Density maximum and finite Darcy–Prandtl number outlooks on Gill's stability problem subject to a lack of thermal equilibrium

When a layer of fluid is heated uniformly from below and cooled from above, a cellular regime of steady convection is set up at values of the Rayleigh number exceeding a critical value. A method is presented here to determine the form and amplitude of this convection. The non-linear equations describing the fields of motion and temperature are expanded in a sequence of inhomogeneous linear equations dependent upon the solutions of the linear stability problem. We find that there are an infinite number of steady-state finite amplitude solutions (having different horizontal plan-forms) which formally satisfy these equations. A criterion for ‘relative stability’ is deduced which selects as the realized solution that one which has the maximum mean-square temperature gradient. Particular conclusions are that for a large Prandtl number the amplitude of the convection is determined primarily by the distortion of the distribution of mean temperature and only secondarily by the self-distortion of the disturbance, and that when the Prandtl number is less than unity self-distortion plays the dominant role in amplitude determination. The initial heat transport due to convection depends linearly on the Rayleigh number; the heat transport at higher Rayleigh numbers departs only slightly from this linear dependence. Square horizontal plan-forms are preferred to hexagonal plan-forms in ordinary fluids with symmetric boundary conditions. The proposed finite amplitude method is applicable to any model of shear flow or convection with a soluble stability problem.

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Cellular convection with finite amplitude in a rotating fluid

Journal of Fluid Mechanics ◽

10.1017/s0022112059000283 ◽

1959 ◽

Vol 5 (3) ◽

pp. 401-435 ◽

Cited By ~ 149

Author(s):

G. Veronis

Keyword(s):

Prandtl Number ◽

Stability Problem ◽

Wave Length ◽

Finite Amplitude ◽

Convection Cell ◽

Special Role ◽

Experimental Conditions ◽

Two Fluids ◽

Square Cells

When a rotating layer of fluid is heated uniformly from below and cooled from above, the onset of instability is inhibited by the rotation. The first part of this paper treats the stability problem as it was considered by Chandrasekhar (1953), but with particular emphasis on the physical interpretation of the results. It is shown that the time-dependent (overstable) motions occur because they can reduce the stabilizing effect of rotation. It is also shown that the boundary of a steady convection cell is distorted by the rotation in such a way that the wave length of the cell measured along the distorted boundary is equal to the wavelength of the non-rotating cell. This conservation of cellular wavelength is traced to the constancy of horizontal vorticity in the rotating and non-rotating systems. In the finite-amplitude investigation the analysis, which is pivoted about the linear stability problem, indicates that the fluid can become unstable to finite-amplitude disturbances before it becomes unstable to infinitesimal perturbations. The finite-amplitude motions generate a non-linear vorticity which tends to counteract the vorticity generated by the imposed constraint of rotation. Under experimental conditions the two fluids, mercury and air, which are considered in this paper, will not exhibit this finite amplitude instability. However, a fluid with a sufficiently small Prandtl number will become unstable to finite-amplitude perturbations. The special role of viscosity as an energy releasing mechanism in this problem and in the Orr-Sommerfeld problem suggests that the occurrence of a finite-amplitude instability depends on this dual role of viscosity (i.e. as an energy releasing mechanism as well as the more familiar dissipative mechanism). The relative stability criterion developed by Malkus & Veronis (1958) is used to determine the preferred type of cellular motions which can occur in the fluid. This preferred motion is a function of the Prandtl number and the Taylor number. In the case of air it is shown that overstable square cells become preferred in finite amplitude, even though steady convective motions occur at a lower Rayleigh number.

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