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**Vigdorovich, Igor and Oberlack, Martin** (2008):

*Analytical study of turbulent Poiseuille flow with wall transpiration.*

In: Physics of Fluids, American Institute of Physics, pp. 055102/1-055102/9, 20, (5), ISSN 1070-6631, [Online-Edition: http://pof.aip.org/resource/1/phfle6/v20/i5/p055102s1],

[Article]

## Abstract

An incompressible, pressure-driven, fully developed turbulent flow between two parallel walls, with an extra constant transverse velocity component, is considered. A closure condition is formulated, which relates the shear stress to the first and the second derivatives of the longitudinal mean velocity. The closure condition is derived without invoking any special hypotheses on the nature of turbulent motion, only taking advantage of the fact that the flow depends on a finite number of governing parameters. By virtue of the closure condition, the momentum equation is reduced to the boundary- value problem for a second-order differential equation, which is solved by the method of matched asymptotic expansions at high values of the logarithm of the Reynolds number based on the friction velocity. There are three characteristic flow regions in the channel: the core region and two wall regions near injection and suction walls. For each region, the solution is constructed. The asymptotic matching gives formulas for the wall shear stress and the maximum mean velocity. A limit transpiration velocity is obtained, such that the shear stress at the injection wall vanishes, while the maximum point on the velocity profile approaches the suction wall. In this case, a sublayer near the suction wall appears where the mean velocity is proportional to the square root of the distance from the wall. A friction law for Poiseuille flow with transpiration is found, which makes it possible to describe the relation between the wall shear stress, the Reynolds number, and the transpiration velocity by a function of one variable. A velocity defect law, which generalizes the classical law for the core region in a channel with impermeable walls to the case of transpiration, is also established. In similarity variables, the mean velocity profiles across the whole channel width outside viscous sublayers can be described by a one-parameter family of curves. The theoretical results obtained are in good agreement with available direct numerical simulation data.

Item Type: | Article |
---|---|

Erschienen: | 2008 |

Creators: | Vigdorovich, Igor and Oberlack, Martin |

Title: | Analytical study of turbulent Poiseuille flow with wall transpiration |

Language: | English |

Abstract: | An incompressible, pressure-driven, fully developed turbulent flow between two parallel walls, with an extra constant transverse velocity component, is considered. A closure condition is formulated, which relates the shear stress to the first and the second derivatives of the longitudinal mean velocity. The closure condition is derived without invoking any special hypotheses on the nature of turbulent motion, only taking advantage of the fact that the flow depends on a finite number of governing parameters. By virtue of the closure condition, the momentum equation is reduced to the boundary- value problem for a second-order differential equation, which is solved by the method of matched asymptotic expansions at high values of the logarithm of the Reynolds number based on the friction velocity. There are three characteristic flow regions in the channel: the core region and two wall regions near injection and suction walls. For each region, the solution is constructed. The asymptotic matching gives formulas for the wall shear stress and the maximum mean velocity. A limit transpiration velocity is obtained, such that the shear stress at the injection wall vanishes, while the maximum point on the velocity profile approaches the suction wall. In this case, a sublayer near the suction wall appears where the mean velocity is proportional to the square root of the distance from the wall. A friction law for Poiseuille flow with transpiration is found, which makes it possible to describe the relation between the wall shear stress, the Reynolds number, and the transpiration velocity by a function of one variable. A velocity defect law, which generalizes the classical law for the core region in a channel with impermeable walls to the case of transpiration, is also established. In similarity variables, the mean velocity profiles across the whole channel width outside viscous sublayers can be described by a one-parameter family of curves. The theoretical results obtained are in good agreement with available direct numerical simulation data. |

Journal or Publication Title: | Physics of Fluids |

Volume: | 20 |

Number: | 5 |

Publisher: | American Institute of Physics |

Uncontrolled Keywords: | boundary layer turbulence; boundary-value problems; channel flow; differential equations; friction; Poiseuille flow; shear turbulence; transpiration |

Divisions: | 16 Department of Mechanical Engineering > Fluid Dynamics (fdy) 16 Department of Mechanical Engineering |

Date Deposited: | 24 Aug 2011 18:00 |

Official URL: | http://pof.aip.org/resource/1/phfle6/v20/i5/p055102s1 |

Additional Information: | doi:10.1063/1.2919111 |

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