# The Scales Of Turbulence And The Energy Cascade – Why Do They Matter?

Before we dive deeper into turbulence modelling in CFD, it is crucial to understand the different scales of turbulence and the energy cascade. Let’s look at a brief overview of the energy cascade and the turbulent energy spectrum.

## The Scales of Turbulence

A turbulent flow consists of eddies, which are swirling regions of fluid motion characterized by vortices within the overall chaotic movement of the fluid. These eddies are of different sizes or varying length scales, which are heavily dependent on the flow dimensions of the system. The largest eddies are comparable to the characteristic length of the system and are geometry dependent. The smallest eddies are several orders or magnitude smaller and are geometry independent. For example, consider the flow over an airplane wing. The largest eddies can be similar to size or even larger than the airplane wing, ranging from several meters (for a smaller plane) to tens of meters (for a large plane). The smallest eddies in the same turbulent flow are as small as few micrometers. In between the largest and smallest scales, you can find eddies of intermediate sizes.

## Energy Transfer in Turbulent Flow: Energy Cascade

In a turbulent flow, the largest eddies extract energy from the mean flow and contain the most amount of kinetic energy. These largest eddies are highly unstable and break down into smaller eddies. As a result, energy is transferred to the smaller eddies. Subsequently, these smaller eddies transfer energy to even smaller eddies. This process continues, till the smallest eddies dissipate energy in the form of heat due to molecular viscosity of the fluid. This is commonly known as the energy cascade, where the energy is transferred from the largest scales to the smallest scales. The energy cascade was first introduced by Lewis. F. Richardson (1922), where he summarized the process with a beautiful poem:

“Big whorls have little whorls

That feed on their velocity,

And little whorls have lesser whorls

And so on to viscosity.

— Lewis F. Richardson “

Schematically, this process is shown in below.

The three major scales of turbulent energy transfer are as follows:

1. Integral length scale: This scale contains the largest eddies in the energy spectrum and lies within the energy containing range. Turbulent eddies in this range are highly anisotropic, heavily geometry dependent, and contain the most energy.
2. Taylor micro scale: This contains eddies of sizes smaller than the integral length scale, but bigger than the Kolmogorov scale, and lies within the inertial sub-range. Here, energy is transferred from the large eddies in the integral length scale to the small eddies in the dissipation range.
3. Kolmogorov scale: This contains eddies of the smallest size, beyond which, the kinetic energy of the eddies is dissipated in the form of heat due to viscosity.

The figure below illustrates the turbulent energy spectrum (E(k)), which is the sum of energies of turbulent eddies of different sizes, is plotted against wave number (k~1/eddy size). It can be observed that turbulent energy decays as the size of the eddies decreases, and energy is transferred from the integral length scale to the Kolmogorov scales, where dissipation occurs.

## Modelling Turbulence using CFD

To accurately resolve a turbulent flow using CFD, the full turbulent energy spectrum must be accurately represented to capture the correct physics of fluid flow. This is accomplished by either resolving the length and time scales all the way to Kolmogorov scale, or resolving some of the energy containing length scales, and modelling the rest of the length scales. Some of the most common methodologies are listed below.

Direct Numerical Simulation (DNS): In DNS, the time dependent Navier-Stokes equations are solved, where the spatial and temporal discretization is able to resolve the full spectrum of the turbulent energy, from integral length scales all the way to dissipation in the Kolmogorov length scale.

Large Eddy Simulation (LES): In LES, most of the energy spectrum is resolved, and only the smallest of turbulent scales are modelled. A ‘good’ LES is able to resolve about 80% of the turbulent energy spectrum.

Reynolds-Averaged Navier-Stokes (RANS): In RANS, only the energy containing scales, or the integral length scale, is resolved, and the rest of the length scales are modelled.

## Final Thoughts

In this post, we summarized how turbulence energy cascades in a turbulent flow, and the different scales of turbulence. We also briefly mentioned how different turbulence modelling techniques resolve the turbulence energy spectrum. In future blog posts, we will describe how this is done in CFD, and do a deeper dive into various turbulence modelling techniques such as DNS, LES, and RANS.

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