Finite Reynolds number effect and the 4/5 law

Kolmogorov [A. N. Kolmogorov, Dokl. Akad. Nauk SSSR 30, 299 (1941)] formulated a theory of small-scale turbulence (K41), valid at extremely large Reynolds numbers, based on two similarity hypotheses and on an exact result derived from the transport equation for the second-order structure function, known as the 4/5 law. Although K41 was praised for its simplicity and elegance, Kolmogorov [A. N. Kolmogorov, J. Fluid Mech. 13, 82 (1962).] proposed a new refined similarity hypothesis (K62) mainly to account for the effect of the large scales on the small scales. It has been widely interpreted in the literature as a correction to K41 arising from the intermittency of the instantaneous energy dissipation rate $\epsilon$. In this paper we argue that since K62 retains the 4/5 law, it must satisfy the same constraints as K41, viz., extremely large Reynolds number and flow stationarity. The retention of the 4/5 law is not however consistent with the presence of nonstationarity due to the effect of the large scales, as postulated by K62. A relatively extensive survey of published data shows that, indeed, the 4/5 law has not yet been observed in either experiments or simulations due to the Reynolds number not being sufficiently large. The use of the transport equation for the second-order structure function, together with an empirical model for the Kolmogorov-normalized second-order velocity structure function, confirms that the 4/5 law is established only after this structure function becomes independent of the Reynolds number.

Figure 1

Distributions of $\overline{{\left(\delta u^{*}\right)}^3}/r^{*}$ in various flows. (a) Shearless grid turbulence: black curves, ${\text{Re}}_{\lambda }=26–99$ [40]; $\blacksquare , {\text{Re}}_{\lambda }=72$ [46]; $•, {\text{Re}}_{\lambda }=144$ [46]; $◯, {\text{Re}}_{\lambda }=99$ [47]; $\square , {\text{Re}}_{\lambda }=134$ [47]; $▵, {\text{Re}}_{\lambda }=319$ [47]; and $\diamond , {\text{Re}}_{\lambda }=448$ [47]. Also shown are the data along the axis of the ONERA wind tunnel (similar to grid turbulence) at ${\text{Re}}_{\lambda }=2260, ⧫$ [46]. (b) Sheared grid turbulence: $⧫, {\text{Re}}_{\lambda }=170$ [48]; $⧫, {\text{Re}}_{\lambda }=660$ [48]; $◯, {\text{Re}}_{\lambda }=196$ [49]; $\square , {\text{Re}}_{\lambda }=254$ [49]; $\diamond , {\text{Re}}_{\lambda }=875$ [49]; and $▵, {\text{Re}}_{\lambda }=938$ [49]. (c) Along the axis of a circular jet: $\blacksquare , {\text{Re}}_{\lambda }=835$ [50]; $\square , {\text{Re}}_{\lambda }=966$ [51]; black curves, ${\text{Re}}_{\lambda }=235–545$ [52]; $•, {\text{Re}}_{\lambda }=485$ [41]; pink curves, ${\text{Re}}_{\lambda }=122–310$ [53]; $▴, {\text{Re}}_{\lambda }=350$ [46]; $▾, {\text{Re}}_{\lambda }=695$ [46]; $◯, {\text{Re}}_{\lambda }=200$ [54]; and $◯, {\text{Re}}_{\lambda }=430$ [54]. (d) Along the axis of a plane jet: ${\text{Re}}_{\lambda }=550–1067$ [44]. (e) SFPBT: red curves, ${\text{Re}}_{\lambda }=38–240$ [43]; green curves, ${\text{Re}}_{\lambda }=177–435$ [36]; blue curves, ${\text{Re}}_{\lambda }=70–460$ [42]; black curves, ${\text{Re}}_{\lambda }=167–1131$ [22]; pink curve, ${\text{Re}}_{\lambda }=805$ [55]; $•, {\text{Re}}_{\lambda }=1300$ [56]; $\times , {\text{Re}}_{\lambda }=650$ [57]; and $+, {\text{Re}}_{\lambda }=240–700$ [58]. (f) In a cylindrical container between counterrotating disks ($\blacksquare , {\text{Re}}_{\lambda }=120$; $•, {\text{Re}}_{\lambda }=300$; and pink curve, ${\text{Re}}_{\lambda }=1170$) [59] and at $y/\delta \approx 0.5$ in a high Reynolds number boundary layer: ${\text{Re}}_{\lambda }=600$ (blue curve) and ${\text{Re}}_{\lambda }=1450$ (black curve) [26]. The dashed horizontal line in each plot corresponds to the value of 4/5.

Figure 2

Local slope $[{\text{LS}}_3\left(r^{*}\right)=dlog\overline{{\left(\delta u^{*}\right)}^3}/dlog{r}^{*}]$ corresponding to the highest ${\text{Re}}_{\lambda }$ in each plot of Fig. 1: $⧫, {\text{Re}}_{\lambda }=2260$ [shearless grid turbulence; see Fig. 1]; $▵, {\text{Re}}_{\lambda }=938$ [sheared grid turbulence; see Fig. 1]; $\square , {\text{Re}}_{\lambda }=966$ [circular jet; see Fig. 1]; red curve, ${\text{Re}}_{\lambda }=1067$ [plane jet; see Fig. 1]; $•, {\text{Re}}_{\lambda }=1300$ [SFPBT; see Fig. 1]; black curve, ${\text{Re}}_{\lambda }=1450$ [boundary layer; see Fig. 1]; and pink curve, ${\text{Re}}_{\lambda }=1170$ [flow between counterrotating disks; see Fig. 1]. Also shown are ASL data ($◯, {\text{Re}}_{\lambda }=10,304$ [62]; $\times , {\text{Re}}_{\lambda }={10}^4$ [64]). The black dashed line corresponds to the value of 1.

Figure 3

(a) Analytical predictions of $\overline{{\left(\delta u^{*}\right)}^3}/r^{*}$ based on Eqs. (11) and (2) at ${\text{Re}}_{\lambda }=51$ (pink curve), ${\text{Re}}_{\lambda }=89$ (green curve), ${\text{Re}}_{\lambda }=144$ (cyan curve), ${\text{Re}}_{\lambda }=2260$ (red curve), ${\text{Re}}_{\lambda }={10}^4$ (blue solid curve), and ${\text{Re}}_{\lambda }={10}^5$ (black solid curve). The symbols are the grid turbulence data [see Fig. 1] at ${\text{Re}}_{\lambda }=51$ ($◯$), ${\text{Re}}_{\lambda }=89$ ($◯$), ${\text{Re}}_{\lambda }=144$ ($◯$), and ${\text{Re}}_{\lambda }=2260$ ($⧫$). Also shown are the analytical predictions of $\overline{{\left(\delta u^{*}\right)}^3}/r^{*}$ based on Eqs. (11) and (13) at ${\text{Re}}_{\lambda }=200$ (blue dashed curve) and ${\text{Re}}_{\lambda }=430$ (black dashed curve), respectively. The symbols are the circular jet data [see Fig. 1] at ${\text{Re}}_{\lambda }=200$ ($◯$) and ${\text{Re}}_{\lambda }=430$ ($◯$). The dashed horizontal line corresponds to 4/5. (b) Corresponding local slope $[{\text{LS}}_3\left(r^{*}\right)=dlog\overline{{\left(\delta u^{*}\right)}^3}/dlog{r}^{*}]$ at ${\text{Re}}_{\lambda }=51$ (pink curve), ${\text{Re}}_{\lambda }=89$ (green curve), ${\text{Re}}_{\lambda }=144$ (cyan curve), ${\text{Re}}_{\lambda }=200$ (blue dashed curve), ${\text{Re}}_{\lambda }=430$ (black dashed curve), ${\text{Re}}_{\lambda }=2260$ (red curve), ${\text{Re}}_{\lambda }={10}^4$ (blue solid curve), and ${\text{Re}}_{\lambda }={10}^5$ (black solid curve). The dashed horizontal line corresponds to the value of 1.

Figure 4

(a) Distributions of $\overline{{\left(\delta u^{*}\right)}^2}/{\left(r^{*}\right)}^{2/3}$ in various flows. (a) Shearless grid turbulence: black curves, ${\text{Re}}_{\lambda }=26–99$ [40]; red curve, ${\text{Re}}_{\lambda }=144$ [71]; and $•$, Modane data of Frisch [12] at ${\text{Re}}_{\lambda }=2260$ (the distributions were renormalized with a more appropriate value of $\overline{\epsilon }$; see the text). (b) Sheared grid turbulence: black curves, ${\text{Re}}_{\lambda }=170–660$ [48]; $◯, {\text{Re}}_{\lambda }=875$ [69]; $\blacksquare , {\text{Re}}_{\lambda }=938$ [69]; $\square , {\text{Re}}_{\lambda }=196$ [49]; and $\diamond , {\text{Re}}_{\lambda }=254$ [49]. (c) Along the axis in the far field of the circular jet: red curve, ${\text{Re}}_{\lambda }=835$ [50]; $\blacksquare , {\text{Re}}_{\lambda }=536$ [50]; $◯, {\text{Re}}_{\lambda }=966$ [51]; black curves, ${\text{Re}}_{\lambda }=235–545$ [52]; $•, {\text{Re}}_{\lambda }=485$ [41]; pink curves, ${\text{Re}}_{\lambda }=122–310$ [53]; blue curve, ${\text{Re}}_{\lambda }=695$ [71]; $◯, {\text{Re}}_{\lambda }=200$ [54]; and $◯, {\text{Re}}_{\lambda }=430$ [54]. (d) Along the axis in the far field of the plane jet: ${\text{Re}}_{\lambda }=550–1067$ [44]. (e) SFPBT: red curves, ${\text{Re}}_{\lambda }=38–240$ [43]; green curves, ${\text{Re}}_{\lambda }=177–435$ [36]; blue curves, ${\text{Re}}_{\lambda }=70–460$ [42]; black curves, ${\text{Re}}_{\lambda }=167–1131$ [22]. Note that thick lines are used for the distributions at the lowest and highest ${\text{Re}}_{\lambda }$. (f) Boundary layer at ${\text{Re}}_{\lambda }=600$ (blue curve) and 1450 (green curve) [26].

Figure 5

Local slope $[{\text{LS}}_2\left(r^{*}\right)=dlog\overline{{\left(\delta u^{*}\right)}^2}/dlog{r}^{*}]$ corresponding to the highest ${\text{Re}}_{\lambda }$ in each plot of Fig. 4: $•, {\text{Re}}_{\lambda }=2260$ [shearless grid turbulence; see Fig. 4]; $\blacksquare , {\text{Re}}_{\lambda }=938$ [sheared grid turbulence; see Fig. 4]; $◯, {\text{Re}}_{\lambda }=966$ [circular jet; see Fig. 4]; red curve, ${\text{Re}}_{\lambda }=1067$ [plane jet; see Fig. 4]; black curve, ${\text{Re}}_{\lambda }=1131$ [SFPBT; see Fig. 4]; and green curve, ${\text{Re}}_{\lambda }=1450$ [boundary layer; see Fig. 4]. The black dashed line corresponds to the value of $2/3$.

Figure 6

(a) Distributions of $\overline{{\left(\delta u\right)}^2}/{\left(\overline{\epsilon }r\right)}^{2/3}$ based on Eq. (2) at ${\text{Re}}_{\lambda }=51$ (red curve), ${\text{Re}}_{\lambda }=89$ (black solid curve), ${\text{Re}}_{\lambda }=144$ (cyan curve), ${\text{Re}}_{\lambda }=200$ (blue dashed curve), ${\text{Re}}_{\lambda }=430$ (black dashed curve), ${\text{Re}}_{\lambda }=2260$ (pink curve), ${\text{Re}}_{\lambda }={10}^4$ (green curve), and ${\text{Re}}_{\lambda }={10}^5$ (blue solid curve). The symbols are the grid turbulence data [see Fig. 4] at ${\text{Re}}_{\lambda }=51$ ($◯$), ${\text{Re}}_{\lambda }=89$ ($◯$), ${\text{Re}}_{\lambda }=144$ ($◯$), and ${\text{Re}}_{\lambda }=2260$ ($•$). Also shown are the circular jet data [see Fig. 4] at ${\text{Re}}_{\lambda }=200$ ($◯$) and ${\text{Re}}_{\lambda }=430$ ($◯$). Note that $C_{\epsilon }\approx 1.4$ is used for circular jet whereas $C_{\epsilon }\approx 1.2$ is used for grid turbulence; see the text. (b) Corresponding local slope $[{\text{LS}}_2\left(r^{*}\right)=dlog\overline{{\left(\delta u^{*}\right)}^2}/dlog{r}^{*}]$ at ${\text{Re}}_{\lambda }=51$ (red curve), ${\text{Re}}_{\lambda }=89$ (black solid curve), ${\text{Re}}_{\lambda }=144$ (cyan curve), ${\text{Re}}_{\lambda }=200$ (blue dashed curve), ${\text{Re}}_{\lambda }=430$ (black dashed curve), ${\text{Re}}_{\lambda }=2260$ (pink curve), ${\text{Re}}_{\lambda }={10}^4$ (green curve), and ${\text{Re}}_{\lambda }={10}^5$ (blue curve). The dashed horizontal line corresponds to $2/3$.

Figure 7

Range of $r^{*}$ over which $\overline{{\left(\delta u^{*}\right)}^2}{r}^{*-2/3}$ (blue line) and $\overline{{\left(\delta u^{*}\right)}^3}{r}^{*-1}$ (red line) depart from 2 and 4/5, respectively, by no more than 2.5%. The locations of the vertical arrows give an approximate indication of the values of ${\text{Re}}_{\lambda }$ needed to attain an IR, of two decades in extent, for $\overline{{\left(\delta u^{*}\right)}^2}$ and $\overline{{\left(\delta u^{*}\right)}^3}$, respectively.