Fractal dimension and scale entropy applications in a spray

Multi-scale structure of spray images is investigated for varying ranges of pressure and temperature in quiescent air. For spray images a standard PIV set is used consisting basically on a CCD camera and a laser sheet. A deviation to fractality is evidenced: the scale analysis has a parabolic form. A scale-dependent fractal dimension is measured which displays a linear variation with scale-logarithm. The classical fractal dimension usually measured so far is reinterpreted as a mean slope for scales close to the outer cut-off scale. This multi-scale behaviour is described by a diffusion equation of a new geometrical quantity called scale entropy related to the wrinkling of a set over scales. This equation is based on the conservation of a scale entropy flux through scale-space which is interpreted as the evolutive potential of the system at a given scale. This gives access to the scale-dependency of fractal dimension and points to the importance of the variations through scale space of this evolutive potential and namely its gradient. It has been shown that for sprays, the evolution of the evolutive potential gradient is constant through scale space which corresponds to a parabolic behaviour for scale analysis.