Chapter 1 Turbulence modeling using fractional derivatives B´ ela J. Szekeres Abstract We propose a new turbulence model in this work. The main idea of the model is that the shear stresses are considered to be random variables and we assume that their differences with respect to time are L´ evy-type distribu- tions. This is a generalization of the classical Newton’s law of viscosity. We tested the model on the classical backward facing step benchmark problem. The simulation results are in a good accordance with real measurements. 1.1 Introduction Turbulence is a velocity fluctuation of the mean flow in fluid dynamics. For this phenomenon there is no any exact definition, we can hardly quantify it and its numerical simulation is also challenging. Its study has a long history, it is enough to refer to the famous wish of Albert Einstein: “After I die, I hope God will explain turbulence to me.” Our study is based on the Navier–Stokes equations as a widely accepted model for fluid dynamics. Starting from this point there are many variant ways to modeling this phenomenon, for example the direct numerical simu- lation, the large eddy simulation and modeling with the Reynolds averaged equations. We propose here a new model and a new way for modeling turbu- lence. We consider the quantity obtained from the Newton’s law of viscosity as a special expected value for the shear stresses. According to our approach, in the simulation we should take into account not only the actual velocity field but also the history of the velocity field to calculate this expected value. We generalize the Navier–Stokes equation, using this hypothesis and get a probabilistic–deterministic model. B´ ela J. Szekeres Department of Applied Analysis and Computational Mathematics, E¨otv¨os Lor´ and Univer- sity H-1117, Budapest, P´azm´any P. stny. 1/C, Hungary, e-mail: szbpagt@cs.elte.hu 1