Some observations on Reynolds number scaling in wall-bounded flows

High Reynolds number wind tunnels are essential tools for testing theories of turbulence. This need has led to the construction of tunnels that use compressed gases as the working fluid, and such facilities have given new insights into the behavior of turbulence in wall-bounded flows. Here, we focus on results obtained at Princeton using the Superpipe and the High Reynolds number Testing Facility that have given us new insights into the behavior of wall-bounded flows, in particular, fully developed pipe flow, and turbulent boundary layers in zero pressure gradients.

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Figure 1

Langley Variable Density Tunnel, completed 1923 [5, 6]. Operating conditions: $p=0.1\text{to}20\text{bar}\text{(abs)},D=1.5\text{m,}L=1.8\text{m},V_m=20\mathrm{m}/\mathrm{s},{\mathrm{Re}}_m=26\times {10}^6/\text{m}.$

Figure 2

Göttingen high pressure tunnel, completed 1981. Operating conditions: $p=1\text{to}100\text{bar}\text{(abs)},D=0.6\text{m}\text{diam.,}L=1\text{m},V_m=35\mathrm{m}/\mathrm{s},{\mathrm{Re}}_m\approx 200\times {10}^6/\text{m}$. Reproduction with permission from [24].

Figure 3

Left: Princeton High Reynolds number Testing Facility (HRTF), completed 2007. Operating conditions: $p=1\text{to}200\text{bar}\text{(abs)},D=0.5\text{m}\text{dia.,}L=3.6\text{m},V_m=10\text{m/s},{\mathrm{Re}}_m\approx 90\times {10}^6/\text{m}$. Right: Princeton Superpipe, completed 1994. Operating conditions: $p=1\text{to}200\text{bar}\text{(abs)},D=0.129\text{m}\text{dia.,}L=22\text{m},V_m=35\text{m/s},{\mathrm{Re}}_m\approx 270\times {10}^6/\text{m}.$ Reproduction with permission from D. Quinn.

Figure 4

Princeton Supertank, expected completion 2021. Reproduction with permission from [18]. Operating conditions: $p=1\text{to}80\text{bar}\text{(abs)},H=0.88\text{m}\text{diam.,}L=7.5\text{m},V_m=14\mathrm{m}/\mathrm{s},{\mathrm{Re}}_D\approx 60\times {10}^6/\text{m}.$

Figure 5

Turbulent boundary layer visualizations, flow from left to right. Left: ${\mathrm{Re}}_{\tau }\approx 150$. Reproduction with permission from [35]. Right: ${\mathrm{Re}}_{\tau }\approx 1100$ (${\mathrm{Re}}_{\theta }\approx 2600$). Reproduction with permission from [36].

Figure 6

Coherent motions in wall-bounded turbulence. Illustrations of sublayer streaks [41]; hairpin vortices [42]; vortex packets or large scale motions (LSM) [30, 43, 44]; very large scale motions (VLSM) or superstructures [38, 40, 45]. All figures reproduced with permission from respective sources.

Figure 7

Superpipe mean velocity results for $850\le {\mathrm{Re}}_{\tau }\le 500000$ in inner scaling. Reproduction with permission from [51].

Figure 8

Superpipe mean velocity results for $850\le {\mathrm{Re}}_{\tau }\le 500000$. (a) Classic outer scaling. (b) Zagarola outer scaling. Reproduction with permission from [51].

Figure 9

Left: Townsend roller eddies. Reproduction with permission from [29]. Middle: Theodorsen hairpin eddies. Reproduction with permission from [42]. Right: Perry and Chong's wall-attached eddies. Reproduction with permission from [30].

Figure 10

Perry and Chong's hierarchy of attached wall eddies. Reproduction with permission from [30].

Figure 11

NSTAP probe design [60, 61, 62].

Figure 12

Scaling of the streamwise turbulence component in pipe flow. Left: inner scaling. Right: outer scaling. Reproduction with permission from [65].

Figure 13

Inner peak magnitude growth with Reynolds number. Dashed line is from DNS [68]. Solid line is ${\overline{{u^2}^{+}}}_{max}=3.54+0.646ln{\mathrm{Re}}^{+}$ [Eq. (11)]. Reproduction with permission from [67].

Figure 14

A universal log law for turbulence? Streamwise turbulence intensity: Melbourne wind tunnel ${\mathrm{Re}}_{\tau }=18010$ (2.5 $\mu \mathrm{m}$ hot wire); Large Cavitation Channel ${\mathrm{Re}}_{\tau }=68780$ (LDV); Princeton Superpipe ${\mathrm{Re}}_{\tau }=98190$ (NSTAP); SLTEST ${\mathrm{Re}}_{\tau }\approx 628000$ (Sonics). The solid straight lines correspond to Eq. (3) with $A_1=1.26$. Reproduction with permission from [69].

Figure 15

A universal log law for turbulence? Outer scaled streamwise Reynolds stress profiles of four highest Reynolds numbers of (a) smooth- and (b) rough-wall Superpipe datasets. The solid line has a slope of $-2.0$. The Gaussian expectation for the slope is $1.26\sqrt{3}=-2.18$. Reproduction with permission from [71].

Figure 16

Scaling in the equilibrium region. Left: pipe flow at ${\mathrm{Re}}_{\tau }=98000$. Right: boundary layer flow at ${\mathrm{Re}}_{\tau }=73000$. Reproduction with permission from [70, 71]. The grey region marks the mesolayer.

Figure 17

Kolmogorov spectra for ${\mathrm{Re}}_{\tau }\approx 3\times {10}^3, 5\times {10}^3, 10\times {10}^3, 20\times {10}^3, 40\times {10}^3$, and $70\times {10}^3$, at $y/\delta \approx 0.05$ (left) and $y/\delta \approx 0.5$ (right). ——, boundary layer; - - - -, pipe. Reproduction with permission from [77].

Figure 18

Premultiplied spectra at $y/\delta =0.1$ in pipe for $2000\le {\mathrm{Re}}_{\tau }\le 98000$. Colors go from blue to red as the Reynolds number increases. Left: premultiplied by $k_1^{5/3}$. Right: premultiplied by $k_1^{1.52}$. Reproduction with permission from [77].

Figure 19

Inertial layer slope. Here, ${\mathrm{\Phi }}_{uu}\sim k_x^{-\frac{5}{3}+\mu }$, where $\mu \sim {ln}^{-1}\mathrm{Re}$. Data marked MW are from [80], theory marked GG is from [78]. Reproduction with permission from [81].

Figure 20

Premultiplied spectra with varying Reynolds number at fixed wall-normal locations $y/\delta \approx 0.05$, 0.1, 0.15, 0.5 for pipe (left) and boundary layer (right). Arrows indicate the increasing Reynolds number from ${\mathrm{Re}}^{+}=3000$ to ${\mathrm{Re}}^{+}=70000$. The region of $k_x^{-1}$ should appear as a plateau. Reproduction with permission from [77].

Figure 21

Turbulence kinetic energy production. Left: premultiplied form, where equal areas represent equal contributions to the total production. Adapted with permission from [37]. Right: fraction of total production contributed by the region for $y^{+}\le 100$ (data from [68]).

Figure 22

Contributions to the Reynolds shear stress at ${\mathrm{Re}}_D=93000$ (${\mathrm{Re}}^{+}=2260$). Bold lines show the shear stress contained in the reconstructed flow, using $····$, 10; $·-·-$, 30; $---$, 80 snapshot POD modes; ——, all modes. The straight line indicates the distribution of the total shear stress. Reproduction with permission from [86].

Figure 23

Contour plots of the streamwise velocity fluctuations at $y/R=0.2$ and ${\mathrm{Re}}_D=12500$, constructed using Taylor's hypothesis. Flow is from left to right. Top: instantaneous fluctuations. Bottom: superposition of the first four POD modes, showing how long, meandering structures can be formed from a small number of modes. Adapted with permission from [85].

Figure 24

The radial behavior of the classical POD modes, averaged over all frequencies. Left: Azimuthal modes $m=1–10$ for the first POD mode, $n=1$. $---, {\mathrm{Re}}_D=$ 47 000 (${\mathrm{Re}}^{+}=1210$); ——, ${\mathrm{Re}}_D=$ 93 000 (${\mathrm{Re}}^{+}=2260$). Right: POD modes for $m=3$ and ——, $n=1$; $---, n=2$; and $····, n=3$. Adapted with permission from [86].

Figure 25

Modal peak location for the first POD mode $(n=1)$ and azimuthal mode numbers $m\in [1,64]$. $▴{\mathrm{Re}}^{+}=1330$; $\blacksquare {\mathrm{Re}}^{+}=2460$; $····y_p/R=2\pi C{\left(k_{\theta }R\right)}^{-1}$, with $C=0.2$. Modes with a peak location $y_p^{+}<75$ are identified with open symbols. The lower abscissa indicates the azimuthal wave number, while the upper abscissa shows the corresponding azimuthal mode number, for ${\mathrm{Re}}^{+}=2460$. Reproduction with permission from [88].