The Scalings of Scalar Structure Functions in a Velocity Field with Coherent Vortical Structures

Abstract

In planar turbulence modelled as an isotropic and homogeneous collection of 2-D non-interacting compact vortices, the structure functions S_p(r) of a statistically stationary passive scalar field have the following scaling behaviour in the limit where the P\'eclet number Pe -> \infty S_p(r) ~ constant+\ln({\frac{r}{LPe^{-1/3}}}) for LPe^{-1/3} << L, S_p(r) ~ ({\frac{r}{LPe^{-1/3}}})^{6(1-D)} for LPe^{-1/2} << LPe^{-1/3}, where L is a large scale and D is the fractal co-dimension of the spiral scalar structures generated by the vortices (1/2 <= D < 2/3). Note that LPe^{-1/2} is the scalar Taylor microscale which stems naturally from our analytical treatment of the advection-diffusion equation. The essential ingredients of our theory are the locality of inter-scale transfer and Lundgren's time average assumption. A phenomenological theory explicitly based only on these two ingredients reproduces our results and a generalisation of this phenomenology to spatially smooth chaotic flows yields (k\ln k)^{-1} generalised power spectra for the advected scalar fields.