The variation of S, the velocity derivative skewness, with the Taylor microscale Reynolds number Re λ is examined for different turbulent flows by considering the locally isotropic form of the transport equation for the mean energy dissipation rate iso . In each flow, the equation can be expressed in the form S + 2G/Re λ = C/Re λ , where G is a non-dimensional rate of destruction of iso and C is a flow-dependent constant. Since 2G/Re λ is found to be very nearly constant for Re λ 70, S should approach a universal constant when Re λ is sufficiently large, but the way this constant is approached is flow dependent. For example, the approach is slow in grid turbulence and rapid along the axis of a round jet. For all the flows considered, the approach is reasonably well supported by experimental and numerical data. The constancy of S at large Re λ has obvious ramifications for small-scale turbulence research since it violates the modified similarity hypothesis introduced by Kolmogorov (J. Fluid Mech., vol. 13, 1962, pp. 82-85) but is consistent with the original similarity hypothesis (Kolmogorov,

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The effect of large-scale forcing on the second- and third-order longitudinal velocity structure functions, evaluated at the Taylor microscale $r=\unicode[STIX]{x1D706}$, is assessed in various turbulent flows at small to moderate values of the Taylor microscale Reynolds number $R_{\unicode[STIX]{x1D706}}$. It is found that the contribution of the large-scale terms to the scale by scale energy budget differs from flow to flow. For a fixed $R_{\unicode[STIX]{x1D706}}$, this contribution is largest on the centreline of a fully developed channel flow but smallest for stationary forced periodic box turbulence. For decaying-type flows, the contribution lies between the previous two cases. Because of the difference in the large-scale term between flows, the third-order longitudinal velocity structure function at $r=\unicode[STIX]{x1D706}$ differs from flow to flow at small to moderate $R_{\unicode[STIX]{x1D706}}$. The effect on the second-order velocity structure functions appears to be negligible. More importantly, the effect of $R_{\unicode[STIX]{x1D706}}$ on the scaling range exponent of the longitudinal velocity structure function is assessed using measurements of the streamwise velocity fluctuation $u$, with $R_{\unicode[STIX]{x1D706}}$ in the range 500–1100, on the axis of a plane jet. It is found that the magnitude of the exponent increases as $R_{\unicode[STIX]{x1D706}}$ increases and the rate of increase depends on the order $n$. The trend of published structure function data on the axes of an axisymmetric jet and a two-dimensional wake confirms this dependence. For a fixed $R_{\unicode[STIX]{x1D706}}$, the exponent can vary from flow to flow and for a given flow, the larger $R_{\unicode[STIX]{x1D706}}$ is, the closer the exponent is to the value predicted by Kolmogorov (Dokl. Akad. Nauk SSSR, vol. 30, 1941a, pp. 299–303) (hereafter K41). The major conclusion is that the finite Reynolds number effect, which depends on the flow, needs to be properly accounted for before determining whether corrections to K41, arising from the intermittency of the energy dissipation rate, are needed. We further point out that it is imprudent, if not incorrect, to associate the finite Reynolds number effect with a consequence of the modified similarity hypothesis introduced by Kolmogorov (J. Fluid Mech., vol. 13, 1962, pp. 82–85) (K62); we contend that this association has misled the vast majority of post K62 investigations of the consequences of K62.

We first analytically show, starting with the Navier–Stokes equations, that the value of the derivative flatness is controlled by pressure diffusion of energy, viscous destructive effects and large-scale effects (decay and/or production). The latter two terms tend to zero when the Taylor-microscale Reynolds number $Re_{\unicode[STIX]{x1D706}}$ is sufficiently large. We argue that the pressure-diffusion term should also tend to a constant at large $Re_{\unicode[STIX]{x1D706}}$. Available data for the velocity derivative flatness, $F$, in different turbulent flows are re-examined and interpreted in the light of the finite-Reynolds-number effect. It is found that $F$ can differ from flow to flow at moderate $Re_{\unicode[STIX]{x1D706}}$; for a given flow, $F$ may also depend on the initial conditions. The data for $F$ in various flows, e.g. along the axis in the far field of plane and circular jets, and grid turbulence, show that it approaches a constant, with a value slightly larger than 10, when $Re_{\unicode[STIX]{x1D706}}$ is sufficiently large. This behaviour for $F$ is supported, at least qualitatively, by our analytical considerations. The constancy of $F$ at large $Re_{\unicode[STIX]{x1D706}}$ violates the refined similarity hypothesis introduced by Kolmogorov (J. Fluid Mech., vol. 13, 1962, pp. 82–85) to account for the intermittency of the energy dissipation rate. It is not, however, inconsistent with Kolmogorov’s original similarity hypothesis (Dokl. Akad. Nauk SSSR, vol. 30, 1941, pp. 299–303), although we contend that the power-law relation $F\sim Re_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D6FC}_{4}}$ (Kolmogorov 1962), which is widely accepted in the literature, has in reality been almost invariably used to ‘model’ the finite-Reynolds-number effect for the laboratory data and has been strongly influenced by the weighting given to the atmospheric surface layer data. The inclusion of the latter data has misled previous investigations of how $F$ varies with $Re_{\unicode[STIX]{x1D706}}$.

WOS:000359643100010International audienceThe transport equation for the mean turbulent energy dissipation rate (epsilon) over bar along the centreline of a fully developed channel flow is derived by applying the limit at small separations to the two-point budget equation. Since the ratio of the isotropic energy dissipation rate to the mean turbulent energy dissipation rate (epsilon) over bar (iso)/(epsilon) over bar is sufficiently close to 1 on the centreline, our main focus is on the isotropic form of the transport equation. It is found that the imbalance between the production of (epsilon) over bar due to vortex stretching and the destruction of (epsilon) over bar caused by the action of viscosity is governed by the diffusion of (epsilon) over bar by the wall-normal velocity fluctuation. This imbalance is intrinsically different from the advection-driven imbalance in decaying-type flows, such as grid turbulence, jets and wakes. In effect, the different types of imbalance represent different constraints on the relation between the skewness of the longitudinal velocity derivative S-1,S-1 and the destruction coefficient G of enstrophy in different flows, thus resulting in non-universal approaches of S-1,S-1 towards a constant value as the Taylor microscale Reynolds number, R-lambda, increases. For example, the approach is slower for the measured values of S-1,S-1 along either the channel or pipe centreline than along the axis in the self-preserving region of a round jet. The data for S-1,S-1 collected in different flows strongly suggest that, in each flow, the magnitude of S-1,S-1 is bounded, the value being slightly larger than 0.5

Failure to recognize the importance of the finite Reynolds number effect on small scale turbulence has, by and large, resulted in misguided assessments of the first two hypotheses of Kolmogorov [“Local structure of turbulence in an incompressible fluid for very large Reynolds numbers,” Dokl. Akad. Nauk SSSR 30, 299–303 (1941)] or K41 as well as his third hypothesis [A. N. Kolmogorov, “A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number,” J. Fluid Mech. 13, 82–85 (1962)] or K62. As formulated by Kolmogorov, all three hypotheses require local isotropy to be valid and the Reynolds number to be very large. In the context of the first hypothesis, there is now strong evidence to suggest that this requirement can be significantly relaxed, at least for dissipative scales and relatively low order moments of the velocity structure function. As the scale increases, the effect of the large scale motion on these moments becomes more prominent and higher Reynolds numbers are needed before K41 and K62 can be tested unambiguously.

Self-preservation (SP) analyses are applied to the mean momentum and the scale-by-scale energy budget equations in the far wake of a circular cylinder. The scale-by-scale SP analysis, which is a two-point analysis, complements the SP analysis of the mean momentum equation. Power-law variations are derived for different length scales (e.g. the Taylor microscale and the Kolmogorov length scale) and velocity scales (e.g. the root mean square and the Kolmogorov velocity scale). Further, the SP solutions for the scale-by-scale energy budget equation are exploited to develop an exact relation to estimate the mean turbulent kinetic energy dissipation rate¯ on the wake axis. These SP solutions and the new¯ relation are well supported by hot-wire data in the far wake at a Reynolds number of 2000 based on the free stream velocity and the cylinder diameter. On the far-wake axis, both the energy spectra and the structure functions exhibit an almost perfect collapse over all wavenumbers and separations, irrespective of the set of scaling variables used for normalisation. This is consistent with a complete self-preservation (i.e. SP is satisfied at all scales of motion) in the far wake.

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