Scale invariance in finite Reynolds number homogeneous isotropic turbulence

The problem of homogeneous isotropic turbulence (HIT) is revisited within the analytical framework of the Navier–Stokes equations, with a view to assessing rigorously the consequences of the scale invariance (an exact property of the Navier–Stokes equations) for any Reynolds number. The analytical development, which is independent of the 1941 (K41) and 1962 (K62) theories of Kolmogorov for HIT for infinitely large Reynolds number, is applied to the transport equations for the second- and third-order moments of the longitudinal velocity increment, $(\unicode[STIX]{x1D6FF}u)$. Once the normalised equations and the constraints required for complying with the scale-invariance property of the equations are presented, results derived from these equations and constraints are discussed and compared with measurements. It is found that the fluid viscosity, $\unicode[STIX]{x1D708}$, and the mean kinetic energy dissipation rate, $\overline{\unicode[STIX]{x1D716}}$ (the overbar denotes spatial and/or temporal averaging), are the only scaling parameters that make the equations scale-invariant. The analysis further leads to expressions for the distributions of the skewness and the flatness factor of $(\unicode[STIX]{x1D6FF}u)$ and shows that these distributions must exhibit plateaus (of different magnitudes) in the dissipative and inertial ranges, as the Taylor microscale Reynolds number $Re_{\unicode[STIX]{x1D706}}$ increases indefinitely. Also, the skewness and flatness factor of the longitudinal velocity derivative become constant as $Re_{\unicode[STIX]{x1D706}}$ increases; this is supported by experimental data. Further, the analysis, backed up by experimental evidence, shows that, beyond the dissipative range, the behaviour of $\overline{(\unicode[STIX]{x1D6FF}u)^{n}}$ with $n=2$, 3 and 4 cannot be represented by a power law of the form $r^{\unicode[STIX]{x1D701}_{n}}$ when the Reynolds number is finite. It is shown that only when $Re_{\unicode[STIX]{x1D706}}\rightarrow \infty$ can an $n$-thirds law (i.e. $\overline{(\unicode[STIX]{x1D6FF}u)^{n}}\sim r^{\unicode[STIX]{x1D701}_{n}}$, with $\unicode[STIX]{x1D701}_{n}=n/3$) emerge, which is consistent with the onset of a scaling range.

Reynolds number effect on the velocity derivative flatness factor

This paper investigates the effect of a finite Reynolds number (FRN) on the flatness factor ($F$) of the velocity derivative in decaying homogeneous isotropic turbulence by applying the eddy damped quasi-normal Markovian (EDQNM) method to calculate all terms in an analytic expression for $F$ (Djenidi et al., Phys. Fluids, vol. 29 (5), 2017b, 051702). These terms and hence $F$ become constant when the Taylor microscale Reynolds number, $Re_{\unicode[STIX]{x1D706}}$ exceeds approximately $10^{4}$. For smaller values of $Re_{\unicode[STIX]{x1D706}}$, $F$, like the skewness $-S$, increases with $Re_{\unicode[STIX]{x1D706}}$; this behaviour is in quantitative agreement with experimental and direct numerical simulation data. These results indicate that one must first ensure that $Re_{\unicode[STIX]{x1D706}}$ is large enough for the FRN effect to be negligibly small before the hypotheses of Kolmogorov (Dokl. Akad. Nauk SSSR, vol. 30, 1941a, pp. 301–305; Dokl. Akad. Nauk SSSR, vol. 32, 1941b, pp. 16–18; J. Fluid Mech., vol. 13, 1962, pp. 82–85) can be assessed unambiguously. An obvious implication is that results from experiments and direct numerical simulations for which $Re_{\unicode[STIX]{x1D706}}$ is well below $10^{4}$ may not be immune from the FRN effect. Another implication is that a power-law increase of $F$ with respect to $Re_{\unicode[STIX]{x1D706}}$, as suggested by the Kolmogorov 1962 theory, is not tenable when $Re_{\unicode[STIX]{x1D706}}$ is large enough.

Derivative Statistics of Axial Velocity and Passive Scalar in the Jet Diffusion Field of High-Schmidt-Number Matter

In this study, a water solution of dye (whose Schmidt number is about 3,800) was issued into the quiescent water as an axisymmetric turbulent jet and the simultaneous measurements of axial velocity and concentration have been performed using the combined probe of I-type hot-film and fiber-optic concentration sensor based on the Lambert-Beer’s law. Then we calculated the PDF (Probability Density Function) for the streamwise velocity derivative ∂u/∂x and streamwise concentration derivative ∂c/∂x. It was confirmed that the PDFs for ∂u/∂x skew negatively, and the values of skewness (S∂u/∂x) and flatness factor (F∂u/∂x) are consistent with the other data (see Sreenivasan and Antonia, 1997). However, with regard to the PDFs for ∂c/∂x, the skewness (S∂c/∂x) show the values very close to zero, unlikely the past other data which show the magnitude of 0.5∼1.0. On the other hand, the flatness factor (F∂c/∂x) show the values of 7.0∼8.0 which are consistent with other data. This result suggests that the fine-scale structure of a high-Schmidt-number diffusion field is almost isotropic although it is intermittent.

Lateral vorticity measurements in a turbulent wake

The accurate measurement of vorticity has proven difficult because of the difficulty of estimating spatial derivatives of velocity fluctuations reliably. A method is proposed for correcting the lateral vorticity spectrum measured using a four-wire probe. The attenuation of the measured spectrum increases as the wavenumber increases but does not vanish when the wavenumber is zero. Although the correction procedure assumes local isotropy, the major contributor to the high-wavenumber part of the vorticity spectrum is the streamwise derivative of the lateral velocity fluctuation, and the correction of this latter quantity does not depend on local isotropy. Satisfactory support for local isotropy is provided by the high-wavenumber parts of the velocity, velocity derivative and vorticity spectra measured on the centreline of a turbulent wake. Second- and fourth-order moments of vorticity show departures from local isotropy but the degree of departure seems unaffected by the turbulence Reynolds number Rλ. The vorticity probability density function is approximately exponential and has tails which stretch out to larger amplitudes as Rλ increases. The vorticity flatness factor, which is appreciably larger than the flatness factor of the streamwise velocity derivative, also increases with Rλ. When Rλ is sufficiently large for velocity structure functions to indicate a r2/3 inertial range, two-point longitudinal correlations of lateral vorticity fluctuations give encouraging support for the theoretical r−4/3 behaviour.

Measurement of streamwise vorticity fluctuations in a turbulent channel flow

In a fully developed turbulent channel flow, measurements of the streamwise vorticity fluctuations ωx have been made. A newly designed probe provides simultaneously in addition to the vorticity signal all three velocity signals. The new probe bears a likeness to the Kovasznay-type vorticity probe, but consists of four electrically independent hot wires, each mounted separately on a total of eight supporting prongs. A new calibration technique had to be developed for this probe.In addition to various statistical properties of the three velocity components, the distributions of vorticity fluctuations and of skewness and flatness factors are given up to wall distances as close as y+ = 19. A pronounced maximum of the streamwise vorticity fluctuations was found at y+ ≈ 20. Large values of the flatness factor characterize the outer flow region.