The spectral velocity tensor for homogeneous boundary-layer turbulence

Abstract

We have postulated a simple model for the spectral tensor Φ ij (k) of an anisotropic, but homogeneous turbulent velocity field. It is a simple generalization of the spectral tensor Φ ij iso (k) for isotropic turbulence and we show how in the limit of isotropy, Φij(k) becomes equal to Φ ij iso (k). Whereas Φ ij iso (k) is determined entirely by one scalar function of k = |k|, namely the energy spectrum, we need three independent scalar functions of k to specify Φ ij (k). We show how it is possible by means of the three stream-wise velocity component spectra to determine the three scalar functions in Φ ij (k) by solving two uncoupled, ordinary linear differential equations of first and second order. The analytic form of the component spectra each has a set of three parameters: the variance and the integral length scale of the velocity component and a dimensionless parameter, which governs the curvature of the spectrum in the transition domain from the inertial subrange towards lower wave numbers. When the three sets of parameters are the same, the three spectra correspond to isotropic turbulence and they are all interrelated and related to the energy spectrum. We show how it is possible to obtain these spectral forms in the neutral surface layer and in the convective boundary layer from data reported in the literature. The spectral tensor is used to predict the lateral coherences for all three velocity components and these predictions are compared with coherences obtained in two experiments, one using three masts at a horizontally homogeneous site in Denmark and one employing two aircraft flying in formation over eastern Colorado. Comparison shows reasonable agreement although with considerable experimental scatter.