Lévy statistics and beta model: A new solution of “wall” turbulence with a critical phenomenon

By applying statistical mechanics Brown's motion leads to the macroscopic diffusion equation, and similar Lévy flights and the β model are the specific flux origin to derive the Difference-Quotient Turbulence Model (DQTM). It is applied to a turbulent flow in the overlap region close to a wall and leads to a mean velocity deficit power law, which may serve as a substitute for the “logarithmic law” with the von Karman constant. Its exponent is slightly dependent on the stress parameter (e.g. the Reynolds number) and is confirmed by the Princeton super pipe data. The solution of this fluid dynamic standard example reveals a physical order concept, just as known in the theory of critical phenomena.