Nature of turbulence and energy cascade

Nature of turbulence

Turbulence is an unsteady, three-dimensional, chaotic flow in which velocity and pressure fluctuate randomly around a mean value. Its main features:

• Irregularity: random fluctuations that cannot be predicted in detail; treated statistically.
• Diffusivity: rapid mixing of momentum, heat and mass, much faster than molecular diffusion.
• Rotationality and 3D vortex stretching: it is made of eddies of many sizes.
• Dissipation: viscous action turns kinetic energy into heat, so turbulence needs a continuous energy supply.
• High Reynolds number: it appears when $Re$ exceeds a critical value.
• It is a property of the flow, not of the fluid.

Energy cascade (Richardson, Kolmogorov)

 mean flow
    | energy input
    v
 [ large eddies  L ]   (size of the geometry)
    |  vortex stretching, breakup
    v
 [ medium eddies   ]   inertial range
    |  no dissipation here
    v
 [ small eddies eta ]  viscous dissipation
    |
    v
   heat

• Large eddies, of size comparable to the flow geometry ($\ell_0$), extract energy from the mean flow.
• They are unstable and break up into smaller eddies, passing energy down the scales (the inertial range) with little loss.
• At the smallest scales, the Kolmogorov scales, viscosity dissipates the energy as heat:

$$\eta=\left(\frac{\nu^3}{\varepsilon}\right)^{1/4},\qquad \tau_\eta=\left(\frac\nu\varepsilon\right)^{1/2},\qquad u_\eta=(\nu\varepsilon)^{1/4}$$

where $\varepsilon$ is the dissipation rate per unit mass. The ratio of the largest to the smallest scale grows as $\ell_0/\eta\sim Re^{3/4}$.

Engineering implications

• Increases drag, pressure losses, wall friction and heat transfer coefficients (useful in heat exchangers, harmful in pipelines).
• Improves mixing (combustors, reactors) and delays flow separation (golf ball dimples).
• Causes noise and vibration.
• For CFD: direct simulation (DNS) needs about $Re^{9/4}$ grid points, which is impractical, so modelling (RANS, LES) is used.