Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
2015 | 1 | 1 |
Tytuł artykułu

On von Kármán spectrum from a view of fractal

Treść / Zawartość
Warianty tytułu
Języki publikacji
Von Kármán originally deduced his spectrum of wind speed fluctuation based on the Stockes-Navier equation. That derivation, however, is insufficient to exhibit the fractal information of time series, such as wind velocity fluctuation. This paper gives a novel derivation of the von Kármán spectrum based on fractional Langevin equation, aiming at establishing the relationship between the conventional von Kármán spectrum and fractal dimension. Thus, the present results imply that a time series that follows the von Kármán spectrum can be taken as a specifically fractional Ornstein-Uhlenbeck process with the fractal dimension 5/3, providing a new view of the famous spectrum of von Kármán’s from the point of view of fractals. More importantly, that also implies a novel relationship between two famous spectra in fluid mechanics, namely, the Kolmogorov’s spectrum and the von Kármán’s. Consequently, the paper may yet be useful in practice, such as ocean engineering and shipbuilding.

Opis fizyczny
  • Ocean College, Zhejiang University, Yuhangtang Rd.
    866, Hangzhou 310058, China;
  • Ocean College, Zhejiang University, Yuhangtang Rd.
    866, Hangzhou 310058, China;
  • [1] T. von Karman, Progress in the statistical theory of turbulence,Proc. N. A. S., vol. 34, no. 11, pp. 530–539, 1948.
  • [2] G. H. Goedecke, V. E. Ostashev, D. K. Wilson, and H. J. Auvermann,Quasi-wavelet model of von Kármán spectrum of turbulentvelocity fluctuations, Boundary-Layer Meteorology, vol.112, no. 1, pp. 33–56, 2004.
  • [3] E. E. Morfiadakis, G. L. Glinou, and M. J. Koulouvari, The suitabilityof the von Karman spectrum for the structure of turbulencein a complex terrain wind farm, Journal ofWind Engineering andIndustrial Aerodynamics, vol. 62, no. 2–3, pp. 237–257, 1996.
  • [4] M. C. H. Hui, A. Larsen, and H. F. Xiang,Wind turbulence characteristicsstudy at the stonecutters bridge site: part IIwind powerspectra, integral length scales and coherences, Journal of WindEngineering and Industrial Aerodynamics, vol. 97, no. 1, pp. 48–59, 2009.[Crossref]
  • [5] G. Huang and X. Chen, Wavelets-based estimation of multivariateevolutionary spectra and its application to nonstationarydownburst winds, Engineering Structures, vol. 31, no. 4, pp.976–989, 2009.[Crossref]
  • [6] D. K.Wilson, V. E. Ostashev, and G. H. Goedecke, Quasi-waveletformulations of turbulence and other random fields with correlatedproperties, Probabilistic Engineering Mechanics, vol. 24,no. 3, pp. 343–357, 2009.[Crossref]
  • [7] G. Li and Q. Li, Theory of Time-Varying Reliability for EngineeringStructures and Its Applications, Science Press, 2001. (In Chinese)
  • [8] J. Pang, Z. Lin, and Y. Lu, Discussion on the simulation of atmosphericboundary layer with spires and roughness elements inwind tunnels, Experiments and Measurements in Fluid Mechanics,vol. 18, no. 2, pp. 32–37, 2004. (In Chinese)
  • [9] Y.-Q. Xiao, J.-C. Sun, and Q. Li, Turbulence integral scale andfluctuation wind speed spectrumof typhoon: an analysis basedon field measurements, Journal of Natural Disasters, vol. 15, no.5, pp. 45–53, 2006. (In Chinese)
  • [10] J. C. Kaimal, J. C. Wyngaard, Y. Yzumi, and O. R. Cote, Spectralcharacteristics of surface layer turbulence, Quarterly Journal ofthe Royal Meteorological Society, vol. 98, no. 417, pp. 563–589,1972.[Crossref]
  • [11] H. Panofsky, D. Larko, R. Lipschutz, and G. Stone, Spectra of velocitycomponents over complex terrain, Quarterly Journal of theRoyal Meteorological Society, vol. 108, no. 455, pp. 215–230,1982.[Crossref]
  • [12] Gaoan Xiushu (Japan), Fractal, Seism Press (China), 1989,(Translation in Chinese by B.-M. Shen and Z.-W. Chang)
  • [13] K. S. Miller and B. Ross, An Introduction to the Fractional Calculusand Fractional Differential Equations, John Wiley, 1993.
  • [14] M. D. Ortigueira, Introduction to fractional linear systems. parti: continuous-time systems, IEE Proc. Vis. Image Sig. Proc., vol.147, no. 1, pp. 62–70, 2000.
  • [15] M. D. Ortigueira, Introduction to fractional linear systems. partii: discrete-time systems, IEE Proc. Vis. Image Sig. Proc., vol.147, no. 1, pp. 71–78, 2000.
  • [16] I. Podlubny, Fractional-order systems and PIlDμ-controllers,IEEE Trans. Automatic Control, vol. 44, no. 1, pp. 208–213, 1999.
  • [17] J. A. Tenreiro Machado and A. M. S. Galhano, Statistical FractionalDynamics, ASME Journal of Computational and NonlinearDynamics, vol. 3, no. 2, pp. 021201-1–021201-5, April 2008.
  • [18] Y. Q. Chen and K. L. Moore, Discretization schemes for fractionalorder differentiators and integrators, IEEE Trans. Circuits andSystems I Fundamental Theory and Applications, vol. 49, no. 3,pp. 363–367, 2002.
  • [19] B. M. Vinagre, Y. Q. Chen, and I. Petras, Two direct Tustindiscretization methods for fractional-order differentiator/integrator, Journal of The Franklin Institute, vol. 340,no. 5, pp. 349–362, 2003.
  • [20] J. Van de Vegte, Fundamentals of Digital Signal Processing,Prentice Hall, 2003.
  • [21] W. T. Coffey, Yu. P. Kalmykov, and J. T. Waldron, The LangevinEquation, 2nd ed., World Scientific, Singapore, 2004.
  • [22] S. C. Lim and S. V. Muniandy, Generalized Ornstein-Uhlenbeckprocesses and associated self-similar processes, Journal ofPhysics A: Mathematical and General, vol 36, no. 14, pp. 3961–3982, 2003.
  • [23] I. M. Gelfand and K. Vilenkin, Generalized Functions, Vol. 1, AcademicPress, New York, 1964. (Translation in Chinese)
  • [24] A. Papoulis, The Fourier Integral and Its Applications, McGraw-Hill, 1962.
  • [25] M. Li and S. C. Lim, A rigorous derivation of power spectrum offractional Gaussian noise, Fluctuation and Noise Letters, vol. 6,no. 4, pp. C33–C36, 2006.[Crossref]
  • [26] P. Hall and R. Roy, On the relationship between fractal dimensionand fractal index for stationary stochastic processes, TheAnnals of Applied Probability, vol. 4, no. 1, pp. 241–253, 1994.[Crossref]
  • [27] M. Li, Fractal time series-a tutorial review,Mathematical Problemsin Engineering, vol. 2010, 2010.
  • [28] B. B. Mandelbrot, Gaussian Self-Aflnity and Fractals, Springer,2001.
  • [29] J. Beran, Statistics for Long-Memory Processes, Chapman &Hall, 1994.
  • [30] S. C. Lim, M. Li, and L. P. Teo, Locally self-similar fractional oscillatorprocesses, Fluctuation and Noise Letters, vol. 7, no. 2,pp. L169–L179, 2007.[Crossref]
  • [31] M. Li and S. C. Lim, Modeling network traflc using generalizedCauchy process, Physica A, vol. 387, no. 11, pp. 2584–2594,2008.
  • [32] M. Li and W. Zhao, Detection of variations of local irregularityof traflc under DDOS flood attack, Mathematical Problems inEngineering, vol. 2008, 2008.
  • [33] A. Kolmogorov, The local structure of turbulence in incompressiblefluid for very large Reynolds numbers, Proc. Roy. Soc. London,Ser. A, vol. 434, no. 1890, pp. 9–13, 1991.
Typ dokumentu
Identyfikator YADDA
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.