  Fully Developed Turbulence (FDT) is a theoretical asymptotic phenomenon which
can only be approximated experimentally or computationally, so its defining
characteristics are hypothetical. It is considered to be a chaotic stationary
flow field, with self-similar fractalline features. A number of approximate
models exist, often exploiting this self-similarity. The idealized mathematical
model of Fractal Potential Flows is hereby presented, and linked
philosophically to the phenomenon of FDT on a free surface, based on its
experimental characteristics. The model hinges on the recursive iteration of a
fluid dynamical transfer operator. The existence of its unique attractor -
called the invariant flow - is shown in an appropriate function space, which
will serve as our suggested model for the FDT flow field. Its sink
singularities are shown to form an IFS fractal, explicitly resolving
Mandelbrot's Conjecture. Meanwhile an isometric isomorphism is defined between
flows and probability measures, hinting at a wealth of future research. The
inverse problem of representing turbulent flow fields with this model is
discussed in closing, along with explicit practical considerations for
experimental verification and visualization.
