### Abstract

One-point statistics of velocity gradients and Eulerian and Lagrangian accelerations are studied by analysing the data from high-resolution direct numerical simulations (DNS) of turbulence in a periodic box, with up to 4096^{3} grid points. The DNS consist of two series of runs; one is with k_{max}η ∼ 1 (Series 1) and the other is with k_{max}η ∼ 2 (Series 2), where k_{max} is the maximum wavenumber and η the Kolmogorov length scale. The maximum Taylor-microscale Reynolds number R_{λ} in Series 1 is about 1130, and it is about 675 in Series 2. Particular attention is paid to the possible Reynolds number (Re) dependence of the statistics. The visualization of the intense vorticity regions shows that the turbulence field at high Re consists of clusters of small intense vorticity regions, and their structure is to be distinguished from those of small eddies. The possible dependence on Re of the probability distribution functions of velocity gradients is analysed through the dependence on R_{λ} of the skewness and flatness factors (S and F). The DNS data suggest that the R_{λ} dependence of S and F of the longitudinal velocity gradients fit well with a simple power law: S ∼ -0.32 R_{λ}^{0.11} and F ∼ 1.14 R_{λ}^{0.34}, in fairly good agreement with previous experimental data. They also suggest that all the fourth-order moments of velocity gradients scale with R_{λ} similarly to each other at R_{λ} > 100, in contrast to R_{λ} < 100. Regarding the statistics of time derivatives, the second-order time derivatives of turbulent velocities are more intermittent than the first-order ones for both the Eulerian and Lagrangian velocities, and the Lagrangian time derivatives of turbulent velocities are more intermittent than the Eulerian time derivatives, as would be expected. The flatness factor of the Lagrangian acceleration is as large as 90 at R_{λ} ≈ 430. The flatness factors of the Eulerian and Lagrangian accelerations increase with R_{λ} approximately proportional to R_{λ}^{αE} and R_{λ}^{αL}, respectively, where α_{E} ≈ 0.5 and α_{L} ≈ 1.0, while those of the second-order time derivatives of the Eulerian and Lagrangian velocities increases approximately proportional to R_{λ}^{βE} and R_{λ}^{βL}, respectively, where β_{E} ≈ 1.5 and β _{L} ≈ 3.0.

Original language | English |
---|---|

Pages (from-to) | 335-366 |

Number of pages | 32 |

Journal | Journal of Fluid Mechanics |

Volume | 592 |

DOIs | |

Publication status | Published - Dec 10 2007 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Condensed Matter Physics
- Mechanics of Materials
- Mechanical Engineering

### Cite this

*Journal of Fluid Mechanics*,

*592*, 335-366. https://doi.org/10.1017/S0022112007008531

**Small-scale statistics in high-resolution direct numerical simulation of turbulence : Reynolds number dependence of one-point velocity gradient statistics.** / Ishihara, Takashi; Kaneda, Y.; Yokokawa, M.; Itakura, K.; Uno, A.

Research output: Contribution to journal › Article

*Journal of Fluid Mechanics*, vol. 592, pp. 335-366. https://doi.org/10.1017/S0022112007008531

}

TY - JOUR

T1 - Small-scale statistics in high-resolution direct numerical simulation of turbulence

T2 - Reynolds number dependence of one-point velocity gradient statistics

AU - Ishihara, Takashi

AU - Kaneda, Y.

AU - Yokokawa, M.

AU - Itakura, K.

AU - Uno, A.

PY - 2007/12/10

Y1 - 2007/12/10

N2 - One-point statistics of velocity gradients and Eulerian and Lagrangian accelerations are studied by analysing the data from high-resolution direct numerical simulations (DNS) of turbulence in a periodic box, with up to 40963 grid points. The DNS consist of two series of runs; one is with kmaxη ∼ 1 (Series 1) and the other is with kmaxη ∼ 2 (Series 2), where kmax is the maximum wavenumber and η the Kolmogorov length scale. The maximum Taylor-microscale Reynolds number Rλ in Series 1 is about 1130, and it is about 675 in Series 2. Particular attention is paid to the possible Reynolds number (Re) dependence of the statistics. The visualization of the intense vorticity regions shows that the turbulence field at high Re consists of clusters of small intense vorticity regions, and their structure is to be distinguished from those of small eddies. The possible dependence on Re of the probability distribution functions of velocity gradients is analysed through the dependence on Rλ of the skewness and flatness factors (S and F). The DNS data suggest that the Rλ dependence of S and F of the longitudinal velocity gradients fit well with a simple power law: S ∼ -0.32 Rλ0.11 and F ∼ 1.14 Rλ0.34, in fairly good agreement with previous experimental data. They also suggest that all the fourth-order moments of velocity gradients scale with Rλ similarly to each other at Rλ > 100, in contrast to Rλ < 100. Regarding the statistics of time derivatives, the second-order time derivatives of turbulent velocities are more intermittent than the first-order ones for both the Eulerian and Lagrangian velocities, and the Lagrangian time derivatives of turbulent velocities are more intermittent than the Eulerian time derivatives, as would be expected. The flatness factor of the Lagrangian acceleration is as large as 90 at Rλ ≈ 430. The flatness factors of the Eulerian and Lagrangian accelerations increase with Rλ approximately proportional to RλαE and RλαL, respectively, where αE ≈ 0.5 and αL ≈ 1.0, while those of the second-order time derivatives of the Eulerian and Lagrangian velocities increases approximately proportional to RλβE and RλβL, respectively, where βE ≈ 1.5 and β L ≈ 3.0.

AB - One-point statistics of velocity gradients and Eulerian and Lagrangian accelerations are studied by analysing the data from high-resolution direct numerical simulations (DNS) of turbulence in a periodic box, with up to 40963 grid points. The DNS consist of two series of runs; one is with kmaxη ∼ 1 (Series 1) and the other is with kmaxη ∼ 2 (Series 2), where kmax is the maximum wavenumber and η the Kolmogorov length scale. The maximum Taylor-microscale Reynolds number Rλ in Series 1 is about 1130, and it is about 675 in Series 2. Particular attention is paid to the possible Reynolds number (Re) dependence of the statistics. The visualization of the intense vorticity regions shows that the turbulence field at high Re consists of clusters of small intense vorticity regions, and their structure is to be distinguished from those of small eddies. The possible dependence on Re of the probability distribution functions of velocity gradients is analysed through the dependence on Rλ of the skewness and flatness factors (S and F). The DNS data suggest that the Rλ dependence of S and F of the longitudinal velocity gradients fit well with a simple power law: S ∼ -0.32 Rλ0.11 and F ∼ 1.14 Rλ0.34, in fairly good agreement with previous experimental data. They also suggest that all the fourth-order moments of velocity gradients scale with Rλ similarly to each other at Rλ > 100, in contrast to Rλ < 100. Regarding the statistics of time derivatives, the second-order time derivatives of turbulent velocities are more intermittent than the first-order ones for both the Eulerian and Lagrangian velocities, and the Lagrangian time derivatives of turbulent velocities are more intermittent than the Eulerian time derivatives, as would be expected. The flatness factor of the Lagrangian acceleration is as large as 90 at Rλ ≈ 430. The flatness factors of the Eulerian and Lagrangian accelerations increase with Rλ approximately proportional to RλαE and RλαL, respectively, where αE ≈ 0.5 and αL ≈ 1.0, while those of the second-order time derivatives of the Eulerian and Lagrangian velocities increases approximately proportional to RλβE and RλβL, respectively, where βE ≈ 1.5 and β L ≈ 3.0.

UR - http://www.scopus.com/inward/record.url?scp=36349022569&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=36349022569&partnerID=8YFLogxK

U2 - 10.1017/S0022112007008531

DO - 10.1017/S0022112007008531

M3 - Article

AN - SCOPUS:36349022569

VL - 592

SP - 335

EP - 366

JO - Journal of Fluid Mechanics

JF - Journal of Fluid Mechanics

SN - 0022-1120

ER -