A dynamical model for turbulence .2. Shear-driven flowsq

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, E(12)(k) = - C epsilon(1/3)Sk(-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 Pi(ij)(r) and Pi(ij)(s). Within the second-order closure models, the closure of Pi(ij)(s) (third-order moment) in terms of second-order moments continues to be particularly difficult. The general expression for Pi(ij) are then specialized to the three flows discussed above. Pi(ij)(s) is written in the form first suggested by Rotta, we show that the Rotta constant is a When IT nonconstant tenser. Fourth, we discuss the dissipation tensor epsilon(ij). In standard turbulence models, one not only assumes that epsilon(ij) = 2/3 epsilon delta(ij) + f(<(u(i)u(j))over bar>), where f(x) is a empirical function of the one-point Reynolds stress <(u(i)u(j))over bar>, but, in addition, one employs a highly parametrized equation for epsilon. In the present model, neither of the two assumptions is required nor adjustable parameters are needed since epsilon(ij) is computed directly. The model provides the k-dependent R(ij)(k) as one of the primary quantities. (C) 1996 American Institute of Physics.