Abstract
An economical representation of effects of turbulence on the time-evolving structure of diffusive scalar fields is obtained by introducing a hierarchical (tree) network connecting fluid parcels, with effects of turbulent advection represented by swapping pairs of sub-trees at rates determined by turbulence time scales associated with the sub-trees. The fluid parcels reside at the base of the tree. The tree structure partitions the fluid parcels into adjacent pairs (or more generally, p-tuples). Adjacent parcels intermix at rates governed by diffusion time scales based on molecular diffusivities and parcel sizes. This simple procedure efficiently accomplishes long-standing objectives of turbulent mixing model development, such as generating physically based time histories of fluid-parcel nearest-neighbor encounters and the associated spatial structure of turbulent scalar fields. Correspondences between features of the hierarchical formulation and turbulent mixing phenomenology, both generic and case-specific, are noted.
Highlights
Guided by turbulent cascade phenomenology, the present study introduces a minimal representation of the time-evolving occurrences of adjacency of parcel pairs in order to capture the leading-order effects of this phenomenology on mixing time-histories, multi-parcel joint statistics, and other such observables
Details of these and other features of the mixing representation can be formulated in various ways, resulting in choices to be made in order to instantiate the described mixing representation as a complete model suitable for analysis or numerical implementation
Much of the phenomenology of scalar mixing in turbulence is reflected in the scalar spectral array, i.e., the family of power spectra of scalar fluctuations parameterized by Reynolds number Re and Prandtl number Pr or Schmidt number Sc
Summary
Mixing closure in computational models of turbulent combustion is typically implemented by partially or fully intermixing pairs or groups of notional fluid parcels selected from a parcel population that discretely instantiates the joint probability distribution function (PDF) of the thermochemical variables that are time advanced by the model [10]. In the absence of specially formulated constraints, this approach allows the intermixing of parcels with highly dissimilar states, which is both unphysical in principle and detrimental to model performance in practice One such constraint that has proven effective is to intermix only parcel pairs that are close, by some criterion, in a metric space defined on the manifold of thermochemical states [26]. The role of this representation is to partition the parcel population into pairs at each instant, such that the parcels in each pair are deemed to be adjacent and subject to intermixing at a rate that is based on the molecular diffusivity and a specified local length scale Details of these and other features of the mixing representation can be formulated in various ways, resulting in choices to be made in order to instantiate the described mixing representation as a complete model suitable for analysis or numerical implementation.
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