Finite Amplitude Longitudal Convection Rolls in an Inclined Layer

1973 ◽  
Vol 95 (3) ◽  
pp. 407-408 ◽  
Author(s):  
R. M. Clever
Keyword(s):  
Fluid Layer ◽  
Horizontal Layer ◽  
Finite Amplitude ◽  
Two Dimensional ◽  
Governing Equations ◽  
Inclined Layer ◽  
Experimental Heat ◽  
Inclined Fluid Layer ◽  
Longitudinal Coordinate ◽  
Convection Rolls

For the case of a large Prandtl number, buoyancy driven flow in an inclined fluid layer, it is shown that all longitudinal-coordinate-independent solutions of the governing equations are obtainable from a knowledge of the existing results for two-dimensional convection in a horizontal layer, heated from below. The rescaling here yields results which compare favorably with those of existing experimental heat transport values.

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

1979 ◽  
Vol 94 (4) ◽  
pp. 609-627 ◽  
Author(s):  
R. M. Clever ◽  
F. H. Busse
Keyword(s):  
Rotation Rate ◽  
Fluid Layer ◽  
Three Dimensional ◽  
Vertical Axis ◽  
Horizontal Layer ◽  
Finite Amplitude ◽  
Two Dimensional ◽  
Nonlinear Properties ◽  
Prandtl Numbers ◽  
The Galerkin Method

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 Küppers & 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.

Nonlinear interaction of magnetic field and convection

1975 ◽  
Vol 71 (1) ◽  
pp. 193-206 ◽  
Author(s):  
F. H. Busse
Keyword(s):  
Magnetic Field ◽  
Magnetic Energy ◽  
Fluid Layer ◽  
Boussinesq Equations ◽  
Finite Amplitude ◽  
Magnetic Reynolds Number ◽  
Convection Rolls ◽  
Rayleigh Numbers ◽  
Horizontal Fluid Layer ◽  
Chandrasekhar Number

The interaction between convection in a horizontal fluid layer heated from below and an ambient vertical magnetic field is considered. The analysis is based on the Boussinesq equations for two-dimensional convection rolls and the assumption that the amplitude A of the convection and the Chandrasekhar number Q are small. It is found that the magnetic energy is amplified by a factor of order R½m, where Rm is the magnetic Reynolds number. The ratio between the magnetic and kinetic energies can reach values much larger than unity. Although the magnetic field always inhibits convection, this influence decreases with increasing amplitude of convection. Thus finite amplitude onset of steady convection becomes possible at Rayleigh numbers considerably below the values predicted by linear theory.

scholarly journals On nonlinear overstable convection rolls in a rotating system

1984 ◽  
Vol 25 (3) ◽  
pp. 406-418 ◽  
Author(s):  
N. Riahi
Keyword(s):  
Prandtl Number ◽  
Travelling Waves ◽  
Vertical Axis ◽  
Horizontal Layer ◽  
Finite Amplitude ◽  
Oscillatory Convection ◽  
Nonlinear Analyses ◽  
Rotating System ◽  
The Stability ◽  
Convection Rolls

AbstractFinite amplitude oscillatory convection rolls in the form of travelling waves are studied for a horizontal layer of a low Prandtl number fluid heated from below and rotating rapidly about a vertical axis. The results of the stability and nonlinear analyses indicate that there is no subcritical instability and that the oscillatory rolls are unstable for the ranges of the Prandtl number and the rotation rate considered in this paper.

scholarly journals Finite amplitude thermal convection with spatially modulated boundary temperatures

1995 ◽  
Vol 449 (1937) ◽  
pp. 459-478 ◽  
Keyword(s):  
Stability Analysis ◽  
Thermal Convection ◽  
Fluid Layer ◽  
Perturbation Technique ◽  
Lower Boundary ◽  
Finite Amplitude ◽  
Resonant Wavelength ◽  
Two Dimensional ◽  
Dimensional Solution ◽  
One Dimensional

Finite amplitude thermal convection in a fluid layer between two horizontal walls with different fixed mean temperatures is considered when spatially modulated temperatures with amplitudes L 1 * and L u * are prescribed at the lower and upper walls, respectively. The nonlinear steady problem is solved by a perturbation technique, and the preferred mode of convection is determined by a stability analysis. In the case of a resonant wavelength excitation, regular or non-regular multi-modal pattern convection can be preferred for some ranges of L 1 * and L u *, provided the wave vectors for such patterns are contained in a certain subset of the wave vectors representing a linear combination of modulated upper and lower boundary temperatures. In the case of non-resonant wavelength excitation, a three (two) dimensional solution in the form of multi-modal (rolls) pattern convection can be preferred, even if the boundary modulations are one (two) or two (one) dimensional, provided the wavelengths of the modulations are not too small. Heat transported by convection can be enhanced by boundary modulations in some ranges of L 1 * and L u *.

Instabilities of convection rolls in a fluid of moderate Prandtl number

1979 ◽  
Vol 91 (2) ◽  
pp. 319-335 ◽  
Author(s):  
F. H. Busse ◽  
R. M. Clever
Keyword(s):  
Prandtl Number ◽  
Fluid Layer ◽  
Experimental Methods ◽  
Two Dimensional ◽  
Theoretical Predictions ◽  
Convection Rolls ◽  
Rayleigh Numbers ◽  
Horizontal Fluid Layer

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.

Transition to time-dependent convection

1974 ◽  
Vol 65 (4) ◽  
pp. 625-645 ◽  
Author(s):  
R. M. Clever ◽  
F. H. Busse
Keyword(s):  
Equations Of Motion ◽  
Three Dimensional ◽  
Horizontal Layer ◽  
Two Dimensional ◽  
Free Boundaries ◽  
Prandtl Numbers ◽  
Steady Solutions ◽  
The Stability ◽  
Convection Rolls ◽  
Good Agreement

Steady solutions in the form of two-dimensional rolls are obtained for convection in a horizontal layer of fluid heated from below as a function of the Rayleigh and Prandtl numbers. Rigid boundaries of infinite heat conductivity are assumed. The stability of the two-dimensional rolls with respect to three-dimensional disturbances is analysed. It is found that convection rolls are unstable for Prandtl numbers less than about 5 with respect to an oscillatory instability investigated earlier by Busse (1972) for the case of free boundaries. Since the instability is caused by the momentum advection terms in the equations of motion the Rayleigh number for the onset of instability increases strongly with Prandtl number. Good agreement with various experimental observations is found.

Nonlinear thermal convection with finite conducting boundaries

1985 ◽  
Vol 152 ◽  
pp. 113-123 ◽  
Author(s):  
N. Riahi
Keyword(s):  
Stability Analysis ◽  
Prandtl Number ◽  
Thermal Convection ◽  
Perturbation Technique ◽  
Three Dimensional ◽  
Horizontal Layer ◽  
Finite Amplitude ◽  
Two Dimensional ◽  
Convection Pattern ◽  
Square Cells

Finite-amplitude thermal convection in a horizontal layer with finite conducting boundaries is investigated. The nonlinear steady problem is solved by a perturbation technique, and the preferred mode of convection is determined by a stability analysis. Square cells are found to be the preferred form of convection in a semi-infinite three-dimensional region Ω in the (γb,γt, P)-space (γb and γt are the ratios of the thermal conductivities of the lower and upper boundaries to that of the fluid and P is the Prandtl number). Two-dimensional rolls are found to be the preferred convection pattern outside Ω. The dependence on γb, γt and P of the heat transported by convection is computed for the various solutions analysed in the paper.

The effect of initial conditions and lateral boundaries on convection

1969 ◽  
Vol 37 (1) ◽  
pp. 81-94 ◽  
Author(s):  
Theodore D. Foster
Keyword(s):  
Nusselt Number ◽  
Initial Conditions ◽  
Fluid Layer ◽  
Double Fourier Series ◽  
Finite Amplitude ◽  
Lateral Boundary ◽  
Two Dimensional ◽  
Lateral Boundary Conditions ◽  
Linear Differential ◽  
Steady State Solutions

A theoretical analysis of two-dimensional, finite-amplitude, thermal convection is made for a fluid which has an infinite Prandtl number. The vertical velocity disturbance is expanded in a double Fourier series which satisfies the horizontal and lateral boundary conditions. The resulting coupled sets of non-linear differential equations are solved numerically. It is found that for a particular Rayleigh number the number and size of the convection cells that form depend upon the ratio of the distance between the lateral boundaries to the depth of the fluid layer and on the initial conditions. The steady-state solutions are not unique and the solution for which the heat transport is a maximum is not necessarily the solution that results. Where there are no lateral boundaries, the lateral edges of the cells tend to tilt and the Nusselt number increases slightly.

Natural Convection in an Inclined Fluid Layer With a Transverse Magnetic Field: Analogy With a Porous Medium

1995 ◽  
Vol 117 (1) ◽  
pp. 121-129 ◽  
Author(s):  
P. Vasseur ◽  
M. Hasnaoui ◽  
E. Bilgen ◽  
L. Robillard
Keyword(s):  
Magnetic Field ◽  
Transverse Magnetic Field ◽  
Hartmann Number ◽  
Fluid Layer ◽  
Critical Rayleigh Number ◽  
Integral Form ◽  
Transverse Magnetic ◽  
Horizontal Layer ◽  
Governing Equations ◽  
Onset Of Convection

In this paper the effect of a transverse magnetic field on buoyancy-driven convection in an inclined two-dimensional cavity is studied analytically and numerically. A constant heat flux is applied for heating and cooling the two opposing walls while the other two walls are insulated. The governing equations are solved analytically, in the limit of a thin layer, using a parallel flow approximation and an integral form of the energy equation. Solutions for the flow fields, temperature distributions, and Nusselt numbers are obtained explicitly in terms of the Rayleigh and Hartmann numbers and the angle of inclination of the cavity. In the high Hartmann number limit it is demonstrated that the resulting solution is equivalent to that obtained for a porous layer on the basis of Darcy’s model. In the low Hartmann number limit the solution for a fluid layer in the absence of a magnetic force is recovered. In the case of a horizontal layer heated from below the critical Rayleigh number for the onset of convection is derived in term of the Hartmann number. A good agreement is found between the analytical predictions and the numerical simulation of the full governing equations.

The effect of distant sidewalls on the transition to finite amplitude Benard convection

1978 ◽  
Vol 358 (1693) ◽  
pp. 173-197 ◽  
Keyword(s):  
Heat Transfer ◽  
Theoretical Model ◽  
Rayleigh Number ◽  
Fluid Layer ◽  
Critical Value ◽  
Finite Amplitude ◽  
Smooth Transition ◽  
Two Dimensional ◽  
Bénard Convection ◽  
Benard Convection

The theory of the nonlinear development of Benard convection in an infinite fluid layer confined between horizontal boundaries predicts that the amplitude of the motion undergoes a bifurcation as the Rayleigh number passes through the critical value for instability predicted by linear theory. Segel (1969) has shown that this is also the case if the flow is confined laterally by rigid perfectly insulating sidewalls. In the present paper it is shown that if there is a small heat transfer through these walls (so that the boundary conditions there are inconsistent with a state of no motion) the bifurcation is in general replaced by a smooth transition to finite amplitude convection. This effect also ensures that the motion is stable. Although an idealized theoretical model is assumed in which the flow is two-dimensional and stress-free at the horizontal boundaries, the results apply qualitatively to more realistic models.

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