Calculation of scalar structure functions from a vortex model of turbulent passive scalar transport

Abstract

A Saffman and Pullin [Phys. Fluids 8, 3072 (1996)] type vortex model for passive scalar structure functions is formulated. The intermittent turbulent fine-scale dynamics in the model is represented by numerical solutions of the advection-diffusion and Navier–Stokes equations in the form of axially strained vortex-scalar structures. The use of these structures is motivated by Pullin and Lundgren’s [Phys. Fluids 13, 2553 (2001)] asymptotic strained spiral vortex model of turbulent passive scalar transport. Ensemble-averaged scalar structure functions, of even orders 2–10, are calculated from a range of vortex-scalar structures using Monte Carlo integration. For axisymmetric strained scalar fields, acceptable agreement of the second-order structure function with experimental data reported by Antonia and Van Atta [J. Fluid Mech. 84, 561 (1978)] is obtained. Structure functions are also calculated for a range of passive scalar spiral structures. These are generated by the winding of single and double scalar patches in single strained vortex patches and in merging strained vortices. Power-law scaling of the second- and higher-order structure functions is obtained from cases involving the winding of single scalar patches in an axisymmetric strained vortex patch. The second-order scaling exponents from these cases are in reasonable agreement with Kolmogorov–Oboukhov–Corrsin scaling and the experimental results of Antonia et al. [Phys. Rev. A 30, 2704 (1984)] and Gylfason and Warhaft [Phys. Fluids 16, 4012 (2004)]. However, the higher-order scaling exponents from these cases fall below theoretical predictions and experimental results. Higher-order moments are sensitive to the composition of the vortex-scalar structures, and various improvements are suggested that could enhance the performance of the model. The present approach is promising, and it is the first demonstration that a vortex model using simplified Navier–Stokes dynamics can produce some scalar structure functions that compare favorably with experimental observations.