Axial motion and scalar transport in stretched spiral vortices

We consider the dynamics of axial velocity and of scalar transport in the s tretched-spiral vortex model of turbulent fine scales. A large-time asympto tic solution to the scalar advection-diffusion equation, with an azimuthal swirling velocity field provided by the stretched spiral vortex, is used to gether with appropriate stretching transformations to determine the evoluti on of both the axial velocity and a passive scalar. This allows calculation of the shell-integrated three-dimensional spectra of these quantities for the spiral-vortex flow. The dominant term in the velocity (energy) spectrum contributed by the axial velocity is found to be produced by the stirring of the initial distribution of axial velocity by the axisymmetric component of the azimuthal velocity. This gives a k(-7/3) spectrum at large wave num bers, compared to the k(-5/3) component for the azimuthal velocity itself. The spectrum of a passive scalar being mixed by the vortex velocity field i s the sum of two power laws. The first is a k(-1) Batchelor spectrum for wa ve numbers up to the inverse Batchelor scale. This is produced by the axisy mmetric component of the axial vorticity but is independent of the detailed radial velocity profile. The second is a k(-5/3) Obukov-Corrsin spectrum f or wave numbers less than the inverse Kolmogorov scale. This is generated b y the nonaxisymmetric axial vorticity and depends on initial correlations b etween this vorticity and the initial scalar field. The one-dimensional sca lar spectrum for the composite model is in satisfactory agreement with expe rimental measurement. (C) 2001 American Institute of Physics.