Scale-invariance in three-dimensional isotropic turbulence
Seminar Room 1, Newton Institute
We present a critical review of the Kolmogorov (1941) theory of isotropic turbulence, with particular reference to the `2/3' power-law for the second-order structure function (and the corresponding `-5/3' law for the energy spectrum). We begin by noting that the recent resolution of an associated paradox allows the inertial range to be defined in terms of the scale-invariance of the energy flux (David McComb, J. Phys. A: Math. Theor. 41, 075501 (2008)), thus permitting the Kolmogorov arguments to be presented independently of concepts such as localness which are themselves counter-intuitive when interpreted in terms of vortex-stretching. If this approach is pursued further, then a simple phenomenological analysis shows that we can regard turbulence as a statistical field theory possessing one nontrivial fixed point (corresponding to the top of the inertial range) and two trivial fixed points at the origin and infinity, respectively, in wavenumber space. Using this framework, various schools of criticism, ranging from the original criticism by Landau (1959), through `intermittency corrections' to present-day analogies with the theory of critical phenomena, with the introduction of `anomalous exponents', are analysed. In particular, we examine the conflict between the recent work of Lundgren (2002), which uses mathematical arguments to show that the `2/3' law must be asymptotically true in the limit of infinite Reynolds numbers; and that of Falcovich which uses mathematical arguments to show that the `2/3' law is incompatible with the observed values for higher-order moments. We conclude by attempting to put forward a picture in which various long-standing contentious issues may be seen as either resolved or at least potentially resolvable.