Transfer of energy from large to small scales in turbulent flows is described as a flux of energy from small wave numbers to large wave numbers in the spectral representation of the Navier–Stokes equation (1.17). The problem of resolving the relevant scales in the flow corresponds in the spectral representation to determining the spectral truncation at large wave numbers. The effective number of degrees of freedom in the flow depends on the Reynolds number. The Kolmogorov scale η depends on Reynolds number as η ∼ Re−¾ (1.11), so the number of waves N necessary to resolve scales larger than η grows with Re as N ∼ η−3 ∼ Re9/4. This means that even for moderate Reynolds numbers ∼ 1000, the effective number of degrees of freedom is of the order of 107. A numerical simulation of the Navier–Stokes equation for high Reynolds numbers is therefore impractical without some sort of reduction of the number of degrees of freedom. Such a calculation with a reduced set of waves was first carried out by Lorenz (1972) in the case of the vorticity equation for 2D turbulence.

The idea is to divide the spectral space into concentric spheres, see Figure 3.1. The spheres may be given exponentially growing radii kn = λn, where λ > 1 is a constant. The set of wave numbers contained in the nth sphere not contained in the (n − 1)th sphere is called the nth shell.

Fluids have always fascinated scientists and their study goes back at least to the ancient Greeks. Archimedes gave in “On Floating Bodies” (c. 250 BC) a surprisingly accurate account of basic hydrostatics. In the fifteenth century, Leonardo da Vinci was an excellent observer and recorder of natural fluid flows, while Isaac Newton experimented with viscosity of different fluids reported in Principia Mathematica (1687); it was his mechanics that formed the basis for describing fluid flow. Daniel Bernoulli established his principle (of energy conservation) in a laminar inviscid flow in Hydrodynamica (1738). The mathematics of the governing equations was treated in the late eighteenth century by Euler, Lagrange, Laplace, and other mathematicians. By including viscosity the governing equations were put in their final form by Claude-Louis Navier (1822) and George Gabriel Stokes (1842) in the Navier–Stokes equation. This has been the basis for a vast body of research since then.

The engineering aspects range from understanding drag and lift in connection with design of airplanes, turbines, ships and so on to all kinds of fluid transports and pipeflows. In weather and climate predictions accurate numerical solutions of the governing equations are important. In all specific cases when the Reynolds number is high, turbulence develops and the kinetic energy is transferred to whirls and waves on smaller and smaller scales until eventually it is dissipated by viscosity. This is the energy cascade in turbulence.

Fully developed turbulence is the notion of the general or universal behavior in any physical situation of a violent fluid flow, be it a dust devil or a cyclone in the atmosphere, the water flow in a white-water river, the rapid mixing of the cream and the coffee when stirring in a coffee cup, or perhaps even the flow in gigantic interstellar gas clouds. It is generally believed that the developments of these different phenomena are describable through the Navier–Stokes equation with suitable initial or boundary conditions. The governing equation has been known for almost two centuries, and a lot of progress has been achieved within practical engineering in fields like aerodynamics, hydrology, and weather forecasting with the ability to perform extensive numerical calculations on computers. However, there are still fundamental questions concerning the nature of fully developed turbulence which have not been answered. This is perhaps the biggest challenge in classical physics. The literature on the subject is vast and very few people, if any, have a full overview of the subject. In the updated version of Monin and Yaglom's classic book the bibliography alone covers more than 60 pages (Monin & Yaglom, 1981).

The phenomenology of turbulence was described by Richardson (1922) and quantified in a scaling theory by Kolmogorov (1941b). This description stands today, and has been shown to be basically correct by numerous experiments and observations.

Chaos can be observed in simple nonlinear Hamiltonian systems. This is a dynamical system governed by Hamilton's equations where the energy is conserved, such as a physical pendulum or double pendulum. The phase space portrait of the trajectories of this kind of system can, even with few degrees of freedom, be very complicated. The phase space flow fulfils Liouville's theorem, which states that phase space volume is conserved. Another type of chaotic dynamics can arise in non-autonomous systems, like the Duffing oscillator, where a simple nonlinear system is influenced by an external periodic force. A third kind of chaotic system is nonlinear dissipative systems, such as the Lorenz (1963) model, which has only three degrees of freedom. The Lorenz model was derived from the set of ordinary differential equations describing development of wave amplitudes in the spectral representation of Rayleigh–Bernard convective flow. The Lorenz model is equivalent to a spectral truncation where only the first three wave numbers are represented. The phase space portrait of dissipative systems is different from that of Hamiltonian systems because the energy dissipation implies a shrinking of phase space volume. The dynamics of such a system is described in phase space by strange attractors. Strange attractors are sets in phase space of states un which are invariant with respect to the dynamical equation. This means that an initial state un(0) belonging to the attractor will develop along a trajectory which will stay within the attractor.

Intermittency in turbulence is a topic which has been actively investigated for several decades, and a major part of Frisch's book (1995) is devoted to the subject.

Dynamical systems are often characterized by long quiescent periods interrupted by bursts of activity. This kind of dynamics is called intermittent. A way of quantifying this could be by high pass filtering the dynamical signal. If the signal has purely Gaussian statistics, which would be natural for a system of many degrees of freedom, high pass filtering is a linear operation and the high pass filtered signal would be Gaussian as well. If the high pass signal differs from the Gaussian by having heavier tails, it is intermittent. Thus intermittency could be formally defined by the deviation from Gaussian statistics. In this case, intermittency could be a sign of dynamics not merely governed by simple statistics given by the central limit theorem or equilibrium statistical mechanics. In the case of a turbulent velocity field, high pass filtering roughly corresponds to extracting information on velocity differences below or at some cutoff length scale. As described previously, the statistics of velocity increments in turbulence is found not to be Gaussian. The self-similarity of the flow assumed in K41 theory is not valid and there will be corrections to the scaling exponents for the moments of the velocity increments as expressed in (1.64).

Kolmogorov's lognormal correction

The dynamical origin of the deviation from the K41 value for the scaling exponents is a very non-uniform distribution of the energy dissipation.