Vanishing enstrophy dissipation in two-dimensional Navier–Stokes turbulence in the inviscid limit

Tran, Chuong V.; Dritschel, David G.

doi:10.1017/s0022112006000577

Cited by 45 publications

(57 citation statements)

References 21 publications

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“…This is consistent with the analytical estimate (2.5) for Batchelor's spectrum (3.1), for which the ratio | ω| 2 1/2 / |∇ω| 2 diverges as Re → ∞. On the other hand, the ratio | ω| 2 / |∇ω| 2 should be proportional to k 2 ν ∝ ν −1 , for enstrophy spectra shallower than k −3 (see Tran & Dritschel 2006). That is, this ratio defines the upper limit of the 'inertial range', and it is expected to increase with Re.…”

Section: Numerical Simulationssupporting

confidence: 90%

Revisiting Batchelor's theory of two-dimensional turbulence

2007

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Recent mathematical results have shown that a central assumption in the theory of two-dimensional turbulence proposed by Batchelor (Phys. Fluids, vol. 12, 1969, p. 233) is false. That theory, which predicts a χ2/3k−1 enstrophy spectrum in the inertial range of freely-decaying turbulence, and which has evidently been successful in describing certain aspects of numerical simulations at high Reynolds numbers Re, assumes that there is a finite, non-zero enstrophy dissipation χ in the limit of infinite Re. This, however, is not true for flows having finite vorticity. The enstrophy dissipation in fact vanishes.We revisit Batchelor's theory and propose a simple modification of it to ensure vanishing χ in the limit Re → ∞. Our proposal is supported by high Reynolds number simulations which confirm that χ decays like 1/ln Re, and which, following the time of peak enstrophy dissipation, exhibit enstrophy spectra containing an increasing proportion of the total enstrophy 〈ω2〉/2 in the inertial range as Re increases. Together with the mathematical analysis of vanishing χ, these observations motivate a straightforward and, indeed, alarmingly simple modification of Batchelor's theory: just replace Batchelor's enstrophy spectrum χ2/3k−1 with 〈ω2〉 k−1 (ln Re)−1).

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Section: Numerical Simulationssupporting

confidence: 90%

Revisiting Batchelor's theory of two-dimensional turbulence

2007

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“…Such properties constitute the key hypothesis of Kolmogorov's theory of three-dimensional turbulence. On the other hand, for two-dimensional turbulence in the same limit, the maximum enstrophy dissipation rate (the parallel of T ) vanishes and T diverges (Tran & Dritschel 2006;Dritschel, Tran & Scott 2007). For the present case, Ohkitani & Yamada (1997) suggested similar behaviours for T and T, thus favouring the possibility of no finite-time singularities in the inviscid dynamics.…”

Section: Introductionsupporting

confidence: 51%

Number of degrees of freedom and energy spectrum of surface quasi-geostrophic turbulence

2011

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We study both theoretically and numerically surface quasi-geostrophic turbulence regularized by the usual molecular viscosity, with an emphasis on a number of classical predictions. It is found that the system's number of degrees of freedom N, which is defined in terms of local Lyapunov exponents, scales as Re, where Re is the Reynolds number expressible in terms of the viscosity, energy dissipation rate and system's integral scale. For general power-law energy spectra k for the exponential dissipation rate at the dissipation wavenumber. These results have been predicted on the basis of Kolmogorov's theory. Our approach thus recovers these classical predictions and is an analytic alternative to the traditional phenomenological method. The implications of the present findings are discussed in conjunction with related results in the literature. Support for the analytic results is provided through a series of direct numerical simulations.

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“…It may be in order to recall that in the case of 2D Navier-Stokes equations, the dissipation rate η of enstrophy is estimated from above [18] as η ∝ (log Re) −1/2 , where Re is the Reynolds number. In the large-Re limit, η decays to zero, but does so very slowly (as a transcendental function).…”

Section: Discussionmentioning

confidence: 99%

Burgers equation with a passive scalar: Dissipation anomaly and Colombeau calculus

Ohkitani

Dowker

2010

Journal of Mathematical Physics

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A connection between dissipation anomaly in fluid dynamics and Colombeau's theory of products of distributions is exemplified by considering Burgers equation with a passive scalar. Besides the well-known viscosity-independent dissipation of energy in the steadily propagating shock wave solution, the lesser known case of passive scalar subject to the shock wave is studied. An exact dependence of the dissipation rate ǫ θ of the passive scalar on the Prandtl number P r is given by a simple analysis: we show in particular ǫ θ ∝ 1/ √ P r for large P r . The passive scalar profile is shown to have a form of a sum of tanh n x with suitably scaled x, thereby implying the necessity to distinguish H from H n when P r is large, where H is the Heaviside function. An incorrect result of ǫ θ ∝ 1/P r would otherwise be obtained. This is a typical example where Colombeau calculus for products of weak solutions is required for a correct interpretation. A Cole-Hopf-like transform is also given for the case of unit Prandtl number.

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Vanishing enstrophy dissipation in two-dimensional Navier–Stokes turbulence in the inviscid limit

Cited by 45 publications

References 21 publications

Revisiting Batchelor's theory of two-dimensional turbulence

Revisiting Batchelor's theory of two-dimensional turbulence

Number of degrees of freedom and energy spectrum of surface quasi-geostrophic turbulence

Burgers equation with a passive scalar: Dissipation anomaly and Colombeau calculus

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