The energy decay in self-preserving isotropic turbulence revisited

Speziale, Charles G.; Bernard, Peter S.

doi:10.1017/s0022112092002180

Cited by 113 publications

(93 citation statements)

References 18 publications

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“…This is a more robust type of similarity von Kármán self-preservation for magnetohydrodynamic turbulence 301 decay, originally suggested by von Kármán & Lin (1949). Also, see Speziale & Bernard (1992) for a discussion of the so-called 'partial' similarity solutions versus the 'complete' similarity solutions in which the viscous term is retained.…”

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confidence: 85%

von Kármán self-preservation hypothesis for magnetohydrodynamic turbulence and its consequences for universality

et al. 2012

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AbstractWe argue that the hypothesis of preservation of shape of dimensionless second- and third-order correlations during decay of incompressible homogeneous magnetohydrodynamic (MHD) turbulence requires, in general, at least two independent similarity length scales. These are associated with the two Elsässer energies. The existence of similarity solutions for the decay of turbulence with varying cross-helicity implies that these length scales cannot remain in proportion, opening the possibility for a wide variety of decay behaviour, in contrast to the simpler classic hydrodynamics case. Although the evolution equations for the second-order correlations lack explicit dependence on either the mean magnetic field or the magnetic helicity, there is inherent implicit dependence on these (and other) quantities through the third-order correlations. The self-similar inertial range, a subclass of the general similarity case, inherits this complexity so that a single universal energy spectral law cannot be anticipated, even though the same pair of third-order laws holds for arbitrary cross-helicity and magnetic helicity. The straightforward notion of universality associated with Kolmogorov theory in hydrodynamics therefore requires careful generalization and reformulation in MHD.

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confidence: 85%

von Kármán self-preservation hypothesis for magnetohydrodynamic turbulence and its consequences for universality

et al. 2012

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“…Most of these investigations have focused on decaying isotropic turbulence (e.g. von Kármán & Howarth 1938;Batchelor 1948;George 1992;Speziale & Bernard 1992) or homogeneous shear Two-point similarity in the round jet 311 turbulence (e.g. George & Gibson 1992).…”

Section: Introductionmentioning

confidence: 99%

Two-point similarity in the round jet

Ewing

Frohnapfel

George³

et al. 2007

J. Fluid Mech.

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The governing equations for the two-point velocity correlations in the far field of the axisymmetric jet are examined and it is shown that these equations can have equilibrium similarity solutions for jets with finite Reynolds number that retain a dependence on the growth rate of the jet. The two-point velocity correlation can be written as the product of a scale that depends on the downstream position of the two points and a function that only depends on the similarity variables. Physically, this result implies that the turbulent processes producing and dissipating energy at the different scales of motion, as well as transferring energy between the different scales of motion, are in equilibrium as the flow evolves downstream. A particularly interesting prediction from the analysis is that the two-point similarity solutions depend only on the separation distance between the points in the streamwise similarity coordinate (i.e. υ = ξ − ξ ), that is, the logarithm of the streamwise coordinate itself (i.e. ξ = ln x 1 , where x 1 is measured from a virtual origin). Thus, the measures of the turbulence are homogeneous in the streamwise similarity coordinate.The predictions from the similarity analysis for the streamwise two-point velocity correlation were compared with combined hot-wire and LDA measurements on the centreline of a round jet at a Reynolds number of 33 000, and with two-point velocity correlations computed from PIV measurements in a round jet at a Reynolds number of 2000 performed by Fukushima et al. In both cases, the measured twopoint velocity correlations in the streamwise direction collapsed when they were scaled in the manner predicted by the similarity analysis. The results provide further evidence that the equilibrium similarity hypothesis does describe the development of the flow in fully developed turbulent round jets and that the two-point correlations are statistically homogeneous in the streamwise similarity coordinate.

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“…Estimates on the energy decay law for the 3D Euler equations are rare (see ref. 25 and references therein). Moreover, all the estimates concern the infinite space case while we use periodic boundary conditions.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…Moreover, all the estimates concern the infinite space case while we use periodic boundary conditions. For the infinite space case, under the assumption of complete self-preservation, i.e., selfsimilarity for all scales from 0 to ϱ, one finds that the energy should decay as t Ϫ1 (25). If the assumption of complete selfpreservation is not satisfied, the energy is expected to decay as t Ϫ␣ , where ␣ Ͼ 1.…”

Section: Numerical Resultsmentioning

confidence: 99%

Optimal prediction and the rate of decay for solutions of the Euler equations in two and three dimensions

Hald

Stinis

2007

Proc. Natl. Acad. Sci. U.S.A.

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The ''t-model'' for dimensional reduction is applied to the estimation of the rate of decay of solutions of the Burgers equation and of the Euler equations in two and three space dimensions. The model was first derived in a statistical mechanics context, but here we analyze it purely as a numerical tool and prove its convergence. In the Burgers case, the model captures the rate of decay exactly, as was previously shown. For the Euler equations in two space dimensions, the model preserves energy as it should. In three dimensions, we find a power law decay in time and observe a temporal intermittency.dimensional reduction ͉ finite time singularity T he advent of powerful computers has enhanced our ability to study complex systems and understand their dynamics, yet there are many problems that are too complex for computer solution. The task facing the numerical analyst working on such problems is to reduce their complexity to something manageable, yet preserve, maybe only in a statistical sense, their salient features. Turbulence and, more generally, flow at a high Reynolds number provide a prime example of a problem where direct solution is impractical and some simplification is needed. Although a lot of knowledge about this problem has accumulated over the years (1-7), a suitable effective model has not yet emerged.In earlier work (8, 9), we and others have derived methods for reducing the number of variables one needs to solve for in complex systems, using a statistical projection formalims borrowed from statistical mechanics. A special case, the ''t-model'' based on a ''long memory'' assumption (i.e., the assumption that the autocorrelations of the noise decay slowly), seemed to be promising for problems in fluid mechanics (10,11). An earlier application of this model to the Burgers equation (12) yielded remarkably accurate estimates of the rate of decay of solutions.In the present paper, we use the t-model to reduce the number of variables in spectral approximations of the Euler equations and calculate the rate of decay of solutions. We expect the methodology to be of broader interest yet, and to be a step toward the development of reduction methods of general applicability. We first prove the convergence of the model as the number of Fourier components increases. We do not address here the claim implicit in previous work that the t-model may yield acceptable results even when the number of variables remains finite. The results we obtain are, however, surprisingly accurate, and a full analysis may well have to go through some version of the arguments on the basis of which the t-model was originally derived. Note that unlike previous damping methods for allowing spectral calculations to proceed to significant time spans (e.g., refs. 13-19), the t-model equations contain no adjustable parameters and is guaranteed to remain stable.The paper is organized as follows. First we present some properties of systems of equations that have constant energy. We then present the derivation of the t-model. We prove some resul...

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The energy decay in self-preserving isotropic turbulence revisited

Cited by 113 publications

References 18 publications

von Kármán self-preservation hypothesis for magnetohydrodynamic turbulence and its consequences for universality

von Kármán self-preservation hypothesis for magnetohydrodynamic turbulence and its consequences for universality

Two-point similarity in the round jet

Optimal prediction and the rate of decay for solutions of the Euler equations in two and three dimensions

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