THE DIMENSION OF TURBULENCE

We suggest a scenario for turbulence where the fractal dimension D increases with Reynolds number: D = 2 at the onset of turbulence, and D --> 3 at large Reynolds number. The picture is based on a new random-cascade model where the length scales of active eddies are randomly chosen from a probability distribution P(r) of length-scale ratios. Exact expressions for the exponents associated with the velocity field are derived for the distributions P(r) = (gamma + 1)r-gamma. Our picture provides a quantitative explanation for recent measurements of pipe and grid flow at the onset of turbulence.