Spatial dependence of correlation functions in the decay problem for a passive scalar in a large-scale velocity field
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AL, NONLINEAR, AND SOFT MATTER PHYSICS
Spatial Dependence of Correlation Functions in the Decay Problem for a Passive Scalar in a Large-Scale Velocity Field S. S. Vergeles Landau Institute for Theoretical Physics, Russian Academy of Sciences, Moscow, 119334 Russia e-mail: [email protected] Received October 6, 2005
Abstract—Statistical characteristics of a passive scalar advected by a turbulent velocity field are considered in the decay problem with a low scalar diffusivity κ (large Prandtl number ν/κ, where ν is kinematic viscosity). A regime in which the scalar correlation length remains smaller than the velocity correlation length is analyzed. The equal-time correlation functions of the scalar field are found to vary according to power laws and have angular singularities reflecting locally layered distribution of the scalar in space. PACS numbers: 05.20.Jj, 47.27.Gs, 47.27.-i DOI: 10.1134/S1063776106040194
1. INTRODUCTION Statistical description of turbulent advection of a passive scalar quantity ϑ is a classical problem in turbulence theory. Examples of scalar field include deviations of tracer concentration and temperature from their mean values. A scalar is passive when the effect of the scalar field evolution on the flow is negligible. With regard to the aforementioned examples, this means that the variations of flow velocity due to tracer concentration fluctuations or thermal expansion can be ignored. This paper presents an analysis of the decay problem in which an initial scalar distribution is given and statistical characteristics of the time-varying scalar field are to be determined. In this study, two- and three-dimensional turbulent flows are considered (d = 2, 3). A well-developed threedimensional turbulent flow at a Reynolds number Re 1 is briefly described as follows (e.g., see [1, 2]). Energy is injected into the fluid through eddies of approximate size L generated by external forcing at a rate of v per unit mass. The fluid viscosity is negligible on length scales much larger than the viscous length 1/4
η = ( ν / v ) , 3
where ν is kinematic viscosity. In the inertial range η r L, kinetic energy is transferred from larger to smaller eddies via a steady cascade process. The corresponding mean velocity difference between fluid elements separated by a distance r was estimated by Kolmogorov [3] as δv ( r ) ≈ ( v r ) . 1/3
The cascade process transfers energy to the smallest eddies of size on the order of η. On the scale of η, viscosity is essential and kinetic energy dissipates into
heat. Intermittency corrections to Kolmogorov’s estimate [4] can be neglected in an approximate analysis. Two-dimensional turbulence [5–7] is different from three-dimensional turbulence in that both kinetic energy and enstrophy are conserved in inviscid flow. The latter quantity is defined as Ω =
∫ d rω , 2
2
where ω = curl v is vorticity. Accordingly, well-developed two-dimensional turbulence involves a downscale enstrophy cascade and an upscale kinetic-energy cascade, both starting from the forcing scale. T
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