Fractional Turbulence Models

  • Peter W. EgolfEmail author
  • Kolumban Hutter
Conference paper
Part of the Springer Proceedings in Physics book series (SPPHY, volume 196)


In this article we propose to generalize Reynolds shear stresses in local zero-equation turbulence models to nonlocal and fractional forms. In the well-accepted general method, starting with a Kraichnanian convolution-integral as Reynolds shear stress, different weighting functions are possible candidates to serve this purpose; e.g. the Liouville weighting function leads to the left-handed Riemann–Liouville fractional derivative and the Heaviside distribution to a mean velocity difference, respectively a difference quotient. Therefore, this weighting function transforms the first gradient (the one in the eddy diffusivity) of Prandtl’s 1925 mixing-length model to an eddy diffusivity with a mean velocity difference and, thereby, directly leads to the (modified) Prandtl shear-layer model of 1942. Prandtl’s intuitive development—which is in agreement with fractional calculus—does not serve as a proof of correctness, but is a welcome coincidence. By further following Prandtl’s intuition and applying the Heaviside distribution also to the remaining driving gradient, yields the Difference-Quotient Turbulence Model (DQTM), which was discovered by other means and had been published by Egolf in 1991. As a result, it becomes clear that the DQTM is a natural nonlocal extension of Prandtl’s models and contains a special case of a simple fractional derivative, namely a difference quotient, which stands for the highest possible nonlocality and minimum calculation time to solve a turbulent flow problem.


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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.University of Applied Sciences of Western SwitzerlandYverdonSwitzerland
  2. 2.Swiss Federal Institute of TechnologyETHZurichSwitzerland

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