Scaling of Near-Wall Flows in Quasi-Two-Dimensional Turbulent Channels

D. Samanta, F. Ingremeau, R. Cerbus, T. Tran, W. I. Goldburg, P. Chakraborty, and H. Kellay

Phys. Rev. Lett. 113, 024504 – Published 9 July 2014

Abstract

The law of the wall and the log law rule the near-wall mean velocity profile of three-dimensional turbulent flows. These well-known laws, which are validated by legions of experiments and simulations, may be universal. Here, using a soap-film channel, we report the first experimental test of these laws in quasi-two-dimensional turbulent channel flows under two disparate turbulent spectra. We find that despite the differences with three-dimensional flows, the laws prevail, albeit with notable distinctions: the two parameters of the log law are markedly distinct from their three-dimensional counterpart; further, one parameter (the von Kármán constant) is independent of the spectrum whereas the other (the offset of the log law) depends on the spectrum. Our results suggest that the classical theory of scaling in wall-bounded turbulence is incomplete wherein a key missing element is the link with the turbulent spectrum.

Received 13 February 2014

DOI:https://doi.org/10.1103/PhysRevLett.113.024504

Authors & Affiliations

D. Samanta², F. Ingremeau¹, R. Cerbus^2,3, T. Tran⁴, W. I. Goldburg³, P. Chakraborty², and H. Kellay¹

¹University of Bordeaux, LOMA (UMR 5798 CNRS), 351 cours de la Libération, 33405 Talence, France
²Okinawa Institute of Science and Technology, Okinawa, Japan 904-0495
³Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA
⁴Division of Thermal and Fluids Engineering, School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore 639798, Singapore

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Vol. 113, Iss. 2 — 11 July 2014

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Images

Figure 1
Soap-film channel. (a) Schematic of the channel. Soap water (2% dish detergent in water) from the upper reservoir feeds the soap film that is supported between a smooth wall and a rough wall. (The walls are steel blades.) After draining at the bottom, the pump $P$ brings the soap water from the lower reservoir to the upper one, thereby establishing steady flow in the channel. The injection of turbulence is controlled in two ways. (b) Experimental setup with a rod at the channel entrance. The resultant energy spectra $E (k)$ corresponds to the inverse energy cascade (where the spectral exponent is $- 5 / 3$ ). The spectral exponent is the same for both the longitudinal (black) and transverse (gray) components of $E (k)$ , and the same across the width of the channel (see dotted and solid lines). We compute the spectra from velocity fluctuations measured using a LDV probe sampling at about 5 kHz acquisition rate and use Taylor’s hypothesis to transform from frequency space to wave number space (see Methods section in Ref. [20]). In the adjacent photograph we show interference fringes (in white light) close to the rough wall; a large eddy shed from rod interacts with the rough wall. (c) Experimental setup with a comb at the channel entrance. The resultant energy spectra $E (k)$ corresponds to the enstrophy cascade (where the spectral exponent is $- 3$ ). In measuring and plotting $E (k)$ , we adopt the same convention as described in (b). In the adjacent photograph we show interference fringes (in white light) close to the rough wall; eddies of different scales are shed from the comb.
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Figure 2
Representative MVPs for both turbulent spectra. The MVPs are measured using LDV. The MVP is linear near the wall (see black lines). The inset (semilog coordinates) shows a zoomed view of the log layer (the logarithmic variation is marked by black lines).
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Figure 3
MVPs as $U^{+}$ versus $y^{+}$ in semilog coordinates for both turbulent spectra (left, inverse energy cascade; right, enstrophy cascade) and for a range of Re. The solid black lines correspond to the viscous sublayer $U^{+} = y^{+}$ . The solid gray lines, which correspond to the log law, are the best fit to the log layer with $κ$ as 0.14 for the inverse energy cascade and 0.13 for the enstrophy cascade (see Fig. 4). The gray dashed line in the enstrophy cascade profiles is a fit to the low Re profile with a value of $κ = 0.09$ . The intercept of these lines at $y^{+} = 1$ (see arrow) yields an estimate of the value of $B$ , which is $\approx 20$ for the inverse energy cascade and $\approx 7$ for the enstrophy cascade. Tests of power-law behavior are shown in the insets (log-log coordinates), where the line next to the wall corresponds to the viscous sublayer and the line farther away is the power-law region.
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Figure 4
Re dependence of $κ$ , for the inverse energy cascade (a) and the enstrophy cascade (b), over the full measured range of Re. $κ$ is computed from the plateau region of the MVP plotted as $y^{+} (d U^{+}) / (d y^{+})$ versus $y^{+}$ ; the error bars represent the scatter around the plateau. The mean value (averaged over Re) of $κ$ , shown with black lines, for the inverse energy cascade is 0.14 and for the enstrophy cascade (for $Re ≳ 20 000$ ) is 0.13.
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