Can small-scale turbulence approach a quasi-universal state?

For the past 50 years or so, Kolmogorov's (1962) correction (K62) to his 1941 hypotheses (K41) has been embraced by an overwhelming majority of turbulence researchers. However, we show in this paper that there are no valid reasons for abandoning K41, a similarity framework known for its simplicity and elegance. In particular, analytical considerations, based on the Navier-Stokes equations, which take into account the finite Reynolds number (FRN) effect, together with all available experimental laboratory data, confirm a tendency towards the universal predictions of K41 as the Reynolds number continues to increase. This is especially true when the focus is on the energy spectrum and velocity structure function in the dissipative range. Incorrectly accounting for the FRN effect, which has been almost invariably mistaken for the intermittency effect, and the inclusion of the atmospheric surface layer data are the major factors which have contributed to the heretofore almost unrivalled acceptance of K62.

Figure 1

(a) $S_n$ ($n=3$ and 4) in a plane jet ($\blacksquare$ [32]) and a circular jet ($◯$ [34]; $•$ [35]; $•$ [36]). The dashed lines indicate values of $S_3$ and $S_4$ averaged over the range ${\text{Re}}_{\lambda }>230$. (b) $\overline{{\left(\delta u^{*}\right)}^n}$ ($n=2$, 3, 4) in a plane jet [32] for ${\text{Re}}_{\lambda }=550$ ($\blacksquare$), 696 ($\blacksquare$), 826 ($\blacksquare$), 914 ($\blacksquare$), and 1067 ($\blacksquare$) and a circular jet [35] for ${\text{Re}}_{\lambda }=235$ ($•$), 305 ($•$), 495 ($•$), and 545 ($•$); note that $S_3$ and $S_4$, as shown in (a), are constant in these two flows for ${\text{Re}}_{\lambda }>230$. The blue dashed lines correspond to ${15}^{-n/2}{S}_n{r}^n[S_n$ is the mean value shown in (a)]. Note that $\overline{{\left(\delta u^{*}\right)}^3}$ has been divided by 10 to avoid interfering with the $\overline{{\left(\delta u^{*}\right)}^2}$ distributions.

Figure 2

Laboratory data for $S_3$ and $S_4$ reproduced from Figs. 5 and 6 of Sreenivasan and Antonia [11] without identifying the data sources. The insets reproduce all (laboratory and ASL) data from Figs. 5 and 6 of Sreenivasan and Antonia [11]. Note that a “red” contour is drawn around the ASL data to distinguish these from the laboratory data. The data for $S_3$ and $S_4$ shown in Fig. 1 are reproduced here (red symbols).

Figure 3

(a) $\overline{{\left(\delta u^{*}\right)}^2}$ in grid turbulence [44] (${\text{Re}}_{\lambda }=25-99$), a circular cylinder wake [45] (${\text{Re}}_{\lambda }=160-280$), and a plane jet [46] (${\text{Re}}_{\lambda }=550-1067$). The arrow indicates the direction of increasing ${\text{Re}}_{\lambda }$. Also shown are distributions of $\overline{{\left(\delta u^{*}\right)}^2}{r}^{*-2/3}$ for a plane jet [46] (${\text{Re}}_{\lambda }=550$ and 1067, green curve and pink curve, respectively) and a circular jet [47] (${\text{Re}}_{\lambda }=966$, blue square symbols). The green and red dashed lines correspond to 2.35 and $r^{*2/3}$, respectively. (b) Compensated one-dimensional (1D) and 3D energy spectra for DNS (box turbulence) and plane jet data at high ${\text{Re}}_{\lambda }$: pink and green curves correspond to the plane jet data reported in (a); the corresponding exponent $q$ is 1.66 (pink) and 1.61 (green). The blue [48] and red [49] curves correspond to DNS data at ${\text{Re}}_{\lambda }=2300$ and 805, respectively. The black and green dashed horizontal lines correspond to 1.83 and 0.66, respectively.

Figure 4

FRN effect on the scaling range behavior of various quantities in different flows. The dashed horizontal lines correspond to the predictions of K41. ${\text{Re}}_{\lambda }$ variation of (a) ${\zeta }_2$ along the axes of a plane jet ($•$ [46]) and circular jet ($\blacksquare$ [47]), and for forced turbulence ($\blacksquare$ [51]); (b) $q[{\varphi }_u\left(k_1\right)\sim k_1^{-q}]$ in grid turbulence ($•$ [52]) and along the axis of a plane jet ($\blacksquare$ [46]); black empirical curve, $q=5/3-8{\text{Re}}_{\lambda }^{-3/4}$ [52]; ($⧫$), DNS reported in Fig. 3; (c) $p=\text{max}[-\overline{{\left(\delta u\right)}^3}/\overline{\epsilon }r]$ in forced turbulence ($•$, the black curve is a fit to the data) and decaying turbulence ($\blacksquare$, the red curve is the analytical prediction [27]; $\blacksquare$, plane jet [46]); (d) $m[{\varphi }_p\left(k_1\right)\sim k_1^{-m}]$ along the axis of a circular jet [53]; (e),(f) ${\zeta }_4$ and ${\zeta }_6$ [see Eq. (6)] along the axis of a plane jet ($•$) and circular jet ($◯$) [46].

Figure 5

Distributions of $\overline{{\left(\delta u^{*}\right)}^3}/r^{*}$ in forced turbulence. flow between counter-rotating disks at ${\text{Re}}_{\lambda }=1170$: $△$ [54]; others are the DNS data (blue, black, and red curves corresponding to ${\text{Re}}_{\lambda }=471$, 732, and 1131 respectively [55]; green curve, ${\text{Re}}_{\lambda }=805$ [49]). Dashed horizontal line: 4/5.