Fine Structure of Scalar Fields Mixed by Turbulence. I. Zero-Gradient Points and Minimal Gradient Surfaces

Abstract

Turbulent mixing of passive scalar properties such as temperature or concentration is discussed. From physical and geometrical considerations it is concluded that the smallest scale features of a steady-state turbulent scalar distribution are primarily determined by the number and distribution of points in the fluid for which the scalar gradient vector is zero, and surfaces for which the gradient magnitude is minimal. Mechanisms for the production, destruction, and motion of such “zero gradient points” and “minimal gradient surfaces” are examined. Initially, zero gradient points must be produced from regions of uniform scalar gradient, but the vast majority result from secondary splitting due to strain induced eccentricities of the closed isoscalar surfaces surrounding maximum or minimum points. An expression for the velocity of surfaces of constant scalar value is derived and used to interpret the Obukhov-Corrsin length scale (D3/ε)14 as the minimum size eddy capable of generating a zero gradient point from a region of uniform gradient, where D is the molecular diffusivity of the scalar and ε is the mean viscous dissipation rate. A steady-state separation distance between zero gradient points of order (D/γ)12 is inferred for scalar fields of arbitrary diffusivity where γ is the root mean square rate of strain of the fluid. Previous theories such as that of Batchelor, Howells, and Townsend have assumed the local strain rate has no effect on scalar structure for large Prandtl number fields.