Corrections to Taylor's frozen turbulence approximation

Taylor's frozen turbulence approximation relates spatial statistics to temporal statistics. Lumley's two-term approximation gives corrections for the effect of fluctuating convection velocity. Such corrections are derived for every turbulence statistic. The statistic may be a tensor of any rank, and may be a correlation or structure function or spectrum. The practicality of other approximations, such as assuming the convecting velocity to be a Gaussian random variable, is considered as are the inaccuracies of approximations for the case of the dissipation range of spectra. Extensions to cases of multiple-position and space-averaging sensors are noted. Local homogeneity and stationarity and the statistical independence of large-scale and small-scale turbulence quantities are necessary approximations. Local isotropy is shown to be the simplest case of a requirement for local symmetry. Specific expressions are given for isotropic tensors of rank up to fourth rank. Specific expressions are given for the inertial range, for which the correction is shown to vanish to second order for the triple-velocity correlation and structure function. Previous correction formulas are verified, with the exception of the correction for the stress cospectrum, which is shown to have been erroneous. The incompressibility condition is used to simplify the corrections. Fluctuating convection causes erroneous measured compressibility.