Abstract
We investigate the two-point averaging over space of fluctuations arising from a multiscale hierarchy of interacting particles. We assume this will satisfy a condition of homogeneity with respect to scale. We consider the second-order correlation of fluctuations arising from particles of a single scale in the hierarchy; we then form an average over the set of such single-scale correlations. As the hierarchy is refined, a condition of scale continuum is approached. We use the limiting value of this procedure to define a two-point correlation function for the multiscale system as a whole, and identify this with the experimental measurement of correlation in such a multiscale context. In the energy spectrum which emerges in this limit, one term comes to dominate the spectrum for large k; this term has the form ‘k −2 ln k’. In fact, a variety of different shape functions (intended to represent correlation functions) leads to this energy spectrum, which bears a qualitative resemblance to a Kolmogorov power-law. In this sense a degree of universality is exhibited. The ideas are illustrated for two simple one-dimensional test cases before a more general treatment in one and three dimensions is developed.
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References
Abramowitz, M., Stegan, I. (eds.): Handook of Mathematical Functions. National Bureau of Standards, Washington, USA (1964)
Box G.E.P., Muller M.E.: A note on the generation of random normal deviates. Ann. Math. Stat. 29, 610–611 (1958)
Bracewell R.N.: The Fourier Transform and its Applications. McGraw Hill, NY (1986)
Chorin A.J.: Vorticity and Turbulence. Springer Verlag, NY (1994)
Chu C.R., Parlange M.B., Katuk G.G., Albertson J.D.: Probability density functions of turbulent velocity and temperature in the atmospheric surface layer. Water Resour. Res. 32, 1681–1688 (1996)
Feller W.: An Introduction to Probability Theory and its Applications, vol. 1, 3rd edn. Wiley, NY (1968)
Gradshteyn I.S., Ryzhik I.M.: Table of Integrals, Series, and Products. Academic Press, San Diego (1980)
Lebedev N.N.: Special Functions and Their Applications. Prentice-Hall, Englewood Cliffs, NJ (1965)
Lighthill M.J.: An Introduction to Fourier Analysis and Generalised Functions. Cambridge University Press, Cambridge (1958)
Richardson L.F.: The supply of energy from and to atmospheric eddies. Proc. Roy. Soc. Lond. A 97, 354–373 (1920)
Roberts, D.E., Summers, D.M.: Velocity correlation considered as a ‘scale average’ in a hierarchical particle model of turbulence: Part II. The role of viscosity. J. Turbul. 10, Art. N3 (2009)
She, Zh.-S.: Hierarchical structures and scalings in turbulence. In: Boratav et al. (eds.) Turbulence Modeling and Vortex Dynamics, Lecture Notes in Physics, vol. 491. Springer-Verlag (1997)
Summers D.M.: Energy and structure of inertial range turbulence deduced from an evolution of fluid impulse. Phys. Rev. E 65, 036314 (2002)
Summers, D.M., Roberts, D.E.: Velocity correlation considered as a ‘scale average’ in a hierarchical particle model of turbulence: Part I. Unbounded hierarchy. J. Turbul. 10, Art. N2 (2009)
Taylor G.I.: Eddy motion in the atmosphere. Phil. Trans. R. Soc. Lond. A 215, 1–26 (1915)
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Communicated by R. Grimshaw
Dedicated to the memory of David Summers who died suddenly on 5th October 2009.
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Roberts, D.E., Summers, D.M. Analysis of fluctuations induced by a scale hierarchy of interacting particles. Theor. Comput. Fluid Dyn. 24, 437–464 (2010). https://doi.org/10.1007/s00162-009-0168-8
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DOI: https://doi.org/10.1007/s00162-009-0168-8