Notes on Computational Fluid Dynamics: General Principles

6.15 Summary of turbulence

Turbulence is a spatially complex distribution of vorticity which advects itself in a chaotic manner, Sec. 6.2 .
Vorticity is mostly generated at solid boundaries and evolves by advection, diﬀusion and stretching/compression, Sec. 6.3 .
Boundary layers are thus the main source of vorticity for turbulence, Sec. 6.4 .
Turbulence occurs when instabilities cause the vorticity to become chaotic, characterised by Reynolds Number, Sec. 6.1 .

Length and time scales of turbulent eddies initially relate to the characteristic scales of the generated vortices, e.g. by shedding, Sec. 6.5 .
The eddies become progressively smaller until they reach a size where the dissipation of their kinetic energy is signiﬁcant, Sec. 6.6 .
The process of eddy breakup involves a cascade of kinetic energy from larger to smaller eddies in which eddy structures become increasingly isotropic, Sec. 6.7 .

The Kolmogorov microscales represent the smallest scales of turbulence at which eddies die out by viscous dissipation.
They are surprisingly small, e.g. a factor of $\text{[math]}$ separates the length scales of the largest and smallest eddies.
Usually, it is prohibitively costly to capture the smallest scales in a CFD simulation, Sec. 6.8 .
Large-eddy simulation captures the large eddies, while using modelling for the unresolved scales.
Reynolds-averaged simulations provide faster solutions by solving equations for averaged ﬁeld variables, Sec. 6.9 .
The averaged momentum equation includes the Reynolds stress which requires modelling.

Momentum exchange at the molecular scale is characterised by viscosity and kinetic theory provides good quantitative predictions of viscosity for gases, Sec. 6.10 .
Comparing the random motion of eddies in a turbulent ﬂuid with particles at a molecular scale, Boussinesq introduced the concept of eddy viscosity, Sec. 6.11 .
Eddy viscosity is not a ﬂuid property but is proportional to a characteristic speed and length scale of the turbulence; models are required which represent each of these scales.
The speed scale is represented by turbulent kinetic energy $\text{[math]}$ , described by a transport equation Eq. (6.28 ), Sec. 6.13 .
The $\text{[math]}$ -equation includes a term for its rate of dissipation $\text{[math]}$ ; a transport equation for $\text{[math]}$ , e.g. Eq. (6.32 ), provides a model for that term — which also represents the length scale of turbulence, Sec. 6.14 .
The concept of mixing length to characterise the length scale of turbulence provides additional modelling, particularly for the near-wall region, Sec. 6.12 .

Notes on CFD: General Principles - 6.15 Summary of turbulence