Convection in the presence of restraints

1964 ◽  
Vol 256 (1067) ◽  
pp. 99-147 ◽  
Keyword(s):  
Magnetic Field ◽  
Rayleigh Number ◽  
Critical Rayleigh Number ◽  
Convective Transport ◽  
Turbulent Convection ◽  
Rotating System ◽  
Onset Of Convection ◽  
Cell Width ◽  
Similar Treatment ◽  
Energy Dissipated

In the presence of rotation or a magnetic field, the linearized convection problem reduces to a cubic characteristic equation. In part I, general methods are given for determining the onset of convection; in particular, the transition from oscillatory to steady modes is considered. The importance of this transition arises from evidence that oscillatory modes are inefficient at transporting heat. These methods are then applied to a rotating system where the critical Rayleigh number can be expressed in terms of a Taylor number. It is found that overstable modes develop into steady unstable modes before the exchange of stabilities for Prandtl numbers less than one-third. The nature of the motions is discussed and a similar treatment is provided for convection in a magnetic field. In part II, criteria for the onset of instability are derived from physical arguments. Convection can be treated by balancing the work done by buoyancy forces against the energy dissipated. In a rotating system, the effect of the Coriolis forces is to restrict the cell width and thus to enhance dissipation and promote stability. A magnetic field similarly attenuates the cells and prevents steady convection until the liberated kinetic energy exceeds the energy in the field. In part III, a cellular model is proposed for turbulent convection in a fluid of negligible viscosity, where the motion is limited by the non-linear transfer of energy to smaller-scale motions. If the Rayleigh number R the convective transport varies as R, while it varies as R. The discussion is extended to convection in the presence of rotation or a magnetic field; it is shown that overstable perturbations cannot develop into steady turbulent convection unless the system is already unstable to non-oscillatory modes. The transition from overstable to steady modes should therefore correspond to a sharp increase in convective transport.

Heat transport by parallel-roll convection in a rectangular container

1987 ◽  
Vol 185 ◽  
pp. 205-234 ◽  
Author(s):  
R. W. Walden ◽  
Paul Kolodner ◽  
A. Passner ◽  
C. M. Surko
Keyword(s):  
Rayleigh Number ◽  
Heat Transport ◽  
Critical Rayleigh Number ◽  
Equation Model ◽  
Onset Of Convection ◽  
Infinite Layer ◽  
Lateral Extent ◽  
Prandtl Numbers ◽  
Rayleigh Numbers ◽  
Good Agreement

Heat-transport measurements are reported for thermal convection in a rectangular box of aspect’ ratio 10 x 5. Results are presented for Rayleigh numbers up to 35Rc, Prandtl numbers between 2 and 20, and wavenumbers between 0.6 and 1.0kc, where Rc and kc are the critical Rayleigh number and wavenumber for the onset of convection in a layer of infinite lateral extent. The measurements are in good agreement with a phenomenological model which combines the calculations of Nusselt number, as a function of Rayleigh number and roll wavenumber for two-dimensional convection in an infinite layer, with a nonlinear amplitude-equation model developed to account for sidewell attenuation. The appearance of bimodal convection increases the heat transport above that expected for simple parallel-roll convection.

Stability and Convection in Impulsively Heated Porous Layers

2008 ◽  
Vol 130 (11) ◽  
Author(s):  
M. J. Kohl ◽  
M. Kristoffersen ◽  
F. A. Kulacki
Keyword(s):  
Porous Medium ◽  
Rayleigh Number ◽  
Heated Surface ◽  
Critical Rayleigh Number ◽  
Axisymmetric Flow ◽  
Temperature Fluctuations ◽  
Lower Surface ◽  
Onset Of Convection ◽  
Upward Moving ◽  
The Stability

Experiments are reported on initial instability, turbulence, and overall heat transfer in a porous medium heated from below. The porous medium comprises either water or a water-glycerin solution and randomly stacked glass spheres in an insulated cylinder of height:diameter ratio of 1.9. Heating is with a constant flux lower surface and a constant temperature upper surface, and the stability criterion is determined for a step heat input. The critical Rayleigh number for the onset of convection is obtained in terms of a length scale normalized to the thermal penetration depth as Rac=83/(1.08η−0.08η2) for 0.02<η<0.18. Steady convection in terms of the Nusselt and Rayleigh numbers is Nu=0.047Ra0.91Pr0.11(μ/μ0)0.72 for 100<Ra<5000. Time-averaged temperatures suggest the existence of a unicellular axisymmetric flow dominated by upflow over the central region of the heated surface. When turbulence is present, the magnitude and frequency of temperature fluctuations increase weakly with increasing Rayleigh number. Analysis of temperature fluctuations in the fluid provides an estimate of the speed of the upward moving thermals, which decreases with distance from the heated surface.

Experiments on the inhibition of thermal convection by a magnetic field

1957 ◽  
Vol 240 (1220) ◽  
pp. 108-113 ◽  
Keyword(s):  
Magnetic Field ◽  
Electrical Conductivity ◽  
Rayleigh Number ◽  
Magnetic Fields ◽  
Thermal Convection ◽  
Kinematic Viscosity ◽  
Critical Rayleigh Number ◽  
Theoretical Relation ◽  
Horizontal Layers

Experiments on the magnetic inhibition of thermal convection in horizontal layers of mercury heated from below are described. A large 36½ in. cyclotron magnet reconditioned for hydromagnetic studies was used in these experiments. By using layers of mercury of depth 3 to 6 cm and magnetic fields of strength 500 to 8000 gauss, it has been possible to determine the dependence of the critical Rayleigh number for the onset of instability on the parameter Q 1 ( = σH 2 d 2 / π 2 ρν , where H denotes the strength of the field, σ the electrical conductivity, ν the coefficient of kinematic viscosity, ρ the density and d the depth of the layer) for Q 1 varying between 40 and 1·6 × 10 6 . The experiments fully confirm the theoretical relation derived by Chandrasekhar.

Numerical simulation of two-dimensional compressible convection

1975 ◽  
Vol 70 (4) ◽  
pp. 689-703 ◽  
Author(s):  
Eric Graham
Keyword(s):  
Rayleigh Number ◽  
Numerical Solutions ◽  
Fluid Velocity ◽  
Critical Rayleigh Number ◽  
Finite Amplitude ◽  
Two Dimensions ◽  
Onset Of Convection ◽  
Local Sound ◽  
Constant Viscosity ◽  
Convection Problem

A procedure for obtaining numerical solutions to the equations describing thermal convection in a compressible fluid is outlined. The method is applied to the case of a perfect gas with constant viscosity and thermal conductivity. The fluid is considered to be confined in a rectangular region by fixed slippery boundaries and motions are restricted to two dimensions. The upper and lower boundaries are maintained at fixed temperatures and the side boundaries are thermally insulating. The resulting convection problem can be characterized by six dimension-less parameters. The onset of convection has been studied both by obtaining solutions to the nonlinear equations in the neighbourhood of the critical Rayleigh number Rc and by solving the linear stability problem. Solutions have been obtained for values of the Rayleigh number up to 100Rc and for pressure variations of a factor of 300 within the fluid. In some cases the fluid velocity is comparable to the local sound speed. The Nusselt number increases with decreasing Prandtl number for moderate values of the depth parameter. Steady finite amplitude solutions have been found in all the cases considered. As the horizontal dimension A of the rectangle is increased, the length of time needed to reach a steady state also increases. For large values of A the solution consists of a number of rolls. Even for small values of A, no solutions have been found where one roll is vertically above another.

Three-dimensional instability of axisymmetric buoyant convection in cylinders heated from below

1996 ◽  
Vol 326 ◽  
pp. 399-415 ◽  
Author(s):  
M. Wanschura ◽  
H. C. Kuhlmann ◽  
H. J. Rath
Keyword(s):  
Rayleigh Number ◽  
Linear Stability ◽  
Critical Rayleigh Number ◽  
Three Dimensional ◽  
Base Flow ◽  
Onset Of Convection ◽  
Buoyant Convection ◽  
Thermal Boundary Conditions ◽  
Aspect Ratios ◽  
Prandtl Numbers

The stability of steady axisymmetric convection in cylinders heated from below and insulated laterally is investigated numerically using a mixed finite-difference/Chebyshev collocation method to solve the base flow and the linear stability equations. Linear stability boundaries are given for radius to height ratios γ from 0.9 to 1.56 and for Prandtl numbers Pr = 0.02 and Pr = 1. Depending on γ and Pr, the azimuthal wavenumber of the critical mode may be m = 1, 2, 3, or 4. The dependence of the critical Rayleigh number on the aspect ratio and the instability mechanisms are explained by analysing the energy transfer to the critical modes for selected cases. In addition to these results the onset of buoyant convection in liquid bridges with stress-free conditions on the cylindrical surface is considered. For insulating thermal boundary conditions, the onset of convection is never axisymmetric and the critical azimuthal wavenumber increases monotonically with γ. The critical Rayleigh number is less then 1708 for most aspect ratios.

Thermal Convective Instabilities in a Porous Medium

1984 ◽  
Vol 106 (1) ◽  
pp. 137-142 ◽  
Author(s):  
M. Kaviany
Keyword(s):  
Porous Medium ◽  
Steady State ◽  
Temperature Distribution ◽  
Rayleigh Number ◽  
Saturated Porous Medium ◽  
Critical Rayleigh Number ◽  
Linear Stability Theory ◽  
Onset Of Convection ◽  
Long Time ◽  
Nonlinear Temperature

The onset of convection due to a nonlinear and time-dependent temperature stratification in a saturated porous medium with upper and lower free surfaces is considered. The initial parabolic temperature distribution is due to uniform internal heating. The medium is then cooled by decreasing the upper surface temperature linearly with time. Linear stability theory is applied to the more formally developed governing equations. In order to obtain an asymptotic solution for transient problems involving very long time scales, the critical Rayleigh number for steady-state, nonlinear temperature distribution is also obtained. The effects of porosity, permeability, and Prandtl number on the time of the onset of convection are examined. The steady-state results show that the critical Rayleigh number depends only on the ratio of porosity to permeability and when this ratio exceeds a value of one thousand, the critical Rayleigh number is directly proportional to this ratio.

Effect of a stabilizing gradient of solute on thermal convection

1968 ◽  
Vol 34 (2) ◽  
pp. 315-336 ◽  
Author(s):  
George Veronis
Keyword(s):  
Rayleigh Number ◽  
Temperature Gradient ◽  
Prandtl Number ◽  
Critical Rayleigh Number ◽  
Effective Diffusion ◽  
Stratified Fluid ◽  
Finite Amplitude ◽  
Oscillatory Motion ◽  
Linear Stability Theory ◽  
Onset Of Convection

A stabilizing gradient of solute inhibits the onset of convection in a fluid which is subjected to an adverse temperature gradient. Furthermore, the onset of instability may occur as an oscillatory motion because of the stabilizing effect of the solute. These results are obtained from linear stability theory which is reviewed briefly in the following paper before finite-amplitude results for two-dimensional flows are considered. It is found that a finite-amplitude instability may occur first for fluids with a Prandtl number somewhat smaller than unity. When the Prandtl number is equal to unity or greater, instability first sets in as an oscillatory motion which subsequently becomes unstable to disturbances which lead to steady, convecting cellular motions with larger heat flux. A solute Rayleigh number, Rs, is defined with the stabilizing solute gradient replacing the destabilizing temperature gradient in the thermal Rayleigh number. When Rs is large compared with the critical Rayleigh number of ordinary Bénard convection, the value of the Rayleigh number at which instability to finite-amplitude steady modes can set in approaches the value of Rs. Hence, asymptotically this type of instability is established when the fluid is marginally stratified. Also, as Rs → ∞ an effective diffusion coefficient, Kρ, is defined as the ratio of the vertical density flux to the density gradient evaluated at the boundary and it is found that κρ = √(κκs) where κ, κs are the diffusion coefficients for temperature and solute respectively. A study is made of the oscillatory behaviour of the fluid when the oscillations have finite amplitudes; the periods of the oscillations are found to increase with amplitude. The horizontally averaged density gradients change sign with height in the oscillating flows. Stably stratified fluid exists near the boundaries and unstably stratified fluid occupies the mid-regions for most of the oscillatory cycle. Thus the step-like behaviour of the density field which has been observed experimentally for time-dependent flows is encountered here numerically.

scholarly journals Rayleigh–Bénard stability and the validity of quasi-Boussinesq or quasi-anelastic liquid approximations

2017 ◽  
Vol 817 ◽  
pp. 264-305 ◽  
Author(s):  
Thierry Alboussière ◽  
Yanick Ricard
Keyword(s):  
Stability Analysis ◽  
Equation Of State ◽  
Rayleigh Number ◽  
Equations Of State ◽  
Critical Rayleigh Number ◽  
Ideal Gas ◽  
Dimensionless Temperature ◽  
Boussinesq Model ◽  
Onset Of Convection ◽  
Rayleigh Benard

The linear stability threshold of the Rayleigh–Bénard configuration is analysed with compressible effects taken into account. It is assumed that the fluid under investigation obeys a Newtonian rheology and Fourier’s law of thermal transport with constant, uniform (dynamic) viscosity and thermal conductivity in a uniform gravity field. Top and bottom boundaries are maintained at different constant temperatures and we consider here mechanical boundary conditions of zero tangential stress and impermeable walls. Under these conditions, and with the Boussinesq approximation, Rayleigh (Phil. Mag., vol. 32 (192), 1916, pp. 529–546) first obtained analytically the critical value $27\unicode[STIX]{x03C0}^{4}/4$ for a dimensionless parameter, now known as the Rayleigh number, at the onset of convection. This paper describes the changes of the critical Rayleigh number due to the compressibility of the fluid, measured by the dimensionless dissipation parameter ${\mathcal{D}}$ and due to a finite temperature difference between the hot and cold boundaries, measured by a dimensionless temperature gradient $a$. Different equations of state are examined: ideal gas equation, Murnaghan’s model (often used to describe the interiors of solid but convective planets) and a generic equation of state with adjustable parameters, which can represent any possible equation of state. In the perspective to assess approximations often made in convective models, we also consider two variations of this stability analysis. In a so-called quasi-Boussinesq model, we consider that density perturbations are solely due to temperature perturbations. In a so-called quasi-anelastic liquid approximation model, we consider that entropy perturbations are solely due to temperature perturbations. In addition to the numerical Chebyshev-based stability analysis, an analytical approximation is obtained when temperature fluctuations are written as a combination of only two modes, one being the original symmetrical (between top and bottom) mode introduced by Rayleigh, the other one being antisymmetrical. The analytical solution allows us to show that the antisymmetrical part of the critical eigenmode increases linearly with the parameters $a$ and ${\mathcal{D}}$, while the superadiabatic critical Rayleigh number departs quadratically in $a$ and ${\mathcal{D}}$ from $27\unicode[STIX]{x03C0}^{4}/4$. For any arbitrary equation of state, the coefficients of the quadratic departure are determined analytically from the coefficients of the expansion of density up to degree three in terms of pressure and temperature.

Experimental Study on the Stabilization of the No-Motion State in the Rayleigh-Benard Convection Problem

2006 ◽  
Author(s):  
Marcel C. Remillieux
Keyword(s):  
Rayleigh Number ◽  
Cooling System ◽  
Critical Rayleigh Number ◽  
Flow Patterns ◽  
Test Cell ◽  
Experimental Investigations ◽  
Pd Controller ◽  
Onset Of Convection ◽  
Motion State ◽  
Proportional Controller

We demonstrate experimentally that through the use of proportional-differential control, it is possible to stabilize the no-motion state of a fluid layer heated from below, cooled from above, and confined in an upright, circular cylinder (the Rayleigh-Be´nard problem). An array of 24 independently controlled heaters (thermal actuators), microfabricated on a silicon wafer, constitutes the bottom boundary of the test cell. A cooling system maintains the top boundary at a constant temperature. Silicon diodes located at the mid-height of the cell, above the actuators, measure the fluid's temperature. The multi-input, multi-output controller adjusts the heaters' power in proportion to the deviation of the fluid's temperatures, as recorded by the diodes, from preset values associated with the no-motion, conductive state. First, a set of experiments was conducted in the absence of a controller to determine the uncontrolled, reference state. Advantage is taken of the linear dependence of the mid-height temperature on the power input in the no-motion state. The preset temperatures are determined by extrapolating the mid-height temperatures to the desired input power values. A proportional controller is then engaged. We show that as the controller's gain increases so does the critical Rayleigh number for the onset of convection. The proportional controller allows us to increase the critical Rayleigh number by as much as a factor of 1.4. When the controller's gain is larger than a critical value, the system becomes time-wise oscillatory (Hopf bifurcation) and the controller's performance deteriorates. The oscillatory convection can be significantly damped out by engaging a proportional-differential (PD) controller. The PD controller allows us to further increase the critical Rayleigh number for the onset of convection to as much as a factor or 1.7 compared to the uncontrolled case. Further increases in the critical Rayleigh number were not possible due to the actuators' saturation. We also compared the supercritical flow patterns at the mid-height of the test cell in the presence of the controller with the flow patterns in the absence of a controller. The proportional controller modified the flow pattern from a single convective cell with ascending fluid in one half of the cell and descending in the other half, to fluid ascending at the center of the cell and descending at near the lateral wall. Our work represents an improvement over previous experimental investigations on the stabilization of Rayleigh-Be´nard convection in which the critical Rayleigh number was increased by only a factor of 1.2. Almost uniform temperature distribution at the mid-height is obtained through the combined action of proportional and derivative controllers. The Rayleigh-Be´nard convection is suppressed under conditions when, in the absence of a controller, flow would persist.

Non-linear Rayleigh-Benard Magnetoconvection in Temperature-sensitive Newtonian Liquids with Variable Heat Source

2021 ◽  
Vol 88 (1-2) ◽  
pp. 08
Author(s):  
A. S. Aruna ◽  
V. Ramachandramurthy ◽  
N. Kavitha
Keyword(s):  
Rayleigh Number ◽  
Heat Source ◽  
Variable Viscosity ◽  
Critical Rayleigh Number ◽  
Lorenz Model ◽  
Onset Of Convection ◽  
Variable Heat ◽  
Non Linear ◽  
Source Sink ◽  
Rayleigh Benard

The present paper aims at weak non-linear stability analysis followed by linear analysis of nite-amplitude Rayleigh-Benard magneto convection problem in an electrically conducting Newtonian liquid with heat source/sink. It is shown that the internal Rayleigh number, ther- morheological parameter, and the Chandrasekhar number in uence the onset of convection. The generalized Lorenz model derived for the prob- lem is essentially the classical Lorenz model but with some coecient depending on the variable heat source (sink), viscosity, and the applied magnetic eld. The result of the parameters' in uence on the critical Rayleigh number explains their in uence on the Nusselt number. It is found that an increasing strength of the magnetic eld is to stabilize the system and diminishes heat transport whereas the heat source and variable viscosity in-tandem to work system unstable and enhances heat transfer.

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