Abstract
The transition to turbulence is a wide subject impossible to set out in few lectures. Here we review some selected topics and, in each case, present a small set of experiments chosen to bring into light a new facet of the problem [1]. Chapter 1 is mainly devoted to setting the general frame, introducing indispensable prerequisites about instability mechanisms, discussing briefly the roles of confinement in closed flows and advection in open flows, and outlining specific difficulties involved in case of “direct” transition to turbulence. In Chapter 2 we consider “plain convection” best illustrating the connection between chaos and turbulence. Both the instability mechanism and confinement effects are appealingly intuitive. We first examine the case of confined systems with frozen spatial structure, which makes the theory of dissipative dynamical systems relevant. Then we turn to extended systems where key-words are modulations and patterns. This presentation is further completed by a brief introduction to convection in binary mixtures (Chapter 3) and centrifugal instabilities (Chapter 4). The first topic adds the possibility of propagating waves and related new features of the nonlinear processes leading to weak turbulence. The second topic is illustrated by the case of a Couette flow between coaxial cylinders rotating at different angular speeds, which introduce the effects of shear in a seemingly simple context. At last we examine plane parallel shear flows (Chapter 5). We first discuss the instability mechanisms and introduce the basic distinction between “absolute” and “convective” instabilities dealing with the specificities of downstream advection. Then we review the phenomenology of the transition to turbulence from the early nonlinear stages to the late stages, including the dynamics of turbulent spots in flows of engineering interest. The importance of the recent advances reviewed is assessed in the conclusion.
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
For a more detailed presentation of sections 1–3, consult: Manneville, P.: Dissipative Structures and Weak Turbulence, Academic Press, 1990.
Landau, L.D.: Akad. Nauk. Doklady 44 (1944), 339;
Landau, L.D.: English translation, in: Collected Papers of L.D. Landau, (Ed. D. ter Haar ), Pergamon Press, 1965.
Ruelle, D. and Takens, F.: Comm. Math. Phys. 20 (1971), 167;
Ruelle, D. and Takens, F.: Comm. Math. Phys. 23 (1971), 344.
A general introduction to fluid flow problems is given in: Tritton, D.J.: Physical fluid dynamics, 2nd Edition, Clarendon Press, 1988.
Huerre, P.: Spatio-Temporal Instabilities in Closed and Open Flows, in: Instabilities and Nonequilibrium Structures, (Eds. E. Tirapegui and D. Villaroel ), Reidel, 1987.
Huerre, P. and Monkewitz, P.A.: Local and global instabilities in spatially developing flows, Annual Review of Fluid Mechanics 22 (1990), 473.
Coles, D.: J. Fluid Mech. 21 (1965), 285.
Carlson, D.R., Widnall, S.E., and Peeters, M.F.: J. Fluid Mech 121 (1982), 487.
Riley, J.J. and Gad-el-Hak, M.: The dynamics of turbulent spots, in [10].
Davis, S.H. and Lumley, J.L., Eds.: Frontiers in fluid dynamics, Springer, 1985.
Pomeau, Y.: Physica D 23 (1986), 3.
Swinney, H.L., Gollub, J.P., Eds.: Hydrodynamic instabilities and the transition to turbulence, Topics in Applied Physics vol.45, Springer, 1985.
Platten, J.K. and Legros, J.C.: Convection in liquids, Springer, 1984
Drazin, P.G. and Reid W.H.: Hydrodynamic stability, Cambridge, 1981.
Krishnamurti, R.: J. Fluid Mech. 60 (1973), 285.
Libchaber, A. and Maurer, J.: J. Physique Coll. 41-C3 (1980), 51.
Dubois, M. and Bergé, P.: J. Physique 42 (1981), 167.
Bergé, P., Dubois, M., Manneville, P., and Pomeau, Y.: J. Physique Lettres 40 (1979), L-505.
Gollub, J.P. and Benson, S.V.: J. Fluid Mech. 100 (1980), 449.
Walden, R.W., Kolodner, P., Passner, A., and Surko, C.M.: Phys. Rev. Lett. 53 (1984), 242.
Grebogi, C., Ott, E., and Yorke, J.A.: Physica D 15 (1985), 354.
Gershenfeld, N.: An experimentalist’s approach to the observation of dynamical systems, in: Directions in Chaos, vol. II, (Ed. Hao Bailin ), World Scientific, 1988.
Takens, F.: Detecting strange attractors in turbulence, in: Dynamical systems and turbulence, Warwick, 1980, (Eds. D.A. Rand and L.S. Young), Lect. Notes in Mathematics Vol. 898, p. 366, Springer, 1981.
Packard, N.H., Crutchfield, J.P., Farmer, J.D., and Shaw, R.S.: Phys. Rev. Lett. 45 (1980), 712.
Sano, M. and Sawada, Y., in [26].
Tatsumi, T., Ed.: Turbulence and Chaotic Phenomena in Fluids, Elsevier, 1984.
Grassberger, P. and Procaccia, I.: Phys. Rev. Lett. 50 (1983), 346;
Grassberger, P. and Procaccia, I.: Physica D 9 (1983), 189.
Malraison, B., Atten, P., Bergé, P., and Dubois, M.: C.R. Acad. Sc. Paris, Série II, 297 (1983), 209.
Eckmann, J.P. and Ruelle, D.: Rev. Mod. Phys. 57 (1985), 617.
Wolf, A., Swift, J.B., Swinney, H.L., and Vastano, J.A.: Physica D 16 (1985), 285.
Sano, M. and Sawada, Y.: Phys. Rev. Lett. 55 (1985), 1082.
Conte R. and Dubois M.: Lyapunov exponents of experimental systems, in Nonlinear evolutions, (Ed. J.P. Leon ), World Scientific, 1988.
Broomhead, D.S. and King, G.P.: Physica D 20 (1986), 217.
Newell, A.C. and Whitehead, J.A.: J. Fluid Mech. 38 (1969), 279;
Segel, L.A.: J. Fluid Mech. 38 (1969), 203.
Cross, M.C.: Phys. Rev. A 25 (1982), 1065.
Cross, M.C. and Newell, A.C.: Physica D 10 (1984), 299.
Heutmaker, M.S., Fraenkel, P.N., and Gollub, J.P.: Phys. Rev. Lett. 54 (1985), 1369.
Ahlers, G. and Behringer, R.P.: Phys. Rev. Lett. 40 (1978), 12.
Libchaber, A. and Maurer, J.: J. Physique-Lettres 39 (1978), L-369.
Pocheau, A., Croquette, V., and Le Gal, P.: Phys. Rev. Lett. 55 (1985), 1094.
Siggia, E.D. and Zippelius, A.: Phys. Rev. Lett. 47 (1981), 835.
Pomeau, Y. and Manneville, P.: J. Physique-Lettres 40 (1980), L-609.
Eckhaus, V.: Studies in nonlinear stability theory, Springer Tracts in natural Philosophy vol.6, Springer, 1965),.
Cross, M.C.: Phys. Rev. A 27 (1983), 490;
Manneville, P. and Piquemal, J.M.: Phys. Rev. A 28 (1983), 1774.
Newell, A.C.: The Dynamics of Patterns, a Survey, in [46].
Wesfreid, J.E. et al, Eds.: Propagation in Systems far from Equilibrium, Springer Series in Synergetics, Vol. 41, Springer, 1988.
Kuramoto, Y.: Prog. Theor. Phys. 71 (1984), 1182.
Brand, H.R.: Phase Dynamics, a Review and a Perspective, in [46].
Ohta, T.: Prog. Theor. Phys. 73 (1985), 1377.
Pocheau, A.: J. Physique 50 (1989), 2050.
Newell, A.C., Passot, T., and Souli, M., Phase-mean drift equations for convection in large aspect ratio containers, in: Nonlinear evolution of statio-temporal structures in dissipative continuous systems, (Eds. F.H. Busse and L. Kramer ), NATO ASI Series, Series B: Physics, Plenum press, 1990.
Bergé, P.: Nucl. Phys. B (Proc.Suppl.), 2 (1987), 247;
Ciliberto, S. and Bigazzi, P.: Phys. Rev. Lett. 60 (1988), 286.
Daviaud, F., Dubois, M., and Berge, P.: Europhys. Lett. 9 (1989), 441.
Kaneko, K.: Prog. Theor. Phys. 74 (1985), 1033.
Kinzel, W.: Directed Percolation, in: Percolation Structures and Processes, (Eds. G. Deutscher et al.) Annals of the Israel Phys Soc., vol. 5, 1983;
Kinzel, W.: Z. Phys. B (Condensed Matter) 58 (1985), 229.
Chaté, H. and Manneville, P.: Transition to turbulence via spatio-temporal intermittency, modeling and critical properties, in [57].
Coullet, P. and Huerre, P., Eds.: New Trends in Nonlinear Dynamics and Pattern Forming Phenomena: the Geometry of Nonequilibrium, Plenum Press, to appear.
Moses, E. and Steinberg, V., Phys. Rev. Lett. 57 (1986), 2018.
Surko, C.M., Kolodner, P., Passner, A., and Walden, R.W.: Physica D 23 (1986), 220;
Kolodner, P., Passner, A., Williams, H.L., and Surko, C.M.: Nuclear Physics B (Proc. Suppl.), 2 (1987), 97.
Bensimon, D., Pumir, A., Shraiman, B.I.: J. Physique 50 (1989), 3089.
Coullet, P., Gil, L., and Legs, J.: Transitions in Systems far from equilibrium, in: Chaos and complexity, (Eds. R. Livi et al.) World Scientific, 1988.
Kolodner, P., Surko, C.M., and Williams, H.L.: Physica. D 37 (1989), 319.
Steinberg, V., Feinberg, J., Moses, E., and Rehberg, I.: Physica D 37 (1989), 359;
Steinberg, V., Moses, E., and Fineberg, J.: Nuclear Physics B (Proc. Suppl.), 2 (1987), 109.
Cross, M.C.: Phys. Rev. A 38 (1988), 3593.
Coullet, P., Fauve, S., and Tira.pegi, E.: J. Physique Lett. 46 (1985), L-787.
Mullin, T.: J. Fluid Mech. 121 (1982), 207.
Fenstermacher, P.R., Swinney, H.L., and Gollub, J.P.: J. Fluid Mech. 94 (1979), 103.
Newhouse, S., Ruelle, D., and Takens, F.: Commun. Math. Phys. 64 (1978), 35.
Brandstäter, A., Swift, J., Swinney, H.L., Wolf, A., Farmer, J.D., Jen, E., and Crutchfield, P.J.: Phys. Rev. Lett. 51 (1983), 1442.
Andereck, C.D., Liu, S.S., and Swinney, H.L.: J. Fluid Mech. 164 (1986), 155.
see, e.g., Chossat, P., and Iooss, G.: Japan J. of Appl. Math. 2 (1985), 37. A book entitled: The Taylor-Couette problem is in preparation by the same authors.
Hegseth, J.J., Andereck, C.D., Hayot, F., and Pomeau, Y.: Phys. Rev. Lett. 62 (1989), 257.
Bayly, B.J., Orszag, S.A., and Herbert, T.: Instability mechanisms in shear flow transition, Annual Review of Fluid Mechanics 20 (1988), 359.
Chomaz, J.M., Huerre, P., and Redekopp, L.G.: Phys. Rev. Lett. 60 (1988), 25.
Ho, C.-H. and Huerre, P.: Perturbed free shear layers, Annual Review of Fluid Mechanics 16 (1984), 365.
Oertel Jr., H.: Wakes behind blunt bodies, Annual Review of Fluid Mechanics 22 (1990), 539.
Sreenivasan, K.R.: Transition and turbulence in fluid flows and low dimensional chaos, in [10].
Van Atta, C.W. and Gharib, M.: J.Fluid Mech. 174 (1987), 113.
Bonetti, M., Meynart, R., Boon, J.P., and Olivari, D.: Phys. Rev. Lett. 55 (1985), 492.
Orszag, S.A. and Patera, A.T.: J. Fluid Mech. 128 (1983), 347.
Schlichting, H.: Boundary Layer Theory, 7th edition, McGraw-Hill, 1979.
Herbert, T.: Secondary instability of boundary layers, Annual Review of Fluid Mechanics 20 (1988), 487.
Saric, W.S. and Thomas, A.S.W.: Experiments on the subharmonic route to turbulence in boundary layers, in [26].
Aubry, N., Holmes, P., Lumley, J.L., and Stone, E.: J. Fluid ech. 192 (1988), 115.
Widnall, S.E.: Growth of turbulent spots in plane Poiseuille flow, in [26].
Itoh, N.: Landau coefficient of the Blasius boundary-layer flow, in [26].
Ottino, J.M.: Mixing, chaotic advection, and turbulence, Annual Review of Fluid Mechanics 22 (1990), 207.
see, e.g., Gollub, J.P. and Solomon, T.H.: Complex particle trajectories and transport in stationary and periodic convective flows, in Chaos and related nonlinear phenomena: where do we go from here? (Ed. I. Procaccia) Plenum Press, 1987.
Van Dyke, M.: An album of fluid motion, The Parabolic Press, 1982.
Tennekes H. and Lumley, J.L.: A first course in turbulence, MIT-Press, 1972.
Wesfreid, J.E. and Zaleski, S., Eds.: Cellular structures in instabilities, Lect. Notes Phys. vol. 210 (Springer, 1984),.
Newell, A.C.: Chaos and turbulence: is there a connection? in: Perspective in nonlinear dynamics, (Eds. M.F. Shlesinger et al.), World Scientific, 1986.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1991 Springer-Verlag Wien
About this chapter
Cite this chapter
Manneville, P. (1991). From Chaos to Turbulence in Fluid Dynamics. In: Szemplinska-Stupnicka, W., Troger, H. (eds) Engineering Applications of Dynamics of Chaos. International Centre for Mechanical Sciences, vol 319. Springer, Vienna. https://doi.org/10.1007/978-3-7091-2610-3_2
Download citation
DOI: https://doi.org/10.1007/978-3-7091-2610-3_2
Publisher Name: Springer, Vienna
Print ISBN: 978-3-211-82328-6
Online ISBN: 978-3-7091-2610-3
eBook Packages: Springer Book Archive