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Review Articles

Two-Dimensional Navier–Stokes Turbulence in Bounded Domains

[+] Author and Article Information
H. J. H. Clercx1

J.M. Burgers Centre for Fluid Dynamics, Department of Physics, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlandsh.j.h.clercx@tue.nl

G. J. F. van Heijst

J.M. Burgers Centre for Fluid Dynamics, Department of Physics, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands

1

Also at Department of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands.

Appl. Mech. Rev 62(2), 020802 (Feb 13, 2009) (25 pages) doi:10.1115/1.3077489 History: Received June 04, 2008; Revised June 18, 2008; Published February 13, 2009

In this review we will discuss recent experimental and numerical results of quasi-two-dimensional decaying and forced Navier–Stokes turbulence in bounded domains. We will give a concise overview of developments in two-dimensional turbulence research, with emphasis on the progress made during the past 10 years. The scope of this review concerns the self-organization of two-dimensional Navier–Stokes turbulence, the quasi-stationary final states in domains with no-slip boundaries, the role of the lateral no-slip walls on two-dimensional turbulence, and their role on the possible destabilization of domain-sized vortices. The overview of the laboratory experiments on quasi-two-dimensional turbulence is restricted to include only those carried out in thin electromagnetically forced shallow fluid layers and in stratified fluids. The effects of the quasi-two-dimensional character of the turbulence in the laboratory experiments will be discussed briefly. As a supplement, the main results from numerical simulations of forced and decaying two-dimensional turbulence in rectangular and circular domains, thus explicitly taking into account the lateral sidewalls, will be summarized and compared with the experimental observations.

Copyright © 2009 by American Society of Mechanical Engineers
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References

Figures

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Figure 2

Enstrophy flux Z(k) normalized by enstrophy dissipation η versus k. The inset is the energy spectrum. Courtesy of Chen (40).

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Figure 4

Entropy of the main modes in (a) a rectangular domain with δ=2 and (b) a circular domain versus the control parameter Λ. The solution with the highest entropy is along the solid line, and the others are along the dashed line. The dipole solution has in both cases the highest entropy and joins continuously the monopole solution. Courtesy of Chavanis and Sommeria (93).

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Figure 5

Time evolution of the compensated enstrophy Ω0(τ) for (a) Re=2000 and (b) Re=5000, with Reλ∊{10,20,25,33.3,50,100,∞}. The dashed lines indicate power laws (a) τ−1.35 and (b) τ−1.05. Both figures are taken from Clercx (95).

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Figure 15

Evolution of the normalized angular momentum L̃(t)=L(t)/Lsb(t), showing distinct phases of spin-up. For this run Re≃3000, A0=6.0, and σ=0.98. This figure is taken from Molenaar (105).

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Figure 6

Time evolution of the average vortex density ρ(τ) for (a) Re=2000 and (b) Re=5000. Filled circles, Reλ=∞; open squares, Reλ=50; open diamonds, Reλ=20; filled triangles, Reλ=10. The dashed lines indicate power laws (a) τ−1.10 and (b) τ−0.80 for τ≲50. Late-time power laws are approximated by τ−0.50. Both figures are taken from Clercx (95).

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Figure 7

Kinetic energy spectrum for the experiments with top-layer thickness of (a) 6 mm and (b) 5 mm. The dashed line in each panel represents the k−3 prediction. In the respective insets, the spectrum compensated with k−3−χ (solid line) and with the Kraichnan prediction k−3 (dashed line). In (a) χ≃0.8±0.1, and in (b) χ≃1.0±0.1. Courtesy of Boffetta (98).

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Figure 8

Sequence of vorticity contour plots for decaying 2D turbulence on a square domain with no-slip boundaries. The flow was initialized by a slightly disturbed array of 10×10 vortices with alternating sense of rotation, with Re=2000. The contour level increments are (a) 3, (b) 1.5, (c) 0.8, (d) 0.4, (e) 0.2, and (f) 0.1. This figure is taken from van Heijst (125).

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Figure 9

(a) Graphical representation of the normalized net angular momentum L(t)/Lsb(t) found in a number of experiments of decaying 2D turbulence. The initial states are characterized by L0≈0 (dashed and dashed-dotted lines) and |L0|>0 (solid lines). (b) The angular momentum L(t) for six runs from numerical simulations as in Fig. 8, and (c) the normalized angular momentum L(t)/Lsb(t) for these runs. These figures are taken from van Heijst (125).

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Figure 10

Stream function contour plots of an experiment with L0≈0 in a circular container. Dashed contours represent negative stream function values, and solid contours represent positive values. The contour level increments are (a) 0.02, (b)–(d) 0.01, (e) 0.008, and (f) 0.004. Courtesy of Maassen (126).

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Figure 11

(a) and (b) Vorticity contour plots of the late-time states emerging in the experiment shown in Fig. 1 (circular container, L0≈0) (126). (c) and (d) Vorticity contour plots of the late-time states emerging in numerical simulations of decaying 2D turbulence on a circular domain with no-slip boundaries (L0≈0)(131). For all panels: dashed contours represent negative values of vorticity, and solid contours represent positive values. Courtesy of Maassen (126) and Li (131).

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Figure 12

Streak images of an experiment with L0≈0 in a square container. The tails of the streaks represent the displacements of tracer particles during an intervals of (a) and (b) 0.1, (c) and (d) 0.2, and (e) and (f) 4 dimensionless time units. Courtesy of Maassen (126).

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Figure 13

(a) The ensemble-averaged 1D energy spectra for runs with no-slip walls and with periodic boundary conditions (the steeper spectrum in the inertial range). (b) The time evolution of 1D spectra for runs with no-slip walls and (c) the growth of the average boundary-layer thickness δav (drawn spiky lines) compared with the position of the kink between k−5/3 and k−3 behaviors. The symbols denote the position of the kink in the spectra (−−+−− for Re=5000, −−×−− for Re=104, and −−∗−− for Re=2×104). Taken from Clercx and van Heijst (143).

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Figure 14

Vorticity plots of simulation with an initial Reynolds numer Re=105 and 40962 Fourier modes.

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Figure 16

Snapshots of the vorticity evolution during the sign reversal of a large monopolar vortex structure with vorticity levels ranging from ω<−5 (black) to ω>5 (white). Same run as shown in Fig. 1. This figure is taken from Molenaar (105).

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Figure 17

Similarity in the behavior of the kinetic energy (solid line) and the absolute angular momentum (dotted) in the upper graph. The lower graph shows the enstrophy (solid line) and the lower bound (dashed). These figures are taken from Molenaar (105).

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Figure 1

Energy spectrum E(k) versus k at steady state obtained by (a) DNS and (b) shallow fluid-layer experiments. The dashed line has a slope of −5/3. Insets: spectral energy flux Π(k)/ϵ from (a) DNS and (b) experiment, with ϵ as the net energy dissipation at all scales. Courtesy of Chen (29).

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Figure 3

Vortex-thinning mechanism: (a) small-scale circular vortex in a large-scale strain field S(0); (b) small-scale vortex drawn out into a narrow shear layer with weakened velocity u(n)<U(n). The axes of the small-scale stress tensor τ(n) are aligned with those of the large-scale strain S(0), whereas the small-scale strain S(n) axes are rotated at ±π/4 rad with respect to S(0). Courtesy of Chen (40).

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