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Large-eddy simulation of passive scalar with phase relaxation time in isotropic turbulence

Hiromichi Kobayashi1,2,* and Toshiyuki Gotoh2,3

  • 1Department of Physics, Hiyoshi Campus, Keio University, 4-1-1 Hiyoshi, Kohoku-ku, Yokohama 223-8521, Japan
  • 2Research and Education Center for Natural Sciences, Hiyoshi Campus, Keio University, 4-1-1 Hiyoshi, Kohoku-ku, Yokohama 223-8521, Japan
  • 3Department of Engineering, Nagoya Institute of Technology, Nagoya 466-8555, Japan

  • *Contact author: hkobayas@keio.jp
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Phys. Rev. Fluids 11, 064613 – Published 18 June, 2026

DOI: https://doi.org/10.1103/zzrf-bclm

Abstract

Supersaturation in clouds was modeled as a passive scalar with the phase-relaxation time under mean uniform vertical gradient. The theoretical prediction that there exist two −5/3 power-law ranges in the supersaturation variance spectrum was examined using direct numerical simulation (DNS) and large-eddy simulation (LES). Although DNS at a Taylor microscale Reynolds number of 946 suggested the existence of two −5/3 ranges, the results were not conclusive because of the limited extent of the inertial-convective range. To achieve a wider inertial-convective range, LES was employed, and a modified eddy diffusivity model for a passive scalar without a phase-relaxation term was developed to yield a k−5/3-spectrum across a broad range of wave numbers. LES results obtained by varying the phase-relaxation time demonstrated the coexistence of two k−5/3 spectral ranges with different amplitudes. As the phase-relaxation time decreased, the transition between the two ranges shifted toward higher wave numbers, which is consistent with theoretical predictions. Furthermore, the probability density function (PDF) of the band-pass-filtered supersaturation indicated that the PDF tails at small scales became increasingly elongated depending on the ratio of the turbulence timescale to the phase-relaxation time.

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Phys. Rev. Fluids 11, 064613– Published 18 June, 2026
  • Received 7 January 2026
  • Revised 22 May 2026
  • Accepted 26 May 2026
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