Pullin, D. I. and Buntine, James D. and Saffman, P. G. (1994) On the spectrum of a stretched spiral vortex. Physics of Fluids, 6 (9). pp. 30103027. ISSN 10706631. http://resolver.caltech.edu/CaltechAUTHORS:PULpof94b

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Abstract
Corrections are found to the k^–5/3 spectrum of Lundgren [Phys. Fluids 25, 2193 (1982)] for a stretched spiral vortex model (a is the stretching strain rate and k the scalar wave number) of turbulent fine scales. These take the form of additional terms arising from the early time evolution, when the stretching of vortex lines is small. For the special case when the spiral takes the form of a rolledup shear layer, it is shown that the composite spectrum is divergent, thus requiring the introduction of a finite early cutoff time tau1 in the time integral for the nonaxisymmetric contribution. The identity nuomega2 = 2nu[integral]<sub>0</sub><sup>[infinity]</sup>k^2E(k)dk which gives the dissipation is then satisfied selfconsistently. Direct numerical calculation of the energy spectrum from the approximate vorticity field for a special choice of spiral structure nevertheless indicates that the oneterm k^–5/3spectrum result is asymptotically valid in the inertial range provided atau1 is O(1) but that the numerically calculated dissipation spectrum appears to lie somewhere between an exp(–B1k2) and an exp(–B2k) form. It is also shown that the stretched, rolledup shearlayer model predicts asymptotic shellsummed spectra of the energy dissipation and of the square of the vorticity, each asymptotically constant, with no powerlaw dependence, for k smaller than the Kolmogorov wave number.The corresponding onedimensional spectra each show –log(k1) behavior for small k1. The extension of the model given by Pullin and Saffman [Phys. Fluids A 5, 126 (1993)] is reformulated by the introduction of a longtime cutoff in the vortex lifetime and an additional requirement that the vortex structures be approximately space filling. This gives a reduction in the number of model freeparameters but introduces a dependence of the calculated Kolmogorov constant and skewness on the ratio of the initial vortex radius to the equivalent Burgersvortex radius. A scaling for this ratio in terms of the Taylor microscale Reynolds number is proposed in which the stretching strain is assumed to be provided by the large scales with spatial coherence limited to the maximum stretched length of the structures. Postdictions of the fourthorder flatness factor and of higher moments of the longitudinal velocity gradient statistics are compared with numerical simulation.
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Additional Information:  Copyright © 1994 American Institute of Physics (Received 2 December 1993; accepted 9 May 1994) The authors wish to thank Professor D. W. Moore and Professor A. Leonard for helpful discussions. D.I.P. was partially supported by NSF Grant No. CTS9311811 and P.G.S. was partially supported by the Department of Energy under Grant No. DEFG0389ER25073. We wish to thank Dr. Maurice Meneguzzi for providing unpublished data on velocityderivative moments. 
Subject Keywords:  TURBULENT FLOW; VORTEX FLOW; ENERGY SPECTRA; ENERGY LOSSES; VISCOUS FLOW; SCALING LAWS; REYNOLDS NUMBER; INCOMPRESSIBLE FLOW; FINE STRUCTURE; CUT–OFF 
Record Number:  CaltechAUTHORS:PULpof94b 
Persistent URL:  http://resolver.caltech.edu/CaltechAUTHORS:PULpof94b 
Alternative URL:  http://dx.doi.org/10.1063/1.868127 
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