Instability of stratified fluid in a vertical cylinder

Abstract

In a previous paper we analysed the stability to small disturbances of stationary stratified fluid which is unbounded. Various forms of the undisturbed density distribution were considered, including a sinusoidal profile and a function of the vertical coordinate z which is constant outside a central horizontal layer. Both these types of stratification are so unstable that the critical Rayleigh number is zero. In this sequel we make the study more complete and more useful by taking account of the effect of a vertical circular cylindrical boundary of radius a which is rigid and impermeable. As in the previous paper we assume that the undisturbed density distribution is steady.The case of fluid in a vertical tube with a uniform density gradient is useful for comparison, and so we review and extend the available results, in particular obtaining growth rates for a disturbance which is neither z-independent nor axisymmetric. A numerical finite-difference method is then developed for the case in which dρ/dz = ρ0kA cos kz. When ka [Lt ] 1 the relation between growth rate and Rayleigh number approximates to that for a uniform density gradient of magnitude ρ0kA; and when ka [Gt ] 1 the tilting-sliding mechanism identified in the previous paper is relevant and the results approximate to those for an unbounded fluid, except that the smallest Rayleigh number for a neutral disturbance is not zero but is of order (ka-1. In the case of an undisturbed density which varies only in a central layer of thickness l, the same mechanism is at work when the horizontal lengthscale of the disturbance is large compared with l, resulting in high growth rates and a critical Rayleigh number which vanishes with l/a. Estimates of the growth rate are given for some particular density profiles.