Finite Reynolds Number Effect on Small-Scale Statistics in Decaying Grid Turbulence
Abstract
:1. Introduction
2. FRN Effect on the Small Scales
3. Dual Scaling and Its Constraints
3.1. Dual Scaling of the Energy Spectrum
3.2. Dual Scaling of the Scale-By-Scale Energy Budget
4. Dual Scaling for Higher-Order Even Moments of
5. Concluding Remarks
- (i)
- Both K41 and K62 were postulated for very large values of . Analytical considerations in the context of the transport equations of and in decaying grid turbulence (Equations (5) and (6) and Figure 1) indicate that values of between and are required before the effect of the large scales (or equivalently, the FRN effect) disappears. Consequently, Equation (2), i.e., the 4/5 law, will be validated when the large scale (or non-stationary) term in (5) is no longer important. This is confirmed by the experimental data and EDQNM results in Figure 2.
- (ii)
- Results, inferred from the dual scaling approach, based on either the energy spectra or the scale-by-scale energy budget are consistent with those in (i). For the scale-by-scale energy budget in decaying grid turbulence, the (, ) scaling at small scales should be effective since the two dimensionless parameters in Equation (16) are universal (by definition). The (, L) scaling should be also tenable for both large scales and scales within the scaling range since both and should approach constant values as increases. The dual scaling approach, which satisfies incomplete similarity of (5), is supported by the experimental data and the EDQNM results (Figure 4 and Figure 5); it is also supported by the () distributions in grid turbulence over a significant range, in the context of laboratory measurements, of [9]. When is sufficiently large, both scalings should overlap, thus leading to the power-law relations and in the overlap region over which the inertial range is established. This is consistent with the constraints imposed by the Hölder and Cauchy-Schwarz inequalities [80]. The EDQNM results of Meldi et al. [34] (Figure 7), the empirical fit of Mydlarski and Warhaft [35,36] for the energy spectra (Figure 6) and the extrapolation of Tang et al. [9] for () (Equation (18); see also Figure 9 for ) indicate that values of between and are required before an overlap range begins to emerge. Evidently, the maximum values of that are achievable in the laboratory experiments and direct numerical simulations are, as yet, insufficient to observe a power-law behavior of significant extent in energy spectra and, more especially, . Also, we note that Equation (9) of McComb [88] (i.e., the energy flux is equal to the dissipation rate and also the transfer spectrum is zero) requires that an IR exists and that the FRN effect is negligible, i.e., the Reynolds number must be large, if not very large. This is consistent with what we find in this review, based on the KH equation when is infinitely large. McComb [11] concluded the discussion in chapter 6 of his monograph with “our view is that K41 is basically correct and that, in particular, the work of Gamard and George [52] and of Lundgren [24], when taken together, leave little room for doubt on this matter”. The EDQNM results (Figure 7) at very high and those obtained by extrapolation, via Equation (18) to comparably high values (Figure 9), reinforce McComb’s conclusion.
- (i)
- the (, ) scaling, inferred from the N-S equation, e.g. [50] (see also Section 3.2);
- (ii)
- (iii)
- the overwhelming support for the (, ) scaling from experimental and EDQNM data in various other flows, as reviewed in this paper.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Tang, S.; Danaila, L.; Antonia, R.A. Finite Reynolds Number Effect on Small-Scale Statistics in Decaying Grid Turbulence. Atmosphere 2024, 15, 540. https://doi.org/10.3390/atmos15050540
Tang S, Danaila L, Antonia RA. Finite Reynolds Number Effect on Small-Scale Statistics in Decaying Grid Turbulence. Atmosphere. 2024; 15(5):540. https://doi.org/10.3390/atmos15050540
Chicago/Turabian StyleTang, Shunlin, Luminita Danaila, and Robert Anthony Antonia. 2024. "Finite Reynolds Number Effect on Small-Scale Statistics in Decaying Grid Turbulence" Atmosphere 15, no. 5: 540. https://doi.org/10.3390/atmos15050540
APA StyleTang, S., Danaila, L., & Antonia, R. A. (2024). Finite Reynolds Number Effect on Small-Scale Statistics in Decaying Grid Turbulence. Atmosphere, 15(5), 540. https://doi.org/10.3390/atmos15050540