A simple model for the streamwise fluctuations in the log-law region of a boundary layer

We build on the work of Davidson [Phys. Fluids 18, 065112 (2006)] and propose an elementary model for the log-law region of a boundary layer. The model is remarkably simple, contains only one free parameter (which we set equal to unity) and, unlike Davidson , assumes very little about the shape of the boundary-layer eddies. The physical content of the model is simple: we assume that the two-point statistics of the streamwise velocity fluctuations know about the presence of the wall only to the extent that, over a range of eddy sizes, it imposes a kinetic energy scale proportional to the square of the shear velocity. Little else is assumed, other than classic Kolmogorov phenomenology for the small scales. Despite its naivety, and the minimal number of free parameters, the model is a good fit to experimental data for the k(-1) law of the one-dimensional, longitudinal spectrum, Phi(uu)(k), and also to Phi(uu)(k) in the inertial range. It is also an excellent fit to experimental data for the real-space analog of the k(-1) law; that is, the logarithmic law for the longitudinal structure function. In addition, the model predicts the cross-stream variation of the variance of the streamwise velocity fluctuations, < u(x)(2)>, and, to within an additive constant, it too is a reasonable fit to the data. Our model differs from the classic analyses of Townsend [J. Fluid Mech. 165, 163 (1986)] and Perry [J. Fluid Mech. 165, 163 (1986)] in three respects. First, we do not invoke the attached eddy hypothesis, which assumes that the vortical structures in the log-layer extend to the wall. Second, we predict the precise shape of Phi(uu)(k) for all values of k and not just the k(-1) region. Third, there are three adjustable parameters in the theory of Perry , whereas we have only one. It is instructive that a model which contains very little information about the morphology of the vorticity field, and has only one free parameter, is able to reproduce both the structural form for, and magnitude of, many features of the one-dimensional spectrum and the second-order structure function. This suggests that these two measures of the scale-by-scale energy distribution are insensitive to the detailed structure of the vorticity field. We argue that this is indeed the case.