Enhanced diffusion coefficients arising from the theory of periodic homogenized
averaging for a passive scalar diffusing in the presence of a large-scale, fluctuating
mean wind superimposed upon a small-scale, steady flow with non-trivial topology are
studied. The purpose of the study is to assess how the extreme sensitivity of enhanced
diffusion coefficients to small variations in large-scale flow parameters previously
exhibited for steady flows in two spatial dimensions is modified by either the presence
of temporal fluctuation, or the consideration of fully three-dimensional steady flow.
We observe the various mixing parameters (Péclet, Strouhal and periodic Péclet
numbers) and related non-dimensionalizations. We document non-monotonic Péclet
number dependence in the enhanced diffusivities, and address how this behaviour is
camouflaged with certain non-dimensional groups. For asymptotically large Strouhal
number at fixed, bounded Péclet number, we establish that rapid wind fluctuations
do not modify the steady theory, whereas for asymptotically small Strouhal number
the enhanced diffusion coefficients are shown to be represented as an average over
the steady geometry. The more difficult case of large Péclet number is considered
numerically through the use of a conjugate gradient algorithm. We consider
Péclet-number-dependent Strouhal numbers, S = QPe−(1+γ), and
present numerical evidence documenting critical values of γ which distinguish the enhanced diffusivities as arising
simply from steady theory (γ < −1) for which fluctuation provides no averaging, fully
unsteady theory (γ ∈ (−1, 0)) with closure coefficients plagued by non-monotonic
Péclet number dependence, and averaged steady theory (γ > 0). The transitional case
with γ = 0 is examined in detail. Steady averaging is observed to agree well with
the full simulations in this case for Q [les ] 1, but fails for larger Q. For non-sheared
flow, with Q [les ] 1, weak temporal fluctuation in a large-scale wind is shown to reduce
the sensitivity arising from the steady flow geometry; however, the degree of this
reduction is itself strongly dependent upon the details of the imposed fluctuation.
For more intense temporal fluctuation, strongly aligned orthogonal to the steady
wind, time variation averages the sensitive scaling existing in the steady geometry,
and the present study observes a Pe1 scaling behaviour in the enhanced diffusion
coefficients at moderately large Péclet number. Finally, we conclude with the numerical
documentation of sensitive scaling behaviour (similar to the two-dimensional steady
case) in fully three dimensional ABC flow.
The drag of a body moving transversely in a confined stratified fluid