Perturbed turbulent flow, eddy viscosity and the generation of turbulent stresses

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Abstract
The perturbation of a turbulent flow by an organized wavelike disturbance is examined using a dynamical, rather than phenomenological, approach. On the basis of the assumption that an infinitesimal perturbation results in a linear change in the statistics of the turbulence, and that the turbulence is either weak or that the turbulent perturbations are quasi-Gaussian, a method for predicting the perturbation turbulent Reynolds stresses is developed. The novel aspect of the analysis is that all averaging is delayed until the dynamical equations have been solved rather than attempting to find, apriori, equations for averaged quantities. When applied to long-wave perturbations the analysis indicates that the perturbation shear stress is of primary dynamical importance, and that this stress is determined by the principal component of mean shear through a relation which depends on the spectrum of the turbulent velocity component parallel to the gradient of the undisturbed mean velocity (the component perpendicular to the wall in a turbulent boundary layer). Theoretical arguments and observations are used to estimate the form of this spectrum in a constant stress shear layer. This results in a prediction of the constitutive law relating turbulent stress and the mean flow. The law is visco-elastic in nature, and is in agreement with the known constitutive relation for stress perturbations to a constant stress boundary layer; it resembles the eddy viscosity relation used successfully by others in describing perturbations in turbulent flows. The details of the constitutive law depend on how well the turbulence obeys Taylor's hypothesis that phase velocity equals mean flow velocity, and some insight into this question is given.