Sensitive enhanced diffusivities for flows with fluctuating mean winds: a two-parameter study

Enhanced diffusion coefficients arising from the theory of periodic homogen ized averaging for a passive scalar diffusing in the presence of a large-sc ale, fluctuating mean wind superimposed upon a small-scale, steady flow wit h non-trivial topology are studied. The purpose of the study is to assess h ow the extreme sensitivity of enhanced diffusion coefficients to small vari ations in large-scale flow parameters previously exhibited for steady flows in two spatial dimensions is modified by either the presence of temporal f luctuation, or the consideration of fully three-dimensional steady flow. We observe the various mixing parameters (Peclet, Strouhal and periodic Pecle t numbers) and related non-dimensionalizations. We document non-monotonic P eclet number dependence in the enhanced diffusivities, and address how this behaviour is camouflaged with certain non-dimensional groups. For asymptot ically large Strouhal number at fixed, bounded Peclet number, we establish that rapid wind fluctuations do not modify the steady theory, whereas for a symptotically small Strouhal number the enhanced diffusion coefficients are shown to be represented as an average over the steady geometry. The more d ifficult case of large Peclet number is considered numerically through the use of a conjugate gradient algorithm. We consider Peclet-number-dependent Strouhal numbers, S = Q Pe(-(1+gamma)), and present numerical evidence docu menting critical values of gamma which distinguish the enhanced diffusiviti es as arising simply from steady theory (gamma < -1) for which fluctuation provides no averaging, fully unsteady theory (gamma is an element of (- 1,0 )) with closure coefficients plagued by non-monotonic Peclet number depende nce, and averaged steady theory (gamma > 0). The transitional case with gam ma = 0 is examined in detail. Steady averaging is observed to agree well wi th the full simulations in this case for Q < 1, but fails for larger Q. For non-sheared flow, with Q < 1, weak temporal fluctuation in a large-scale w ind is shown to reduce the sensitivity arising from the steady flow geometr y; however, the degree of this reduction is itself strongly dependent upon the details of the imposed fluctuation. For more intense temporal fluctuati on, strongly aligned orthogonal to the steady wind, time variation averages the sensitive scaling existing in the steady geometry, and the present stu dy observes a Pe(1) scaling behaviour in the enhanced diffusion coefficient s at moderately large Peclet number. Finally, we conclude with the numerica l documentation of sensitive scaling behaviour (similar to the two-dimensio nal steady case) in fully three dimensional ABC flow.