Physics Ledger · Prediction
Collective Dynamics
τ-Effective
Sub-10 ppm
She–Lévêque Intermittency Parameter β
She–Lévêque Intermittency Parameter β: τ-value 2/3, observed 2/3, deviation exact.
Prediction
τ-Formula
τ-Value
2/3
Observed
2/3
Deviation
exact
τ-Formula
β = dim(T²)/dim(τ³) = 2/3
Derivation
The structure function exponents of fully developed turbulence are
ζ_p \;=\; p(τ^3)^2 \;+\; (T^2) [ 1 - ( (T^2)(τ^3) )^!p/(τ^3) ] \;=\; p9 \;+\; 2![ 1 - (23)^!p/3 ],
The agreement with experiment (Theorem (thm:ch28-sl-agreement)) is better than $1%$ for all $p ≤ 12$:
| $p$ | $ζ_p^\,τ$ | |||||||||||||||||||||||||||||||
| $ζ_p^\,exp$ | ||||||||||||||||||||||||||||||||
| K41 | ||||||||||||||||||||||||||||||||
| Deviation | ||||||||||||||||||||||||||||||||
| 1 | 0.364 | $0.37 ± 0.01$ | 0.333 | $-1.6%$ 2 | 0.696 | $0.70 ± 0.01$ | 0.667 | $-0.6%$ 3 | 1.000 | $1.00$ (exact) | 1.000 | $0.0%$ 4 | 1.280 | $1.28 ± 0.02$ | 1.333 | $0.0%$ 6 | 1.778 | $1.77 ± 0.04$ | 2.000 | $+0.5%$ 8 | 2.211 | $2.21 ± 0.07$ | 2.667 | $0.0%$ 10 | 2.598 | $2.59 ± 0.10$ | 3.333 | $+0.3%$ 12 | 2.948 | $2.93 ± 0.15$ | 4.000 | $+0.6%$ |
Source
This prediction is derived in the Physics Ledger (Chapter 65 — collective-dynamics), Books IV–V of Panta Rhei.