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.
Issue Section:
Review Articles
1.
Kraichnan
, R. H.
, and Montgomery
, D.
, 1980, “Two-Dimensional Turbulence
,” Rep. Prog. Phys.
0034-4885, 43
, pp. 547
–619
.2.
Danilov
, S. D.
, and Gurarie
, D.
, 2000, “Quasi-Two-Dimensional Turbulence
,” Phys. Usp.
1063-7869, 43
, pp. 863
–900
.3.
Tabeling
, P.
, 2002, “Two-Dimensional Turbulence: A Physicist Approach
,” Phys. Rep.
0370-1573, 362
, pp. 1
–62
.4.
Kellay
, H.
, and Goldburg
, W. I.
, 2002, “Two-Dimensional Turbulence: A Review of Some Recent Experiments
,” Rep. Prog. Phys.
0034-4885, 65
, pp. 845
–894
.5.
Doering
, C. H.
, and Newton
, P. K.
, 2007, “Introduction to Special Issue: Mathematical Fluid Dynamics
,” J. Math. Phys.
0022-2488, 48
, p. 065101
(see also other contributions in this issue).6.
Onsager
, L.
, 1949, “Statistical Hydrodynamics
,” Nuovo Cimento
0029-6341, 6
, pp. 279
–287
.7.
Fjørtoft
, R.
, 1953, “On the Changes in the Spectral Distribution of Kinetic Energy for Two-Dimensional Non-Divergent Flow
,” Tellus
, 5
, pp. 225
–230
. 0040-28268.
Lee
, T. D.
, 1952, “On Some Statistical Properties of Hydrodynamical and Magneto-Hydrodynamical Fields
,” Q. Appl. Math.
, 10
, pp. 69
–74
. 0033-569X9.
Similar statements can be made for ideal 3D turbulence and ideal 2D and 3D magnetohydrodynamic turbulence.
10.
Kraichnan
, R. H.
, 1967, “Inertial Ranges in Two-Dimensional Turbulence
,” Phys. Fluids
0031-9171, 10
, pp. 1417
–1423
.11.
Shebalin
, J. V.
, 1989, “Broken Ergodicity and Coherent Structure in Homogeneous Turbulence
,” Physica D
0167-2789, 37
, pp. 173
–191
.12.
Shebalin
, J. V.
, 1998, “Phase Space Structure in Ideal Homogeneous Turbulence
,” Phys. Lett. A
0375-9601, 250
, pp. 319
–322
.13.
Kraichnan
, R. H.
, 1971, “Inertial-Range Transfer in Two and Three-Dimensional Turbulence
,” J. Fluid Mech.
0022-1120, 47
, pp. 525
–535
.14.
Leith
, C. E.
, 1968, “Diffusion Approximation for Two-Dimensional Turbulence
,” Phys. Fluids
0031-9171, 11
, pp. 671
–673
.15.
Batchelor
, G. K.
, 1969, “Computation of the Energy Spectrum in Homogeneous Two-Dimensional Turbulence
,” Phys. Fluids
0031-9171, 12
, pp. II
-233–II-239
.16.
Lilly
, D.
, 1969, “Numerical Simulation of Two-Dimensional Turbulence
,” Phys. Fluids
0031-9171, 12
, pp. II
-240–II-249
.17.
In this review we do not devote much space to the role of different forcing functions to drive 2D turbulence. The most obvious distinction is between random forcing in space and fixed steady-state forcing in some wave number band. These do not necessarily lead to the same flow configurations. This difference is particularly relevant for the question regarding what constitutes and what is to be expected from the forcing in real experimental configurations as can be found in the laboratory and in large-scale geophysical and planetary flows.
18.
Boffetta
, G.
, 2007, “Energy and Enstrophy Fluxes in the Double Cascade of Two-Dimensional Turbulence
,” J. Fluid Mech.
, 589
, pp. 253
–260
. 0022-112019.
Fyfe
, D.
, Montgomery
, D.
, and Joyce
, G.
, 1977, “Dissipative, Forced Turbulence in Two-Dimensional Magnetohydrodynamics
,” J. Plasma Phys.
, 17
, pp. 369
–398
. 0022-377820.
Hossain
, M.
, Matthaeus
, W. H.
, and Montgomery
, D.
, 1983, “Long-Time States of Inverse Cascades in the Presence of a Maximum Length Scale
,” J. Plasma Phys.
, 30
, pp. 479
–493
. 0022-377821.
Frisch
, U.
, and Sulem
, P. -L.
, 1984, “Numerical Simulation of the Inverse Cascade in Two-Dimensional Turbulence
,” Phys. Fluids
0031-9171, 27
, pp. 1921
–1923
.22.
Herring
, J. R.
, and McWilliams
, J. C.
, 1985, “Comparison of Direct Numerical Simulation of Two-Dimensional Turbulence With Two-Point Closure: The Effects of Intermittency
,” J. Fluid Mech.
0022-1120, 153
, pp. 229
–242
.23.
Siggia
, E. D.
, and Aref
, H.
, 1981, “Point-Vortex Simulation of the Inverse Energy Cascade in Two-Dimensional Turbulence
,” Phys. Fluids
0031-9171, 24
, pp. 171
–173
.24.
Smith
, L. M.
, and Yakhot
, V.
, 1993, “Bose Condensation and Small-Scale Structure Generation in a Random Force Driven 2D Turbulence
,” Phys. Rev. Lett.
0031-9007, 71
, pp. 352
–355
.25.
Chertkov
, M.
, Connaughton
, C.
, Kolokolov
, I.
, and Lebedev
, V.
, 2007, “Dynamics of Energy Condensation in Two-Dimensional Turbulence
,” Phys. Rev. Lett.
0031-9007, 99
, p. 084501
.26.
Borue
, V.
, 1994, “Inverse Energy Cascade in Stationary Two-Dimensional Homogeneous Turbulence
,” Phys. Rev. Lett.
0031-9007, 72
, pp. 1475
–1478
.27.
Danilov
, S.
, and Gurarie
, D.
, 2001, “Nonuniversal Features of Forced Two-Dimensional Turbulence in the Energy Range
,” Phys. Rev. E
1063-651X, 63
, p. 020203
.28.
Boffetta
, G.
, Celani
, A.
, and Vergassola
, M.
, 2000, “Inverse Energy Cascade in Two-Dimensional Turbulence: Deviations From Gaussian Behavior
,” Phys. Rev. E
1063-651X, 61
, pp. R29
–R32
.29.
Chen
, S.
, Ecke
, R. E.
, Eyink
, G. L.
, Rivera
, M.
, Wan
, M.
, and Xiao
, Z.
, 2006, “Physical Mechanism of the Two-Dimensional Inverse Energy Cascade
,” Phys. Rev. Lett.
0031-9007, 96
, p. 084502
.30.
Saffman
, P. G.
, 1995, Vortex Dynamics
, Cambridge University Press
, Cambridge, UK
.31.
Moffatt
, H. K.
, 1986, Advances in Turbulence
, G.
Comte-Bellot
and J.
Mathieu
, eds., Springer
, Berlin
, p. 284
.32.
Legras
, B.
, Santangelo
, P.
, and Benzi
, R.
, 1988, “High-Resolution Numerical Experiments for Forced Two-Dimensional Turbulence
,” Europhys. Lett.
0295-5075, 5
, pp. 37
–42
.33.
Ohkitani
, K.
, 1991, “Wave Number Space Dynamics of Enstrophy Cascade in a Forced Two-Dimensional Turbulence
,” Phys. Fluids A
0899-8213, 3
, pp. 1598
–1611
.34.
Maltrud
, M. E.
, and Vallis
, G. K.
, 1991, “Energy Spectra and Coherent Structures in Forced Two-Dimensional and Beta-Plane Turbulence
,” J. Fluid Mech.
, 228
, pp. 321
–342
. 0022-112035.
Borue
, V.
, 1993, “Spectral Exponents of Enstrophy Cascade in Stationary Two-Dimensional Homogeneous Turbulence
,” Phys. Rev. Lett.
0031-9007, 71
, pp. 3967
–3970
.36.
Gotoh
, T.
, 1998, “Energy Spectrum in the Inertial and Dissipation Ranges of Two-Dimensional Steady Turbulence
,” Phys. Rev. E
1063-651X, 57
, pp. 2984
–2991
.37.
Lindborg
, E.
, and Älvelius
, K.
, 2000, “The Kinetic Energy Spectrum of the Two-Dimensional Enstrophy Turbulent Cascade
,” Phys. Fluids
1070-6631, 12
, pp. 945
–947
.38.
Ishihara
, T.
, and Kaneda
, Y.
, 2001, “Energy Spectrum in the Enstrophy Transfer Range of Two-Dimensional Forced Turbulence
,” Phys. Fluids
1070-6631, 13
, pp. 544
–547
.39.
Pasquero
, C.
, and Falkovich
, G.
, 2002, “Stationary Spectrum of Vorticity Cascade in Two-Dimensional Turbulence
,” Phys. Rev. E
1063-651X, 65
, p. 056305
.40.
Chen
, S.
, Ecke
, R. E.
, Eyink
, G. L.
, Wang
, X.
, and Xiao
, Z.
, 2003, “Physical Mechanism of the Two-Dimensional Enstrophy Cascade
,” Phys. Rev. Lett.
0031-9007, 91
, p. 214501
.41.
Tran
, C. V.
, and Shepherd
, T. G.
, 2002, “Constraints on the Spectral Distribution of Energy and Enstrophy Dissipation in Forced Two-Dimensional Turbulence
,” Physica D
0167-2789, 165
, pp. 199
–212
.42.
Tran
, C. V.
, and Bowman
, J. C.
, 2003, “On the Dual Cascade in Two-Dimensional Turbulence
,” Physica D
0167-2789, 176
, pp. 242
–255
.43.
Alexakis
, A.
, and Doering
, C. R.
, 2006, “Energy and Enstrophy Dissipation in Steady State 2D Turbulence
,” Phys. Lett. A
0375-9601, 359
, pp. 652
–657
.44.
Eyink
, G. L.
, 1996, “Exact Results on Stationary Turbulence in 2D: Consequences of Vorticity Conservation
,” Physica D
0167-2789, 91
, pp. 97
–142
.45.
Dubos
, T.
, and Babiano
, A.
, 2002, “Two-Dimensional Cascades and Mixing: A Physical Space Approach
,” J. Fluid Mech.
, 467
, pp. 81
–100
. 0022-112046.
Babiano
, A.
, and Provenzale
, A.
, 2007, “Coherent Vortices and Tracer Cascades in Two-Dimensional Turbulence
,” J. Fluid Mech.
, 574
, pp. 429
–448
. 0022-112047.
Rivera
, M. K.
, Daniel
, W. B.
, Chen
, S. Y.
, and Ecke
, R. E.
, 2003, “Energy and Enstrophy Transfer in Decaying Two-Dimensional Turbulence
,” Phys. Rev. Lett.
0031-9007, 90
, p. 104502
.48.
Kraichnan
, R. H.
, 1975, “Statistical Dynamics of Two-Dimensional Flow
,” J. Fluid Mech.
0022-1120, 67
, pp. 155
–175
.49.
Eyink
, G. L.
, 2006, “Multi-Scale Gradient Expansion of the Turbulent Stress Tensor
,” J. Fluid Mech.
0022-1120, 549
, pp. 159
–190
.50.
Eyink
, G. L.
, 2006, “A Turbulent Constitutive Law for the Two-Dimensional Inverse Energy Cascade
,” J. Fluid Mech.
0022-1120, 549
, pp. 191
–214
.51.
Kraichnan
, R. H.
, 1976, “Eddy Viscosity in Two and Three Dimensions
,” J. Atmos. Sci.
0022-4928, 33
, pp. 1521
–1536
.52.
Lilly
, D. K.
, 1971, “Numerical Simulation of Developing and Decaying Two-Dimensional Turbulence
,” J. Fluid Mech.
0022-1120, 45
, pp. 395
–415
.53.
For bounded domains with no-slip walls, additional boundary contributions should be added to the right hand side of Eq. 14. In particular, the condition Ω(t)≤Ω(t=0) for flows in domains with no-slip boundaries is invalidated, as we will see later on in this review. Note that the equation describing the rate of change of kinetic energy remains unaffected by the presence of walls. However, the relation for dΩ/dt will change.
54.
Fornberg
, B.
, 1977, “Numerical Study of 2-D Turbulence
,” J. Comput. Phys.
0021-9991, 25
, pp. 1
–31
.55.
Matthaeus
, W. H.
, and Montgomery
, D.
, 1980, “Selective Decay Hypothesis at High Mechanical and Magnetic Reynolds Numbers
,” Ann. N.Y. Acad. Sci.
0077-8923, 357
, pp. 203
–222
.56.
Basdevant
, C.
, Legras
, B.
, and Sadourny
, R.
, 1981, “A Study of Barotropic Model Flows: Intermittency, Waves and Predictability
,” J. Atmos. Sci.
0022-4928, 38
, pp. 2305
–2326
.57.
McWilliams
, J. C.
, 1984, “The Emergence of Isolated Coherent Vortices in Turbulent Flow
,” J. Fluid Mech.
0022-1120, 146
, pp. 21
–43
.58.
Santangelo
, P.
, Benzi
, R.
, and Legras
, B.
, 1989, “The Generation of Vortices in High-Resolution, Two-Dimensional Decaying Turbulence and the Influence of Initial Conditions on the Breaking of Self-Similarity
,” Phys. Fluids A
0899-8213, 1
, pp. 1027
–1034
.59.
Kida
, S.
, 1985, “Numerical Simulation of Two-Dimensional Turbulence With High Symmetry
,” J. Phys. Soc. Jpn.
0031-9015, 54
, pp. 2840
–2854
.60.
Brachet
, M. E.
, Meneguzzi
, M.
, and Sulem
, P. -L.
, 1986, “Small-Scale Dynamics of High-Reynolds Number Two-Dimensional Turbulence
,” Phys. Rev. Lett.
0031-9007, 57
, pp. 683
–686
.61.
Brachet
, M. E.
, Meneguzzi
, M.
, Politano
, H.
, and Sulem
, P. -L.
, 1988, “The Dynamics of Freely Decaying Two-Dimensional Turbulence
,” J. Fluid Mech.
0022-1120, 194
, pp. 333
–349
.62.
Benzi
, R.
, Patarnello
, S.
, and Santangelo
, P.
, 1987, “On the Statistical Properties of Two-Dimensional Decaying Turbulence
,” Europhys. Lett.
0295-5075, 3
, pp. 811
–818
.63.
Benzi
, R.
, Patarnello
, S.
, and Santangelo
, P.
, 1988, “Self-Similar Coherent Structures in Two-Dimensional Decaying Turbulence
,” J. Phys. A
0305-4470, 21
, pp. 1221
–1237
.64.
McWilliams
, J. C.
, 1990, “A Demonstration of the Suppression of Turbulent Cascades by Coherent Vortices in Two-Dimensional Turbulence
,” Phys. Fluids A
0899-8213, 2
, pp. 547
–552
.65.
Tran
, C. V.
, and Dritschel
, D. G.
, 2006, “Vanishing Enstrophy Dissipation in Two-Dimensional Navier–Stokes Turbulence in the Inviscid Limit
,” J. Fluid Mech.
0022-1120, 559
, pp. 107
–116
.66.
Leith
, C. E.
, 1984, “Minimum Enstrophy Vortices
,” Phys. Fluids
0031-9171, 27
, pp. 1388
–1395
.67.
Carnevale
, G. F.
, McWilliams
, J. C.
, Pomeau
, Y.
, Weiss
, J. B.
, and Young
, W. R.
, 1991, “Evolution of Vortex Statistics in Two-Dimensional Turbulence
,” Phys. Rev. Lett.
0031-9007, 66
, pp. 2735
–2737
.68.
Carnevale
, G. F.
, McWilliams
, J. C.
, Pomeau
, Y.
, Weiss
, J. B.
, and Young
, W. R.
, 1992, “Rates, Pathways, and End States of Nonlinear Evolution in Decaying Two-Dimensional Turbulence: Scaling Theory Versus Selective Decay
,” Phys. Fluids A
0899-8213, 4
, pp. 1314
–1316
.69.
Weiss
, J. B.
, and McWilliams
, J. C.
, 1993, “Temporal Scaling Behavior of Decaying Two-Dimensional Turbulence
,” Phys. Fluids A
0899-8213, 5
, pp. 608
–621
.70.
Cardoso
, O.
, Marteau
, D.
, and Tabeling
, P.
, 1994, “Quantitative Experimental Study of the Free Decay of Quasi-Two-Dimensional Turbulence
,” Phys. Rev. E
1063-651X, 49
, pp. 454
–461
.71.
Hansen
, A. E.
, Marteau
, D.
, and Tabeling
, P.
, 1998, “Two-Dimensional Turbulence and Dispersion in a Freely Decaying System
,” Phys. Rev. E
1063-651X, 58
, pp. 7261
–7271
.72.
Bracco
, A.
, McWilliams
, J. C.
, Murante
, G.
, Provenzale
, A.
, and Weiss
, J. B.
, 2000, “Revisiting Freely Decaying Two-Dimensional Turbulence at Millennial Resolution
,” Phys. Fluids
1070-6631, 12
, pp. 2931
–2941
.73.
McWilliams
, J. C.
, 1990, “The Vortices of Two-Dimensional Turbulence
,” J. Fluid Mech.
0022-1120, 219
, pp. 361
–385
.74.
Dritschel
, D. G.
, 1993, “Vortex Properties of Two-Dimensional Turbulence
,” Phys. Fluids A
0899-8213, 5
, pp. 984
–997
.75.
Chasnov
, J. R.
, 1997, “On the Decay of Two-Dimensional Homogeneous Turbulence
,” Phys. Fluids
1070-6631, 9
, pp. 171
–180
.76.
Dmitruk
, P.
, and Montgomery
, D. C.
, 2005, “Numerical Study of the Decay of Enstrophy in a Two-Dimensional Navier-Stokes Fluid in the Limit of Very Small Viscosities
,” Phys. Fluids
1070-6631, 17
, p. 035114
.77.
Bartello
, P.
, and Warn
, T.
, 1996, “Self-Similarity of Decaying Two-Dimensional Turbulence
,” J. Fluid Mech.
0022-1120, 326
, pp. 357
–372
.78.
Clercx
, H. J. H.
, and Nielsen
, A. H.
, 2000, “Vortex Statistics for Turbulence in a Container With Rigid Boundaries
,” Phys. Rev. Lett.
0031-9007, 85
, pp. 752
–755
.79.
Clercx
, H. J. H.
, Nielsen
, A. H.
, Torres
, D. J.
, and Coutsias
, E. A.
, 2001, “Two-Dimensional Turbulence in Square and Circular Domains With No-Slip Walls
,” Eur. J. Mech. B/Fluids
0997-7546, 20
, pp. 557
–576
.80.
van Bokhoven
, L. J. A.
, Trieling
, R. R.
, Clercx
, H. J. H.
, and van Heijst
, G. J. F.
, 2007, “Influence of Initial Conditions on Decaying Two-Dimensional Turbulence
,” Phys. Fluids
1070-6631, 19
, p. 046601
.81.
Matthaeus
, W. H.
, Stribling
, W. T.
, Martinez
, D.
, Oughton
, S.
, and Montgomery
, D.
, 1991, “Decaying, Two-Dimensional, Navier-Stokes Turbulence at Very Long Times
,” Physica D
0167-2789, 51
, pp. 531
–538
.82.
Yin
, Z.
, Montgomery
, D. C.
, and Clercx
, H. J. H.
, 2003, “Alternative Statistical-Mechanical Descriptions of Decaying Two-Dimensional Turbulence in Terms of ‘Patches’ and ‘Points’
,” Phys. Fluids
1070-6631, 15
, pp. 1937
–1953
.83.
Eyink
, G. L.
, and Sreenivasan
, K. R.
, 2006, “Onsager and the Theory of Hydrodynamic Turbulence
,” Rev. Mod. Phys.
0034-6861, 78
, pp. 87
–135
.84.
Matthaeus
, W. H.
, Stribling
, W. T.
, Martinez
, D.
, Oughton
, S.
, and Montgomery
, D.
, 1991, “Selective Decay and Coherent Vortices in Two-Dimensional Incompressible Turbulence
,” Phys. Rev. Lett.
0031-9007, 66
, pp. 2731
–2734
.85.
Montgomery
, D.
, Matthaeus
, W. H.
, Stribling
, W. T.
, Martinez
, D.
, and Oughton
, S.
, 1992, “Relaxation in Two Dimensions and the ‘Sinh-Poisson’ Equation
,” Phys. Fluids A
0899-8213, 4
, pp. 3
–6
.86.
Joyce
, G. R.
, and Montgomery
, D.
, 1973, “Negative Temperature States for the Two-Dimensional Guiding Centre Plasma
,” J. Plasma Phys.
, 10
, pp. 107
–121
. 0022-377887.
Montgomery
, D.
, and Joyce
, G. R.
, 1974, “Statistical Mechanics of Negative Temperature States
,” Phys. Fluids
0031-9171, 17
, pp. 1139
–1145
.88.
Pointin
, Y. B.
, and Lundgren
, T. S.
, 1976, “Statistical Mechanics of Two-Dimensional Vortices in a Bounded Domain
,” Phys. Fluids
0031-9171, 19
, pp. 1459
–1470
.89.
Ting
, A. C.
, Chen
, H. H.
, and Lee
, Y. C.
, 1987, “Exact Solution of a Nonlinear Boundary Value Problem: The Vortices of the Two-Dimensional Sinh-Poisson Equation
,” Physica D
0167-2789, 26
, pp. 37
–66
.90.
Miller
, J.
, 1990, “Statistical Mechanics of Euler’s Equation in Two Dimensions
,” Phys. Rev. Lett.
0031-9007, 65
, pp. 2137
–2140
.91.
Miller
, J.
, Weichman
, P. B.
, and Cross
, M. C.
, 1992, “Statistical Mechanics, Euler’s Equation, and Jupiter’s Red Spot
,” Phys. Rev. A
1050-2947, 45
, pp. 2328
–2359
.92.
Robert
, R.
, and Sommeria
, J.
, 1991, “Statistical Equilibrium States for Two-Dimensional Flow
,” J. Fluid Mech.
0022-1120, 229
, pp. 291
–310
.93.
Chavanis
, P. H.
, and Sommeria
, J.
, 1996, “Classification of Self-Organized Structures in Two-Dimensional Turbulence: The Case of a Bounded Domain
,” J. Fluid Mech.
0022-1120, 314
, pp. 267
–297
.94.
Danilov
, S.
, Dolzhanskii
, F. V.
, Dovzhenko
, V. A.
, and Krymov
, V. A.
, 2002, “Experiments on Free Decay of Quasi-Two-Dimensional Turbulent Flows
,” Phys. Rev. E
1063-651X, 65
, p. 036316
.95.
Clercx
, H. J. H.
, van Heijst
, G. J. F.
, and Zoeteweij
, M. L.
, 2003, “Quasi-Two-Dimensional Turbulence in Shallow Fluid Layers: The Role of Bottom Friction and Fluid Layer Depth
,” Phys. Rev. E
1063-651X, 67
, p. 066303
.96.
Wells
, J.
, and Afanasyev
, Ya. D.
, 2004, “Decaying Quasi-Two-Dimensional Turbulence in a Rectangular Container: Laboratory Experiments
,” Geophys. Astrophys. Fluid Dyn.
0309-1929, 98
, pp. 1
–20
.97.
Rivera
, M. K.
, and Ecke
, R. E.
, 2005, “Pair Dispersion and Doubling Time Statistics in Two-Dimensional Turbulence
,” Phys. Rev. Lett.
0031-9007, 95
, p. 194503
.98.
Boffetta
, G.
, Cenedese
, A.
, Espa
, S.
, and Musacchio
, S.
, 2005, “Effects of Friction on 2D Turbulence: An Experimental Study of the Direct Cascade
,” Europhys. Lett.
0295-5075, 71
, pp. 590
–596
.99.
Shats
, M. G.
, Xia
, H.
, and Punzmann
, H.
, 2005, “Spectral Condensation of Turbulence in Plasma and Fluids and Its Role in Low-to-High Phase Transitions in Toroidal Plasmas
,” Phys. Rev. E
1063-651X, 71
, p. 046409
.100.
Akkermans
, R. A. D.
, Kamp
, L. P. J.
, Clercx
, H. J. H.
, and van Heijst
, G. J. F.
, 2008, “Intrinsic Three-Dimensionality in Electromagnetically Driven Shallow Flows
,” EPL
0295-5075, 83
, p. 24001
.101.
Akkermans
, R. A. D.
, Cieslik
, A. R.
, Kamp
, L. P. J.
, Trieling
, R. R.
, Clercx
, H. J. H.
, and van Heijst
, G. J. F.
, 2008, “The Three-Dimensional Structure of an Electromagnetically Generated Dipolar Vortex in a Shallow Fluid Layer
,” Phys. Fluids
1070-6631, 20
, p. 116601
.102.
Sommeria
, J.
, 1986, “Experimental Study of the Two-Dimensional Inverse Energy Cascade in a Square Box
,” J. Fluid Mech.
0022-1120, 170
, pp. 139
–168
.103.
We adopt for this and other experiments a Cartesian coordinate frame, with the x- and y-axis spanning a plane parallel to the bottom of the tank, and the z-axis is taken vertically upward.
104.
Verron
, J.
, and Sommeria
, J.
, 1987, “Numerical Simulation of a Two-Dimensional Turbulence Experiment in Magnetohydrodynamics
,” Phys. Fluids
0031-9171, 30
, pp. 732
–739
.105.
Molenaar
, D.
, Clercx
, H. J. H.
, and van Heijst
, G. J. F.
, 2004, “Angular Momentum of Forced 2D Turbulence in a Square No-Slip Domain
,” Physica D
0167-2789, 196
, pp. 329
–340
.106.
Tabeling
, P.
, Burkhart
, S.
, Cardoso
, O.
, and Willaime
, H.
, 1991, “Experimental Study of Freely Decaying Two-Dimensional Turbulence
,” Phys. Rev. Lett.
0031-9007, 67
, pp. 3772
–3775
.107.
Dolzhanskii
, F. V.
, Krymov
, V. A.
, and Manin
, D. Yu.
, 1992, “An Advanced Experimental Investigation of Quasi-Two-Dimensional Shear Flows
,” J. Fluid Mech.
0022-1120, 241
, pp. 705
–722
.108.
Jüttner
, B.
, Marteau
, D.
, Tabeling
, P.
, and Thess
, A.
, 1997, “Numerical Simulations of Experiments on Quasi-Two-Dimensional Turbulence
,” Phys. Rev. E
1063-651X, 55
, pp. 5479
–5488
.109.
Marteau
, D.
, Cardoso
, O.
, and Tabeling
, P.
, 1995, “Equilibrium States of Two-Dimensional Turbulence: An Experimental Study
,” Phys. Rev. E
1063-651X, 51
, pp. 5124
–5127
.110.
Paret
, J.
, Marteau
, D.
, Paireau
, O.
, and Tabeling
, P.
, 1997, “Are Flows Electromagnetically Forced in Thin Stratified Layers Two-Dimensional?
” Phys. Fluids
1070-6631, 9
, pp. 3102
–3104
.111.
Clercx
, H. J. H.
, Maassen
, S. R.
, and van Heijst
, G. J. F.
, 1999, “Decaying Two-Dimensional Turbulence in Square Containers With No-Slip or Stress-Free Boundaries
,” Phys. Fluids
1070-6631, 11
, pp. 611
–626
.112.
Satijn
, M. P.
, Cense
, A. W.
, Verzicco
, R.
, Clercx
, H. J. H.
, and van Heijst
, G. J. F.
, 2001, “Three-Dimensional Structure and Decay Properties of Vortices in Shallow Fluid Layers
,” Phys. Fluids
1070-6631, 13
, pp. 1932
–1945
.113.
Note that limλ→0t∗=t and t∗≈t if t≲Reλ.
114.
It is found more appropriate to use the typical eddy turnover time of the initial vortices in the discussion of the numerical results. For the simulations discussed in this section, τ≈4t.
115.
Paret
, J.
, and Tabeling
, P.
, 1997, “Experimental Observation of the Two-Dimensional Inverse Energy Cascade
,” Phys. Rev. Lett.
0031-9007, 79
, pp. 4162
–4165
.116.
Paret
, J.
, and Tabeling
, P.
, 1998, “Intermittency in the Two-Dimensional Inverse Cascade of Energy: Experimental Observations
,” Phys. Fluids
1070-6631, 10
, pp. 3126
–3136
.117.
The notation, although commonly used, might be somewhat confusing. Strictly spoken it is the statistical average that depends solely on r=|r| and not on x; thus ⟨|δu∥(x,r)|n⟩=f(r).
118.
Dubos
, T.
, Babiano
, A.
, Paret
, J.
, and Tabeling
, P.
, 2001, “Intermittency and Coherent Structures in the Two-Dimensional Inverse Energy Cascade: Comparing Numerical and Laboratory Experiments
,” Phys. Rev. E
1063-651X, 64
, p. 036302
.119.
Paret
, J.
, Jullien
, M. -C.
, and Tabeling
, P.
, 1999, “Vorticity Statistics in the Two-Dimensional Enstrophy Cascade
,” Phys. Rev. Lett.
0031-9007, 83
, pp. 3418
–3421
.120.
Falkovich
, G.
, and Lebedev
, V.
, 1994, “Universal Direct Cascade in Two-Dimensional Turbulence
,” Phys. Rev. E
1063-651X, 50
, pp. 3883
–3899
.121.
Eyink
, G. L.
, 1995, “Exact Results on Scaling Exponents in the 2D Enstrophy Cascade
,” Phys. Rev. Lett.
0031-9007, 74
, pp. 3800
–3803
.122.
Nam
, K.
, Ott
, E.
, Antonsen
, T. M.
, Jr., and Guzdar
, P. N.
, 2000, “Lagrangian Chaos and the Effect of Drag on the Enstrophy Cascade in Two-Dimensional Turbulence
,” Phys. Rev. Lett.
0031-9007, 84
, pp. 5134
–5137
.123.
Boffetta
, G.
, Celani
, A.
, Musacchio
, S.
, and Vergassola
, M.
, 2002, “Intermittency in Two-Dimensional Ekman-Navier-Stokes Turbulence
,” Phys. Rev. E
1063-651X, 66
, p. 026304
.124.
Wells
, M. G.
, Clercx
, H. J. H.
, and van Heijst
, G. J. F.
, 2007, “Vortices in Oscillating Spin-Up
,” J. Fluid Mech.
0022-1120, 573
, pp. 339
–369
.125.
van Heijst
, G. J. F.
, Clercx
, H. J. H.
, and Molenaar
, D.
, 2006, “The Effects of Solid Boundaries on Confined Two-Dimensional Turbulence
,” J. Fluid Mech.
0022-1120, 554
, pp. 411
–431
.126.
Maassen
, S. R.
, Clercx
, H. J. H.
, and van Heijst
, G. J. F.
, 2002, “Self-Organization of Quasi-2D Turbulence in Stratified Fluids in Square and Circular Containers
,” Phys. Fluids
1070-6631, 14
, pp. 2150
–2169
.127.
Clercx
, H. J. H.
, Maassen
, S. R.
, and van Heijst
, G. J. F.
, 1998, “Spontaneous Spin-Up During the Decay of 2D Turbulence in a Square Container With Rigid Boundaries
,” Phys. Rev. Lett.
0031-9007, 80
, pp. 5129
–5132
.128.
Maassen
, S. R.
, Clercx
, H. J. H.
, and van Heijst
, G. J. F.
, 2003, “Self-Organization of Decaying Quasi-2D Turbulence in Stratified Fluids in Rectangular Containers
,” J. Fluid Mech.
0022-1120, 495
, pp. 19
–33
.129.
Li
, S.
, and Montgomery
, D.
, 1996, “Decaying Two-Dimensional Turbulence With Rigid Walls
,” Phys. Lett. A
0375-9601, 218
, pp. 281
–291
.130.
Li
, S.
, Montgomery
, D.
, and Jones
, W. B.
, 1996, “Inverse Cascades of Angular Momentum
,” J. Plasma Phys.
, 56
, pp. 615
–639
. 0022-3778131.
Li
, S.
, Montgomery
, D.
, and Jones
, W. B.
, 1997, “Two-Dimensional Turbulence With Rigid Circular Walls
,” Theor. Comput. Fluid Dyn.
0935-4964, 9
, pp. 167
–181
.132.
Equivalence of both terms can also be shown by expressing the pressure boundary condition for the present problem in terms of the normal vorticity gradient at the no-slip boundary.
133.
Clercx
, H. J. H.
, 1997, “A Spectral Solver for the Navier-Stokes Equations in the Velocity-Vorticity Formulation for Flows With Two Non-Periodic Directions
,” J. Comput. Phys.
0021-9991, 137
, pp. 186
–211
.134.
Daube
, O.
, 1992, “Resolution of the 2D Navier-Stokes Equations in Velocity-Vorticity Form by Means of an Influence Matrix Technique
,” J. Comput. Phys.
0021-9991, 103
, pp. 402
–414
.135.
Clercx
, H. J. H.
, and Bruneau
, C. -H.
, 2006, “The Normal and Oblique Collision of a Dipole With a No-Slip Boundary
,” Comput. Fluids
0045-7930, 35
, pp. 245
–279
.136.
Orszag
, S. A.
, 1969, “Numerical Methods for the Simulation of Turbulence
,” Phys. Fluids
0031-9171, 12
, pp. II
-250–II-257
.137.
Kress
, B. T.
, and Montgomery
, D. C.
, 2000, “Pressure Determinations for Incompressible Fluids and Magnetofluids
,” J. Plasma Phys.
, 64
, pp. 371
–377
. 0022-3778138.
Maassen
, S. R.
, Clercx
, H. J. H.
, and van Heijst
, G. J. F.
, 1999, “Decaying Quasi-2D Turbulence in a Stratified Fluid With Circular Boundaries
,” Europhys. Lett.
0295-5075, 46
, pp. 339
–345
.139.
Yap
, C. T.
, and van Atta
, C. W.
, 1993, “Experimental Studies of the Development of Quasi-Two-Dimensional Turbulence in Stably Stratified Fluid
,” Dyn. Atmos. Oceans
0377-0265, 19
, pp. 289
–323
.140.
Fincham
, A. M.
, Maxworthy
, T.
, and Spedding
, G. R.
, 1996, “Energy Dissipation and Vortex Structure in Freely Decaying Stratified Grid Turbulence
,” Dyn. Atmos. Oceans
0377-0265, 23
, pp. 155
–169
.141.
Schneider
, K.
, and Farge
, M.
, 2005, “Decaying Two-Dimensional Turbulence in a Circular Container
,” Phys. Rev. Lett.
0031-9007, 95
, p. 244502
.142.
Clercx
, H. J. H.
, and van Heijst
, G. J. F.
, 2002, “Dissipation of Kinetic Energy in Two-Dimensional Bounded Flows
,” Phys. Rev. E
1063-651X, 65
, p. 066305
.143.
Clercx
, H. J. H.
, and van Heijst
, G. J. F.
, 2000, “Energy Spectra for Decaying 2D Turbulence in a Bounded Domain
,” Phys. Rev. Lett.
0031-9007, 85
, pp. 306
–309
.144.
Angot
, P.
, Bruneau
, C. -H.
, and Fabrie
, P.
, 1999, “A Penalization Method to Take Into Account Obstacles in Viscous Flows
,” Numer. Math.
0029-599X, 81
, pp. 497
–520
.145.
Arquis
, E.
, and Caltagirone
, J. P.
, 1984, “Sur les conditions hydrodynamique au voisinage d’une interface milieu fluide-milieu poreux: Application à la convection naturelle
,” C. R. Acad. Sci., Ser. II: Mec., Phys., Chim., Sci. Terre Univers
0764-4450, 299
, pp. 1
–4
.146.
Keetels
, G. H.
, D’Ortona
, U.
, Kramer
, W.
, Clercx
, H. J. H.
, Schneider
, K.
, and van Heijst
, G. J. F.
, 2007, “Fourier Spectral and Wavelet Solvers for the Incompressible Navier–Stokes Equations With Volume-Penalization: Convergence of a Dipole-Wall Collision
,” J. Comput. Phys.
0021-9991, 227
, pp. 919
–945
.147.
Keetels
, G. H.
, Clercx
, H. J. H.
, and van Heijst
, G. J. F.
, 2009, “The Origin of Spin-Up Processes in Decaying Two-Dimensional Turbulence
,” Eur. J. Mech. B/Fluids
0997-7546, submitted.Copyright © 2009
by American Society of Mechanical Engineers
You do not currently have access to this content.
