Showing papers on "Rayleigh number published in 1982"

Onset of convection in a variable-viscosity fluid

Abstract: The Rayleigh number R, in a horizontal layer with temperature-dependent viscosity can be based on the viscosity at T0, the mean of the boundary temperatures The critical Rayleigh number Roc for fluids with exponential and super-exponential viscosity variation is nearly constant at low values of the ratio of the viscosities at the top and bottom boundaries; increases at moderate values of the viscosity ratio, reaching a maximum at a ratio of about 3000, and then decreases This behaviour is explained by a simple physical argument based on the idea that convection begins first in the sublayer with maximum Rayleigh number The prediction of Palm (1960) that certain types of temperature-dependent viscosity always decrease Roc is confirmed by numerical results but is not relevant to the viscosity variations typical of real liquids The infinitesimal-amplitude state assumed by linear theory in calculating Roc does not exist because the convection jumps immediately to a finite amplitude at R0c We observe a heat-flux jump at R0c exceeding 10% when the viscosity ratio exceeds 150 However, experimental measurements of R0c for glycerol up to a viscosity ratio of 3400 are in good agreement with the numerical predictions when the effects of a temperature-dependent expansion coefficient and thermal diffusivity are included

Numerical calculation of two-dimensional natural convection in isothermal open cavities

Abstract: Numerical computations of free convection flow inside an isothermal open square cavity are presented The effects of Grashof number and inclination of cavity are examined Unsteady solutions are observed for Grashof number larger than 105 and cavity aperture facing upward

Thermal Convection in Non-Newtonian Fluids*

Abstract: Publisher Summary This chapter presents the field of thermal convection in non-Newtonian fluids. The thermal convection in external flows of inelastic fluids is relatively well understood in the chapter. Thermal convection in viscoelastic fluids has been studied only in a restricted way. An approximate correlating equation for mixed convection in external flows on non-Newtonian fluids is presented. The chapter explores the theoretical indications to show that there are likely to be tremendous advantages in adding drag-reducing polymers to impart viscoelastic properties to materials undergoing the process of thermal convection under turbulent conditions. The problem of buoyancy-induced secondary flow in heated horizontal tubes appears to be most challenging from a theoretical viewpoint. There are certain equations that correlate heat transfer data empirically, but the situation is not wholly satisfactory. The possibilities of oscillatory convection in viscoelastic fluids appear to exist under certain circumstances. The chapter concludes that thermal convection experiments are difficult to conduct, therefore there appears to be a massive generation of data correlating overall heat transfer coefficients with process variables but practically very little information on the velocity and temperature fields.

The effects of significant viscosity variation on convective heat transport in water-saturated porous media

Abstract: Weakly nonlinear theory and finite-difference calculations are used to describe steadystate and oscillatory convective heat transport in water-saturated porous media. Two-dimensional rolls in a rectangular region are considered when the imposed temperature difference between the horizontal boundaries is as large as 200 K, corresponding to a viscosity ratio of about 6·5. The lowest-order weakly nonlinear results indicate that the variation of the Nusselt number with the ratio of the actual Rayleigh number to the corresponding critical value R/Rc, is independent of the temperature difference for the range considered. Results for the Nusselt number obtained from finite-difference solutions contain a weak dependence on temperature difference which increases with the magnitude of R/Rc. When R/Rc = 8 the constantviscosity convection pattern is steady, while those with temperature differences of 100 and 200 K are found to oscillate.

Thermal boundary layer convection in silicic magma chambers: Effects of temperature‐dependent rheology and implications for thermogravitational chemical fractionation

Abstract: Solution of the nonlinear boundary value problem describing the thermomechanical structure of a boundary layer adjacent to an isothermal cooled vertical wall in a strongly temperature dependent rheological medium is presented. The analysis and boundary conditions chosen are applicable to large Rayleigh number convection in a magma chamber. Numerical solution of the boundary layer equations by parallel shooting have been obtained for viscosity contrast up to 1011. Parameterization of these results allows extrapolation to higher viscosity contrasts. The non-Arrhenius viscosity function μ = μ∞ exp (−a(T∞ - T), where a is the rheological parameter and T is the absolute temperature, is based on experimental data and accounts for the effects of temperature and crystallinity on magma viscosity. The calculations clearly show the importance of explicit consideration of the viscosity-temperature relationship in determining the quantitative features of boundary layer convection. For example, for anhydrous rhyolite (μ∞ = 109 P), the thermal boundary layer thickness (δT), wall heat flux (q0), wall viscous shear stress (τ0), and maximum vertical convective velocity (umax) are 10 m, 700 HFU, 5×10−3 bar, and 2.5 km yr−1 when μ is taken as constant (i.e., a = 0). However, for a viscosity contrast across the thermal layer of 1012 (i.e., a = 0.07 K−1), δT = 120 m, q0 = 350 HFU, τ0 = 0.36 bar, and umax = 400 m yr−1. Extreme caution must be exercised in using results from isoviscous boundary layer theories for the prediction of the thermomechanical and heat transfer parameters of magma chamber convection. For a wall temperature of 600°C (country rock temperature far from intrusion is ∼200°C) and a core temperature of 1000°C and adopting rheological parameters characteristic of rhyolitic magma containing 3 wt % dissolved H2O, we find at distances of 1 and 10 km from the top of the convecting zone viscous shear stresses, τ0 around 6×10−2 and 3×10−2 bar, maximum vertical velocities wmax of 10 and 25 km/yr, boundary layer thicknesses δT of 20 and 50 m, and marginal heat flows q0 of 1200 and 700 HFU, respectively. Transport parameters depend markedly on H2O content; for anhydrous rhyolite with identical boundary conditions q0 ∼ 600 HFU, umax ∼0.5 km yr−1 and τ0 ∼ 10−1 bar at a distance of 1 km along the vertical wall. Maximum vertical convective velocities usually exceed crystal settling rates by several orders of magnitude; crystal settling cannot be important in vigorously convecting chambers except in local regions. The petrological and geothermal implications of the calculations are discussed in terms of an extreme (but plausible) type of magmatic system, the constant enthalpy open magma chamber. In this case, heat losses due to dissipation of magmatic heat to the country rock are precisely balanced by heat input by injection of hot, mafic magma into the roots of the chamber. The requirements of the thermal steady-state chamber permit an estimation of the mass flow rate into the roots of the chamber. Results agree well with known rates of basaltic magmatism along mid-ocean ridges and at intraplate ‘hot spot’ sites. Semi-quantitative evaluation of the magnitude of chemical fractionation in a constant enthalpy magma chamber due to coupling of rapid (km yr−1) vertical convective flow with slow horizontal Soret diffusion across thin marginal thermal layers suggests that on a 106 year time scale, significant chemical gradients can be generated. An analytical approach is suggested for answering the question of whether or not such fractionated melt can maintain its integrity (i.e., not become remixed).

Nonlinear Marangoni convection in bounded layers. Part 2. Rectangular cylindrical containers

Abstract: We consider liquid in a circular cylinder that undergoes nonlinear Marangoni insta- bility. The upper free surface of the liquid is taken to have large-enough surface tension that surface deflections are neglected. The side walls are adiabatic and impenetrable, and for mathematical simplicity the liquid is allowed to slip on the side walls. The linearized stability theory for heating from below gives the critical Marangoni number Mc as a function of cylinder dimensions, surface-cooling condition and Rayleigh number. The steady nonlinear convective states near Mc are calculated using an asymptotic theory, and the stability of these states is examined. At simple eigenvalues Mc the finite-amplitude states are determined. We find th at the Prandtl number of the liquid influences the stability of axisymmetric states, distinguishing upflow at the centre from downflow. Near those aspect ratios corresponding to double eigenvalues Me, where two convective states of linear theory are equally likely, the nonlinear theory predicts sequences of transitions from one steady convective state to another as the Marangoni number is increased. These transitions are determined and discussed in detail. Time-periodic convection is possible in certain cases.

Thermal Convection in a Horizontal Layer of Magnetic Fluids

Abstract: The linear instability is investigated for a horizontal magnetic fluid layer confined between two ferromagnetic boundaries and heated from below in the presence of a vertical magnetic field Galerkin method is used for solving the disturbance equations The resultant eigenvalue problem is solved numerically The critical Rayleigh number varies with the magnetic number, N , as R a c =1067(1- N / N 0 ), where N 0 is a function of M 3 , a measure of the nonlinear magnetization of fluid, and decreases from 1417 to 1067 as M 3 increases from 0 to ∞ The linear relation between R a c and N is proved analytically for very large values of M 3 It is concluded that the magnetization of the boundaries and the non-linearity of fluid magnetization both reduces R a c , and that the effects of magnetic force and buoyancy compensate each other

Transition from periodic to chaotic thermal convection

Abstract: The transition to turbulence in Benard convection in a layer of air bounded by rigid conducting walls is studied by numerical solution of the three-dimensional time-dependent Boussinesq equations. The wavy instability of rolls is compared with available experimental and theoretical results. The subsequent transition to chaotic convection is shown to occur for Rayleigh numbers larger than about 9000. The role of symmetry-breaking perturbations in the production of chaos is clarified.

Direct numerical simulation of laminar and turbulent Bénard convection

Abstract: The TURBIT-3 computer code has been used for the direct numerical simulation of Benard convection in an infinite plane channel filled with air. The method is based on the three-dimensional non-steady-state equations for the conservation of mass, momentum and enthalpy. Subgrid-scale models of turbulence are not required, as calculations with different grids show that the spatial resolution of grids with about 322 × 16 nodes provides sufficient accuracy for Rayleigh numbers up to Ra = 3·8 × 105. Hence this simulation model contains no tuning parameters.The simulations start from nearly random initial conditions. This has been found to be essential for calculating flow patterns and statistical data insensitive to grid parameters and agreeing with experimental experience. The numerical results show the theoretically predicted ‘skewed varicose’ instability at Ra = 4000. Warm and cold ‘blobs’ are identified as causing temperature-gradient reversals for all the high Rayleigh numbers under consideration. The calculated wavelengths and the corresponding flow regimes observed in the transition range confirm the stability maps determined theoretically. In the turbulent range the wavelengths agree qualitatively with low-aspect-ratio experiments. Accordingly, the Nusselt numbers lie at the upper end of the scatter band of experimental data, as these also depend on the aspect ratio. Appropriately normalized, the velocity and temperature fluctuation peaks are independent of the Rayleigh number. The vertical profiles agree largely with experimental data and, especially in case of temperature statistics, exhibit comparable or less scatter.

On the Thermal Instability of Superposed Porous and Fluid Layers

Abstract: A solution is presented for the problem of predicting the onset of convection for a system consisting of a volumetrically heated porous bed saturated with and overlaid with a fluid, heated or cooled from below. Results are presented in graphical form in terms of the external Rayleigh number based on the fluid layer, which is shown to be the sole stability parameter of the problem. A wide range of independent parameters are investigated and physical justification for the behavior of the instability with respect to them is given. The results are compared with the 2 bounding cases of the problem and are found to be in agreement with them.

Numerical Simulation of Natural Convection in Concentric and Eccentric Horizontal Cylindrical Annuli

Abstract: Natural convection heat transfer in concentric and eccentric annuli made of two isothermal horizontal circular cylinders is numerically investigated for Rayleigh numbers less than 50 × 104 which is based on the difference of radii Bipolar coordinates are used for eccentric annuli, and it is found that for very small eccentricity the overall thermal behavior of the annuli exhibits that of the exactly concentric cylinders The maximum deviation of the local heat-transfer coefficient of the cylinder walls remains, for example, for e = 001, Ri /R0 = 03846 and RaL = 10 × 104 , within a meager 5 percent The parametric effect on the heat-transfer characteristics is discussed with respect to the diameter ratio for concentric cylinders, and eccentricity and azimuthal angular location of the inner cylinder for eccentric annuli Output is displayed in terms of streamlines, isothermal contours, radial temperature distribution and equivalent thermal conductivities Convection patterns are explained in detail

Anisotropic modelling of thermal convection in multilayered porous media

Abstract: The principle of large-scale anisotropy due to small-scale layering is applied to thermal convection. The motion takes place in a bounded porous medium heated from below. The medium is periodically layered with respect to permeability and thermal conductivity. The onset of convection as well as slightly supercritical convection are investigated. Anisotropic modelling proves useful even for small numbers of layers as long as the motion is of ‘large-scale convection’ type (Masuoka et al. 1978). The modelling always fails for motion of ‘local convection’ type.

Heat transfer by free convection between two parallel flat plates

Abstract: A few numerical analyses of the free convection between two heated parallel plates have been carried out without using the boundary-layer approximation. In this paper, an approach that can determine the flow rate with consideration of the pressure drop is proposed. As an example of the calculation, the method is applied to Kettleborough's model. In addition, a comparison is made with Kettleborough's and Aihara's results.

Numerical experiments on the onset of convective instability in the Earth's mantle

Abstract: Summary. A simplified model of convection in the mantle is used to investigate the transient effect of cooling a fluid layer from above, The model, representing the mantle overlain by the lithosphere, consists of a two- dimensional fluid layer overlain by a solid conducting lid. The initial temperature of both layers is the same, with the top surface of the lid kept at 0°C throughout. We observe the onset of small-scale flow in the model. In the absence of internal heating the behaviour of the system is controlled by the Rayleigh number, R, and the ratio of the thicknesses of the two layers, a. The onset time of convection as defined by reference to conduction temperature profiles is related simply to a boundary layer critical Rayleigh number. The mean temperature profiles for the convection model are also compared with the observed depth-age relation for oceanic lithosphere and the results are used to estimate the viscosity of the mantle.

Effect of Thermal Boundary Conditions on Natural Convection in Vertical and Inclined Air Layers

Abstract: Measurements of the heat transfer by natural convection across vertical and inclined air layers are reported. The air layer is bounded by two parallel isothermal flat plates and around the edges by a low thermal conductivity wall of thickness B. A critical wall thickness Bc was determined by solving the two-dimensional conduction equation in the bounding wall region: for B>Bc the convective heat transfer should be insensitive to B. Measurements are reported for air layers with B>Bc and an aspect ratio of 5. They cover a range in Rayleigh number from 103 to 108 , and a range in orientation from horizontal to vertical. The effect of the emissivity of the bounding wall on the heat transfer across the air layer was evaluated from measurements obtained respectively with a low and a high value of the wall emissivity. The paper proposes terminology required to define the thermal conditions, and discusses the impact of these boundary conditions on the total heat transfer.

A Rayleigh Bénard Experiment: Helium in a Small Box

Abstract: This is a limited excursion in the field of hydrodynamical instabilities, in itself an infinite domain of research. It is first restricted to a Rayleigh Benard experiment, and we will study the case of a small Prandtl number fluid (0.4 < P < 1). To simplify the problem some more we shall restrict ourselves to the geometry of a small rectangular box with two or three convective rolls present. This somewhat artificial case allows us to truncate the degrees of freedom of the system and thus to define some simple bifurcations to turbulence.

Heat transport and temporal evolution of fluid flow near the Rayleigh-Bénard instability in cylindrical containers

Abstract: First this paper describes in detail an apparatus for heat-transport measurements in shallow horizontal layers of fluid at low temperatures Then high-precision results of convective heat transport as a function of the Rayleigh number R are presented for cylindrical cells of aspect ratio L = 208,472 and 57 The present paper concentrates on the long-time behaviour of Boussinesq systems Non-Boussinesq effects, transient effects near the convective onset, and time-dependent states are described elsewhere (Walden & Ahlers 1981 Ahlers et al 1981 Ahlers 1980b and references therein) The measurements show that the convective onset near the critical Rayleigh number Rc is sharp within the experimental resolution of about 01 % of the Nusselt number N even in laterally finite containers Values of R and of the initial slopes of N(R), are obtained and compared with predictions for different flow patterns Over a wider range of R and for L = 57 and 472, N was found within experimental resolution to be a unique, continuous function of R For L = 208, hysteretic transitions are revealed by N(R) near R ≈ 3 and R ≈ 10 For L = 472, the effect of impulsive heating was studied and revealed complicated, long-lived, but surprisingly repro- ducible transients