Abstract
We present a method of how to estimate from experimental data of a turbulent velocity field the drift and the diffusion coefficient of a Fokker-Planck equation. It is shown that solutions of this Fokker-Planck equation reproduce with high accuracy the statistics of velocity increments in the inertial range. Using solutions with different initial conditions at large scales we show that they converge. This can be interpreted as a signature of the universality of small scale turbulence in the limit of large inertial ranges.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
A.S. Monin, A.M. Yaglom, Statistical Fluid Mechanics, (MIT Press, Cambridge 1975).
U. Frisch, Turbulence (Cambridge 1995).
R. Benzi, S. Ciliberto, C. Baudet, G.R. Chavarria, Physica D 80, 385 (1995).
A. Arneodo, et. al. Europhys. Lett. 34, 411 (1996).
K.R. Sreenivasan, R.A. Antonia, Annu. Rev. Fluid Mech. 29, 435 (1997).
R. Benzi, L. Biferale, G. Paladin, A. Vulpiani, M. Vergassola, Phys. Rev. 67, 2299 (1991); P. Kailasnath, K.R. Sreenivasan, G. Stolovitzky, Phys. Rev. 68, 2767 (1992).
B. Castaing, Y. Gagne, E. Hopfinger, Physica D 46, 177 (1990).
H. Tennekes, J.C. Wyngaard, J. Fluid Mech. 55, 93 (1972); F. Anselmat, Y. Gagne, E.J. Hopfinger, R.A. Antonia, J. Fluid. Mech 149,63 (1984); J. Peinke, B. Castaing, B. Chabaud, F. Chilla, B. Hebral, A. Naert, in Fractals in the Natural and Applied Sciences, edt.: M.M. Novak (North Holland, Amsterdam 1994) p.295.
R.A. Antonia, B.R. Satyaprakash, A.K.M.F. Hussain, J. Fluid Mech. 119, 55 (1982); B. Castaing, Y. Gagne, E.J. Hopfinger, A new View of Developed Turbulence, in New Approaches and Concepts in Turbulence, edts. Th. Dracos and A. Tsinober (Birkhäuser, Basel 1993) see also discussion p. 47-60 therein; B. Castaing, Y. Gagne, M. Marchand, Physica D 68, 387 (1993); B. Chabaud, A. Naert, J. Peinke, F. Chilla, B. Castaing, B. Hebral, Phys. Rev. Lett. 73 (1994) 3227.
R. Friedrich, J. Peinke, Phys. Rev. Lett. 78, 863 (1997); A. Naert, R. Friedrich, J. Peinke, Phys. Rev. E 56, 6719 (1997).
R. Friedrich, J. Peinke, Physica D 102, 147 (1997).
R. Friedrich, J. Zeller, J. Peinke, Europhys. Lett. 41, 143 (1998).
It has recently been shown that in an analogous way it is possible to analyse dynamical systems, and to extract the Langevin equation directly from a given data set; S. Siegert, R. Friedrich, J. Peinke, Phys. Lett. A 243, 275 (1998).
N. Rajartnan, Turbulent Jets (Elsevier, Amsterdam 1976); Ch. Renner, Diplomarbeit (Bayreuth 1997).
B. Biimel and H.E. Fiedler (Berlin) priv. communication.
D. Aronson, L. Lofdahl, Phys. Fluids A5, 1433 (1993).
J. Peinke, R. Friedrich, F. Chilla, B. Chabaud, and A. Naert, Z. Phys. B 101, 157 (1996).
H. Risken, The Fokker-Planck Equation, (Springer-Verlag Berlin, 1984); P. Hänggi and H. Thomas, Physics Reports 88, 207 (1982).
J. Peinke, R. Friedrich, A. Naert, Z. Naturforsch. 52a, 588 (1997).
The Fokker-Planck equations for P even and P odd have the same form as (5), only if the coefficient δ = 0, this may clear up the question of the sense of evaluating moments of the absolute values of v i, see for example footnote on page 446 in [5].
A.M. Oboukhov, J. Fluid Mech. 13, 77 (1962); A.N. Kolmogorov, J. Fluid Mech. 13, 82 (1962).
F. Chilla et. al. to be published
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer Basel AG
About this paper
Cite this paper
Renner, C., Reisner, B., Lück, S., Peinke, J., Friedrich, R. (1999). On the statistics of small-scale turbulence and its universality. In: Gyr, A., Kinzelbach, W., Tsinober, A. (eds) Fundamental Problematic Issues in Turbulence. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8689-5_36
Download citation
DOI: https://doi.org/10.1007/978-3-0348-8689-5_36
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9730-3
Online ISBN: 978-3-0348-8689-5
eBook Packages: Springer Book Archive