On the Application of the Abel Transformation in Statistically Axisymmetric Turbulent Flows

The application of the inverse Abel transformation to statistically axisymmetric data is described in this work. The general theory of Abel inversion tomography is discussed, and it is shown that in order to properly apply the Abel inversion, an ensemble operator must be axisymmetric and commute with path integration. For statistically axisymmetric data where the individual realizations are asymmetric (such as turbulent flows issuing from circular nozzles), only the planar mean can be properly recovered. Higher-order moments, such as the standard deviation (root mean square) cannot be recovered because path integration is not commutative with the statistical moment operators. It is further shown that the Abel transformation cannot be used to recover the one-sided Fourier spectrum for the same reason. For the mean, the Abel inversion can be applied to the ensemble quantity, or the ensemble operator can be applied to the Abel inversion of individual asymmetric realizations. These findings are rigorously verified with numerical simulations and demonstrated on experimental data by comparing planar particle image velocimetry data to path-averaged schlieren image velocimetry results from a turbulent jet of issuing into ambient air.