still air above loses heat to the air, the cooled water in a region just under the surface converges along lines and then plunges down in sheets. These sheets of falling water dissipate as they move into the body of the water, which is in turbulent motion. The vertical profiles of the horizontally averaged temperature and its standard deviation agree fairly closely with theoretical profiles based on time averages of the solution to the heat diffusion equation. The differences between observed and thus predicted profile shapes are consistent with the expected effects of the falling cold thermals and the warm return flow, which are neglected in the theories. The profiles of the standard deviation have large values up to the interface and lie between predictions based on boundary conditions of constant surface temperature and constant heat flux, in keeping with the experimental conditions.The relation between the net heat flux and the temperature difference across the boundary layer is given in non-dimensional form by N = 0[sdot ]156R0[sdot ]33, which is in good agreement with the asymptotic similarity prediction N [vprop ] R1/3 but lower than theoretical calculations of the upper bound of N vs. R.

Thermal convection of water in a porous medium: Effects of temperature- and pressure-dependent thermodynamic and transport properties

Abstract: Natural convection of water in thick geothermal layers, across which there are temperature differences as large as 345 K and pressure differences as great as 1 kbar, is investigated. Complete account is taken of the variable thermodynamic and transport properties of water as well as of non-Boussinesq effects. The increase of thermal expansivity with temperature by as much as a factor of 70 and the decrease of viscosity by more than 1 order of magnitude are primarily responsible for the enhanced instability to convection of a water-saturated porous layer compared with a porous layer saturated with a Boussinesq fluid having the constant properties of surface water. The critical Rayleigh number, critical wave number, and streamline and isotherm patterns are determined at the onset of convection of temperature gradients of 25, 50, 75, 100, 150, and 200 K/km in layers as thick as 10 km. The critical surface Rayleigh number is reduced by as much as a factor of 31 below the value of 4..pi../sup 2/ appropriate for a constant property Boussinesq fluid. Variable water properties thus allow convection to occur for smaller vertical temperature differences in rock of a given permeability or for smaller permeability at a given temperature difference.more » The horizontal scale of convection is somewhat reduced, and the flow is concentrated toward the bottom of the porous layer by effects of variable expansivity and viscosity.« less

Concentration dependent solute redistribution at the ice–water phase boundary. III. Spontaneous convection. Chloride solutions

Abstract: Spontaneous convection in the ice–water system was investigated with temperature and concentration measurements. Earlier work had shown that mild convection causes an apparent concentration dependence of the distribution coefficient that masks any real concentration dependence. Convection was found to be determined primarily by the density inversion at the 4 C isotherm. A simple method of evaluating the height of convection cells was developed which enables one to obtain an order‐of‐magnitude estimate of the distribution coefficient. Forced convection (stirring at 300 or 1000 rpm) and improved sampling and analysis techniques were used to redetermine the distribution coefficients of chlorides in ice. Solutions of HCl, LiCl, NaCl, KCl, RbCl, CsCl, and NH4Cl were investigated in the concentration range 10−6–10−2M. For the hydrogen and alkali chlorides no effect of the different cations on the chloride distribution coefficient was evident. Its average value is 2.7×10−3. For the ammonium chloride, the distrib...

Natural Convection in Porous Media with Heat Generation

Abstract: The heat transfer characteristics of a fluid-saturated porous media are investigated for the case of uniform internal heat generation with cooling from above. Analytical models of conduction and si...

Steady and transient thermal convection in a fluid layer with uniform volumetric energy sources

Abstract: Measurements of the overall heat flux in steady convection have been made in a horizontal layer of dilute aqueous electrolyte. The layer is bounded below by a rigid zero-heat-flux surface and above by a rigid isothermal surface. Joule heating by an alternating current passing horizontally through the layer provides a uniformly distributed volumetric energy source. The Nusselt number at the upper surface is found to be proportional to Ra0[sdot ]227 in the range 1[sdot ]4 ≤ Ra/Rac ≤ 1[sdot ]6 × 109, which covers the laminar, transitional and turbulent flow regimes. Eight discrete transitions in the heat flux are found in this Rayleigh number range. Extrapolation of the heat-transfer correlation to the conduction value of the Nusselt number yields a critical Rayleigh number which is within -6[sdot ]7% of the value given by linearized stability theory. Measurements have been made of the time scales of developing convection after a sudden start of volumetric heating and of decaying convection when volumetric heating is suddenly stopped. In both cases, the steady-state temperature difference across the layer appears to be the controlling physical parameter, with both processes exhibiting the same time scale for a given steady-state temperature difference, or [mid ]ΔRa[mid ]. For step changes in Ra such that [mid ]ΔRa[mid ] > 100Rac, the time scales for both processes can be represented by Fo [vprop ] [mid ]ΔRa[mid ]m, where Fo is the Fourier number of the layer. Temperature profiles of developing convection exhibit a temperature excess in the upper 15–20 % of the layer in the early stages of flow development for Rayleigh numbers corresponding to turbulent convection. This excess disappears when the average core temperature becomes large enough to permit eddy transport and mixing processes near the upper surface. The steady-state limits in the transient experiments yield heat-transfer data in agreement with the results of the steady-state experiments.

Thermal convection in a Hele-Shaw cell

Abstract: We derive the Rayleigh number RHS for thermal convection in a Hele-Shaw cell with gap width d and full width (gap plus walls) Y. For the state of marginal stability, the system of equations is found to be formally identical to that describing flow through a uniform porous medium, if d3/12Y is identified as the Hele-Shaw permeability. Thus Lapwood's (1948) thermal-instability analysis should apply, and the critical Rayleigh number should be 4π2 when the cell has impermeable isothermal boundaries.Baker's (1966) pH-indicator method for visualizing fluid flow has been adapted for use in a Hele-Shaw cell. In addition to revealing the convection pattern clearly, this technique proves to be an especially sensitive detector of incipient flow, and a highly accurate means of verifying the onset of convection. Our experiments confirm that the critical Hele-Shaw Rayleigh number is 40 ± 2, thereby validating our theoretically derived expression for the Rayleigh number. We also measure the vertical flow velocity wm and find that wm∝ (R2HS−402)½ closely fits our data for 40 < RHS < 140.

Observations of convection at rayleigh numbers up to 760,000 in a fluid with large prandtl number

Abstract: Observations are reported on the stability and structure of Rayleigh-Benard convection in a fluid with a Prandtl number of 8,600, and Rayleigh numbers between 50,000 and 760,000. Under carefully initiated and controlled conditions, stationary bimodal convection was observed, even at the highest Rayleigh number. Unstable bimodal patterns broke down in a way that tends to reduce the wavelength of the cells. Oscillating spoke-shaped convection was observed in convection started from random initial conditions. The gradual increase in the occurrence of oscillations and the dependence on the initial conditions probably accounts for previous disagreements about the onset of time dependence. A square convection planform was found which was stable at the highest Rayleigh numbers but unstable for lower Rayleigh numbers. The observations demonstrate the existence of different possible solutions at a given Rayleigh number.

Numerical solutions of single-mode convection equations

Abstract: In the Boussinesq approximation, single-mode equations describing thermal convection are constructed by expanding the fluctuating velocity and temperature fields in a complete set of functions (or planforms) of the horizontal coordinates and retaining just one term. Numerical solutions of the single-mode equations are investigated, chief consideration being given to hexagonal planforms. Extensive surveys of steady solutions are presented for various Rayleigh numbers, Prandtl numbers, and horizontal wavenumbers. The dependences on Rayleigh number and Prandtl number at very large Rayleigh number are in satisfactory agreement with the results of asymptotic expansions.

Heat, Mass, and Momentum Transfer

Abstract: Life depends on heat and mass transfer between organisms and their surroundings. Such processes as carbon dioxide exchange between leaves and the atmosphere, oxygen uptake by microorganisms, oxygen and carbon dioxide exchange in the lungs of animals, or convective heat loss from the surfaces of animal coats are fundamental to the existence of living organisms. A thorough understanding of these exchange processes is therefore a necessary part of the study of physical ecology. In this chapter we will first discuss molecular diffusion. It is by this process that heat and mass are transported in still air or water, as they are in parts of the lungs of animals, in soils, and in the substomatal cavities of leaves. Molecular diffusion is also important in convective heat and mass transfer between surfaces and fluids flowing over them since a thin boundary layer is always formed near the surface through which transport is by diffusion. After diffusion processes are discussed, we will then present convective heat and mass transfer theory as it applies to fluids moving over plates, cylinders, and spheres. Finally, we will discuss momentum exchange and the force of moving fluids on objects in them.

The sea surface temperature deviation at very low wind speeds; is there a limit?

Abstract: Laboratory and field measurements of the temperature drop across the boundary layer at the upper surface of evaporating and cooling water in turbulent free convection show that greater values are possible than previously indicated. An upper limit on the surface temperature deviation is imposed only by the maximum heat loss occurring in the earth's most extreme climates. DOI: 10.1111/j.2153-3490.1977.tb00729.x

Application of a K-ε Turbulence Model to Natural Convection From a Vertical Isothermal Surface

Abstract: A K-e turbulence model similar to that proposed by Jones and Launder is applied to the calculation of the turbulent natural convective boundary layer on a vertical, isothermal surface. Conservation equations for the turbulent kinetic energy, dissipation rate of turbulent kinetic energy, and mean square temperature fluctuations are solved numerically along with the turbulent momentum and energy equations using the Spalding-Patankar boundary layer method. Various model constants and wall functions, and wall terms were tested. The results are compared with available experimental data and found to be in reasonable agreement.