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.