A tour to classical turbulence: scaling arguments, cascade and Kolmogorov spectrum

The theory of classical turbulence starts from the Navier–Stokes equation (1.87) for an incompressible fluid. But it is a long way from this starting point to a full picture of turbulent flows. The linearised Navier–Stokes equation can be solved more or less straightforwardly for various geometries of laminar flows. If the velocity of the flow grows, the non-linear inertial term (v · ∇)v in the Navier–Stokes equation becomes relevant and eventually leads to instability of the laminar flow. The instability threshold is controlled by the Reynolds number (1.88). After the instability threshold is reached, the flow becomes strongly inhomogeneous in space and time in a chaotic manner. This is despite the fully deterministic character of the Navier–Stokes equation. The emergence of chaos from the deterministic description is a fundamental problem in physics and mathematics, but we mostly skip this transient process of turbulence evolution except for a few comments in Section 14.8. We are interested in a discussion of developed turbulence, which arises at rather high Reynolds numbers of the order of a few thousands or more. Usually it is assumed that in the state of developed turbulence the fluid is infinite and homogeneous in space and time, but only on average. At scales smaller than the size L of the fluid there are intensive temporal and spatial fluctuations of the velocity field. The natural description of such a chaotic field is in terms of probability distributions and random correlation functions. The velocity ⟨v⟩ averaged over a scale comparable with the fluid size L is not so important since it can be removed by the Galilean transformation. More important are fluctuations of velocity v = v − ⟨v⟩. Further we omit the ‘prime’, assuming that ⟨v⟩ = 0. The amplitude of velocity fluctuations depends on the scale at which it is considered. Following Landau and Lifshitz (1987, Chapter III) let us introduce the scale-dependent Reynolds number Rel = lv(l)/ν for the velocity fluctuation v(l) at the scale l. The scale-dependent Reynolds number characterises the effectiveness of the viscosity, which becomes important at small scales with Rel ∼ 1.

Thermodynamics of a one-component perfect fluid

In the strict sense of the word, hydrodynamics describes the dynamical behaviour of a fluid. But sometimes the hydrodynamical approach refers to phenomenological theories dealing with various types of condensed media, such as solids, liquid crystals, superconductors, magnetically ordered systems and so on. Two important and interconnected features characterise the hydrodynamical description.

1. • It refers to spatial and temporal scales much longer than any relevant microscopical scale of the medium under consideration.

2. • It does not need the microscopical theory for derivation of dynamical equations but uses as a starting point a set of conservation laws and thermodynamical and symmetry properties of the medium under consideration.

The latter feature gives us the possibility to study condensed matter without waiting for the moment when a closed self-consistent microscopical theory is developed. Sometimes it can be a long time to wait for such a moment. For example, one may recall the microscopical theory of fluid with strong interactions, or as the latest example the microscopical theory of high-Tc superconductivity. In fact, the cases when the hydrodynamical description can be derived rigorously from the ‘first-principle’ theory are more the exceptions rather than the rule. Such exceptions include, for example, weakly non-ideal gases and weak-coupling superconductors. Even if it is possible to derive the hydrodynamical description from the microscopical theory, the former as based on the most global properties (conservation laws and symmetry) is a reliable check of the microscopical theory. If hydrodynamics does not follow from a microscopical theory this is an alarming signal of potential problems with the microscopical theory.

Impressive evidence of the fruitfulness of the hydrodynamical (phenomenological) approach to condensed matter physics is provided by the volumes of Landau and Lifshitz's course addressing continuous media: Electrodynamics of Continuous Media, Theory of Elasticity, and Fluid Mechanics (Landau and Lifshitz, 1984, 1986, 1987). The hydrodynamical approach was very fruitful also for studying properties of rotating superfluids, as will be demonstrated in this book. The hydrodynamical description always deals with the continuous medium even if the medium under consideration is a lattice (an atomic lattice in elasticity theory, for example). Indeed, the lattice constant is a microscopical scale which should be ignored in accordance with the nature of the hydrodynamical description.