Passive scalar mixing: Analytic study of time scale ratio, variance, and mix rate

Some very reasonable approximations, consistent with numerical and experimental evidence, were applied to the skewness and palinstrophy coefficients in the dissipation equations to produce a simple closed moment model for mixing. Such a model, first suggested on the grounds of a Taylor microscale self-similarity of the scalar field, was studied numerically by Gonzalez and Fall [“The approach to self-preservation of scalar fluctuation decay in isotropic turbulence,” Phys. Fluids 10, 654 (1998)]. Here, in a somewhat old fashioned and physically meaningful style, analytic solutions to the four coupled nonlinear moment equations for mixing by decaying and forced stationary turbulence, are given. Analytic expressions for the variance ⟨c2⟩, the mixing rate εc, and the time scale ratio r(t) are derived and compared in different mixing situations. The solutions show the sensitive dependence on the initial relative length ratio as studied experimentally by Warhaft and Lumley [“An experimental study of the decay of temperature fluctuations in grid-generated turbulence,” J. Fluid Mech. 88, 659 (1978)], and simulated by Eswaran and Pope [“Direct numerical simulation of the turbulent mixing of a passive scalar,” Phys. Fluids 31, 506 (1988)]. The length scale ratio saturation effect predicted by Durbin [“Analysis of the decay of temperature fluctuations in isotropic turbulence,” Phys. Fluids 25, 1328 (1982)], resolving the apparent contradiction with the results of Sreenivasan, Tavoularis, and Corrsin [“Temperature fluctuations and scales in grid generated turbulence,” J. Fluid Mech. 100, 597 (1980)] is predicted. For stationary turbulence the solutions indicate, in contradistinction to the power law “stirring” result predicted by a stochastic Lagrangian analysis, that the mixing is asymptotically exponential as shown in the phenomenological analysis of Corrsin [“The isotropic turbulent mixer,” AIChE J. 10, 870 (1964)]. That the time scale ratio solution also depends on Reynolds number is consistent with the DNS observations of Overholt and Pope [“Direct numerical simulation of passive scalar with imposed mean gradient in isotropic turbulence,” Phys. Fluids 31, 506 (1998)]. As a consequence, the customary approximations in k-ε type turbulence moment models for the mixing rate is, on theoretical grounds, not justified. The analysis predicts important phenomenological differences between mixing by stationary forced turbulence and decaying turbulence. Mixing by forced turbulence is asymptotically exponential with long lasting dependence on the initial time scale ratio and features an intermediate time transient. The time scale for the variance ⟨c2⟩ and its mix rate εc are commensurate. Mixing by decaying turbulence appears described by variable power law and only asymptotically as a constant power law. In decaying turbulence the characteristic time scale of ⟨c2⟩ and εc are very different and dependent on Reynolds number. An additional class of decays, seen by Antonia et al. [“Scaling of the mean energy dissipation rate equation in grid turbulence,” J. Turbulence 3, 1 (2002)], in which the palinstrophy coefficient scales as Rλ, is subsumed by this analysis. Solutions for mixing by constant power law decay (k∼t−nc) are given.