Kolmogorov's equation, which relates second- and third-order moments of the velocity increment, provides a simple method for estimating the mean energy dissipation rate 〈ε〉 for homogeneous and isotropic turbulence. However, this equation is usually not verified in small to moderate Reynolds number flows. This is due partly to the lack of isotropy in either sheared or non-sheared flows, and, more importantly, to the influence, which is flow specific, of the inhomogeneous and anisotropic large scales. These shortcomings are examined in the context of the central region of a turbulent channel flow. In this case, we propose a generalized form of Kolmogorov's equation, which includes some additional terms reflecting the large-scale turbulent diffusion acting from the walls through to the centreline of the channel. For moderate Reynolds numbers, the mean turbulent energy transferred at a scale r also contains a large-scale contribution, reflecting the non-homogeneity of these scales. There is reasonable agreement between the new equation and hot-wire measurements in the central region of a fully developed channel flow.

In most real or numerically simulated turbulent flows, the energy dissipated at small scales is equal to that injected at very large scales, which are anisotropic. Despite this injection-scale anisotropy, one generally expects the inertial-range scales to be locally isotropic. For moderate Reynolds numbers, the isotropic relations between second-order and third-order moments for temperature (Yaglom's equation) or velocity increments (Kolmogorov's equation) are not respected, reflecting a non-negligible correlation between the scales responsible for the injection, the transfer and the dissipation of energy. In order to shed some light on the influence of the large scales on inertial-range properties, a generalization of Yaglom's equation is deduced and tested, in heated grid turbulence (Rλ=66). In this case, the main phenomenon responsible for the non-universal inertial-range behaviour is the non-stationarity of the second-order moments, acting as a negative production term.

An equilibrium similarity analysis is applied to the transport equation for $\langle(\delta q)^{2}\rangle$ (${\equiv}\,\langle(\delta u)^{2}\rangle + \langle(\delta v)^{2}\rangle + \langle(\delta w)^{2}\rangle$), the turbulent energy structure function, for decaying homogeneous isotropic turbulence. A possible solution requires that the mean energy $\langle q^{2}\rangle$ decays with a power-law behaviour ($\langle q^{2}\rangle\,{\sim}\,x^{m}$), and the characteristic length scale, which is readily identifiable with the Taylor microscale, varies as $x^{1/2}$. This solution is identical to that obtained by George (1992) from the spectral energy equation. The solution does not depend on the actual magnitude of the Taylor-microscale Reynolds number $R_{\lambda}$ (${\sim}\,{\langle q^{2}\rangle}^{1/2} \lambda/\nu$); $R_{\lambda}$ should decay as $x^{(m+1)/2}$ when $m < -1$. The solution is tested at relatively low $R_{\lambda}$ against grid turbulence data for which $m \simeq -1.25$ and $R_{\lambda}$ decays as $x^{-0.125}$. Although homogeneity and isotropy are poorly approximated in this flow, the measurements of $\langle(\delta q)^{2}\rangle$ and, to a lesser extent, $\langle(\delta u)(\delta q)^{2}\rangle$, satisfy similarity reasonably over a significant range of $r/\lambda$, where $r$ is the streamwise separation across which velocity increments are estimated. For this range, a similarity-based calculation of the third-order structure function $\langle(\delta u)(\delta q)^{2}\rangle$ is in reasonable agreement with measurements. Kolmogorov-normalized distributions of $\langle(\delta q)^{2}\rangle$ and $\langle(\delta u)(\delta q)^{2}\rangle$ collapse only at small $r$. Assuming homogeneity, isotropy and a Batchelor-type parameterization for $\langle(\delta q)^{2}\rangle$, it is found that $R_{\lambda}$ may need to be as large as $10^{6}$ before a two-decade inertial range is observed

A numerical investigation on the effect of the inflow conditions on the self-similar region of a round jetIn this paper, we test the idea of equilibrium similarity, for which all scales evolve in a similar way in a turbulent round jet, for a prescribed set of initial conditions. Similarity requirements of the mean momentum and turbulent energy equations are reviewed briefly but the main focus is on the velocity structure function equation, which represents an energy budget at any particular scale. For similarity of the structure function equation along the jet axis, it is found that the Taylor microscale is the relevant characteristic length scale. Energy structure functions and spectra, measured at a number of locations along the axis of the jet, support this finding reasonably well, i.e., they collapse over a significant range of scales when normalized by and the mean turbulent energy ͗q 2 ͘. Since the Taylor microscale Reynolds number R is approximately constant ͑Ӎ450͒ along the jet axis, the structure functions and spectra also collapse approximately when the normalization uses either the Kolmogorov or integral length scales. Over the dissipative range, the best collapse occurs when Kolmogorov variables are used. The use of ͗q 2 ͘ and the integral length scale L provides the best collapse at large separations. A measure of the quality of collapse is given.

The well-known isotropic relations ͓see Kolmogorov, Dokl. Akad. Nauk. SSSR 30, 301 ͑1941͒; 32, 16 ͑1941͒; A. M. Yaglom, ibid. 69, 743 ͑1949͔͒ between second-order and third-order structure functions are, in general, unlikely to be satisfied in turbulent flows encountered in the laboratory at moderate values of the Reynolds number. The main reason for this is the non-negligible correlation between the length scales at which the initial injection of turbulent energy occurs, those which dominate the transfer of this energy down the ''cascade'' and those which are responsible for dissipating this energy. In the majority of flows, there is a non-negligible inhomogeneity ͑sometimes nonstationarity͒ which may be caused by different physical phenomena. This paper presents an overview of how the equations of Kolmogorov and Yaglom can be ''generalized'' to provide a more realistic description of small-scale turbulence. The focus is mainly on locally isotropic regions of the flow, investigated using one-point measurements and Taylor's hypothesis. We are concerned principally with decaying grid turbulence, for which several results have already been obtained, but other flows, e.g., fully developed channel and jet flows, are also discussed.

For moderate Reynolds numbers, the spectral scaling exponents of both velocity ͓E͑k͒ ϰ k −m u ͔ and the transported passive scalar ͓E ͑k͒ ϰ k −m ͔ fields exhibit departures from the asymptotic prediction m u = m =5/ 3. However, at the same Reynolds number, the passive scalar spectrum for homogeneous isotropic turbulence is closer to the universal asymptotic state than the dynamic velocity field that transports it. This paper provides a possible explanation for this behavior, in the case of a gaseous mixing with Prandtl ͑or Schmidt͒ number PrӍ 1. A scenario of the scalar energy transfer toward higher wavenumbers is proposed and validated using experimental data, in which the velocity field itself is actively involved via its characteristic time. A direct relationship between velocity and scalar spectra and therefore between m u and m is thus established.At large values of Reynolds and Peclet numbers, the similarity hypotheses of Kolmogorov and Obukhov predict that the velocity and scalar spectra exhibit a "Ϫ5/3" inertial range ͑IR͒, viz.,In Eqs. ͑1͒ and ͑2͒, E͑k͒ and E ͑k͒ are three-dimensional ͑3D͒ spectra defined so that ͐ 0 ϱ E͑k͒dk =3uЈ 2 / 2=͗q 2 ͘ / 2, and ͐ 0 ϱ E ͑k͒ = ͗ 2 ͘ / 2, where uЈ is the rms longitudinal velocity, ͗q 2 ͘ϵ͗u i u i ͘, with repeated indices signifying summation and ͗ 2 ͘ is the scalar variance. ͗⑀͘ and ͗͘ are the mean dissipation rates of the turbulent energy and scalar variance, respectively, and C and C are constants, the values of which are ͑for sufficiently high Reynolds numbers͒ C Ϸ 1.52 ͑Ref. 1͒ and C Ϸ 0.68. 2 For the majority of the flows investigated in the laboratory and the computer, asymptotic theoretical predictions of different statistics are not valid because either ͑i͒ the large scales are anisotropic or ͑ii͒ the large scales are very nearly isotropic, but the Reynolds number is finite. Even in the latter case, the applicability of Eqs. ͑1͒ and ͑2͒ is restricted. The wavenumber range over which spectra exhibit a Ϫ5/3 scaling is defined as the IR for which injection and dissipative effects are absent. For moderate Reynolds numbers, a scaling range can be defined by following, e.g., the method proposed in Ref. 3, but its slope is different from Ϫ5/3. This scaling range is called the restricted scaling range ͑RSR͒. In the following, we only focus on locally homogeneous and isotropic turbulence and mixing, and thus limit our attention only to finite Reynolds numbers ͑FRN͒ and the associated RSR.The starting point of our discussion relates to the difference in spectral behavior between the passively advected scalar field and the transporting velocity field, in homogeneous isotropic turbulence ͑HIT͒, as reported by several studies such as in Refs. 3-6 or, for example, Ref. 2, in the larger context of both shearless and sheared flows with a passive scalar. All these studies showed that at a FRN, the scalar spectrum is closer to Ϫ5/3 than the spectrum of the advecting velocity field. This relative behavior is emphasized when the scalar injection is done with a toaster, the mean...

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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