Linear stability of rotating convection in an imposed shear flow

1997 ◽  
Vol 350 ◽  
pp. 271-293 ◽  
Author(s):  
PAUL MATTHEWS ◽  
STEPHEN COX
Keyword(s):  
Rayleigh Number ◽  
Shear Flow ◽  
Eigenvalue Problem ◽  
Linear Stability ◽  
Thermal Convection ◽  
Critical Rayleigh Number ◽  
Rotation Vector ◽  
Linear Stability Theory ◽  
Fixed Temperature ◽  
Differential Motion

In many geophysical and astrophysical contexts, thermal convection is influenced by both rotation and an underlying shear flow. The linear theory for thermal convection is presented, with attention restricted to a layer of fluid rotating about a horizontal axis, and plane Couette flow driven by differential motion of the horizontal boundaries.The eigenvalue problem to determine the critical Rayleigh number is solved numerically assuming rigid, fixed-temperature boundaries. The preferred orientation of the convection rolls is found, for different orientations of the rotation vector with respect to the shear flow. For moderate rates of shear and rotation, the preferred roll orientation depends only on their ratio, the Rossby number.It is well known that rotation alone acts to favour rolls aligned with the rotation vector, and to suppress rolls of other orientations. Similarly, in a shear flow, rolls parallel to the shear flow are preferred. However, it is found that when the rotation vector and shear flow are parallel, the two effects lead counter-intuitively (as in other, analogous convection problems) to a preference for oblique rolls, and a critical Rayleigh number below that for Rayleigh–Bénard convection.When the boundaries are poorly conducting, the eigenvalue problem is solved analytically by means of an asymptotic expansion in the aspect ratio of the rolls. The behaviour of the stability problem is found to be qualitatively similar to that for fixed-temperature boundaries.Fully nonlinear numerical simulations of the convection are also carried out. These are generally consistent with the linear stability theory, showing convection in the form of rolls near the onset of motion, with the appropriate orientation. More complicated states are found further from critical.

Stability of Thermal Convection in a Vertical Porous Layer

1987 ◽  
Vol 109 (4) ◽  
pp. 889-893 ◽  
Author(s):  
L. P. Kwok ◽  
C. F. Chen
Keyword(s):  
Rayleigh Number ◽  
Linear Stability ◽  
Porous Layer ◽  
Thermal Convection ◽  
Variable Viscosity ◽  
Critical Rayleigh Number ◽  
Linear Stability Theory ◽  
Lateral Heating ◽  
The Stability ◽  
Allowable Temperature

Experiments were carried out to study the stability of thermal convection generated in a vertical porous layer by lateral heating in a tall, narrow tank. The porous medium, consisting of glass beads, was saturated with distilled water. It was found that the flow became unstable at a critical ΔT of 29.2°C (critical Rayleigh number of 66.2). Linear stability analysis was applied to study the effects of the Brinkman term and of variable viscosity separately using a quadratic relationship between the density and temperature. It was found that with the Brinkman term, no instability could occur within the allowable temperature difference across the tank. With the effect of variable viscosity included, linear stability theory predicts a critical ΔT of 43.4°C (Rayleigh number of 98.3).

Stability of Gravity Driven Convection in a Cylindrical Porous Layer Subjected to Vibration

2006 ◽  
Author(s):  
Saneshan Govender
Keyword(s):  
Porous Medium ◽  
Rayleigh Number ◽  
Linear Stability ◽  
Density Gradient ◽  
Porous Layer ◽  
Binary Alloy ◽  
Critical Rayleigh Number ◽  
Linear Stability Theory ◽  
Direct Result ◽  
Porous Cylinder

In both pure fluids and porous media, the density gradient becomes unstable and fluid motion (convection) occurs when the critical Rayleigh number is exceeded. The classical stability analysis no longer applies if the Rayleigh number is time dependant, as found in systems where the density gradient is subjected to vibration. The influence of vibrations on thermal convection depends on the orientation of the time dependant acceleration with respect to the thermal stratification. The problem of a vibrating porous cylinder has numerous important engineering applications, the most important one being in the field of binary alloy solidification. In particular we may extend the above results to understanding the dynamics in the mushy layer (essentially a reactive porous medium) that is sandwiched between the underlying solid and overlying melt regions. Alloyed components are widely used in demanding and critical applications, such as turbine blades, and a consistent internal structure is paramount to the performance and integrity of the component. Alloys are susceptible to the formation of vertical channels which are a direct result of the presence convection, so any technique that suppresses convection/the formation of channels would be welcomed by the plant metallurgical engineer. In the current study, the linear stability theory is used to investigate analytically the effects of gravity modulation on convection in a homogeneous cylindrical porous layer heated from below. The linear stability results show that increasing the frequency of vibration stabilizes the convection. In addition the aspect ratio of the porous cylinder is shown to influence the stability of convection for all frequencies analysed. It was also observed that only synchronous solutions are possible in cylindrical porous layers, with no transition to sub harmonic solutions as was the case in Govender (2005a) for rectangular layers or cavities. The results of the current analysis will be used in the formulation of a model for binary alloy systems that includes the reactive porous medium model.

Stability of Heat Pipes in Vapor-Dominated Systems

1999 ◽  
Author(s):  
Pouya Amili ◽  
Yanis C. Yortsos
Keyword(s):  
Rayleigh Number ◽  
Linear Stability ◽  
Oil Recovery ◽  
Critical Rayleigh Number ◽  
Stability Limit ◽  
Wave Number ◽  
Nuclear Waste Repository ◽  
Geothermal Systems ◽  
Dimensionless Numbers ◽  
The Stability

Abstract We study the linear stability of a two-phase heat pipe zone (vapor-liquid counterflow) in a porous medium, overlying a superheated vapor zone. The competing effects of gravity, condensation and heat transfer on the stability of a planar base state are analyzed in the linear stability limit. The rate of growth of unstable disturbances is expressed in terms of the wave number of the disturbance, and dimensionless numbers, such as the Rayleigh number, a dimensionless heat flux and other parameters. A critical Rayleigh number is identified and shown to be different than in natural convection under single phase conditions. The results find applications to geothermal systems, to enhanced oil recovery using steam injection, as well as to the conditions of the proposed Yucca Mountain nuclear waste repository. This study complements recent work of the stability of boiling by Ramesh and Torrance (1993).

Lower Bounds to the Critical Rayleigh Number in Completely Confined Regions

1967 ◽  
Vol 34 (2) ◽  
pp. 308-312 ◽  
Author(s):  
M. Sherman ◽  
S. Ostrach
Keyword(s):  
Rayleigh Number ◽  
Lower Bounds ◽  
Body Force ◽  
Critical Rayleigh Number ◽  
Convective Motion ◽  
The Body ◽  
Maximum Dimension ◽  
Fixed Temperature ◽  
Boussinesq Fluid ◽  
The Given

A method is presented for estimating lower bounds to the minimum Rayleigh number that will induce a state of convective motion in a quasi-incompressible (Boussinesq) fluid where the temperature gradient is in the direction of the body force. The fluid is completely confined by fixed-temperature, rigid bounding walls. For any arbitrary region, the critical Rayleigh number is greater than 1558(h/D)4 where h is the maximum dimension of the given region in the direction of the body force and D is the diameter of an equal volume sphere. In certain simple geometrical configurations, improved lower-bound estimates are calculated.

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.

Convection in a box: linear theory

1967 ◽  
Vol 30 (3) ◽  
pp. 465-478 ◽  
Author(s):  
Stephen H. Davis
Keyword(s):  
Rayleigh Number ◽  
Linear Stability ◽  
Linear Theory ◽  
Cross Sections ◽  
Critical Rayleigh Number ◽  
Three Dimensional ◽  
Wave Number ◽  
Spatial Variables ◽  
Zero Velocity ◽  
The Cross

The linear stability of a quiescent, three-dimensional rectangular box of fluid heated from below is considered. It is found that finite rolls (cells with two non-zero velocity components dependent on all three spatial variables) with axes parallel to the shorter side are predicted. When the depth is the shortest dimension, the cross-sections of these finite rolls are near-square, but otherwise (in wafer-shaped boxes) narrower cells appear. The value of the critical Rayleigh number and preferred wave-number (number of finite rolls) for a given size box is determined for boxes with horizontal dimensions h, ¼ ≤ h/d ≤ 6, where d is the depth.

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.

Sizes and shapes of salt fingers near the marginal state

1976 ◽  
Vol 78 (3) ◽  
pp. 601-607 ◽  
Author(s):  
C. F. Chen ◽  
R. D. Sandford
Keyword(s):  
Rayleigh Number ◽  
Stability Theory ◽  
Critical Rayleigh Number ◽  
Linear Stability Theory ◽  
Two Component System ◽  
Component System ◽  
Number Ratio ◽  
Salt Fingers ◽  
Two Component ◽  
Marginal State

Experiments were carried out to investigate the onset, size and shape of fingers in a salt–sugar two-component system. The results on the critical Rayleigh number ratio for the marginal state agree well with the prediction of the linear stability theory of Stern (1960). The theory also predicts that the fingers at the marginal state should be as wide as the layer thickness. In our experiments, in a layer of approximately 3·5 cm deep, the fingers, when they appeared, were always long and narrow, with typical lateral dimensions of approximately 1 mm.

Sign in / Sign up

Export Citation Format

Share Document