The relation, first written by Kolmogorov, between the third-order moment of the longitudinal velocity increment δu1 and the second-order moment of δu1 is presented in a slightly more general form relating the mean value of the product δu1(δui)2, where (δui)2 is the sum of the square of the three velocity increments, to the secondorder moment of δui. In this form, the relation is similar to that derived by Yaglom for the mean value of the product δu1(δuθ)2 where (δuθ)2 is the square of the temperature increment. Both equations reduce to a ‘four-thirds’ relation for inertialrange separations and differ only through the appearance of the molecular Prandtl number for very small separations. These results are confirmed by experiments in a turbulent wake, albeit at relatively small values of the turbulence Reynolds number.

The previously established similarity between the temperature spectrum and the spectrum corresponding to the mean turbulent energy in a wide variety of turbulent (shear) flows is re-examined within the framework of second-order velocity and temperature structure functions. Measurements in a turbulent wake indicate that Dq, the sum of the three second-order velocity structure functions bears close similarity to Dθ, the second-order temperature structure function, when Dq and Dθ are normalized by the mean turbulent energy and temperature variance, respectively. This similarity also applies to other flows. In the limit of small separations, the Kolmogorov-normalized structure functions differ only by the value of the molecular Prandtl number. In the inertial range, the Obukhov–Corrsin constant differs from the Dq Kolmogorov constant by a factor equal to the dissipation time scale ratio. This ratio is typically about 0.5.

The relationship between fourth-order moments of δu1 and δu2, the longitudinal- and transverse-velocity increments, respectively, is examined in some detail in the context of local isotropy and internal intermittency. Measurements in a plane jet at a sufficiently large value of the turbulence Reynolds number indicate an inertial-range behaviour of the flatness factors of δu1 and δu2 that is in reasonable agreement with the lognormal hypothesis of Kolmogorov and the log-Poisson model. In the inertial range, the fourth-order structure function exhibits a moderately larger departure from local isotropy than the second-order structure function.

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