Finite-size scaling of two-point statistics and the turbulent energy cascade generators

Abstract

Within the framework of random multiplicative energy cascade models of fully developed turbulence, finite-size-scaling expressions for two-point correlators and cumulants are derived, taking into account the observationally unavoidable conversion from an ultrametric to an Euclidean two-point distance. The comparison with two-point statistics of the surrogate energy dissipation, extracted from various wind tunnel and atmospheric boundary layer records, allows an accurate deduction of multiscaling exponents and cumulants, even at moderate Reynolds numbers for which simple power-law fits are not feasible. The extracted exponents serve as input for parametric estimates of the probabilistic cascade generator. Various cascade generators are evaluated.

Hierarchical Parcel-Swapping Representation of Turbulent Mixing. Part 1. Formulation and Scaling Properties

“…Various approaches to modeling turbulence and turbulent mixing by coupling state variables defined on a set of spatial (or equivalently, wavenumber) levels within a geometrical hierarchy have been developed [2,4,5,7,10,13,14,22]. A distinguishing feature of the present hierarchical approach is that the physical state is specified solely at the smallest scale and is thus fully resolved in space and time.…”

“…The level n of the minimal spanning sub-tree is termed the parcel proximity, where larger n corresponds to closer proximity. (Proximity defined in this way is equivalent to the 'ultrametric distance' between tree nodes [7].) As illustrated in Fig.…”

“…Assuming that homogeneous turbulence is represented operationally using finite trees, the formal definition of var n θ in HiPS additionally invokes ensemble averaging. Alternatively, an approach involving a chain of finite trees and a way of relating ultrametric distance to physical distance [7] can be used.…”

“…Notation introduced by Cleve et al [7] is adopted. At each level, nodes i are numbered sequentially from zero going from left to right.…”

Intermittency exponent of the turbulent energy cascade

“…The follow-up effort [6] was able to characterize and understand the functional form of the two-point correlation function beyond the power-law scaling range. Within the theory of (binary) random multiplicative cascade processes, the finite-size parametrization…”