The initial transient and approach to self-similarity of a very viscous buoyant thermal

Abstract

AbstractAn isolated buoyant thermal in very viscous fluid has been shown to attain a self-similar form at large times which grows as $t^{1/2}$ (Whittaker & Lister J. Fluid Mech., vol. 606, 2008, pp. 295–324). For large values of the Rayleigh number $\mathit{Ra}$ (based on the conserved total buoyancy), the similarity solution is slender with a roughly spherical head at the top and a long tail that contains most of the buoyancy and extends down to the origin. We investigate the time-dependent behaviour of the thermal numerically; both the long-time behaviour in terms of perturbations to the similarity solution and the short-time evolution from a spherical initial condition. Using a spectral method, we find the growth rates of the linear perturbations and their spatial structure in similarity space. All eigenmodes decay monotonically for $\mathit{Ra}\lesssim 360$, while for larger $\mathit{Ra}$ the dominant (slowest decaying or fastest growing) eigenmodes are oscillatory with waves propagating up the tail. Above a critical value $\mathit{Ra}_c \approx 10\, 000$, the steady solution becomes unstable to a limit cycle. A one-dimensional reduction to horizontally integrated quantities hints at a theoretical explanation for the oscillatory behaviour, but does not reproduce the loss of stability. Investigation of the initial transient at large $\mathit{Ra}$ reveals that an initially spherical thermal can rise $O(100)$ times its initial diameter before approaching its final self-similar shape. The presence of a rigid horizontal floor below the thermal makes a quantitative difference of around 10 % to the rate of rise at large $\mathit{Ra}$.