Fluid Mechanics for Chemical Engineering

Chapter 11

Small Scales in Turbulence

In Chapter 8, we characterized the large scales of turbulence by introducing the turbulent velocity u_rms and the integral scale ℓ_t. These two quantities are used to quantify two key processes of turbulence, the turbulent diffusion coefficient κ_t ≈ u_rmsℓ_t (equation [8.29] of Chapter 8) and the kinetic energy dissipation rate in a turbulent flow:

[11.1]

(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 ℓ_t 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 ℓ_K, 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 ...