Fine Structure of Scalar Fields Mixed by Turbulence. II. Spectral Theory

Abstract

Universal similarity hypotheses are proposed based on the local straining mechanisms, Kolmogoroff's local isotropy theory, and the mixing theories of Obukhov, Corrsin, and Batchelor. Three sets of similarity coordinates follow from the hypotheses depending on five fundamental parameters of turbulent mixing: ε, the turbulence dissipation rate; χ, the scalar variance dissipation rate; γ, the local strain rate; ν, the kinematic viscosity; and D, the molecular diffusivity of the scalar. Transformations between coordinate systems are shown to depend only on $\mathrm{Pr\; \equiv \; \nu /D}$ as a mapping parameter. A unified spectral array with convergence properties required by the hypotheses is produced when the similarity hypotheses are used to predict the scalar spectrum function Γ. The inertial subrange (${\mathrm{\Gamma \; \sim \; k}}^{\text{−5/3}}$ , k is the wavenumber) of Obukhov and Corrsin and the large Pr value viscous‐convective ${\mathrm{(\Gamma \; \sim \; k}}^{\text{−1}})$ subrange of Batchelor are reproduced. However, for small Pr values, a new inertial‐diffusive subrange arises with ${\mathrm{\Gamma \; \sim \; k}}^{\text{−3}}$ and cutoff at wavenumber ${\mathrm{(\gamma /D)}}^{\frac{1}{2}}$ . Bounds for the universal subrange constants and functional forms for arbitrary Pr values are inferred using Batchelor's diffusive cutoff function, and compared with available experimental measurements.