Abstract
We formulate multifractal models for velocity differences and gradients which describe the full range of length scales in turbulent flow, namely: laminar, dissipation, inertial, and stirring ranges. The models subsume existing models of inertial range turbulence. In the localized ranges of length scales in which the turbulence is only partially developed, we propose multifractal scaling laws with scaling exponents modified from their inertial range values. In local regions, even within a fully developed turbulent flow, the turbulence is not isotropic nor scale invariant due to the influence of larger turbulent structures (or their absence). For this reason, turbulence that is not fully developed is an important issue which inertial range study can not address. In the ranges of partially developed turbulence, the flow can be far from universal, so that standard inertial range turbulence scaling models become inapplicable. The model proposed here serves as a replacement. Details of the fitting of the parameters for the τp and ζp models in the dissipation range are discussed. Some of the behavior of ζp for larger p is unexplained. The theories are verified by comparing to high resolution simulation data.
Highlights
We develop a conceptual framework for turbulent scaling laws across length scales extending beyond the inertial range
With the variables Z3 and a3 calculated from ζ3
We interpret the length scale l/η limit in terms of TaylorGreen vortices continued past the instability point
Summary
We develop a conceptual framework for turbulent scaling laws across length scales extending beyond the inertial range. The piecewise constant values of Tp and Zp appear to be a new discovery These piecewise constant values over the 4 length scale ranges are described as follows: 0T,pdr , 0T,psr , laminar range (LR) dissipation range (DR) inertial range (IR) stirring range (SR). Tpdr and Zpdr are constant in the dissipation range, and Tpsr and Zpsr are constant in the stirring range for the problems we study These Tp and Zp values are verified in the JHTDB and the Tp values in the UMA data shown in section 4.1 addressing the laminar region where the eddy length scale is smaller than the Kolmogorov microscale η. The dissipation defined in the Navier-Stokes equation itself occurs in the laminar range at a rate proportional to the length scale l.
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