Journal of Fluid Mechanics

Small-scale variation of convected quantities like temperature in turbulent fluid Part 1. General discussion and the case of small conductivity

G. K.  Batchelor a1
a1 Cavendish Laboratory, University of Cambridge

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When some external agency imposes on a fluid large-scale variations of some dynamically passive, conserved, scalar quantity θ like temperature or concentration of solute, turbulent motion of the fluid generates small-scale variations of θ. This paper describes a theoretical investigation of the form of the spectrum of θ at large wave-numbers, taking into account the two effects of convection with the fluid and molecular diffusion with diffusivity k. Hypotheses of the kind made by Kolmogoroff for the small-scale variations of velocity in a turbulent motion at high Reynolds number are assumed to apply also to small-scale variations of θ.

Previous contributions to the problem are reviewed. These have established that the spectrum of θ varies as $ n^{- \frac {5} {3}}$ (n being wave-number) at the low wave-number end of the equilibrium range, but there has been some confusion about the wave-number marking the upper end of the range of validity of this relation. The existence of a conduction ‘cut-off’ near $ n = \isin{\sol}k {^3}) {^\frac {1} {4}}$ as put forward by Obukhoff and Corrsin is shown to hold only when ν [double less-than sign] κ, and that near n = (ε/νκ2) $ ^{\frac {1} {4}}$ put forward by Batchelor is shown to apply only when ν [dbl greater-than sign] κ. In the case ν [double less-than sign] κ, the remaining problem is to determine the form of the spectrum of θ beyond the conduction cut-off; this is done in Part 2. In the case ν [dbl greater-than sign] κ, the conduction cut-off occurs at wave-numbers much higher than (ε/ν3)$ ^{\frac {1} {4}}$, which is where the energy spectrum is cut off by viscosity, and where the spectrum of θ ceases to vary as $ n^{- \frac {5} {3}}$.

The form of the spectrum of θ in this latter case is determined over the range n > (ε/ν3)$ ^{\frac {1} {4}}$ by analysing the effect of the velocity field, regarded as effectively a persistent uniform straining motion for these small-scale variations of θ, and of molecular diffusion on a single Fourier component of θ. The wave-number of this sinusoidal variation of θ is changed (and generally increased in magnitude) by the straining motion and the amplitude is diminished by diffusion. By supposing that the level of the spectrum of θ is kept steady at wave-numbers near (ε/ν3)$ ^{\frac {1} {4}}$ by some mechanism of transfer from lower wave-numbers, the linearity of the equation for θ then allows the determination of the spectrum for n > (ε/ν3) $ ^{\frac {1} {4}}$, the result being given by (4.8). The same result is obtained, using essentially the same approximation about the velocity field, from a different kind of analysis in terms of velocity and θ correlations. Finally, the relation between this work and Townsend's model of the small-scale variations of vorticity in a turbulent fluid is discussed.

(Published Online March 28 2006)
(Received June 1 1958)