# Turbulence Scales and Energy Cascade

Turbulence is composed of turbulent eddies of different sizes. At high Reynolds numbers, a scale separation exists between the largest eddies and smallest eddies.

The largest eddies extract kinetic energy from the mean flow as the energy production. The length scale is comparable to the flow dimensions. These processes are highly anisotropic and mostly are not influenced by viscosity. Most transport and mixing happen in this range.

The smallest eddies have universal characters independent of the flow geometry and conditions. Those eddies in this range usually receive energy from the larger eddies and dissipate their energy into heat through the fluid’s molecular viscosity. These eddies are isotropic with length scales as described by Kolmogorov scales. It is assumed that the small scale eddies are determined by viscosity and dissipation.

The intermediate eddies are called Taylor length scale eddies with the length scales in the inertial subrange. Compared to the dissipation range, the turbulence in this region is also isotropic but the eddies in this region are independent of both the largest eddies and the smallest eddies in the dissipation range. It is assumed that the eddies in this region can be characterized by the turbulent dissipation $\epsilon$ and the wave number $\kappa$ (or eddy size l). Through dimensional analysis you obtain (1)
$E={C}_{k}{\kappa }^{-5/3}{\epsilon }^{2/3}$

where the Kolmogorov constant ${C}_{k}$ = 1.5.

If the flow is fully turbulent flow at a high Reynolds number, the energy spectra should exhibit a -5/3 decay in the inertia region. This is called Kolmogorov spectrum law or the -5/3 law.

Richardson (1922) introduced the energy cascade concept, “Big whorls have little whorls that feed on their velocity; And little whorls have lesser and so on to viscosity.”

The largest eddies with integral length scale are unstable and break up, transferring kinetic energy to smaller eddies. A similar break-up process with these smaller eddies transfer their energy to subsequent smaller eddies. These processes continue until the eddy size is reduced to the smallest size and the eddy motion is stable. This process is known as the energy cascade, as shown in Figure 1.

The image above illustrates a typical energy spectrum (E) of turbulent eddies in wave number space (κ). Since the turbulent flow has a wide range of eddies with different length scales, it is convenient to utilize the energy spectrum of the velocity field for the classification of eddies into three representative length scales, the integral length scale, the Taylor microscale and the Kolmogorov length scale.

The integral length scale is the largest eddy size of the energy spectrum. These are the most energetic and highly anisotropic eddies that produce energy via the interaction with the mean flow. Eddies in the integral range (A) are very sensitive to flow conditions and their sizes are close to the characteristic length of the flow (for example hydraulic diameter). Its spectrum is defined as $E\propto {\nu }^{2}l$ . The integral length scale is $l={k}^{3/2}/\epsilon$ and corresponding time scales of $t=k/\epsilon$ and velocity scale $v=\sqrt{k}$ .

The Taylor microscale is an intermediate eddy size of the energy spectrum between the largest eddy and the smallest eddy where the flow is inviscid. In this inertial subrange (B), the kinetic energy is transferred from large turbulent eddies to the small eddies in what is defined as the energy cascade. The occurrence of the energy cascade is related to the process of the vortex stretching. The vortex stretching causes the rotational rate of the turbulent eddies to increase and the radius of their cross sections to decrease. The energy spectrum of the inertial subrange is obtained through dimensional analysis and shows that the energy spectrum is proportional to the product of the wave number and the dissipation rate $E\propto {\kappa }^{-5/3}{\epsilon }^{2/3}$ .

The Kolmogorov scale is the smallest eddy size and is in the dissipative range (C) of the energy spectrum. Turbulent eddies in the dissipative range are isotropic because the diffusive actions of strong viscosity smear out anisotropic characteristics of the larger eddies. In this region the kinetic energy received from larger eddies is dissipated into heat. The length scale of this region is assumed be a function of the kinematic viscosity ( $\nu$ ) and the turbulent dissipation rate per unit mass ( $\epsilon$ ). Using dimensional analysis you obtain

• The Kolmogorov length scale $\eta ={\left(\frac{{\nu }^{3}}{\epsilon }\right)}^{1/4}$
• The Kolmogorov time scale ${\tau }_{\eta }={\left(\frac{\nu }{\epsilon }\right)}^{1/2}$
• The Kolmogorov velocity scale ${v}_{\eta }={\left(\nu \epsilon \right)}^{1/4}$
• The energy spectrum $E\propto {\upsilon }^{5/4}{\epsilon }^{1/4}$ .

The ratio of the integral length scale to the Kolmogorov length scale $l/\eta ~{\left(\frac{\rho vl}{\mu }\right)}^{3/4}$ = ${\text{Re}}^{3/4}$ .

Corresponding time scale and velocity scale ratios are, respectively, given as $t/\tau =\sqrt{\text{Re}}$ and .

Considering these scale ratios, you can see larger scale separation between the integral length scale and the Kolmogorov length scale with an increase of the turbulent Reynolds number. This suggests that the Kolmogorov length scale is much smaller than the integral length scale for high turbulent Reynolds number.