c  2006 Nonlinear Phenomena in Complex Systems What Rayleigh-B´ enard, Taylor-Couette and Pipe Flows Have in Common Bruno Eckhardt * and Siegfried Grossmann † Fachbereich Physik, Philipps-Universit¨at, Renthof 6, D-35032 Marburg, GERMANY Detlef Lohse ‡ Department of Applied Physics, University of Twente, 7500 AE Enschede, THE NETHERLANDS (Received 17 March 2006) The very close correspondence between the three types of thermally or shear driven fluid flows is elucidated. Expressions for the relevant currents of temperature, angular momentum, and axial velocity by transverse convective flow in the profile direction are derived from the Navier-Stokes equations. Also the dissipation rates of the advective flows are calculated. Exact relations between the respective currents, the dissipation rates and the external control parameters are presented. PACS numbers: 47.55.P- , 47.55.pb , 47.20.Qr , 47.27.nd Keywords: pipe flow, Rayleigh-B´ enard thermal convection, Taylor-Couette flow, dissipation and transport currents, Nusselt number, torque, Skin friction 1. Introduction Although Rayleigh-B´ enard, Taylor-Couette, and pipe flow have rather different driving mech- anisms – externally controlled temperature dif- ferences, concentric cylinders rotating at differ- ent speeds, or an externally controlled pressure drop or mean flow – the also different physical quantities of interest – the heat flow, the torque, or the skin friction – have a remarkable feature in common. Instead of the expected scaling be- havior of these quantities of interest in terms of the driving forces, the corresponding scaling ex- ponents turn out to be valid only locally. In fact they depend on the driving force and change for increasing forcing. This striking correspondence prompts the idea, that the corresponding mecha- nism might be very similar. In this talk we report on this. Section 2 deals with the main ideas to calculate the Nusselt number in Rayleigh-B´ enard flow. In the next Section 3 we show that the cor- ∗ E-mail: bruno.eckhardt@physik.uni-marburg.de † E-mail: grossmann@physik.uni-marburg.de ‡ E-mail: d.lohse@utwente.nl responding ideas can be developed also for flow in the gap between independently rotating cylin- ders. In the last step, Section 4, we derive the very corresponding relations also for pipe flow. We close (Sect. 5) by indicating the answer to the question of a forcing dependent scaling: It is the varying weight of the boundary layer relative to the bulk contributions that changes the rel- evant scaling exponent with increasing external forcing! 2. Rayleigh-B´ enard convection Thermal heat convection in fluid layers heated from below, known as Rayleigh-B´ enard (RB) flow, has been intensively studied in the re- cent years, see [1–3] and others. Considerable, impressive experimental progress has been ob- tained. The standards of precision today are very high, with a scatter ≤ 0.1% and a temperature stabilization of ≈ 0.001 o C or better. The quantities of main interest are the heat flux Q and the amplitude U of the convection field. The corresponding dimensionless quanti- ties are the Nusselt number Nu = Q/ΛΔL -1 = Nonlinear Phenomena in Complex Systems, 9:2 (2006) 109 - 114 109