Lagrangian Statistics from Numerically Integrated Turbulent Shear Flow

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Abstract
Lagrangian turbulence statistics were obtained from a three‐dimensional numerical model of plane Poiseuille flow at large Reynolds number. Single‐particle Lagrangian correlations were computed and compared with Eulerian space correlations. Although the curves were not self‐similar, a dimensionless scale ratio defined at the e‐ folding time was comparable to Eulerian‐Lagrangian scale ratios found in the literature. The numerically obtained mean‐square particle displacements exhibited correct short‐ and long‐time behavior. Mean‐square particle separations were analyzed, and two‐particle Lagrangian velocity correlations taken at the same time were more persistent than Lagrangian autocorrelations. A semiempirical functional form was constructed for the two‐particle velocity correlations which yielded two‐particle distance correlations in good agreement with those of the numerical model. The effect of mean shear on downstream separation was examined. Results indicate a t 3 or steeper dependence for downstream mean‐square separation 〈(x a − x b ) 2 〉 , with strong shear and Reynolds stress. Batchelor's similarity law, namely, that 〈(x aj − x bj ) 2 〉 ∝ εt 3 in directions not controlled by shear, is postulated for the direction of shear when shear generates ε. This postulate was tested numerically and found to be consistent.

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