Stationary data lend themselves well to the Fourier decomposition into harmonic components. Conversely, spectral characteristics of non-stationary data vary with time, and hence do not generally admit the application of Fourier transform. In order to investigate the localized time-frequency characteristics of non-stationary data, the notions of instantaneous frequency and amplitude are invoked. These concepts are applied to the von Ka´rma´n vortex shedding observed in the wake of a self-sustained pitching airfoil. For this range of Reynolds numbers (104 – 105), it has been reported that at any given airspeed the shedding frequency of the vortex street varies with angle of attack (AOA), ranging from the Strouhal number St ≈ 0.6 at zero AOA and tending to St ≈ 0.1 for high AOA. For the pitching motion, which originates from a positive energy transfer from the flow to the airfoil due to negative aerodynamic damping, the von Ka´rma´n vortex shedding frequency varies with pitch angle hence with time. Hilbert transform provides a robust estimate of instantaneous frequency through the definition of analytic signals. However, Hilbert transform provides meaningful instantaneous frequency for only monocomponent signals. To overcome this difficulty, the Hilbert-Huang transform is commonly exploited. In this paper, both the Hilbert and Hilbert-Huang transforms are applied in order to capture the instantaneous vortex shedding frequency. For multicomponent signals Empirical Mode Decomposition (EMD) splits the signal to monocomponent signals, namely Intrinsic Mode Functions, through a so-called sifting process. Application of Hilbert transform to these functions produces instantaneous frequencies and amplitudes. Therefore the time-frequency-amplitude representation of the signal appears to be a promising tool for obtaining more physical insight into the time-varying vortex shedding frequency in the wake of a pitching airfoil.
The bilinear Hilbert transform in UMD spaces
