### Abstract

Unsteady flow over a stationary sphere with a small fluctuation in the free-stream velocity is considered at small Reynolds number, Re. A matched asymptotic solution is obtained for the frequency-dependent (or the acceleration-dependent) part of the unsteady flow at very small frequency, omega, under the restriction St ((Re ((1, where St is the Strouhal number. The acceleration-dependent part of the unsteady drag is found to be proportional to St approx omega instead of the omega "SUP 1/2" dependence predicted by Stokes' solution. Consequently, the expression for the Basset history force is incorrect for large time even for very small Reynolds numbers. Present results compare well with the previous numerical results of Mei, Lawrence & Adrian (1991) using a finite-difference method for the same unsteady flow at small Reynolds number. Using the principle of causality, the present analytical results at small Re, the numerical results at finite Re for low frequency, and Stokes' results for high frequency, a modified expression for the history force is proposed in the time domain. It is confirmed by comparing with the finite-difference results at arbitrary frequency through Fourier transformation. The modified history force has an integration kernel that decays as r "SUP 2" , instead of t "SUP -1/2" , at large time for both small and finite Reynolds numbers. (A)

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

Pages (from-to) | 323-341 |

Number of pages | 19 |

Journal | Journal of Fluid Mechanics |

Volume | 237 |

State | Published - 1992 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Computational Mechanics
- Mechanics of Materials
- Physics and Astronomy(all)
- Condensed Matter Physics

### Cite this

*Journal of Fluid Mechanics*,

*237*, 323-341.

**Flow past a sphere with an oscillation in the free-stream velocity and unsteady drag at finite Reynolds number.** / Renwei Mei, Mei; Adrian, Ronald.

Research output: Contribution to journal › Article

*Journal of Fluid Mechanics*, vol. 237, pp. 323-341.

}

TY - JOUR

T1 - Flow past a sphere with an oscillation in the free-stream velocity and unsteady drag at finite Reynolds number

AU - Renwei Mei, Mei

AU - Adrian, Ronald

PY - 1992

Y1 - 1992

N2 - Unsteady flow over a stationary sphere with a small fluctuation in the free-stream velocity is considered at small Reynolds number, Re. A matched asymptotic solution is obtained for the frequency-dependent (or the acceleration-dependent) part of the unsteady flow at very small frequency, omega, under the restriction St ((Re ((1, where St is the Strouhal number. The acceleration-dependent part of the unsteady drag is found to be proportional to St approx omega instead of the omega "SUP 1/2" dependence predicted by Stokes' solution. Consequently, the expression for the Basset history force is incorrect for large time even for very small Reynolds numbers. Present results compare well with the previous numerical results of Mei, Lawrence & Adrian (1991) using a finite-difference method for the same unsteady flow at small Reynolds number. Using the principle of causality, the present analytical results at small Re, the numerical results at finite Re for low frequency, and Stokes' results for high frequency, a modified expression for the history force is proposed in the time domain. It is confirmed by comparing with the finite-difference results at arbitrary frequency through Fourier transformation. The modified history force has an integration kernel that decays as r "SUP 2" , instead of t "SUP -1/2" , at large time for both small and finite Reynolds numbers. (A)

AB - Unsteady flow over a stationary sphere with a small fluctuation in the free-stream velocity is considered at small Reynolds number, Re. A matched asymptotic solution is obtained for the frequency-dependent (or the acceleration-dependent) part of the unsteady flow at very small frequency, omega, under the restriction St ((Re ((1, where St is the Strouhal number. The acceleration-dependent part of the unsteady drag is found to be proportional to St approx omega instead of the omega "SUP 1/2" dependence predicted by Stokes' solution. Consequently, the expression for the Basset history force is incorrect for large time even for very small Reynolds numbers. Present results compare well with the previous numerical results of Mei, Lawrence & Adrian (1991) using a finite-difference method for the same unsteady flow at small Reynolds number. Using the principle of causality, the present analytical results at small Re, the numerical results at finite Re for low frequency, and Stokes' results for high frequency, a modified expression for the history force is proposed in the time domain. It is confirmed by comparing with the finite-difference results at arbitrary frequency through Fourier transformation. The modified history force has an integration kernel that decays as r "SUP 2" , instead of t "SUP -1/2" , at large time for both small and finite Reynolds numbers. (A)

UR - http://www.scopus.com/inward/record.url?scp=0026792388&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0026792388&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0026792388

VL - 237

SP - 323

EP - 341

JO - Journal of Fluid Mechanics

JF - Journal of Fluid Mechanics

SN - 0022-1120

ER -