Scaling of normalized mean energy and scalar dissipation rates in a turbulent channel flow

Abstract

Non-dimensional parameters for the mean energy and scalar dissipation rates Cɛ and Cɛθ are examined using direct numerical simulation (DNS) data obtained in a fully developed turbulent channel flow with a passive scalar (Pr = 0.71) at several values of the Kármán (Reynolds) number h+. It is shown that Cɛ and Cɛθ are approximately equal in the near-equilibrium region (viz., y+ = 100 to y/h = 0.7) where the production and dissipation rates of either the turbulent kinetic energy or scalar variance are approximately equal and the magnitudes of the diffusion terms are negligibly small. The magnitudes of Cɛ and Cɛθ are about 2 and 1 in the logarithmic and outer regions, respectively, when h+ is sufficiently large. The former value is about the same for the channel, pipe, and turbulent boundary layer, reflecting the similarity between the mean velocity and temperature distributions among these three canonical flows. The latter value is, on the other hand, about twice as large as in homogeneous isotropic turbulence due to the existence of the large-scale u structures in the channel. The behaviour of Cɛ and Cɛθ impacts on turbulence modeling. In particular, the similarity between Cɛ and Cɛθ leads to a simple relation for the scalar variance to turbulent kinetic energy time-scale ratio, an important ingredient in the eddy diffusivity model. This similarity also yields a relation between the Taylor and Corrsin microscales and analogous relations, in terms of h+, for the Taylor microscale Reynolds number and Corrsin microscale Peclet number. This dependence is reasonably well supported by both the DNS data at small to moderate h+ and the experimental data of Comte-Bellot [Ph. D. thesis (University of Grenoble, 1963)] at larger h+. It does not however apply to a turbulent boundary layer where the mean energy dissipation rate, normalized on either wall or outer variables, is about 30% larger than for the channel flow.