### Abstract

We present direct numerical simulations of the mixing of the passive scalar at modest Taylor microscale (10≤R_{λ}≤42) and Schmidt numbers larger than unity (2≤Sc≤32). The simulations resolve below the Batchelor scale up to a factor of 4. The advecting turbulence is homogeneous and isotropic, and is maintained stationary by stochastic forcing at low wave numbers. The passive scalar is rendered stationary by a mean scalar gradient in one direction. The relation between geometrical and statistical properties of scalar field and its gradients is examined. The Reynolds numbers and Schmidt numbers are not large enough for either the Kolmogorov scaling or the Batchelor scaling to develop and, not surprisingly, we find no fractal scaling of scalar level sets, or isosurfaces, in the intermediate viscous range. The area-to-volume ratio of isosurfaces reflects the nearly Gaussian statistics of the scalar fluctuations. The scalar flux across the isosurfaces, which is determined by the conditional probability density function (PDF) of the scalar gradient magnitude, has a stretched exponential distribution towards the tails. The PDF of the scalar dissipation departs distinctly, for both small and large amplitudes, from the log-normal distribution for all cases considered. The joint statistics of the scalar and its dissipation rate, and the mean conditional moment of the scalar dissipation, are studied as well. We examine the effects of coarse-graining on the probability density to simulate the effects of poor probe-resolution in measurements.

Original language | English (US) |
---|---|

Article number | 125107 |

Pages (from-to) | 1-9 |

Number of pages | 9 |

Journal | Physics of Fluids |

Volume | 17 |

Issue number | 12 |

DOIs | |

State | Published - Dec 2005 |

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### ASJC Scopus subject areas

- Mechanics of Materials
- Computational Mechanics
- Physics and Astronomy(all)
- Fluid Flow and Transfer Processes
- Condensed Matter Physics

### Cite this

*Physics of Fluids*,

*17*(12), 1-9. [125107]. https://doi.org/10.1063/1.2140024

**Statistics and geometry of passive scalars in turbulence.** / Schumacher, Jörg; Sreenivasan, Katepalli R.

Research output: Contribution to journal › Article

*Physics of Fluids*, vol. 17, no. 12, 125107, pp. 1-9. https://doi.org/10.1063/1.2140024

}

TY - JOUR

T1 - Statistics and geometry of passive scalars in turbulence

AU - Schumacher, Jörg

AU - Sreenivasan, Katepalli R.

PY - 2005/12

Y1 - 2005/12

N2 - We present direct numerical simulations of the mixing of the passive scalar at modest Taylor microscale (10≤Rλ≤42) and Schmidt numbers larger than unity (2≤Sc≤32). The simulations resolve below the Batchelor scale up to a factor of 4. The advecting turbulence is homogeneous and isotropic, and is maintained stationary by stochastic forcing at low wave numbers. The passive scalar is rendered stationary by a mean scalar gradient in one direction. The relation between geometrical and statistical properties of scalar field and its gradients is examined. The Reynolds numbers and Schmidt numbers are not large enough for either the Kolmogorov scaling or the Batchelor scaling to develop and, not surprisingly, we find no fractal scaling of scalar level sets, or isosurfaces, in the intermediate viscous range. The area-to-volume ratio of isosurfaces reflects the nearly Gaussian statistics of the scalar fluctuations. The scalar flux across the isosurfaces, which is determined by the conditional probability density function (PDF) of the scalar gradient magnitude, has a stretched exponential distribution towards the tails. The PDF of the scalar dissipation departs distinctly, for both small and large amplitudes, from the log-normal distribution for all cases considered. The joint statistics of the scalar and its dissipation rate, and the mean conditional moment of the scalar dissipation, are studied as well. We examine the effects of coarse-graining on the probability density to simulate the effects of poor probe-resolution in measurements.

AB - We present direct numerical simulations of the mixing of the passive scalar at modest Taylor microscale (10≤Rλ≤42) and Schmidt numbers larger than unity (2≤Sc≤32). The simulations resolve below the Batchelor scale up to a factor of 4. The advecting turbulence is homogeneous and isotropic, and is maintained stationary by stochastic forcing at low wave numbers. The passive scalar is rendered stationary by a mean scalar gradient in one direction. The relation between geometrical and statistical properties of scalar field and its gradients is examined. The Reynolds numbers and Schmidt numbers are not large enough for either the Kolmogorov scaling or the Batchelor scaling to develop and, not surprisingly, we find no fractal scaling of scalar level sets, or isosurfaces, in the intermediate viscous range. The area-to-volume ratio of isosurfaces reflects the nearly Gaussian statistics of the scalar fluctuations. The scalar flux across the isosurfaces, which is determined by the conditional probability density function (PDF) of the scalar gradient magnitude, has a stretched exponential distribution towards the tails. The PDF of the scalar dissipation departs distinctly, for both small and large amplitudes, from the log-normal distribution for all cases considered. The joint statistics of the scalar and its dissipation rate, and the mean conditional moment of the scalar dissipation, are studied as well. We examine the effects of coarse-graining on the probability density to simulate the effects of poor probe-resolution in measurements.

UR - http://www.scopus.com/inward/record.url?scp=31144436205&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=31144436205&partnerID=8YFLogxK

U2 - 10.1063/1.2140024

DO - 10.1063/1.2140024

M3 - Article

VL - 17

SP - 1

EP - 9

JO - Physics of Fluids

JF - Physics of Fluids

SN - 1070-6631

IS - 12

M1 - 125107

ER -