Showing papers on "Rayleigh number published in 1974"

Transition to time-dependent convection

Abstract: Steady solutions in the form of two-dimensional rolls are obtained for convection in a horizontal layer of fluid heated from below as a function of the Rayleigh and Prandtl numbers. Rigid boundaries of infinite heat conductivity are assumed. The stability of the two-dimensional rolls with respect to three-dimensional disturbances is analysed. It is found that convection rolls are unstable for Prandtl numbers less than about 5 with respect to an oscillatory instability investigated earlier by Busse (1972) for the case of free boundaries. Since the instability is caused by the momentum advection terms in the equations of motion the Rayleigh number for the onset of instability increases strongly with Prandtl number. Good agreement with various experimental observations is found.

Natural Convection Adjacent to Horizontal Surface of Various Planforms

Abstract: Natural convection adjacent to horizontal surfaces of circular, square, rectangular, and right triangular planforms has been studied experimentally. Electrochemical techniques were employed involving a fluid with a Schmidt number of about 2200. The results encompass a wide range of Rayleigh numbers thus providing information on both the laminar and the turbulent regimes. The data for all planforms are reduced to a single correlation in the laminar and turbulent regimes using the characteristic length, as recommended by Goldstein, Sparrow, and Jones. L* = A/p, where A is the surface area and p is the surface perimeter. The laminar data for all planforms are correlated by the expression Sh = 0.54 Ra1/4 (2.2 × 104 ≤ Ra ≤ 8 × 106) and the data for the turbulent regime are correlated by the expression Sh = 0.15 Ra1/3 (8 × 106 ≤ 1.6 × 109) Transition is found to occur at about Ra = 8 × 106 . The present work thus significantly extends the Rayleigh number range of validity for the use of L* through the 1/4 power laminar regime into the turbulent 1/3 power regime. It also demonstrates the validity of the use of L* to correlate natural convection transfer coefficients for highly unsymmetrical planforms, which heretofore had not been demonstrated. Comparisons to analytical solutions and other experimental heat and mass transfer data are presented.

Natural convection in enclosed spaces - A review of application to solar energy collection

Abstract: A useful solar-thermal converter requires effective control of heat losses from the hot absorber to the cooler surroundings. Based upon the theory and some experimental measurements it is shown that the spacing between the tilted hot solar absorber and successive glass covers should be in the range 4 to 8 cm to assure minimum gap conductance. Poor choice of spacing can significantly affect thermal conversion efficiency, particularly when the efficiency is low or when selective black absorbers are used. Recommended data for gap Nusselt number are presented as a function of the Rayleigh number for the high aspect ratios of interest in solar collector designs. It is also shown that a rectangular cell structure placed over a solar absorber is an effective device to suppress natural convection, if designed with the proper cell spacing d, height to spacing ratio L/d and width to spacing ratio W/d needed to give a cell Rayleigh number less than the critical value.

Large amplitude convection in porous media

Abstract: The properties of convective flow driven by an adverse temperature gradient in a fluid-filled porous medium are investigated. The Galerkin technique is used to treat the steady-state two-dimensional problem for Rayleigh numbers as large as ten times the critical value. The flow is found to look very much like ordinary Benard convection, but the Nusselt number depends much more strongly on the Rayleigh number than in Benard convection. The stability of the finite amplitude two-dimensional solutions is treated. At a given value of the Rayleigh number, stable two-dimensional flow is possible for a finite band of horizontal wavenumbers as long as the Rayleigh number is small enough. For Rayleigh numbers larger than about 380, however, no two-dimensional solutions are stable. Comparisons with previous theoretical and experimental work are given.

Oscillatory and collective instabilities in large Prandtl number convection

Abstract: An experimental study of transitions from steady bimodal convection to time-dependent forms of convection is described. Using controlled initial conditions for the onset of bimodal convection two mechanisms of instability can be separated from the effects of random noise. The oscillatory instability of bimodal cells introduces standing waves closely resembling those occurring in low Prandtl number convection. The collective instability introduces spoke-pattern convection which is characteristic for turbulent large Prandtl number convection. Both instabilities originate primarily from the momentum advection terms in the equations of motion, as is evident from the strong Prandtl number dependence of the critical Rayleigh number Rt for the onset of oscillations. The results are discussed in relation to previous experiments and recent theoretical work.

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

Abstract: 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.

Oscillatory convection in a porous medium heated from below

Abstract: The stability of natural convective flow in a porous medium heated both uniformly and non-uniformly from below is studied in order to determine the possibility of oscillatory and other unsteady flows, and to explore the conditions under which they may occur. The results of the numerical work are directly comparable with experiments using a Hele Shaw cell and also, in the uniformly heated case, with the results of Combarnous & Le Fur (1969) and Caltagirone, Cloupeau & Combarnous (1971). It is shown that for the uniformly heated problem there exist, in certain cases, two distinct possible modes of flow, one of which is fluctuating, the other being steady. However in the non-uniformly heated case the boundary conditions force the solution into a unique mode of flow which is regularly oscillatory when there is considerable non-uniformity in the heat input at the lower boundary, provided that the Rayleigh number is sufficiently high.

The evolution of under-ice melt ponds, or double diffusion at the freezing point

Abstract: In an experimental and theoretical study, we model a phenomenon observed in the summer Arctic, where a fresh-water layer at a temperature of 0°C floats both over a sea-water layer at its freezing point and under an ice layer. Our results show that the ice growth in this system takes place in three phases. First, because the fresh-water density decreases upon supercooling, the rapid diffusion of heat relative to salt from the fresh to the salt water causes a density inversion and thereby generates a high Rayleigh number convection in the fresh water. In this convection, supercooled water rises to the ice layer, where it nucleates into thin vertical interlocking ice crystals. When these sheets grow down to the interface, supercooling ceases. Second, the presence of the vertical ice sheets both constrains the temperature T and salinity s to lie on the freezing curve and allows them to diffuse in the vertical. In the interfacial region, the combination of these processes generates a lateral crystal growth, which continues until a horizontal ice sheet forms. Third, because of the T and s gradients in the sea water below this ice sheet, the horizontal sheet both migrates upwards and increases in thickness. From one-dimensional theoretical models of the first two phases, we find that the heat-transfer rates are 5–10 times those calculated for classic thermal diffusion.

Finite-amplitude thermal convection in a spherical shell

Abstract: The properties of finite-amplitude thermal convection for a Boussinesq fluid contained in a spherical shell are investigated. All nonlinear terms are retained in the equations, and both axisymmetric and nonaxisymmetric solutions are studied. The velocity is expanded in terms of poloidal and toroidal vectors. Spherical surface harmonics resolve the horizontal structure of the flow, but finite differences are used in the vertical. With a few modifications, the transform method developed by Orszag (1970) is used to calculate the nonlinear terms, while Green's function techniques are applied to the poloidal equation and diffusion terms.

On transition mechanisms in vertical natural convection flow

Abstract: An experimental investigation has been made of the processes occurring during the natural transition from laminar to turbulent flow of natural convection flow of water adjacent to a flat vertical surface where the surface heat flux is uniform. Measurements of both the velocity and temperature fields were made over wide ranges of the heat flux and at various downstream locations. Of principal interest were the definitions of the boundaries of the transition regime and their determination at several values of the surface heat flux. The interaction of the velocity and temperature fields during transition was measured. Our results show that transition events are not correlated in terms of the Grashof number G*. The form G*/xn, where n is of order ½ was found to give satisfactory correlations. Measurements of the frequency and growth rate of disturbances indicate the primacy of the velocity field during transition and show that the growth of turbulence in the temperature field lags behind that in the velocity field. The study of the turbulence growth, in terms of intermittency factors in both the velocity and temperature fields, resulted in unambiguous criteria for the boundaries of the transition regime. Our results suggest a kinetic energy flux parameter E and a single value closely correlates both our measurements of the onset of transition as well as those from all past studies known to us, for both different fluids and heating conditions.

On Finite Amplitude Convection in a Rotating Magnetic System

Abstract: The instabilities that can arise in a stratified, rapidly rotating, magnetohydrodynamic system such as the Earth’s core are often thought to play a key role in dynamo theory — that is, in the study of how the magnetic field in the system is maintained in the face of ohmic dissipation. An account of such instabilities is to be found in the M.A.C.- wave theory of Braginsky (1967), who, however, laid his greatest emphasis on the dissipationless modes, an idealization which leads to difficulties described below, ohmic and thermal diffusion is therefore restored, and three key dimensionless parameters are isolated: q , the ratio of thermal to ohmic diffusivities; A, a measure of the relative importance of Coriolis and magnetic forces; and R , a Rayleigh number, which is here the ratio of buoyancy to Coriolis forces. This study concentrates on a particular M.A.C.-wave model originally proposed by Braginsky. It consists of a horizontal layer containing a uniform horizontal magnetic field, B0, and rotated about the vertial, an adverse temperature gradient being maintained on the horizontal boundaries to provide the unstable density stratification. In the rotationally dominant case of large A, the principle of the exchange of stabilities holds, and the motions that arise in the marginal state are steady. The planform of the convection is in rolls orthogonal to B0. If q and A are sufficiently small the principle of the exchange of stabilities remains valid, but the planform consists of one or other of two families of rolls oblique to B0, or a combination of each. If q is large but A is small, the modes are again oblique, but overstability occurs, a type of oscillation which also arises when q is large and A takes intermediate values, although the motion is then in rolls transverse to B0. A theory is developed for the weakly nonlinear convection that arises when R exceeds only slightly the critical value Rc{q, A) at which marginal convection occurs. A critical curve q = ^D(A) is located which roughly divides the (qA) plane into regions of small q and of large q , although when q is large it separates the largeAq from the small. On the one side of the curve, where q or Xq are sufficiently small, it is concluded that, starting from an arbitrary initial perturbation, the convection that arises when R exceeds Rc will ultimately become a completely regular tesselated pattern filling the horizontal plane. On the other side of the curve the situation is considerably more complicated but it is argued that, for sufficiently large q and Aq, subcritical instabilities can occur and that supercritical bursting is likely; that is, the instability that arises from the general initial perturbation will focus into a small spot in a finite time. The relevance of the theory to sunspot formation is discussed. In an appendix, the form of the weakly nonlinear convection that arises when q differs only slightly from qB, and R only slightly from R c, is considered in situations in which the exchange of stabilities holds.

Interferometric study of two-dimensional Benard convection cells

Abstract: A Mach-Zehnder interferometer was used to stud two-dimensional Benard convection cells. The experiments were performed with distilled water and sea water in the region where density is a linear function of temperature. Two-dimensional convection rolls were formed with Rayleigh numbers as great as 23400. Reversal in the temperature profile was obtained for R/Rc ≥ 3·8, and an overshoot of about 6% was observed at R/Rc = 9·2 and 13·8. This agrees with the values predicted theoretically by Veronis (1966) for stress-free boundaries and Royal (1969) for rigid boundaries. This disagrees with the experimental results of Gille (1967), who reports an overshoot of only 1 ½% at R/Rc = 16. Many of the other results agree with those of other experimenters, such as the relation between the cell height-to-width ratio and Rayleigh number, the relation between the Nusselt number and Rayleigh number, and the value of the critical Rayleigh number.

Natural Convection about a Heated Sphere in a Porous Medium

Abstract: The natural convection about a heated sphere in an unbounded region of porous medium is studied theoretically when the induced flow by buoyancy is slow and hence conduction dominates over convection. A uniformly valid asymptotic solution is obtained by straightforward expansion in terms of a small parameter which represents the effect of convection. It is shown that the effect of the convection of the heat flux from the sphere is of the order of square of the small parameter.

Investigation of turbulent convection under a rotational constraint

Abstract: Turbulent convection for a rotating layer of fluid heated from below is studied in this paper. The boundaries of the fluid layer are taken to be free. The underlying principle, used is the Malkus hypothesis that the flow tends to transport the maximum amount of heat possible, subject to certain constraints. By taking the Prandtl number to be infinite, a linear differential constraint and an integral constraint are used. The variational problem that follows then depends on two dimensionless parameters, the Taylor number T and the Rayleigh number R.Asymptotic analysis for the turbulent regime shows that the flow arranges itself so as to tend to offset the stabilizing effect of the rotational constraint, at least in so far as the heat flux is concerned. The dimensionless heat flux, or the Nusselt number, has in general different dependence on T and R, depending on the particular region in the parameter space. For T [les ] O(R), the flow is essentially non-rotating. For O(R) [les ]T [les ] O(R4/3), the flow will always have finitely many horizontal wavenumbers, though the total number of modes increases as T increases in this region. For O(R4/3) [les ] T [les ] O (R3/2), the Nusselt number has a functional dependence proportional to R3/T2, having essentially infinitely many horizontal modes as both R and T increase indefinitely in this region. The last expression is particularly interesting, as it agrees qualitatively with results in finite-amplitude laminar convection. It is also linearly dependent on the layer thickness, as one might expect from dimensional argument. It is suggested that, in the context of the maximum principle, the result in this region of the parameter space may be applicable as well to the same fluid layer with rigid boundaries through the existence of an Ekman layer that is thinner than the thermal layer.