The one-dimensional surrogate for the dimensionless energy dissipation rate Cε is measured in shear flows over a range of the Taylor microscale Reynolds number Rλ, 70≲Rλ≲1217. We recommend that Cε should be defined with respect to an energy length scale derived from the turbulent energy spectrum. For Rλ≳300, a value of Cε≈0.5 appears to be a good universal approximation for flow regions free of strong mean shear. The present results for Cε support a key assumption of turbulence—the mean turbulent energy dissipation rate is finite in the limit of zero viscosity.

The dimensionless kinetic energy dissipation rate Cε is estimated from numerical simulations of statistically stationary isotropic box turbulence that is slightly compressible. The Taylor microscale Reynolds number (Re λ ) range is 20Re λ 220 and the statistical stationarity is achieved with a random phase forcing method. The strong Re λ dependence of Cε abates when Re λ ≈ 100 after which Cε slowly approaches ≈ 0.5, a value slightly different to previously reported simulations but in good agreement with experimental results. If Cε is estimated at a specific time step from the time series of the quantities involved it is necessary to account for the time lag between energy injection and energy dissipation. Also, the resulting value can differ from the ensemble averaged value by up to ±30%. This may explain the spread in results from previously published estimates of Cε.

The main focus is the Reynolds number dependence of Kolmogorov normalized low-order moments of longitudinal and transverse velocity increments. The velocity increments are obtained in a large number of flows and over a wide range (40–4250) of the Taylor microscale Reynolds number Rλ. The Rλ dependence is examined for values of the separation, r, in the dissipative range, inertial range and in excess of the integral length scale. In each range, the Kolmogorov-normalized moments of longitudinal and transverse velocity increments increase with Rλ. The scaling exponents of both longitudinal and transverse velocity increments increase with Rλ, the increase being more significant for the latter than the former. As Rλ increases, the inequality between scaling exponents of longitudinal and transverse velocity increments diminishes, reflecting a reduced influence from the large-scale anisotropy or the mean shear on inertial range scales. At sufficiently large Rλ, inertial range exponents for the second-order moment of the pressure increment follow more closely those for the fourth-order moments of transverse velocity increments than the fourth-order moments of longitudinal velocity increments. Comparison with DNS data indicates that the magnitude and Rλ dependence of the mean square pressure gradient, based on the joint-Gaussian approximation, is incorrect. The validity of this approximation improves as r increases; when r exceeds the integral length scale, the Rλ dependence of the second-order pressure structure functions is in reasonable agreement with the result originally given by Batchelor (1951).

Measurements in a turbulent wake indicate that, in the inertial range, the power law exponents inferred from the transverse velocity and temperature increments are nearly equal and significantly smaller than the exponents of the longitudinal velocity increment. The relative magnitudes of the exponents for moments up to the eighth order have been determined using the extended self-similarity method.

It has been suggested that the equilibrium-range properties of high-Reynolds number turbulence are more readily observed in spectral space, using E(k) or T(k), than in real space, using second- or third-order structure functions. For example, the -5/3 law is usually easier to see in experimental data than the equivalent 2/3 law. We argue that this is not an implicit feature of a real-space description of turbulence. Rather, it is because the second-order structure function mixes small and large-scale information. To remedy this problem we adopt a real-space function, the signature function, which plays the role of an energy density, somewhat analogous to E(k). In this Letter we determine the form of the signature function in a variety of turbulent flows. We find that dissipation-range phenomena, such as the so-called bottleneck effect, are evident in the signature function, while absent in the structure function.