Abstract
It seems a safe bet that the understanding of developed turbulence, a long standing challenge for theoretical and mathematical physics, will enter into the third millennium as an unsolved problem. This is an introductory course to the subject. We discuss
-
in Lecture 1:
the Navier Stokes equations, existence of solutions, statistical description, energy balance and cascade picture;
-
in Lecture 2:
the Kolmogorov theory of three-dimensional turbulence versus intermittency, the Kraichnan-Batchelor theory of two-dimensional turbulence;
-
in Lecture 3:
the Richardson dispersion law and the breakdown of the Lagrangian flow;
-
in Lecture 4:
direct and inverse cascades and intermittency in the Kraichnan model of passive advection.
This is a preview of subscription content, log in via an institution to check access.
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
C.L.M.H. Navier, Mémoire sur le lois du mouvement des fluides. Mém. Acad. Roy. Sci. 6 (1823), 389–440.
G.G. Stokes, On some cases of fluid motion. Trans. Camb. Phil. Soc. 8 (1843), 105.
J.M. Burgers, The Nonlinear Diffusion Equation. D. Reidel, Dordrecht, 1974.
G. Gallavotti, Some rigorous results about 3D Navier-Stokes, Lecture Notes, “Turbulence in spatially ordered systems,” Les Houches 1992, mp_arc/92–109.
R. Temam, Navier-Stokes Equations. North-Holland, Amsterdam, 1984.
D. Ruelle, The turbulent fluid as a dynamical system, in New Perspectives in Turbulence, ed. L. Sirovich, Springer, Berlin 1991, 123–138.
T. Bohr, M.H. Jensen, G. Paladin, and A. Vulpiani, Dynamical System Approach to Turbulence. Cambridge University Press, Cambridge 1998.
M.J. Vishik and A.V. Fursikov, Mathematical Problems of Statistical Hydrodynamics. Kluwer, Dordrecht 1988.
G.K. Batchelor, Introduction to Fluid Dynamics. Cambridge Univ. Press, Cambridge 1967.
A.S. Monin and A.M. Yaglom, Statistical Fluid Mechanics I II. MIT Press, Cambridge MA 1971 & 1975.
L.F. Richardson, Weather Prediction by Numerical Process. Cambridge Univ. Press, Cambridge 1922
T. von Karman and L. Horwarth, On the statistical theory of isotropic turbulence, Proc. Roy. Soc. London A164 (1938), 192–215.
A. N. Kolmogorov, The local structure of turbulence in incompressible viscous fluid for very large Reynolds’ numbers, C.R. Acad. Sci. URSS 30 (1941), 301–305.
U. Frisch, Turbulence: the Legacy of A.N. Kolmogorov, Cambridge Univ. Press, Cambridge, 1995.
J. Duchon and R. Robert, Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokes equations, Nonlinear- ity 13 (2000), 249–255.
R. Benzi, S. Ciliberto, C. Baudet, and G. Ruiz Chavaria, On the scaling of three dimensional homogeneous and isotropic turbulence. Physica D80 (1995), 385–398.
R.H. Kraichnan, Inertialranges in two-dimensional turbulence, Phys. Fluids 10 (1967), 1417–1423.
G.K. Batchelor, Computation of the energy spectrum in homogeneous two-dimensional turbulence, Phys. Fluids Suppl. II 12 (1969), 233–239.
D. Bernard, On the three point velocity correlation functions in 2d forced turbulence, chao-dyn/9902010.
J. Paret and P. Tabeling, Intermittency in the 2D inverse cascade of energy: experimental observations, Phys. Fluids 10 (1998), 3126–3136.
J. Paret, M.C. Jullien, and P. Tabeling, Vorticity statistics in the two-dimensional enstrophy cascade, Phys. Rev. Lett. 83 (1999), 3418–3421, physics/9904044.
G. Boffetta, A. Celani, and M. Vergassola, Inverse cascade in two-dimensional turbulence: deviations from Gaussianity, chao-dyn/9906016.
L.F. Richardson, Atmospheric diffusion shown on a distance-neighbour graph, Proc. Roy. Soc. London A110 (1926), 709–737.
Y. Brenier, A homogenized model for vortex sheet, Arch. Rational Mech. Anal. 138 (1997), 319–363.
A. Shnirelman, Weak solutions with decreasing energy of incompressible Euler equations, Commun. Math. Phys. 210 (2000), 541–603.
R.H. Kraichnan, Small-scale structure of a scalar field connected by turbulence, Phys. Fluids 11 (1968), 945–963.
Y. Le Jan and O. Raimond, Solution statistiques fortes des équations différentielles stochastiques, C.R. Acad. Sci. 327 (1998), 893–89.
K. Gawçdzki and M. Vergassola, Phase transition in the passive scalar advection, Physica D138 (2000), 63–90, cond-mat/9811399.
A.M. Obukhov, Structure of the temperature field in a turbulent flow, Izv. Akad. Nauk SSSR, Geogr. Geofiz. 13 (1949), 58–69.
S. Corrsin, On the spectrum of isotropic temperature fluctuations in an isotropic turbulence, J. Appl. Phys. 22 (1951), 469–473.
R.H. Kraichnan, Anomalous scaling of a randomly advected passive scalar, Phys. Rev. Lett. 72 (1994), 1016–1019.
B. Shraiman and E. Siggia, Anomalous scaling of a passive scalar in turbulent flow, C.R. Acad. Sci. 321 (1995), 279–284.
K. Gawedzki and A. Kupiainen, Anomalous scaling of the passive scalar, Phys. Rev. Lett. 75 (1995), 3834–3837.
M. Chertkov, G. Falkovich, I. Kolokolov and V. Lebedev, Normal and anomalous scaling of the fourth-order correlation function of a randomly advected scalar, Phys. Rev. E52 (1995), 4924–4941.
U. Frisch, A. Mazzino, and M. Vergassola, Intermittency in passive scalar advection., Phys. Rev. Lett. 80 (1998), 5532–5535.
M. Chertkov, I. Kolokolov and M. Vergassola, Inverse versus direct cascades in turbulent advection, Phys. Rev. Lett. 80 (1998), 512–515.
K. Gawȩdzki, Intermittency of passive advection, in “Advances in Turbulence VII,” ed. U. Frisch, Kluwer Acad. Publ. 1998, 493–502.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer Science+Business Media New York
About this chapter
Cite this chapter
Gawȩdzki, K. (2002). Easy Turbulence. In: Saint-Aubin, Y., Vinet, L. (eds) Theoretical Physics at the End of the Twentieth Century. CRM Series in Mathematical Physics. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-3671-7_3
Download citation
DOI: https://doi.org/10.1007/978-1-4757-3671-7_3
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-2948-8
Online ISBN: 978-1-4757-3671-7
eBook Packages: Springer Book Archive