Showing papers on "Rayleigh number published in 1979"

Instabilities of convection rolls in a fluid of moderate Prandtl number

Abstract: The instabilities of two-dimensional convection rolls in a horizontal fluid layer heated from below are investigated in the case when the Prandtl number is seven or lower. Two new mechanisms of instability are described theoretically as well as experimentally. The knot instability causes the transition to spoke-pattern convection at higher Rayleigh numbers while the skewed varicose instability accomplishes a change to larger horizontal wavelengths of the convection rolls. Both instabilities disappear in the limits of small and large Prandtl number. Although the experimental methods fail in realizing closely the infinitely conducting boundaries assumed in the theory, the observations agree in all qualitative aspects with the theoretical predictions.

Subsolidus Convection in the Mantles of Terrestrial Planets

Abstract: The role of heat transport by solid state mantle convection in determining the past and present thermal states of terrestrial planets is examined. Mantle convection models have relied on two-dimensional and axisymmetric three-dimensional numerical calculations incorporating the temperature and pressure dependence of mantle rheology and its non-Newtonian nature. Convection at high Rayleigh numbers has been investigated through theoretical scaling arguments and boundary layer theories; nevertheless, computational limits prevent modeling of the fully three-dimensional, time-dependent, very high Rayleigh number convection which probably prevails in terrestrial planets. Radar measurements of Venus, as well as Voyager exploration of the Galilean satellites, should also provide information on mantle convection.

Rayleigh-Bénard experiment in liquid helium ; frequency locking and the onset of turbulence

Abstract: 2014 In a Rayleigh-Bénard experiment in liquid helium, we study the time-dependent regimes, above the onset of convection, in a parallelepipedic cell with two convective rolls of oblate cross-section 03B1 = 1.90, where 03B1 is the rolls wave number. A first frequency f1, associated to oscillating rolls, appears for a Rayleigh number R ~ 2 x 104, then for R ~ 2.7 104 a second frequency f2, much smaller, is observed. Two frequency locking regimes are present, with hysteresis, for the frequency ratios f1/f2 = 6.5 and f1/f2 = 7. The onset of turbulence germinates from the last frequency locking state for R ~ 3.4 104. This transition to turbulence is triggered by the generation of the frequencies f2/2, f2/4, ... and in real time is defined by a phase-chaotic behaviour rather than by an amplitude behaviour. LE JOURNAL DE PHYSIQUE LETTRES TOME 40, 15 AOUT 1979, Classification Physics Abstracls 67.90 47.25Q We present in this Letter some new results on a Rayleigh-Benard experiment whereby a parallelepipedic cell of liquid helium is heated from below, the liquid being at a temperature of 3.00 K under a pressure of 3 atmosphere, its Prandtl number being (J = v/x = 0.5. In such an experiment, as one increases the temperature difference AT between the top and bottom part of the fluid, the first bifurcation is associated with fluid convection. In our geometry, a parallelepipedic cell with a base 1.6 x 2.8 mm and a height 0.85 mm, we have studied the metastable state where two convective rolls, perpendicular to the largest side of the sample, are formed. Figure 1 shows the curve of marginal stability for the onset of convection. Whereas for a sample of infinite transverse dimension the roll wave number [1] is equal to ac = 3.117, our metastable state corresponds to a = 1.90 (a is a normalized wavevector, a = 2IId/~, where d is the cell height). The second bifurcation, for a small Prandtl number fluid, leads to a limit cycle, physically associated to waves propagating along the convective rolls, the so-called oscillatory [1] instability. We detect, with a local bolometer, a temperature oscillation of frequency /i ~ 550 mHz at the onset for a Rayleigh number R ~ 2 x 104, or a temperature difference AT ~ 120 mK. The third bifurcation generates a second frequency, f2, the onset of which is for R = 2.7 x 104 and its starting value f2 = 90 mHz. Then, as one keeps increasing the temperature difference, all the combination frequencies, f = mfl 1 + nf 2, m and n integer, appear as one Fourier analyses the data (Fig. 2). This is followed by two frequencylocking states (Fig. 3), for Rayleigh numbers between 3 x 104 and 3.3 x 104, the first one corresponds to fl1f2 = 6.5, and the second one to fl/f2 = 7 (Fig. 4). Finally, for R = 3.38 x 104, the onset of turbulence is observed, germinating from the last locking state, with a generation of the frequencies f 2/2, f 2/4 and

Thermal and magnetic instabilities in a rapidly rotating fluid sphere

Abstract: A linear analysis is used to study the stability of a rapidly rotating, electrically-conducting, self-gravitating fluid sphere of radius r 0, containing a uniform distribution of heat sources and under the influence of an azimuthal magnetic field whose strength is proportional to the distance from the rotation axis The Lorentz force is of a magnitude comparable with that of the Coriolis force and so convective motions are fully three-dimensional, filling the entire sphere We are primarily interested in the limit where the ratio q of the thermal diffusivity κ to the magnetic diffusivity η is much smaller than unity since this is possibly of the greatest geophysical relevance Thermal convection sets in when the temperature gradient exceeds some critical value as measured by the modified Rayleigh number Rc The critical temperature gradient is smallest (Rc reaches a minimum) when the magnetic field strength parameter Λ ≃ 1 [Rc and Λ are defined in (23)] The instability takes the form of a very

Fast viscous bénard convection

Abstract: We have previously described a method for the precise study of steady two-dimensional convection problems in the asymptotic limit of large Rayleigh number R, assuming a very viscous fluid with the ...

Nonlinear properties of convection rolls in a horizontal layer rotating about a vertical axis

Abstract: Steady finite amplitude two-dimensional solutions are obtained for the problem of convection in a horizontal fluid layer heated from below and rotating about its vertical axis. Rigid boundaries with prescribed constant temperatures are assumed and the solutions are obtained numerically by the Galerkin method. The existence of steady subcritical finite amplitude solutions is demonstrated for Prandtl numbers P < 1. A stability analysis of the finite amplitude solutions is performed by superimposing arbitrary three-dimensional disturbances. A strong reduction in the domain of stable rolls occurs as the rotation rate is increased. The reduction is most pronounced at low Prandtl numbers. The numerical analysis confirms the small amplitude results of Kuppers & Lortz (1969) that all two-dimensional solutions become unstable when the dimensionless rotation rate Ω exceeds a value of about 27 at P ≃ ∞. A brief discussion is given of the three-dimensional time-dependent forms of convection which are realized at rotation rates exceeding the critical value.

Three-dimensional and multicellular steady and unsteady convection in fluid-saturated porous media at high Rayleigh numbers

Abstract: In an effort to determine the characteristics of the various types of convection that can occur in a fluid-saturated porous medium heated from below, a Galerkin approach is used to investigate three-dimensional convection in a cube and two-dimensional convection in a square cross-section. Strictly two-dimensional, single-cell flow in a square cross-section is steady for Rayleigh numbers R between 4π2 and a critical value which lies between 300 and 320; it is unsteady at higher values of R. Double-cell, two-dimensional flow in a square cross-section becomes unsteady when R exceeds a value between 650 and 700, and triple-cell motion is unsteady for R larger than a value between 800 and 1000. Considerable caution must be exercised in attributing physical reality to these flows. Strictly two-dimensional, steady, multicellular convection may not be realizable in a three-dimensional geometry because of instability to perturbations in the orthogonal dimension. For example, even though single-cell, two-dimensional convection in a square cross-section is steady at R = 200, it cannot exist in either an infinitely long square cylinder or in a cube. It could exist, however, in a cylinder whose length is smaller than 0.38 times the dimension of its square cross-section. Three-dimensional convection in a cube becomes unsteady when R exceeds a value between 300 and 320, similar to the unicellular two-dimensional flow in a square cross-section. Nusselt numbers Nu, generally accurate to 1%, are given for the strictly two-dimensional flows up to R = 1000 and for three-dimensional convection in cubes up to R = 500. Single-cell, two-dimensional, steady convection in a square cross-section transports the most heat for R < 97; this mode of convection is also stable in square cylinders of arbitrary length including the cube for R < 97. Steady three-dimensional convection in cubes transports more heat for 97 [lsim ] R [lsim ] 300 than do any of the realizable two-dimensional modes. At R [gsim ] 300 the unsteady modes of convection in both square cylinders and cubes involve wide variations in Nu.

Convective instabilities in vertical fractures and faults

Abstract: Natural convection of water contained in a vertical fracture or fault in which the temperature increases with depth is strongly influenced by the heat transport processes not only within the water itself but also by the heat transferred to and from the surrounding rock mass. The results of a linear stability analysis indicate that the critical Rayleigh number R* is time dependent. For spontaneous neutral stability, R* (t=0) approx. =10(h/a) /sup 2/, where h and a are the fault height and aperture, respectively. Since h>>a, R* (0) is several orders of magnitude greater than the value 4..pi../sup 2/ that would pertain to the same situation without the influence of the surrounding rock masses, e.g., a porous bed with large horizontal dimensions. The resultant cell motion consists of rolls about axes parallel to the aperture. These rolls are of height h and closely spaced in the strike direction. Cases of spontaneous instabilities in fractures or faults are expected to be infrequent, but initially subcritical convection could be fostered by other means such as tectonic displacements at the fault. Because R* diminishes as time /sup -1/2/, eventually, this subcritical convection becomes unstable, and exponential growth ensues. As the heat of the surroundingmore » rock is depleted and an isothermal state is approached, the convection eventually dampens until a period of thermal recovery allows its resumption.« less

Turbulent wall fires

Abstract: Turbulent natural convection fires at the base of a vertical wall were considered. The burning surface was simulated by 51–305 mm high wicks soaked with methanol, ethanol or 1-propanol. Measurements were made of burning rates in the pyrolysis zone; radiative and convective heat fluxes to the wall (and radiative heat flux to the ambiance) in the wall plume above the pyrolysis zone; and profiles of mean velocity, temperature and concentrations in all regions of the flow. The measurements are compared with solutions of the laminar boundary layer equations, solutions of the turbulent boundary layer equations employing a mixing length model, and integral models approximating the more complete theories. The theoreis provided a unified correlation of laminar and turbulent burning rate measurements from both the present and earlier studies. The correlation was relatively insensitive to the radiant heat flux from the flame to the burning surface, which varies in the range 0–86 percent of the total surface heat flux; this behavior is convenient, but has not been fully explained and deserves further attention. Laminar wall flames are 2–3 times longer than turbulent wall flames, when length is normalized by the length of the pyrolysis zone; theory provides good length predictions in both cases. The soot content of the flames was low for the present tests and radiation comprised only 10–20% of the heat flux to the surface; the flux of radiation to the ambiance was larger than the wall component. Predicted wall heat fluxes were within 20% of the measurements. Predictions in the noncombusting portion of the plume employed earlier results for weakly buoyant plumes using average physical properties at each wall position; profiles of mean velocity and temperature approximated local similarity, based on local plume energy flux and height along the wall; however, effects of combustion near the flame tip, variable property effects, and an additional Rayleigh number dependence were observed.

Angular velocity gradients in the solar convection zone

Abstract: We test the hypothesis that the weak influence of rotation upon solar supergranulation, resulting in fluid particles conserving their angular momentum while moving radially, is responsible for the outward decrease in angular velocity inferred from the difference between photospheric plasma and sunspot rotation rates. This test is performed using numerical integrations of a Boussinesq spherical convection model for a thin shell at small Taylor number (implying weak influence of rotation). We find that the convection does maintain an outward decrease in angular velocity, which approaches the limit implied by angular momentum conservation as the Rayleigh number or driving for convection is increased.By examining the energetics of the motion, we verify that the dominant process maintaining the calculated angular velocity profile against viscous diffusion is the inward transport of angular momentum by the convection. Axisymmetric meridional circulation plays virtually no role in this process. We further find there is no tendency for convection weakly influenced by rotation to form an equatorial acceleration. We argue from these and earlier calculations that the origin of the Sun's latitudinal gradient of angular velocity is deep in the convection zone. At these depths there may be a strong tendency for angular velocity to be constantmore » on cylinders, implying a positive radial gradient of angular velocity. The latitude gradient is transmitted to the photosphere by supergranulation which locally produces the negative radial gradient in the top layers. We suggest from the rotation of various magnetic features that the transition from negative to positive radial angular velocity gradient occurs near the bottom of the supergranule layer. We argue that angular momentum conservation in radially moving fluid particles should produce a similar angular velocity profile in compressible convecting fluid layers.« less

The influence of a third diffusing component upon the onset of convection

Abstract: The small amplitude stability analysis for the onset of double-diffusive convection when the density gradient is gravitationally stable is extended to include a third diffusing component. Special attention is given to systems with κ1 [Gt ] κ2, κ3 and Pr [Gt ] κ/κ1, where κi is the molecular diffusivity of the ith component and Pr is the Prandtl number based on the largest of the Ki. It is found that the boundary for the onset of overstability is approximated by two straight lines in a Rayleigh number plane. Small concentrations of a third property with a smaller diffusivity can have a significant effect upon the nature of diffusive instabilities, the magnitude of this effect being proportional to κ1/Ki. Oscillatory and direct ‘salt-finger’ modes are found to be simultaneously unstable under a wide range of conditions when the density gradients due to the components with the greatest and smallest diffusivities are of the same sign.

Combined heat and mass transfer in natural convection on inclined surfaces

Abstract: The combined heat and mass transfer characteristics of natural convection flow along inclined surfaces are studied analytically. The buoyancy forces arise from both temperature and concentration variations in the fluid. In the analysis, the diffusion-thermo and thermo-diffusion effects are neglected, as are the interfacial velocities resulting from mass diffusion. The surfaces are either maintained at a uniform temperature/concentration or subjected to a uniform heat/mass flux. The important parameters of the problem include Prandtl and Schmidt numbers, thermal and concentration Grashof numbers, the relative buoyancy force effect between species and thermal diffusion, and the angle of inclination from the vertical. Numerical results are presented for diffusion of common species into air and water. For both heating/diffusing conditions, the wall shear stress and the local Nusselt number are found to increase and decrease as the buoyancy force from species diffusion assists and opposes, respectively, the th...

Note on Gill's solution for free convection in a vertical enclosure

Abstract: This paper develops an alternative approach to evaluating the arbitrary constants found in Gill's solution for the boundary-layer free-convection regime in a vertical rectangular enclosure. The new method consists of calculating the net upward flow of energy through the enclosure and setting it equal to zero near the top and bottom boundaries of the cavity. The present method takes into account the impermeable and adiabatic properties of the horizontal end walls. The overall Nusselt number derived on this new basis is shown to agree well with available experimental and numerical heat-transfer correlations.

Thermally driven hydromagnetic convection in a rapidly rotating sphere

Abstract: Buoyancy-induced convection is investigated in a rapidly rotating, selfgravitating fluid sphere internally heated by a uniform distribution of heat sources and under the influence of an azimuthal magnetic field whose strength is proportional to the distance z* from the rotation axis. Attention is restricted to relatively small magnetic field strengths (as measured by the parameter A) such that the dominant force balance remains geostrophic. Convection is then confined to a thin cylindrical annulus, radius Wu, about the rotation axis. A linear analysis is used to find the state of marginal stability together with the corresponding minimum critical value of the modified Rayleigh number, Re (a measure of the buoyancy force required to maintain convective motions). Two distinct modes of instability are found to operate: 'Rossby' and 'magnetic'. When no magnetic field is applied (A = 0), and when the Prandtl number o- < 1, the instability takes the form of a thermally driven Rossby wave propagating eastward but with group velocity westward. Modifications to this mode due to a non-zero magnetic field

Free Convection Effects on the Flow Past an Infinite Vertical Oscillating Plate

Abstract: The motion of a semi-infinite incompressible viscous fluid, caused by the oscillation of a plane vertical plate, has been studied, taking into account the presence of free convection currents. Closed form solutions to the velocity, temperature and the penetration distance through which the leading edge effect propagates have been derived on neglecting the transient part. Velocity profiles are shown forGr>0 (Grashof number) (cooling of the plate by the free convection currents),Gr<0 (heating of the plate) on graph. Also the penetration distance has been shown on graphs for different values ofP, the Prandtl number. It has been observed that for ωt=3π/2, greater cooling of the plate may cause the flow to become unstable. Also, the penetration distance is not found to be affected by the frequency of the oscillating vertical plate.

An experimental investigation of natural convection in the melted region around a heated horizontal cylinder

Abstract: Melting from an electrically heated horizontal cylinder embedded in a paraffin (n-octadecane, fusion temperature 301·3 °K) has been studied experimentally. The shape of the solid-liquid interface has been determined photographically, and the local heat transfer coefficients have been measured using a shadowgraph technique. The experiments provide conclusive evidence of the important role played by natural convection in melting a solid due to an embedded cylindrical heat source. The four distinct pieces of quantitative evidence which contribute to this conclusion are the melt shape, surface temperature, local and average heat transfer coefficients and their variation with time.The experimental findings prove the importance of natural convection in phase change problems involving melting and indicate that continued practice of neglecting the effects in the analysis of such problems does not appear reasonable. Natural convection should be considered in analysis and design of systems involving phase change.

Thermal and magnetically driven convection in a rapidly rotating fluid layer

Abstract: The stability of an electrically conducting Boussinesq fluid which is confined between two horizontal planes a distance d apart is investigated. The fluid is heated from below, cooled from above and the whole system rotates rapidly with angular velocity Ω c about a vertical axis. A weak non-uniform horizontal magnetic field, whose strength is measured by the Alfven angular velocity Ω M [[Lt ] Ω c , see (1.2)] permeates the fluid and corresponds to the flow of a uniform electric current parallel to the rotation axis. When the modified Rayleigh number R [see (2.1)] is greater than zero and q = κ/λ d 2 . When R = 0 and Ω M > 2(ν/λ) ½ Ω c , where ν is the viscosity, Roberts & Loper (1979) have isolated an exceptional class of unstable fast inertial waves which grow on the magnetic diffusion time scale τ λ = d 2 /λ. When R λ Ω 2 M /τ c , exceeds some value dependent upon q , a class of unstable slow waves also exists for a range of negative values of R. These waves propagate eastwards (westwards) when q is less (greater) than unity. In this case the fluid is stably stratified and the energy for the disturbance is taken from the magnetic field. The resulting description of the stability boundary for R R plane extends and clarifies the results of Roberts & Loper (1979), which are valid when both Γ and q are large.

Three-dimensional natural convection in a confined porous medium heated from below

Abstract: Previous analyses of natural convection in a porous medium have drawn seemingly contradictory conclusions as to whether the motion is two- or three-dimensional. This investigation uses numerical results to show the relationship between previous contending observations, and demonstrates that there exists more than one mode of convection for any particular physical configuration and Rayleigh number. In some cases, a particular flow situation may be stable even though it does not maximize the energy transfer across the system. The methods used are based on the efficient numerical solution of the governing equations, formulated with the definition of a vector potential. This approach is shown to be superior to formulating the equations in terms of pressure. For a cubic region the flow pattern at a particular value of the Rayleigh number is not unique and is determined by the initial conditions. In some cases there exist four alternatives, two- and three-dimensional, steady and unsteady.

Convective instabilities in a closed vertical cylinder heated from below. Part 1. Monocomponent gases

Abstract: Kr, Xe and SiCI4 have been investigated for convective instabilities in closed vertical cylinders with conductive walls heated from below. Critical Rayleigh numbers NiRa for the onset of various convective modes (including the onset of marginally stable and periodic flow) have been determined with a high resolution differential temperature sensing method. Flow patterns were deduced from a multiple sensor arrangement. For the three lowest modes (i = 1, 2, 3) good quantitative agreement with linear stability theory is found. Stable oscillatory modes (periodic fluctuations of the mean flow) with a period of approximately 5 s are found for a relatively narrow range of NRa. The critical Rayleigh number NoscRa for the onset of oscillatory temperature fluctuations is 1348 ± 50 for an aspect ratio (height/radius) of 6.

Moving lithospheric plates and mantle convection

Abstract: Summary. The coupling between a rigidly moving lithospheric plate and a convecting mantle is investigated using a simple two-dimensional numerical model that incorporates a horizontally moving upper boundary, simulating the effect of a moving plate, over a fluid layer heated from below. The moving boundary strongly controls the horizontal length scale of convection cells when its velocity is greater than the free convective velocity (i.e. the velocity with which the fluid would convect under a stationary boundary). In a box of aspect ratio 4 (width/depth), a transition in flow structure occurs from several equidimensional convection cells under a slowly moving boundary to a single long convection cell under a rapidly moving boundary. The flow structure transition occurs approximately when Pe/RaW3 = 0.04, where the Peclet number, Pe, measures the (prescribed) velocity of the upper boundary, and the Rayleigh number, Ra, measures the heating of the fluid layer. Near the transition, the flow tends to be unsteady; this behaviour can be well understood in terms of the instability of the thermal boundary layers, which can be characterized by a local Rayleigh number. Using conventional estimates of mantle parameters, the mantle is either near or above the transition to single-cell convection, whether upper-mantle or whole-mantle convection is assumed. The net tangential force exerted by the fluid on the upper boundary varies approximately linearly with the boundary velocity above the transition, and it is positive (driving) for boundary velocities ranging from the value at the transition to about three times the transition

Parameterized convection within the moon and the terrestrial planets

Abstract: For a fluid layer or a self-gravitating fluid sphere uniformly heated from within, the internal temperature can be parameterized in terms of the appropriate Rayleigh number. The heat generation term includes both radioactive heat release and transient heating or cooling. This parameterization has been verified by comparisons with laboratory experiments. Thermal history calculations have been carried out for the earth, Venus, Mars, Mercury, and the moon. The results for the earth and Venus indicate that two scales of convection are occurring, one including the surface plates and the second occurring beneath the surface plates. In all cases the present heat flows are between seven and twelve per cent greater than the values obtained assuming a steady state balance between heat flow and internal heat generation.