### Abstract

A fundamental tenet of statistical mechanics is that the rate of collision of two objects is related to the expectation value of their relative velocities. In pioneering work by Saffman and Turner [J. Fluid Mech. 1, 16 (1956)], two different formulations of this tenet are used to calculate the collision kernel Γ between two arbitrary particle size groups in a turbulent flow. The first or spherical formulation is based on the radial component w_{r} of the relative velocity w between two particles: Γ^{sph} = 2πR^{2}〈\w_{r}\〉, where w_{r} = w· R/R, R is the separation vector, and R= |R|. The second or cylindrical formulation is based on the vector velocity itself: Γ^{cyl}=2πR^{2}(|w|), which is supported by molecular collision statistical mechanics. Saffman and Turner obtained different results from the two formulations and attributed the difference to the form of the probability function of w used in their work. A more careful examination reveals that there is a fundamental difference between the two formulations. An underlying assumption in the second formulation is that the relative velocity at any instant is locally uniform over a spatial scale on the order of the collision radius R, which is certainly not the case in turbulent flow. Therefore, the second formulation is not expected to be rigorously correct. In fact, both our analysis and numerical simulations show that the second formulation leads to a collision kernel about 25% larger than the first formulation in isotropic turbulence. For a simple uniform shear flow, the second formulation is about 20% too large. The two formulations, however, are equivalent for treating the collision rates among random molecules and the gravitational collision rates.

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

Pages (from-to) | 2647-2651 |

Number of pages | 5 |

Journal | Physics of Fluids |

Volume | 10 |

Issue number | 10 |

State | Published - 1998 |

Externally published | Yes |

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### ASJC Scopus subject areas

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

### Cite this

*Physics of Fluids*,

*10*(10), 2647-2651.

**Statistical mechanical descriptions of turbulent coagulation.** / Wang, Lian Ping; Wexler, Anthony S.; Zhou, Yong.

Research output: Contribution to journal › Article

*Physics of Fluids*, vol. 10, no. 10, pp. 2647-2651.

}

TY - JOUR

T1 - Statistical mechanical descriptions of turbulent coagulation

AU - Wang, Lian Ping

AU - Wexler, Anthony S.

AU - Zhou, Yong

PY - 1998

Y1 - 1998

N2 - A fundamental tenet of statistical mechanics is that the rate of collision of two objects is related to the expectation value of their relative velocities. In pioneering work by Saffman and Turner [J. Fluid Mech. 1, 16 (1956)], two different formulations of this tenet are used to calculate the collision kernel Γ between two arbitrary particle size groups in a turbulent flow. The first or spherical formulation is based on the radial component wr of the relative velocity w between two particles: Γsph = 2πR2〈\wr\〉, where wr = w· R/R, R is the separation vector, and R= |R|. The second or cylindrical formulation is based on the vector velocity itself: Γcyl=2πR2(|w|), which is supported by molecular collision statistical mechanics. Saffman and Turner obtained different results from the two formulations and attributed the difference to the form of the probability function of w used in their work. A more careful examination reveals that there is a fundamental difference between the two formulations. An underlying assumption in the second formulation is that the relative velocity at any instant is locally uniform over a spatial scale on the order of the collision radius R, which is certainly not the case in turbulent flow. Therefore, the second formulation is not expected to be rigorously correct. In fact, both our analysis and numerical simulations show that the second formulation leads to a collision kernel about 25% larger than the first formulation in isotropic turbulence. For a simple uniform shear flow, the second formulation is about 20% too large. The two formulations, however, are equivalent for treating the collision rates among random molecules and the gravitational collision rates.

AB - A fundamental tenet of statistical mechanics is that the rate of collision of two objects is related to the expectation value of their relative velocities. In pioneering work by Saffman and Turner [J. Fluid Mech. 1, 16 (1956)], two different formulations of this tenet are used to calculate the collision kernel Γ between two arbitrary particle size groups in a turbulent flow. The first or spherical formulation is based on the radial component wr of the relative velocity w between two particles: Γsph = 2πR2〈\wr\〉, where wr = w· R/R, R is the separation vector, and R= |R|. The second or cylindrical formulation is based on the vector velocity itself: Γcyl=2πR2(|w|), which is supported by molecular collision statistical mechanics. Saffman and Turner obtained different results from the two formulations and attributed the difference to the form of the probability function of w used in their work. A more careful examination reveals that there is a fundamental difference between the two formulations. An underlying assumption in the second formulation is that the relative velocity at any instant is locally uniform over a spatial scale on the order of the collision radius R, which is certainly not the case in turbulent flow. Therefore, the second formulation is not expected to be rigorously correct. In fact, both our analysis and numerical simulations show that the second formulation leads to a collision kernel about 25% larger than the first formulation in isotropic turbulence. For a simple uniform shear flow, the second formulation is about 20% too large. The two formulations, however, are equivalent for treating the collision rates among random molecules and the gravitational collision rates.

UR - http://www.scopus.com/inward/record.url?scp=0000999201&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0000999201&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0000999201

VL - 10

SP - 2647

EP - 2651

JO - Physics of Fluids

JF - Physics of Fluids

SN - 0031-9171

IS - 10

ER -