The dependency of the self-similar Rayleigh-Taylor bubble acceleration constant ${\alpha }_b\mathbf{(}\equiv [\left(\text{amplitude}\right)∕2]\times \left(\text{displacement}\right)\times \left(\text{Atwood}\text{number}\right)\mathbf{)}$ on the initial perturbation amplitude $h_{0k}$ is described with a model in which the exponential growth of a small amplitude packet of modes makes a continuous nonlinear transition to its “terminal” bubble velocity $\propto \text{Fr}\left[\text{equal}\text{to}{\left(\text{Froude}\text{number}\right)}^{1∕2}\right]$. Then, by applying self-similarity (diameter $\propto$ amplitude), ${\alpha }_b$ is found to increase proportional to $\text{Fr}$ and logarithmically with $h_{0k}$. The model has two free parameters that are determined from experiments and simulations. The augmentation of long wavelength perturbations by mode coupling is also evaluated. This is found to decrease the sensitivity of ${\alpha }_b$ on the initial perturbations when they are smaller than the saturation amplitude of the most unstable modes. These results show that ${\alpha }_b$ can vary by a factor of 2–3 with initial conditions in reasonable agreement with experiments and simulations.

• Received 6 October 2003

DOI:https://doi.org/10.1103/PhysRevE.69.056305