The dynamics of a subsiding shell at the edges of actively growing shallow cumulus clouds with updrafts is analyzed using direct numerical simulation. The actively growing clouds have a fixed in-cloud buoyancy and velocity. Turbulent mixing and evaporative cooling at the cloud edges generate a subsiding shell which grows with time. A self-similar regime is observed for first and second order moments when normalized with respective maximum values. Internal scales derived from integral properties of the flow problem are identified. Self-similarity analysis conducted by normalizing using these scales reveal that contrary to classical self similar flows, the turbulent kinetic energy budget terms and velocity moments scale according to the buoyancy and not with the mean velocity. The shell thickness is observed to increase linearly with time. The buoyancy scale remains time-invariant and is set by the initial cloud-environment thermodynamics. The shell accelerates ballistically with a magnitude set by the saturation value of the buoyancy of the cloud-environment mixture. In this regime, the shell is buoyancy driven and independent of the in-cloud velocity. Relations are obtained for predicting the shell thickness and minimum velocities by linking the internal scales with external flow parameters. The values thus calculated are consistent with the thickness and velocities observed in typical shallow cumulus clouds. The entrainment coefficient is a function of the initial state of the cloud and the environment, and is shown to be of the same order of magnitude as fractional entrainment rates calculated for large scale models.