On the logarithmic mean profile

Abstract

Elements of the first-principles-based theory of Wei et al. (J. Fluid Mech., vol. 522, 2005, p. 303), Fife et al. (Multiscale Model. Simul., vol. 4, 2005a, p. 936; J. Fluid Mech., vol. 532, 2005b, p. 165) and Fife, Klewicki & Wei (J. Discrete Continuous Dyn. Syst., vol. 24, 2009, p. 781) are clarified and their veracity tested relative to the properties of the logarithmic mean velocity profile. While the approach employed broadly reveals the mathematical structure admitted by the time averaged Navier–Stokes equations, results are primarily provided for fully developed pressure driven flow in a two-dimensional channel. The theory demonstrates that the appropriately simplified mean differential statement of Newton's second law formally admits a hierarchy of scaling layers, each having a distinct characteristic length. The theory also specifies that these characteristic lengths asymptotically scale with distance from the wall over a well-defined range of wall-normal positions, y. Numerical simulation data are shown to support these analytical findings in every measure explored. The mean velocity profile is shown to exhibit logarithmic dependence (exact or approximate) when the solution to the mean equation of motion exhibits (exact or approximate) self-similarity from layer to layer within the hierarchy. The condition of pure self-similarity corresponds to a constant leading coefficient in the logarithmic mean velocity equation. The theory predicts and clarifies why logarithmic behaviour is better approximated as the Reynolds number gets large. An exact equation for the leading coefficient (von Kármán coefficient κ) is tested against direct numerical simulation (DNS) data. Two methods for precisely estimating the leading coefficient over any selected range of y are presented. These methods reveal that the differences between the theory and simulation are essentially within the uncertainty level of the simulation. The von Kármán coefficient physically exists owing to an approximate self-similarity in the flux of turbulent force across an internal layer hierarchy. Mathematically, this self-similarity relates to the slope and curvature of the Reynolds stress profile, or equivalently the slope and curvature of the mean vorticity profile. The theory addresses how, why and under what conditions logarithmic dependence is approximated relative to the specific mechanisms contained within the mean statement of dynamics.

Mean dynamics of transitional channel flow

“…Mean flow theory in the four-layer regime A number of the mathematical and Reynolds number scaling properties associated with the U and T profiles in the four-layer regime have been analytically determined and empirically verified Wei et al 2005a;Klewicki et al 2009). Central to these analyses is the finding that the equations of motion can be…”

Properties of the streamwise velocity fluctuations in the inertial layer of turbulent boundary layers and their connection to self-similar mean dynamics

“…Efforts over the past decade have sought to address the theoretical challenges just described by exploring what can be discerned directly from the equations governing the mean dynamics, e.g., Fife et al, 2005;Fife et al, 2009;Klewicki et al, 2009;. One advantage of this line of inquiry is that the mean velocity, UðyÞ, and Reynolds shear stress, T ¼ ÀqhuviðyÞ, are solutions to these equations, while, more generally, the energetics of the turbulence (as in other dissipative systems) emerge as a consequence of these dynamics.…”

On the universality of inertial energy in the log layer of turbulent boundary layer and pipe flows

“…For example, classically the start of the log region is taken to be at a fixed value of z + , while others (e.g. Klewicki et al 2009) have argued that it scales with Re 1/2 τ in inner variables. Throughout this paper, we will follow Marusic et al (2013) and refer to the 'log region' as the range of wall distances where the mean and variance of the streamwise velocity follow logarithmic laws and use the tentative and conservative limits, 3 Re 1/2 τ < z + < 0.15 Re τ .…”