Small Scales in Turbulence
In Chapter 8, we characterized the large scales of turbulence by introducing the turbulent velocity urms and the integral scale ℓt. These two quantities are used to quantify two key processes of turbulence, the turbulent diffusion coefficient κt ≈ urmsℓt (equation [8.29] of Chapter 8) and the kinetic energy dissipation rate in a turbulent flow:
(equation [8.14] of Chapter 8). Again, the non-inclusion of viscosity in this relation is particularly noteworthy. This property, together with equation [11.1], constitutes the cornerstone of turbulence theories and of the concepts presented in this chapter.
The integral scale ℓt characterizes the size of large-scale motion observed in a turbulent flow. On a weather map, these are observed as cyclonic and anti-cyclonic eddies. Although it is the most visible, large-scale motion does not describe the entirety of turbulence-induced phenomena. Kolmogorov’s theory, presented in this chapter, allows the determination of the Kolmogorov scale ℓK, which characterizes the size of the smallest eddies present in a turbulent flow.1 This scale is used to evaluate the characteristic time of diffusion processes and apply the concepts of macromixing and micromixing introduced in Chapter 10 for a developed turbulence flow.
11.1. Notion of signal processing, expansion of a time signal into Fourier series
Some notions ...