zbMATH — the first resource for mathematics

Exponential decay of the power spectrum of turbulence. (English) Zbl 1081.35505
Summary: The analyticity on a strip of the solutions of Navier-Stokes equations in 2D is shown to explain the observed fast decay of the frequency power spectrum of the turbulent velocity field. Some subtleties in the application of Wiener-Khinchine method to turbulence are resolved by showing that the frequency power spectrum of turbulent velocities is in fact a measure exponentially decaying for frequency \(\to \pm \infty\). Our approach also shows that the conventional procedures used in analyzing data in turbulence experiments are valid even in the absence of the ergodic property in the flow.

35Q30 Navier-Stokes equations
76F99 Turbulence
76M25 Other numerical methods (fluid mechanics) (MSC2010)
Full Text: DOI
[1] E. Ching, P. Constantin, L. P. Kadanoff, A. Libchaber, I. Procaccia, and X.-Z. Wu, Transitions in convective turbulence: The role of thermal plumes,Phys. Rev. A 44:8091–8102 (1991). · doi:10.1103/PhysRevA.44.3622
[2] P. Constantin and C. Foias,Navier-Stokes Equations (University of Chicago Press, Chicago, 1988). · Zbl 0687.35071
[3] N. Dunford and J. T. Schwartz,Linear Operators. Part I: General Theory (Intersiience, New York, 1958).
[4] L. Hörmander,The Analysis of Linear Partial Differential Operators (Springer-Verlag, Berlin, 1983).
[5] N. Krylov and N. N. Bogoliubov, La théorie générale de la mesure dans son application à l’étude des systèmes dinamiques de la mécanique non linèaire,Ann. Math. 38:65–113 (1937). · Zbl 0016.08604 · doi:10.2307/1968511
[6] Y. W. Lee,Statistical Theory of Communication (Wiley, New York, 1960).
[7] J. Leray, Etude de divers équations intégrales non linéaires et de quelques problèmes que posent l’hydrodinamique,J. Math. Pures Appl. 9e Ser. 12:1–82 (1933). · Zbl 0006.16702
[8] W. Rudin,Real and Complex Analysis (McGraw-Hill, New York, 1966). · Zbl 0142.01701
[9] R. Temam,Navier-Stokes Equations: Theory and Numerical Analysis (North-Holland, Amsterdam, 1977). · Zbl 0383.35057
[10] N. Wiener,The Fourier Integral and Certain of Its Applications (Dover, New York, 1958). · Zbl 0082.29101
[11] X.-Z. Wu, L. P. Kadanoff, A. Libchaber, and M. Sano, Frequency power spectrum of temperature fluctuations in free convection,Phys. Rev. Lett. 64:2140–2143 (1990). · doi:10.1103/PhysRevLett.64.2140
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.