Turbulence spectra

In a turbulent flow, kinetic energy \(E\) is usually transfered from largest eddies as they break down into smaller vortices until they reach a critical length to which the viscous dissipation converts this energy into heat. The most common way to visualize this phenomenon is in Fourier space using wave-numbers \(k\) to quantify spatial scales, as illustrated below:

The intermediate range between both largest and smallest scales (inertial range) is statiscally isotropic. Hence the net energy coming from the energy-containing eddies is in equilibrium with the net energy cascading to smaller eddies, where it is dissipated and the slope of energy spectrum remains constant.

In LES simulations, the energy spectra plot indicates up to which frequency the flow is being solved. LES models solve explicitly only the larger scales from turbulence spectrum, with small ones being modeled. A comparison between LES and Direct Numerical Solutions (DNS) is illustrated below:

Turbulence spectra are commonly represented using the Power Spectral Density (PSD) of a probe point. The PSD \(S_{xx}\left(f\right)\) of a discrete time-dependent signal dataset \(x_{n}\left(t_{n}\right)\) is its squared Fourier transform normalized by the frequency resolution, hence:

(1)\[S_{xx}\left(f\right) = \frac{1}{N}\left|\sum_{n=0}^{N-1}x_{n}\left(t_{n}\right)e^{-i2\pi f t_{n}}\right|^{2}\]

where \(f\) is the frequency, \(t_{0}=0\), and \(N\) the number of elements of the dataset. The resulting curve can be used to determine the quality of LES modeling since it shows the cutoff frequency of the CFD solution. The signals used for representation of PSD are usually the velocity and the pressure.

Note

When using the scipy library from python, the PSD can be calculated through the function scipy.signal.periodogram(signal, frequency, scaling = "density").

The use of smaller datasets will result in noisy curves. For such curves, a Gaussian filter can be applied to reduce the noise, as illustrated below: