Symmetries and scaling‐laws in turbulence

Abstract

AbstractA new turbulence approach based on Lie‐group analysis is presented. It unifies a large set of self‐similar solutions for the mean velocity of stationary parallel turbulent shear flows. The theory is derived from the Reynolds averaged Navier‐Stokes equations, the fluctuation equations, and the velocity product equations, which are the dyad product of the velocity fluctuations with the equations for the velocity fluctuations. For the plane case the results include the logarithmic law of the wall, an algebraic law, the viscous sublayer, the linear region in the centre of a Couette flow and in the centre of a rotating channel flow, and a new exponential mean velocity profile that corresponds to the mid‐wake region of high Reynolds number flat‐plate boundary layers. The algebraic scaling law conforms to both the centre and the near wall regions of turbulent channel flows. For a non‐rotating and a moderately rotating pipe about its axis an algebraic law was found for the axial and the azimuthal velocity near the pipe‐axis with both laws having equal scaling exponents. In case of a rapidly rotating pipe a new logarithmic scaling law for the axial velocity is developed.