Everything I ever learned about chemistry, electrochemistry, and the nature of ionic bonds, covalent bonds, etc in molecular materials argues that there is no possible way to recover all of the energy expended in producing "Brown's Gas", let alone provide a effective fuel by itself.
That said, there have been anomalous results reproducibly found in controlled circumstances. Most striking are the results demonstrated in affecting "spontaneous" nuclear reaction rates amongst radioactive materials. Both hydrogen gas and oxygen preferentially combine to form two atom molecules in nature - neither gas is commonly found as a single atom, it is virtually always H2 or O2 - the monatomic state is unstable (which is a function of their electron orbital shells being "unfilled").
So, the energy that must be expended in an electrolytic apparatus to seperate the atoms in the water into its constituent gasses is known, and that amount of energy is what is returned to the system upon combustion - at 100% efficiency. 100% efficiency is impossible to attain - any energy that appears as heat, or sound, or radio frequency noise (and electrolysis creates all three) is "lost" energy, insofar as useful work is concerned.
I've been looking, but I've found nothing credible that leads me to accept water as fuel. Water as a fuel additive, even electrolytic dissociation of water into hydrogen gas and oxygen for recombination as part of a fuel load in an engine may have advantages, but the net cost/benefit in energy expenditure is negative, overall.
