Showing papers on "Rayleigh number published in 1980"
TL;DR: In this paper, the authors used laser-Doppler methods to identify four distinct sequences of instabilities leading to turbulent convection at low Prandtl number (2·5−5·0), in fluid layers of small horizontal extent.
Abstract: Using automated laser-Doppler methods we have identified four distinct sequences of instabilities leading to turbulent convection at low Prandtl number (2·5–5·0), in fluid layers of small horizontal extent. Contour maps of the structure of the time-averaged velocity field, in conjunction with high-resolution power spectral analysis, demonstrate that several mean flows are stable over a wide range in the Rayleigh number R, and that the sequence of time-dependent instabilities depends on the mean flow. A number of routes to non-periodic motion have been identified by varying the geometrical aspect ratio, Prandtl number, and mean flow. Quasi-periodic motion at two frequencies leads to phase locking or entrainment, as identified by a step in a graph of the ratio of the two frequencies. The onset of non-periodicity in this case is associated with the loss of entrainment as R is increased. Another route to turbulence involves successive subharmonic (or period doubling) bifurcations of a periodic flow. A third route contains a well-defined regime with three generally incommensurate frequencies and no broadband noise. The spectral analysis used to demonstrate the presence of three frequencies has a precision of about one part in 104 to 105. Finally, we observe a process of intermittent non-periodicity first identified by Libchaber & Maurer at lower Prandtl number. In this case the fluid alternates between quasi-periodic and non-periodic states over a finite range in R. Several of these processes are also manifested by rather simple mathematical models, but the complicated dependence on geometrical parameters, Prandtl number, and mean flow structure has not been explained.
553 citations
TL;DR: In this paper, the Navier-Stokes and energy equations were solved using an elliptic numerical procedure for a horizontal isothermal cylinder, and the flow approach natural convection from a line heat source as Ra → 0 and laminar boundary-layer flow as Ra→ ∞.
308 citations
TL;DR: In this paper, the authors considered the flow of a uniform stream past an impermeable vertical surface embedded in a saturated porous medium and which is supplying heat to the porous medium at a constant rate.
Abstract: The flow of a uniform stream past an impermeable vertical surface embedded in a saturated porous medium and which is supplying heat to the porous medium at a constant rate is considered. The cases when the flow and the buoyancy forces are in the same direction and when they are in opposite direction are discussed. In the former case, the flow develops from mainly forced convection near the leading edge to mainly free convection far downstream. Series solutions are derived in both cases and a numerical solution of the equations is used to describe the flow in the intermediate region. In the latter case, the numerical solution indicates that the flow separates downstream of the leading edge and the nature of the solution near this separation point is discussed.
200 citations
TL;DR: In this paper, the convective instability of a layer of fluid heated from below is studied on the assumption that the flux of heat through the boundaries is unaffected by the motion in the layer, and when the heat flux is above the critical value for the onset of convection, motion takes place on a horizontal scale much greater than the layer depth.
Abstract: The convective instability of a layer of fluid heated from below is studied on the assumption that the flux of heat through the boundaries is unaffected by the motion in the layer. It is shown that when the heat flux is above the critical value for the onset of convection, motion takes place on a horizontal scale much greater than the layer depth. Following Childress & Spiegel (1980) the disparity of scales is exploited in an expansion scheme that results in a nonlinear evolution equation for the leading-order temperature perturbation. This equation which does not depend on the vertical co-ordinate, is solved analytically where possible and numerically where necessary; most attention is concentrated on solutions representing two-dimensional rolls. It is found that for any given heat flux a continuum of steady solutions is possible for all wave numbers smaller than a given cut off. Stability analysis reveals, however, that each mode is unstable to one of longer wavelength than itself, so that any long box will eventually contain a single roll, even though the most rapidly growing mode on linear theory has much shorter wavelength.
198 citations
TL;DR: In this article, it was shown that the mantle is currently cooling at a rate of 36°K/109 years and that three billion years ago the mean temperature was 150°K higher than it is today; 83% of the present surface heat flow is attributed to the decay of radioactive isotopes and 17% to the cooling of the earth.
166 citations
TL;DR: In this article, a variational energy stability theory for two-dimensional buoyancy-thermocapillary convection in a layer with a free surface is presented. But the authors do not consider the case of planar interfaces.
Abstract: Energy stability theory has been formulated for two-dimensional buoyancy–thermocapillary convection in a layer with a free surface. The theory yields a critical Rayleigh number RE for which R < RE is a sufficient condition for stability of the layer. RE emerges from the variational formulation as an eigenvalue of a nonlinear system of Euler–Lagrange equations. For the case of small capillary number (large mean surface tension) explicit values are obtained for RE. The analogous linear-theory results for this case are obtained in terms of a critical Rayleigh number RL. These are compared. It is found that the existence of the deformable interface can lead to a stabilization relative to the case of a planar interface. This result is explained in physical terms. The energy theory is then generalized to include general flow problems having three-dimensional disturbances, non-Newtonian bulk fluids and general interfacial mechanics such as surface viscosity and elasticity.
135 citations
TL;DR: In this paper, a simple analytical model for mantle convection with mobile surface plates is presented, and the steady-state structure consists of cells with isentropic interiors enclosed by thermal boundary layers.
Abstract: Summary. A simple, analytical model for mantle convection with mobile surface plates is presented. Our aim is to determine under what conditions free convection can account for the observed plate motions, and to evaluate the thermal structure of the mantle existing under these conditions. Boundary layer methods are used to represent two-dimensional cellular convection at large Rayleigh and infinite Prandtl numbers. The steady-state structure consists of cells with isentropic interiors enclosed by thermal boundary layers. Lithospheric plates are represented as upper surfaces on each cell free to move at a uniform speed. Buoyancy forces are concentrated in narrow rising and decending thermal plumes; torques imparted by these plumes drive both the deformable mantle and overlying plate. Solutions are found for a comprehensive range of cell sizes. We derive an expression for the plate speed as a function of its length, the mantle viscosity and surface heat flux. Using mean values for these parameters, we find that thermal convection extending to 700 km depth can move plates at 1 cm yr-’, while convection through the whole mantle can move plates at 4-5 cm yr-’. Analysis of the steady-state temperature field, for the case of heating from below, shows that the upper thermal boundary layer develops a complex structure, including an ‘asthenosphere’ defined by a local maximum in the geotherm occurring at depths of 50-1 50 km.
108 citations
99 citations
TL;DR: In this paper, a two-dimensional time-dependent numerical computation method has been developed to determine laminar free convection in closed cavities and forced convection, both in ducts and open cavities.
85 citations
TL;DR: In this paper, the effects of axial conduction in the flat plate on the interfacial temperature are significant in the constant heat flux case, and correlated by the dimensionless parameter KD.
TL;DR: In this article, the stability of natural convection in a vertical annular enclosure has been studied by the linear theory and it was found that for all Prandtl numbers the instability sets in as a wave travelling upward.
Abstract: The stability of natural convection in a vertical annular enclosure has been studied by the linear theory. It was found that for all Prandtl numbers the instability sets in as a wave travelling upward. For low Prandtl numbers, the larger the curvature the more stable the flow; the reverse is true for high Prandtl numbers. The theoretical predictions of the mode of instability were verified for air. A multicellular flow pattern was observed to drift upward with the predicted wave speed. The measured wavelength of the cells is in good agreement with the linear analysis.
TL;DR: In this article, the evolution of turbulence for a Rayleigh-B\'enard system with aspect ratio ε = 4.72 was studied, with a threshold near the critical Rayleigh number, which consists of a random background time dependence and rare, randomly spaced, major events.
Abstract: New long-term (days) measurements of the evolution of turbulence for a Rayleigh-B\'enard system with aspect ratio $\ensuremath{\Gamma}=4.72$ reveal a turbulent state, with a threshold near the critical Rayleigh number ${R}_{c}$, which consists of a random background time dependence and rare, randomly spaced, major events. These events are discussed in terms of the analog of a particle under the influence of a stochastic driving force and in a potential with two minima.
TL;DR: In this paper, an attempt is made to derive constraints on mantle convection from observed surface fields: plate velocities, gravity, topography, and heat flow, and a mean value of the effective mantle viscosity of approximately 10 to the 23rd g/cm per sec is obtained.
Abstract: An attempt is made to derive constraints on mantle convection from observed surface fields: plate velocities, gravity, topography and heat flow. The spherical harmonic spectra of the fields are expressed in terms of a spectral magnitude and slope, and requirements for the minimal representation of the equations for mantle convection are discussed. The effects of the boundary layer represented by the surface fields on convection at the mantle surface and at deeper levels are then examined, and a mean value of the effective mantle viscosity of approximately 10 to the 23rd g/cm per sec is obtained, together with values of 10 to the 8th and 10 to the 7th for the Rayleigh numbers of whole mantle and upper mantle convection, respectively. Consideration is then given to the compositional, thermal and rheological aspects of mantle convection, and it is pointed out that constraints on the depth and other properties of convection will require more detailed modeling using the relationships between the harmonic coefficients of the surface fields.
TL;DR: In this article, the steady states of two-dimensional convection in a laterally finite rectangular container near threshold were analyzed and it was shown that the presence of sidewalls severely restricts the allowed wave vectors which can occur in the bulk of the container.
Abstract: An analysis is presented of the steady states of two-dimensional convection in a laterally finite rectangular container near threshold. It is shown that the presence of sidewalls severely restricts the allowed wave vectors which can occur in the bulk of the container. This effect provides a possible mechanism to explain the observed wavelength increase of convective rolls with increasing Rayleigh numbers.
TL;DR: In this article, a Galerkin technique is used to calculate the steady-state axisymmetric nonlinear convective motions in an infinite-Prandtl-number Boussinesq fluid in a relatively thick spherical shell heated from below.
Abstract: A Galerkin technique is used to calculate the steady-state axisymmetric nonlinear convective motions in an infinite-Prandtl-number Boussinesq fluid in a relatively thick spherical shell heated from below. A reasonably complete study of the properties of the even and general axisymmetric steady states is carried out for a range of moderately supercritical Rayleigh numbers. In addition, stability analyses are conducted to determine which form of axisymmetric steady convection is the preferred one and whether the axisymmetric steady flows are unstable to azimuthal perturbations.
TL;DR: In this article, the equations describing thermal convection in a binary fluid driven by the Soret and Dufour effects are shown to be identical, except for a relation between the thermal and solutal Rayleigh numbers.
Abstract: The equations describing thermal convection in a binary fluid driven by the Soret and Dufour effects are shown to be identical, except for a relation between the thermal and solutal Rayleigh numbers, to the equations describing thermosolutal convection. A number of results about the former problem are deduced.
01 Aug 1980
TL;DR: In this article, the free-convective flow from a heated sphere, in the Boussinesq approximation, at high Grashof number, is considered, and the characteristics of the boundary layer close to the surface of the sphere are evaluated numerically.
Abstract: We consider the free-convective flow from a heated sphere, in the Boussinesq approximation, at high Grashof number. The characteristics of the boundary layer close to the surface of the sphere are evaluated numerically, and the eruption of the fluid from the boundary layer into the plume above the sphere is discussed.
TL;DR: Feigenbaum et al. as mentioned in this paper showed that a wide class of maps with a single extremum exhibit a sequence of subharmonic bifurcations as a parameter X is varied.
Abstract: It has recently been discovered that the transition to turbulent convection can occur in a variety of qualitatively distinct ways as the temperature difference across a fluid layer is increased.'-' One common route to turbulence involves a succession of instabilities, which cause the fluid to oscillate quasi-periodically a t two, or sometimes three, incommensurate frequencies. These oscillations can exhibit phase locking and the interactions among them can result in broadband spectral noise in the velocity field.',4 The resulting nonperiodic motion may be defined as the onset of turbulence. A second route to turbulence involves successive subharmonic bifurcations, each of which halves the characteristic frequency of a periodic oscillation.',' I t has been known for some time that a one-parameter family of maps on an interval can show a sequence of subharmonic bifurcations.' Furthermore, maps can be generated from continuous flows by observing the intersections of trajectories with a fixed hyperplane in phase space. Thus, it is not far-fetched to imagine a connection between the properties of maps and the physical behavior of a fluid system. Feigenbaum has recently shown that a wide class of maps with a single extremum exhibit a sequence of subharmonic bifurcations as a parameter X is varied, with the bifurcation points, A,, forming a geometric series in the limit,\
TL;DR: In this paper, the boundary-layer structure on the side walls of a rectangular cavity filled with a porous material is determined using an integral relations approach, which leads to results for the core mass flux, for core-temperature gradient and for the heat transfer characteristics which are in excellent agreement with numerical solutions of the boundary layer equations.
TL;DR: In this article, the effects of different combinations of thermally insulated boundaries and nonuniform thermal gradient caused by either sudden heating or cooling at the boundaries or by distributed heat sources on convective stability in a fluid saturated porous medium are investigated using linear theory by considering the Brinkman model.
Abstract: The effects of different combinations of thermally insulated boundaries and nonuniform thermal gradient caused by either sudden heating or cooling at the boundaries or by distributed heat sources on convective stability in a fluid saturated porous medium are investigated using linear theory by considering the Brinkman model. In the case of sudden heating or cooling, solutions are obtained using single-term Galerkin expansion and attention is focused on the situation where the critical Rayleigh number is less than that for uniform temperature gradient and the convection is not maintained. Numerical values are obtained for various basic temperature profiles and some general conclusions about their destabilizing effects are presented. In particular, it is shown that the results of viscous fluid (σ = 0) and the usual Darcy porous medium (σ → ∞ ) emerge from our analysis as special cases. In the case of convection caused by heat source, since the effect of heat source is not brought out by the single-term Galerkin expansion, the critical internal Rayleigh number is determined using higher order expansion by specifying the external Rayleigh number. It is shown that, for values of σ 2 ≥ 2.45 × 10 5 , the different combinations of bounding surfaces give almost the same Rayleigh number and an explanation, following Lapwood, for this surprising behavior is given. It is found that the heat source's effect on convection decreases for wave numbers up to the value 2.2 and drops suddenly around the critical value of 2.4 and then increases up to 2.5.
TL;DR: In this paper, finite difference solutions of the equations governing thermal convection driven by uniform volumetric energy sources are presented for two-dimensional flows in a rectangular domain, and the boundary conditions are a rigid (i.e., zero slip), zero heat-flux lower surface, rigid adiabatic sides, and either a rigid or free (ie, zero shear) isothermal upper surface.
Abstract: Finite difference solutions of the equations governing thermal convection driven by uniform volumetric energy sources are presented for two-dimensional flows in a rectangular domain The boundary conditions are a rigid (ie, zero slip), zero heat-flux lower surface, rigid adiabatic sides, and either a rigid or free (ie, zero shear) isothermal upper surface Computations are carried out for Prandtl numbers from 005 to 20 and Rayleigh numbers from 5 x 10 to the 4th to 5 x 10 to the 8th Nusselt numbers and average temperature profiles within the layer are in good agreement with experimental data for rigid-rigid boundaries For rigid-free boundaries, Nusselt numbers are larger than in the former case The structure of the flow and temperature fields in both cases is dominated by rolls, except at larger Rayleigh numbers where large-scale eddy transport occurs Generally, low velocity upflows over broad regions of the layer are balanced by higher velocity downflows when the flow exhibits a cellular structure The hydrodynamic constraint at the upper surface and the Prandtl number are found to influence only the detailed nature of flow and temperature fields No truly steady velocity and temperature fields are found despite the fact that average Nusselt numbers reach steady values
TL;DR: In this article, numerical experiments have been carried out on two-dimensional thermal convection, in a Boussinesq fluid with infinite Prandtl number, at high Rayleigh numbers.
TL;DR: In this article, the effects of a different thermal boundary condition, in which the flux of heat is held fixed on both layer boundaries, were investigated, and it was shown that if this flux is just greater than that required for the onset of convection, motion takes place on horizontal scales much greater than the layer depth.
TL;DR: In this paper, the influence of hydrodynamic dispersion on thermal convection in porous media is studied theoretically, and the supercritical, steady two-dimensional motion, the heat transport and the stability of the motion are investigated.
Abstract: The influence of hydrodynamic dispersion on thermal convection in porous media is studied theoretically. The fluid-saturated porous layer is homogeneous, isotropic and bounded by two infinite horizontal planes kept at constant temperatures. The supercritical, steady two-dimensional motion, the heat transport and the stability of the motion are investigated. The dispersion effects depend strongly on the Rayleigh number and on the ratio of grain diameter to layer depth. The present results provide new and closer approximations to experimental data of the heat transport.