On a Fractal Model for Turbulence

In characterizing fully developed turbulence by means of statistical methods, it is still an actual problem to explain what happens if one transits from large scales (scale of the system) to small scales (dissipationrange scale or Kolmogorov microscale). From experiments it is known that the shape of the probability density functions (PDF) of velocity differences between two points dur = (y(y + r) — r(y)) changes sensitively with the chosen length scale (r denotes the distance between two points; y denotes the position; v usually stands for the velocity component in direction of r). Gaussian shapes of the P D F are found for large length scales. Coming to smaller length scales, small and large velocity differences become more probable compared to a Gaussian distribution. This effect is called intermittency. There are two common methods to characterize this intermittency effect. One may evaluate directly the changing form of the PDF. Here a set of theoretical P D F s has to be used. The question is, how well are the experimental P D F s fitted by the theoretical PDFs. An other common method is to calculate the structure functions S 9 (r ) := <((5ur)">, where < . . . ) denotes the spatial average. Using the structure functions to evaluate the intermittency in turbulence, one has to be very cautious because the calculations become precarious already for relatively small values of q. This point will be discussed later in this paper. To see the connection between these two methods, one should note that S(r) corresponds to the q-th moment of the probability density function for the chosen r, denoted