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Phys. Fluids 6, 1820 (1994); doi:10.1063/1.868243 (18 pages)
Turbulent mixing of a passive scalar
(Received 28 July 1993; accepted 25 January 1994)
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Add to FacebookThe statistically stationary state of a turbulently advected passive scalar is studied, with an imposed linear mean gradient in two dimensions, via a number of numerical experiments. For a synthetic Gaussian velocity field, which is generated by a linear stochastic process, and whose spectra and Eulerian correlation time follow Kolmogorov scaling on all scales, the exponents of the scalar spectra are consistent with 5/3 or 17/3 depending on the diffusivity. For large Péclet numbers (Pe), the probability density function (PDF) of the scalar gradients perpendicular to the mean is well fit, from about 0.1–10 times the root‐mean‐square value, by a stretched exponential with exponent ∼0.6. The PDF for gradients parallel to the mean has similar tails and a O(1) skewness for all Pe studied. The scalar has a ramp‐and‐cliff structure similar to that first seen in shear‐flow experiments with scalars. A physical picture of the mechanism by which the ramp‐and‐cliff features form is given. A second model with the velocity evolving under the Euler equations restricted to a band of wave numbers produces the k−1 Batchelor spectrum when the scalar is dissipated with a hyperdiffusivity (∝k4). For physical dissipation (∝k2), the PDF of the scalar has exponential tails, and for gradients less than the cutoff set by the maximum strain, the PDF of the gradients is similar to that obtained with the stochastic velocity model. The PDF of the dissipation is approximately stretched exponential like the gradient PDFs and not lognormal. The skewness of the gradients parallel to the mean decreases with decreasing autocorrelation time of the velocity, and the gradient PDFs assume a limiting form in the white‐noise limit.
ERRATUM
- Erratum: ``Turbulent mixing of a passive scalar'' [Phys. Fluids 6, 1820 (1994)]
Mark Holzer et al.
Phys. Fluids 7, 1519 (1995)PHFLE6000007000006001519000001
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The estimation of the scalar derivative skewness and anisotropy is more straightforward than for the velocity, where a skewness is required to transfer energy, and a small anisotropy in
(![[partial-derivative]](http://scitation.aip.org/stockgif3/part.gif)
i
i)2
(no sum on i) can be maintained by the pressure. [K. Yeung and J. G. Brasseur, “The response of isotropic turbulence to isotropic and anisotropic forcing at the large scales,” Phys. Fluids A 3, 884 (1991PFADEB000003000005000884000001)., Also, J. Lumley, “Some comments on turbulence,” Phys. Fluids A 4, 203 (1992PFADEB000004000002000203000001). ]One might think that for real flows, the nonzero (ii)3 could induce a significant scalar derivative skewness since ![[del]](http://scitation.aip.org/stockgif3/nabla.gif)
(
·v) is the source term for scalar gradients in Eq. (2). Formally solving (2) for its gradient by means of a suitable Lagrangian history Green function G, one obtains ·
= ![[integral]](http://scitation.aip.org/stockgif3/int.gif)
G·(·v)dt dr. Taking third moments, the velocity term is estimated on dimensional grounds as ~
(ii)3, where 
is the velocity gradient correlation time on the dissipation scale. Since this expression is order unity, we obtain again a skewness which goes like ~Pe−3/2. See, e.g., R. H. Kraichnan, “Inertial ranges in two-dimensional turbulence,” Phys. Fluids 10, 1417 (1967PFLDAS000010000007001417000001).
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