The statistical relation between Eulerian and Lagrangian systems of turbulence (β-approach) is derived to improve HAY and PASQUILL'S approximation from the equality of energies of turbulence in both systems.
INOUE'S turbulon theory of turbulence is also applied to derive the equation for LagrangianEulerian time-scale ratio, or so-called HAY-PAS-QUILL scale parameter β, which is shown to be evaluated only from Eulerian statistics.
Theoretical discussion shows that Lagrangian correlation function and spectrum can be estimated from Eulerian properties, and that β is the function of Lagrangian time and dependents on intensity of turbulence as well as turbulence scale and approaches the ultimate time-scale ratio β
e at very large values of time.
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