Advertisement

Wavelet Transform Analysis of Turbulence

  • Lokenath Debnath
  • Firdous Ahmad Shah
Chapter

Abstract

Considerable progress has been made over the last three decades in our understanding of turbulence through new developments of theory, experiment, and computation. More and more evidence has been accumulated for the physical description of turbulent motions in both two and three dimensions. Consequently, turbulence is now characterized by a remarkable degree of order even though turbulence is usually defined as disordered fluid flows.

Keywords

Stokes Equation Vortex Tube Inertial Range Coherent Vortex Koch Curve 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Bibliography

  1. Anselmet, F., Gagne, Y., Hopfinger, E.J., and Antonia, R.A. (1984), High-order velocity structure functions in turbulent shear flows, J. Fluid Mech. 140, 63–89.Google Scholar
  2. Argoul, F., Arnéodo, A., Elezgaray, J., Grasseau, G., and Murenzi, R. (1988), Wavelet transform of fractal aggregates, Phys. Lett. A135, 327–333.Google Scholar
  3. Argoul, F., Arnéodo, A., E1ezgaray, J., Grasseau, G., and Murenzi, R. (1990), Wavelet transform analysis of self-similarity of diffusion limited aggregate and electro-deposition clusters, Phys. Rev. A41, 5537–5560.Google Scholar
  4. Batchelor, G.K. (1967), The Theory of Homogeneous Turbulence, Cambridge University Press, Cambridge.Google Scholar
  5. Benzi, R., Paladin, G., Parisi, G., and Vulpiani, A. (1984), On the multifractal nature of fully developed turbulence and chaotic systems, J. Phys. A17, 3521–3531.Google Scholar
  6. Benzi, R. and Vergassola, M. (1991), Optimal wavelet transform and its application to two-dimensional turbulence, Fluid Dyn. Res. 8, 117–126.Google Scholar
  7. Chandrasekhar, S. (1949), On Heisenberg’s elementary theory of turbulence, Proc. Roy. Soc. London A200, 20–33.Google Scholar
  8. Chandrasekhar, S. (1956), A theory of turbulence, Proc. Roy. Soc. London A210, 1–19.Google Scholar
  9. Debnath, L. (1978), Oceanic Turbulence, Memoir Series of the Calcutta Mathematical Society, Calcutta.Google Scholar
  10. Debnath, L. (1998a), Wavelet transforms, fractals, and turbulence, in Nonlinear Instability, Chaos, and Turbulence, Vol. I, (Ed. L. Debnath and D.N. Riahi), Computational Mechanics Publications, WIT Press, Southampton, England.Google Scholar
  11. Debnath, L. (1998b), Brief introduction to history of wavelets, Internat. J. Math. Edu. Sci. Tech. 29, 677–688.Google Scholar
  12. Everson, R., Sirovich, L., and Sreenivasan, K.R. (1990), Wavelet analysis of turbulent jet, Phys. Lett. A145, 314–319.Google Scholar
  13. Falconer, K.J. (1990), Fractal Geometry, Mathematical Foundations and Applications, John Wiley, Chichester.MATHGoogle Scholar
  14. Farge, M. (1992), Wavelet transforms and their applications to turbulence, Ann. Rev. Fluid Mech. 24, 395–457.Google Scholar
  15. Farge, M., Goirand, E., Meyer, Y., Pascal, F., and Wickerhauser, M.V. (1992), Improved predictability of two-dimensional turbulent flows using wavelet packet compression, Fluid Dyn. Res. 10, 229–242.Google Scholar
  16. Farge, M., Guezenne, J., Ho, C.M., and Meneveau, C. (1990), Continuous Wavelet Analysis of Coherent Structures, Proceedings of the Summer Progress Centre for Turbulence Research, Stanford University-NASA Ames, 331–398.Google Scholar
  17. Farge, M. and Holschneider, M. (1989), Analysis of two-dimensional turbulent flow, Proc. Scaling, Fractals, and Nonlinear Variability in Geophysics II, Barcelona.Google Scholar
  18. Farge, M. and Holschneider, M. (1990a), Interpolation of two-dimensional turbulence spectrum in terms of singularity in the vortex cores, Europhys. Lett. 15, 737–743.Google Scholar
  19. Farge, M., Holschneider, M., and Colonna, J.F. (1990b), Wavelet analysis of coherent structures in two-dimensional turbulent flows, in Topological Fluid Mechanics (Ed. A.K. Moffatt and A. Tsinober), Cambridge University Press, Cambridge, 765–766.Google Scholar
  20. Farge, M., Kevlahan, N., Perrier, V., and Goirand, E. (1996), Wavelets and turbulence, Proc. IEEE 84(4), 639–669.Google Scholar
  21. Farge, M., Kevlahan, N., Perrier, V., and Schneider, K. (1999a), Turbulence analysis, modelling and computing using wavelets, Wavelets in Physics (Ed. J.e. van den Berg), Cambridge University Press, Cambridge, 117–200.Google Scholar
  22. Farge, M., Schneider, K., and Kevlahan, N. (1999b), Non-Gaussianity and coherent vortex simulation for two-dimensional turbulence using an adaptive orthonormal wavelet basis, Phys. Fluids 11(8), 2187–2201.CrossRefMATHMathSciNetGoogle Scholar
  23. Farge, M. and Rabreau, G. (1988), Transformé en ondelettes pour detecter et analyser les structures cohérentes dans les ecoulements turbulents bidimensionnels, C.R. Acad. Sci. Paris Ser. II 307, 1479–1486.Google Scholar
  24. Farge, M. and Rabreau, G. (1989), Wavelet transform to analyze coherent structures in two-dimensional turbulent flows, Proceedings on Scaling, Fractals and Nonlinear Variability in Geophysics I, Paris.Google Scholar
  25. Frisch, U. and Parisi, G. (1985), On the singularity structure of fully developed turbulence, in Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics (Ed. M. Ghil, R. Benzi, and G. Parisi), North Holland, Amsterdam, 84–125.Google Scholar
  26. Frisch, U., Sulem, P.L., and Nelkin, M. (1978), A simple dynamical model of intermittent fully developed turbulence, J. Fluid Mech. 87, 719–736.Google Scholar
  27. Frisch, U. and Vergassola, M. (1991), A prediction of the multifractal model; the intermediate dissipation range, Europhys. Lett. 14, 439–450.Google Scholar
  28. Frisch, U. (1995), Turbulence. The Legacy of A.N. Kolmogorov, Cambridge University Press, Cambridge.MATHGoogle Scholar
  29. Fröhlich, J. and Schneider, K. (1997), An adaptive wavelet-vaguelette algorithm for the solution of PDEs, J. Comput. Phys. 130, 174–191.Google Scholar
  30. Gagne, Y. and Castaing, B. (1991), A universal non-globally self-similar representation of the energy spectra in fully developed turbulence,C.R. A cad. Sci. Paris, 312, 414–430.Google Scholar
  31. Grant, H.L., Stewart, R.W., and Moilliet, A. (1962), Turbulence spectra from a tidal channel, J. Fluid Mech. 12, 241–268.Google Scholar
  32. Halsey, T.C., Jensen, M.H., Kadanoff, L.P., Procaccia, I., and Shraiman, B.l. (1986), Fractal measures and their singularities: The characterization of strange sets, Phys. Rev. A33, 1141–1151.Google Scholar
  33. Heisenberg, W. (1948a), Zur statistischen theori der turbulenz, Z. Phys. 124, 628–657.Google Scholar
  34. Heisenberg, W. (1948b), On the theory of statistical and isotropic turbulence, Proc. Roy. Soc. London A195, 402–406.Google Scholar
  35. Hunt, J.C.R. and Vassilicos, J.C. (1991), Kolmogorov’s contributions to the physical and geometrical understanding of small-scale turbulence and recent developments, Proc. Roy. Soc. London A434, 183–210.Google Scholar
  36. Kolmogorov, A.N. (1941a), The local structure of turbulence in an incompressible fluid with very large Reynolds numbers, Dokl. Akad. Nauk. SSSR 30, 301–305.Google Scholar
  37. Kolmogorov, A.N. (1941b), Dissipation of energy under locally isotropic turbulence,Dokl. Akad. Nauk. SSSR 32, 16–18.Google Scholar
  38. Kolmogorov, A.N. (1962), A refinement of previous hypotheses concerning the local structure of turbulence of a viscous incompressible fluid at high Reynolds numbers, J. Fluid Mech. 13, 82–85.Google Scholar
  39. Leonard, A. (1974), Energy cascade in large eddy simulations of turbulent fluid flows, Adv. Geophys. 18A, 237–249.Google Scholar
  40. Lin, C.C. (1948), Note on the law of decay of isotropic turbulence,Proc. Nat. Acad. Sci. U.S.A., 34, 230–233.Google Scholar
  41. Mandelbrot, B.B. (1974), Intermittent turbulence in self-similar cascades: Divergence of high moments and dimension of the carrier. J. Fluid Mech. 62, 331–358.Google Scholar
  42. Mandelbrot, B.B. (1975), On the geometry of homogeneous turbulence with stress on the fractal dimension of the iso-surfaces of scalars, J. Fluid Mech. 72, 401–420.Google Scholar
  43. Mandelbrot, B.B. (1982), The Fractal Geometry of Nature, W.H. Freeman, New York.MATHGoogle Scholar
  44. Meneveau, C. (1991), Analysis of turbulence in the orthonormal wavelet representation, J. Fluid Mech. 232, 469–520.Google Scholar
  45. Meneveau, C. (1993), Wavelet analysis of turbulence: The mixed energy cascade, in Wavelets, Fractals, and Fourier Transforms (Ed. M. Farge, J.C.R. Hunt, and J.e. Vassilicos), Oxford University Press, Oxford, 251–264.Google Scholar
  46. Meneveau, C. and Sreenivasan, K.R. (1987a), Simple multifractal cascade model for fully developed turbulence, Phys. Rev. Lett. 59, 1424–1427.Google Scholar
  47. Meneveau, C. and Sreenivasan, K.R. (1987b), In Physics of Chaos and Systems Far From Equilibrium (Ed. M.D. Van, and B. Nichols), Nucl. Phys. B, North Holland, Amsterdam, 2–49.Google Scholar
  48. Métais, O. and Lesieur, M. (1992), Spectral large-eddy simulation of isotropic and stably stratified turbulence, J. Fluid Mech. 239, 157–194.Google Scholar
  49. Moffatt, H.K. (1984), Simple topological aspects of turbulent vorticity dynamics, in Turbulence and Chaotic Phenomena in Fluids (Ed. T. Tatsumi), Elsevier, New York, 223–230.Google Scholar
  50. Monin, A.S. and Yaglom, A.M. (1975), Statistical Fluid Mechanics: Mechanics of Turbulence, Vol. 2, MIT Press, Cambridge.Google Scholar
  51. Oboukhov, A.M. (1941), On the distribution of energy in the spectrum of turbulent flow, Dokl. Akad. Nauk. SSSR 32, 19–21.Google Scholar
  52. Perrier, V., Philipovitch, T., and Basdevant, C. (1995), Wavelet spectra compared to Fourier spectra, J. Math. Phys. 36, 1506–1519.Google Scholar
  53. Sarker, S.K. (1985), Generalization of singularities in nonlocal dynamics, Phys. Rev. A31, 3468–3472.Google Scholar
  54. Schneider, K. and Farge, M. (1997), Wavelet forcing for numerical simulation of two-dimensional turbulence, C.R. Acad. Sci. Paris Série II 325, 263–270.Google Scholar
  55. Schneider, K. and Farge, M. (2000), Computing and analyzing turbulent flows using wavelets, in Wavelet Transforms and Time-Frequency Signal Analysis (Ed. L. Debnath), Birkhäuser, Boston, 181–216.Google Scholar
  56. Schwarz, K.W. (1990), Evidence for organized small-scale structure in fully developed turbulence, Phys. Rev. Lett. 64, 415–418.Google Scholar
  57. Sen, N.R. (1951), On Heisenberg’s spectrum of turbulence, Bull. Cal. Math. Soc. 43, 1–7.Google Scholar
  58. Sen, N.R. (1958), On decay of energy spectrum of isotropic turbulence,Proc. Natl. Inst. Sci. India A23, 530–533.Google Scholar
  59. She, Z.S., Jackson, E., and Orszag, S.A. (1991), Structure and dynamics of homogeneous turbulence, models and simulations, Proc. Roy. Soc. London A434, 101–124.Google Scholar
  60. Sreenivasan, K.R. (1991), Fractals and multifractals in fluid turbulence, Ann. Rev. Fluid Mech. 23, 539–600.Google Scholar
  61. Sreenivasan, K.R. and Meneveau, C. (1986), The fractal facets of turbulence, J. Fluid Mech. 173, 357–386.Google Scholar
  62. Thomson, Sir William and Tait, P.G. (1879), Treatise on natural philosophy, Cambridge University Press, New edition: Principles of Mechanics and Dynamics (1962), Dover Publications, New York.Google Scholar
  63. Vassilicos, J.C (1992), The multi-spiral model of turbulence and intermittency, in Topological Aspects of the Dynamics of Fluids and Plasmas, Kluwer, Amsterdam, 427–442.Google Scholar
  64. Vassilicos, J.C (1993), Fractals in turbulence, in Wavelets, Fractals and Fourier Transforms: New Developments and New Applications, (Ed. M. Farge, J.CR. Hunt, and J.C Vassilicos), Oxford University Press, Oxford, 325–340.Google Scholar
  65. Vassilicos, J.C and Hunt, J.CR. (1991), Fractal dimensions and spectra of interfaces with applications to turbulence, Proc. Roy. Soc. London A435, 505–534.Google Scholar
  66. Vincent A. and Meneguzzi, M. (1991), The spatial structure and statistical properties of homogeneous turbulence, J. Fluid Mech. 225, 1–20.Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Lokenath Debnath
    • 1
  • Firdous Ahmad Shah
    • 2
  1. 1.Department of MathematicsUniversity of Texas, Pan AmericanEdinburgUSA
  2. 2.Department of MathematicsUniversity of KashmirAnantnagIndia

Personalised recommendations