  The non-dimensional energy of starting vortex rings typically converges to
values around 0.33 when they are created by a cylinder piston or a bluff body
translating at a constant speed. To explore the limits of the universality of
this value and to analyse the variations that occur outside of those limits, we
present an alternative approach to the slug-flow model to predict the
non-dimensional energy of a vortex ring. Our approach is based on the
self-similar vortex sheet roll-up described by Pullin. We derive the vorticity
distribution for the vortex core resulting from a spiralling shear layer
roll-up and compute the associated non-dimensional energy. To demonstrate the
validity of our model, we consider different velocity profiles of the vortex
generator that follow a power-law with a variable exponent m. For a constant
velocity (m=0), our model yields a non-dimensional energy of E*=0.33. For a
constant acceleration (m=1), we find E*=0.19. For a constant velocity, we
obtain realistic vorticity distributions by radially diffusing the vorticity
distribution of the Pullin spiral and predict a decrease of the non-dimensional
energy from 0.33 to 0.28, in accordance with experimental results. Our proposed
model offers a practical alternative to the existing slug flow model to predict
the minimum non-dimensional energy of a vortex ring. The model is applicable to
piston-generated and wake vortex rings and requires only the kinematics of the
vortex generator as input.
