The law of the wake in the turbulent boundary layer

Abstract

SUMMARYAfter an extensive survey of mean-velocity profile measurements in various two-dimensional incompressible turbulent boundarylayer flows, it is proposed to represent the profile by a linear combination of two universal functions. One is the well-known law of the wall. The other, called the law of the wake, is characterized by the profile at a point of separation or reattachment. These functions are considered to be established empirically, by a study of the mean-velocity profile, without reference to any hypothetical mechanism of turbulence. Using the resulting complete analytic representation for the mean-velocity field, the shearing-stress field for several flows is computed from the boundary-layer equations and compared with experimental data.The development of a turbulent boundary layer is ultimately interpreted in terms of an equivalent wake profile, which supposedly represents the large-eddy structure and is a consequence of the constraint provided by inertia. This equivalent wake profile is modified by the presence of a wall, at which a further constraint is provided by viscosity. The wall constraint, although it penetrates the entire boundary layer, is manifested chiefly in the sublayer flow and in the logarithmic profile near the wall.Finally, it is suggested that yawed or three-dimensional flows may be usefully represented by the same two universal functions, considered as vector rather than scalar quantities. If the wall component is defined to be in the direction of the surface shearing stress, then the wake component, at least in the few cases studied, is found to be very nearly parallel to the gradient of the pressure.

Dominant features in three-dimensional turbulence structure: comparison of non-uniform accelerating and decelerating flows

“…(2), relatively consistent values have been found in different literature studies. A range of j = 0.40-0.42 was proposed for the flows over smooth bed investigated by Coles [6] and Cardoso et al [4]; while, similar value (j = 0.40) was also suggested for the rough bed flow by Song et al [30]. More recently, Auel et al [2] summarised from various smooth and rough bed flow studies that j = 0.385-0.435, universally.…”

“…More recently, Auel et al [2] summarised from various smooth and rough bed flow studies that j = 0.385-0.435, universally. For the log-law integration constant B r of smooth bed uniform flows, it was proposed as B r = 4.9 in Mellor and Gibson [15], and Anwar and Atkins [1]; and B r = 5.1 in Coles [6], and Cardoso et al [4]. In comparison to different rough bed flow studies (i.e.…”

“…To systematically represent flow velocity, the Prandtl van Karman type velocity distribution's logarithmic-wall law was utilised by Keulegan [10] in his investigation on rectangular open channel flow. To improve Keulegan's study, Coles [6] proposed the log-wake law with a wake correction to more precisely represent velocity distribution at the outer flow region where the ratio of flow vertical location to full flow depth (z/h) is bigger than 0.2. Coles' method has been proven to give better accuracy compared to the log-wall law, as concluded by Song and Graf [29] and Dey and Raikar [7]; as well as in the modified log-wake law study by Yang [32].…”

An experiment on the adiabatic compressible turbulent boundary layer in adverse and favourable pressure gradients

“…the law of the wall, and the velocity defect law. Both of these are contained within the composite boundary-layer profile used by Coles (1956): where V, = (~, / p ) $ is the friction velocity, Y is the kinematic viscosity, K is the von K&rm&n constant = 0.41,ii is the wake strength, W(y/6) is the wake function, and 6 is the local boundary-layer thickness. Further, although the data show considerable scatter, the wake strength parameter ii appears to be reasonably correlated with Clauser's (1 956) equilibrium pressure gradient parameter / 3 = (&*/T,) (dp/dx), indicating that most of the data represent a state of local equilibrium and are not strongly dependent on their upstream history.…”

Incomplete self‐similarity and flow velocity in gravel bed channels

“…Several authors [Coles, 1956;Finley et al, 1966;Coleman, 1981;Zippe and Graf, 1983] suggested that the deviations of (1) from velocity measurements are not accounted by adjusting A and B but, rather, by adding a divergence function ok(z). Coles [1956] extended the applicability of (1), introducing a purely empirical divergence function [Nezu and Rodi, 1986]: cos: ( tb(z) = b2z2(1 -z) + b3z2(3 -2z),…”