Fractal behaviors of intermittent turbulence-Applications of fractional dimension and fractional derivatives

Fractional dimension was coined by Mandelbrot 30 years ago and the fractional derivatives proposed in 1695 by L 'Hospital has a history of more than 400 years. In this paper, intermittent turbulence problems in physics have been applied to illustrate the physical significance of fractal dimension and fractional derivatives. Due to incomplete occupation of the intermittent turbulent eddies in the space, so the dimension of intermittent turbulence is 2<D<3. Due to the reduced proportion, the power of small vortex is decreased and the slope of power spectrum over the inertial range is increased (i. e. power spectrum index is increased). The coexistence of vortices of various sizes causes that the walk distance and waiting time of turbulent eddies are very different, which results in that the turbulence is a kind of anomalous diffusion. Since the velocity variations resulted from the cascade or collision of turbulent eddies are not temporary transient, this will affect the velocity field in the future. This will lead to the memory of turbulence motions, so the eddy viscosity of turbulence should be represented by fractional Laplacian operator with memory kernel. Due to the incomplete occupation of the intermittent turbulent eddies in the space, related micro point element, micro area element and micro volume element in fluid mechanics calculation must be modified.