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Date
2025-02-01
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Abstract
This chapter is devoted to the advanced modeling of turbulent flows using fractional calculus. Turbulence, characterized by chaotic and unpredictable fluid motion, plays a crucial role in various engineering and natural systems. Traditional models using integer-order derivatives often fall short in capturing the complexity of turbulent flows. Here we apply fractional-order derivatives to enhance the modeling of fluid turbulent flow. By incorporating fractional derivatives we account for the nonlocal and memory effects inherent in turbulent systems. We begin with an overview of turbulent flows, distinguishing them from laminar flows and highlighting their significance in different fields. The chapter then presents the Reynolds-averaged Navier–Stokes (RANS) equations, detailing their derivation and the closure problem introduced by the Reynolds stresses. Various eddy-viscosity models are discussed, including fractional models that introduce nonlocal elements to better capture the complexities of turbulence. The chapter further discusses the fractional Kolmogorov energy spectrum, illustrating how fractional derivatives modify the classical spectrum to account for nonlocal interactions and various scaling properties. We also examine the application of variable-order fractional models, which allow the order of the derivative to vary spatially and temporally, providing a more accurate representation of turbulent flows with changing dynamics.
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Fractional Modeling of Fluid Flow and Transport Phenomena