Fundamentals of Turbulent Flows

The spectral description of turbulence allows us to decompose velocity and pressure fields in terms of wavenumbers and frequencies, or length and time scales. We discuss the notion of scale decomposition and introduce several properties of the Fourier transform between physical (spatial/temporal) space and scale (spectral) space in various dimensions, including complex conjugate relations for real functions and Parseval’s theorem. The Fourier transform allows us to develop useful relations between correlations and energy spectra, which are used extensively in the statistical theory of turbulence. The one-dimensional and three-dimensional energy spectra are specifically discussed in conjunction with Taylor’s hypothesis to enable spectra computation from single-point time-resolved measurements. The discrete version of the transform, or the discrete Fourier series, is then introduced, as it is typically encountered in numerical simulations and postprocessing of discrete experimental data. Treatment of periodic data is first considered, followed by nonperiodic data with the help of windowing. The procedure for the computation of various discrete spectra is outlined.