Application of Taylor's hypothesis to an unsteady convective field for the spectral analysis of turbulence in the aorta

Two types of unsteadiness must be considered when spectral analysis is applied to unsteady turbulence such as that found in the aorta. Firstly, the statistical properties of the turbulence itself change in time and so the definition of spectral density must be reconsidered. Secondly, the turbulent velocity fluctuations, whether they are steady or unsteady, are carried by an unsteady convective velocity which alters their properties as seen by a stationary observer.

In the present study, unsteadiness of turbulence in the latter sense is discussed by applying Taylor's hypothesis of ‘frozen turbulence’ to turbulence with an unsteady convective velocity. If both a ‘frozen’ pattern of turbulence and a constant convective velocity are assumed, measured frequency spectra can be easily transformed into wavenumber (spatial) spectra, usually as a trivial part of normalisation. In the case of unsteady turbulence, however, the convection velocity is no longer constant and the conventional method can not be used. A new method of estimating the spatial properties of unsteady turbulence is proposed in which the temporal fluctuations of the turbulent velocity are transformed into spatial fluctuations using a nonlinear transformation based upon the unsteady convective velocity. The transformed data are then Fourier analysed to yield a wavenumber spectrum directly.

The proposed method is applied to data obtained in the canine ascending aorta. Spectra calculated by the proposed method differ significantly from those obtained by the conventional method, particularly in the high wavenumber (or frequency) range. This difference is discussed as an ‘aliasing’ phenomenon that has also been known in steady turbulence.