Turbulent Hydrodynamic Line Stretching: Consequences of Isotropy

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Abstract
A Lagrangian formalism for treating the stretching of material line and surface elements convected with a turbulent fluid is developed, and consequences of statistical isotropy for the distribution function of the components of a tensor are derived. Incompressibility and isotropy are then used to prove rigorously that the expectation value of the logarithmic change of the length of a line element (and area of a surface element) is greater than zero. A connection between the distribution function of the velocity shear tensor and that of the symmetric time-development tensor used in the Lagrangian formalism is shown.

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