A dynamical model for turbulence. II. Shear-driven flows

Publisher Website
Google Scholar
Abstract
Using the formalism developed in paper I, we treat the case of shear-driven flows. First, we derive the dynamic equations for the Reynolds stress. The equations are expressed in both tensorial and scalar forms, that is, as a set of coupled differential equations for the functions that enter the expansion of the Reynolds stress in terms of basic tensors. We specialize the general results to (a) axisymmetric contraction, (b) plane strain, and (c) homogeneous shear, for which there is a wealth of DNS, LES, and laboratory data to test the predictions of our model. Second, for homogeneous shear, in the inertial range, the equations for the Reynolds stress spectral function can be solved analytically, E12(k)=−Cε1/3Sk−7/3, which is in excellent agreement with recent data. Since the model has no free parameters, we stress that the model yields a numerical coefficient C, which is also in agreement with the data. Third, we derive the general expressions for the rapid and slow parts of the pressure–strain correlation tensors Πrij and Πsij. Within the second-order closure models, the closure of Πsij (third-order moment) in terms of second-order moments continues to be particularly difficult. The general expression for Πij are then specialized to the three flows discussed above. When Πsij is written in the form first suggested by Rotta, we show that the Rotta constant is a nonconstant tensor. Fourth, we discuss the dissipation tensor εij. In standard turbulence models, one not only assumes that εij=2/3εδij+f(uiuj), where f(x) is a empirical function of the one-point Reynolds stress uiuj, but, in addition, one employs a highly parametrized equation for ε. In the present model, neither of the two assumptions is required nor adjustable parameters are needed since εij is computed directly. The model provides the k-dependent Rij(k) as one of the primary quantities.