Renormalization group analysis of anisotropic diffusion in turbulent shear flows

Abstract

The renormalization group is applied to compute anisotropic corrections to the scalar eddy diffusivity representation of turbulent diffusion of a passive scalar. The corrections are linear in the mean velocity gradients. All model constants are computed theoretically. A form of the theory valid at arbitrary Reynolds number is derived. The theory applies only when convection of the velocity–scalar correlation can be neglected. A ratio of diffusivity components, found experimentally to have a nearly constant value in a variety of shear flows, is computed theoretically for flows in a certain state of equilibrium. The theoretical value is well within the fairly narrow range of experimentally observed values. Theoretical predictions of this diffusivity ratio are also compared with data from experiments and direct numerical simulations of homogeneous shear flows with constant velocity and scalar gradients.

On the physical accuracy of scalar transport modeling in inhomogeneous turbulence

“…The mathematical structure of the gradient model given in (26) and (27) or (28) and (29) proved to be adequate for modeling the steady far field of plumes. Since the shear terms in (28) had very little effect on predictions, the m<;>st important issue in the development of gradient models is the choice of time scales.…”

“…For the predicted scalar fields, however, both alternatives are found to have comparable accuracy. Implementation of (28) and (29) in applications may be easier than (26) and (27) since it may be problematical to obtain values of the four Lagrangian integral scales under general flow conditions. In both cases, the Reynolds stresses need to be available, and for the algebraic model, k and E, as well.…”

“…This behavior is attributable to the dependence of (26) and (28) on ~~, which is large and positive at the source of the plume. Since T 11 is much larger than T/Cn, (26) is less accurate than (28) and the plume in (26) over (28) plume, the difficulty in resolving the wall influence becomes quite noticeable in FIG. 15b at x+ = 500 where the plume is now in contact with the wall.…”

“…This appears to be a consequence of the fact that beyond the initial region, the model for vc-which according to (29) has no dependence on shear -is the dominant turbulent influence on the plume. It may be concluded that the success of gradient models in predicting the far field of homogeneous turbulence also extends · to the non-homogeneous case as well, so long as they include the basic dependence on Reynolds stress exhibited by (26) and (27) or (28) and (29). The use of physically accurate scales improves the predictions of the far field fluxes at the expense of greater errors in the near field, while the opposite happens if the modeled scales are used.…”

“…The near field errors associated with the closure models may be attributed to a fundamental failure of (26) and (28) 11 is a plot of the DNS prediction of uc on a streamwise cut at y+ = 15 through they+ = 15 plume, together with evaluations of (26) and (28) using the C field computed from the DNS solution. The latter are the values uc would have if the models had yielded the correct C field.…”