An upper bound for passive scalar diffusion in shear flows

Abstract

This study is concerned with the diffusion of a passive scalar $\Theta (\mathring{,}t)$ advected by general $n$-dimensional shear flows $\breve{=}u(y,z,...,t)\hat{x}$ having finite mean-square velocity gradients. The unidirectionality of the incompressible flows conserves the stream-wise scalar gradient, ${\partial }_x\Theta$, allowing only the cross-stream components to be amplified by shearing effects. This amplification is relatively weak because an important contributing factor, ${\partial }_x\Theta$, is conserved, effectively rendering a slow diffusion process. It is found that the decay of the scalar variance $<{\Theta }^2>$ satisfies $d<{\Theta }^2>/dt\ge -C{\kappa }^{1/3}$, where $C>0$ is a constant, depending on the fluid velocity gradients and initial distribution of $\Theta$, and $\kappa$ is the molecular diffusivity. This result generalizes to axisymmetric flows on the plane and on the sphere having finite mean-square angular velocity gradients.

4 pages, to appear in Phys. Fluids

Cited by in corpus (2)

• Local transfer and spectra of a diffusive field advected by large-scale incompressible flows
• Constraints on scalar diffusion anomaly in three-dimensional flows having bounded velocity gradients