Vortex tubes in turbulence velocity fields at Reynolds numbers ${\mathrm{Re}}_{\ensuremath{\lambda}}\ensuremath{\simeq}300--1300$

Vortex tubes in turbulence velocity fields at Reynolds numbers ${Re}_{λ} ≃ 300 - 1300$

Hideaki Mouri, Akihiro Hori, and Yoshihide Kawashima

Phys. Rev. E 70, 066305 – Published 7 December 2004

Abstract

The most elementary structures of turbulence, i.e., vortex tubes, are studied using velocity data obtained in a laboratory experiment for boundary layers with Reynolds numbers ${Re}_{λ} = 295 - 1258$ . We conduct conditional averaging for enhancements of a small-scale velocity increment and obtain the typical velocity profile for vortex tubes. Their radii are of the order of the Kolmogorov length. Their circulation velocities are of the order of the root-mean-square velocity fluctuation. We also obtain the distribution of the interval between successive enhancements of the velocity increment as the measure of the spatial distribution of vortex tubes. They tend to cluster together below about the integral length and more significantly below about the Taylor microscale. These properties are independent of the Reynolds number and are hence expected to be universal.

Received 5 January 2004

DOI:https://doi.org/10.1103/PhysRevE.70.066305

Authors & Affiliations

Hideaki Mouri^*, Akihiro Hori^†, and Yoshihide Kawashima^†

Meteorological Research Institute, Nagamine 1-1, Tsukuba 305-0052, Japan

^*Electronic address: hmouri@mri-jma.go.jp
^†Also at Meteorological and Environmental Service, Inc., Tama, Tokyo 206-0012, Japan.

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Vol. 70, Iss. 6 — December 2004

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Figure 1
Energy spectrum of the spanwise velocity at ${Re}_{λ} = 295$ , 430, 655, 861, 1054, and 1258 (from bottom to top). The wave number $k$ is in units of $m^{- 1}$ instead of the usual $rad m^{- 1}$ . The dotted line denotes Kolmogorov’s $k^{- 5 ∕ 3}$ law for the inertial range.Reuse & Permissions
Figure 2
Typical profiles for vortex tubes in the streamwise $(u)$ and spanwise $(v)$ velocities at ${Re}_{λ} = 295$ , 430, 655, 861, 1054, and 1258 (from top to bottom). The $u$ profile is shown separately for $u (x + U ∕ f_{s}) - u (x) > 0$ and $u (x + U ∕ f_{s}) - u (x) ⩽ 0$ at $x = 0$ (designated as $u^{+}$ and $u^{-}$ ). The position $x$ is normalized by the Kolmogorov length $η$ . On the ordinate, the unit scale corresponds to one-tenth of the root-mean-square velocity fluctuation ${⟨ v^{2} ⟩}^{1 ∕ 2}$ . The $v$ profiles of Burgers vortices with $Δ = 0$ and $θ = 0$ are shown with dotted lines [see Eqs. (2b, 5)].Reuse & Permissions
Figure 3
Probability density distribution of the absolute velocity increment $∣ v (x + U ∕ f_{s}) - v (x) ∣$ at ${Re}_{λ} = 295$ , 430, 655, 861, 1054, and 1258 (from top to bottom). The distribution is shifted vertically by a factor $10^{3}$ . The velocity increment is normalized by its standard deviation, which is 0.00678, 0.0186, 0.0497, 0.0830, 0.124, and $0.167 m s^{- 1}$ (from top to bottom). The arrows indicate the range of the enhanced velocity increments used in our analyses, which share 1% of the total. The dotted lines denote a Gaussian distribution.Reuse & Permissions
Figure 4
Probability density distribution of the interval $δ x^{'}$ between vortex tubes at ${Re}_{λ} = 295$ , 430, 655, 861, 1054, and 1258 (from top to bottom). The probability density distribution is normalized by the amplitude of its exponential tail (dotted straight lines), and it is shifted vertically by a factor 10. The interval is normalized by the Taylor microscale $λ$ . The dotted curves denote the results of a least-squares fit with a sum of two exponential functions at $δ x^{'} = 5 λ - 25 λ$ [Eq. (6)]. The arrows denote the spanwise integral length $L_{v}$ .Reuse & Permissions
Figure 5
Dependence of tube parameters on ${Re}_{λ}$ . (a) $R_{0} ∕ η$ . (b) $V_{0} ∕ {⟨ v^{2} ⟩}^{1 ∕ 2}$ . (c) $V_{0} ∕ u_{K}$ . (d) ${Re}_{0} ∕ {Re}_{λ}^{1 ∕ 2}$ . (e) ${Re}_{0}$ . (f) $P_{0} (λ)$ . They are individually normalized by the values at ${Re}_{λ} = 430$ . The filled circles denote the data at ${Re}_{λ} = 295$ , 430, 655, 861, 1054, and 1258 from our present experiment. The open circles denote the data at ${Re}_{λ} = 105$ , 165, 225, 292, and 329 from our previous experiment of grid turbulence [18].Reuse & Permissions
Figure 6
Probability density distribution of the interval $δ x^{'}$ between vortex tubes in grid turbulence at ${Re}_{λ} = 105$ obtained in a $3 \times 0.8 \times 0.8 m^{3}$ wind tunnel, grid turbulence at ${Re}_{λ} = 165$ , 225, 292, and 329 obtained in a $18 \times 3 \times 2 m^{3}$ wind tunnel, and boundary-layer turbulence at ${Re}_{λ} = 295$ , 430, 655, 861, 1054, and 1258. We normalize the probability density distribution by the amplitude of its exponential tail. The interval is normalized by the Taylor microscale $λ$ . The data of boundary-layer turbulence are from our present experiment. The data of grid turbulence are from our previous experiment [18]. Among the data of boundary-layer turbulence, those at ${Re}_{λ} = 295$ yield the lowest probability density.Reuse & Permissions

Physical Review E

covering statistical, nonlinear, biological, and soft matter physics

Vortex tubes in turbulence velocity fields at Reynolds numbers ${Re}_{λ} ≃ 300 - 1300$

Hideaki Mouri, Akihiro Hori, and Yoshihide Kawashima

Phys. Rev. E 70, 066305 – Published 7 December 2004

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Physical Review E

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Vortex tubes in turbulence velocity fields at Reynolds numbers Reλ≃300–1300

Hideaki Mouri, Akihiro Hori, and Yoshihide Kawashima

Phys. Rev. E 70, 066305 – Published 7 December 2004

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Vortex tubes in turbulence velocity fields at Reynolds numbers ${Re}_{λ} ≃ 300 - 1300$