Title: The dimensionless dissipation rate and the Kolmogorov (1941) hypothesis of local stationarity in freely decaying isotropic turbulence,
Abstract: An expression for the dimensionless dissipation rate was derived from the
Karman-Howarth equation by asymptotic expansion of the second- and third- order
structure functions in powers of the inverse Reynolds number. The implications
of the time-derivative term for the assumption of local stationarity (or local
equilibrium) which underpins the derivation of the Kolmogorov `4/5' law for the
third-order structure function were studied. It was concluded that neglect of
the time-derivative cannot be justified by reason of restriction to certain
scales (the inertial range) nor to large Reynolds numbers. In principle,
therefore, the hypothesis cannot be correct, although it may be a good
approximation. It follows, at least in principle, that the quantitative aspects
of the hypothesis of local stationarity could be tested by a comparison of the
asymptotic dimensionless dissipation rate for free decay with that for the
stationary case. But in practice this is complicated by the absence of an
agreed evolution time for making the measurements during the decay. However, we
can assess the quantitative error involved in using the hypothesis by comparing
the exact asymptotic value of the dimensionless dissipation in free decay
calculated on the assumption of local stationarity to the experimentally
determined value (e.g. by means of direct numerical simulation), as this
relationship holds for all measuring times. Should the assumption of local
stationarity lead to significant error, then the `4/5' law needs to be
corrected. Despite this, scale invariance in wavenumber space appears to hold
in the formal limit of infinite Reynolds numbers, which implies that the `-5/3'
energy spectrum does not require correction in this limit.