Small-Scale Structure of a Scalar Field Convected by Turbulence
1968/05/01 by Robert H. Kraichnan · 940 citations
Engineering · Environmental Science · #Classical mechanics #Convection #Fluid Dynamics and Turbulent Flows #Geometry #Mechanics #Physics #Plant Water Relations and Carbon Dynamics #Quantum mechanics #Reynolds number #Scalar (mathematics) #Scalar field #Turbulence #Wavenumber #Wind and Air Flow Studies
paper · doi:10.1063/1.1692063
published in The Physics of Fluids 11(5), 945-953 (AIP Publishing)
openalex publication_date 1968/05/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31
Abstract
Batchelor's theory of the turbulent straining of small-spatial-scale amplitude variations of a convected scalar field is re-examined to see the effects of fluctuation of the rates of strain in space and time. The k−1 viscous-convective-range spectrum is unaltered, except for the constant of proportionality, but spectrum level in the viscous-diffusive range displays a sensitivity to fluctuations which increases with wavenumber. The Gaussian cutoff found by Batchelor is replaced by more gently decreasing dependences of spectrum level on wavenumber. The scalar spectrum is also treated by the Lagrangian-history direct-interaction approximation. The k−5/3 inertial-convective range of Obukhov and Corrsin, the k−1 viscous-convective range, and the k−17/3 inertial-diffusive range of Batchelor, Howells, and Townsend all are recovered. For a given rate of spectral transport of scalar variance, the predicted spectrum levels in the k−5/3 and k−1 ranges are too small, in comparison with experiment, by numerical factors in the neighborhood of one to three. The largest error is for the k−1 range at large Reynolds number.
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