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Turbulent Poiseuille flow with wall transpiration

DOI: 10.1615/ICHMT.2009.TurbulHeatMassTransf.620
13 pages

Igor Vigdorovich
Lomonosov Moscow State University

Martin Oberlack
Chair of Fluid Dynamics, Dept. Mech. Eng., TU Darmstadt, Otto-Berndt-Str. 2, 64287 Darmstadt, Germany; Center of Smart Interfaces, TU Darmstadt, Germany; GS Computational Engineering, TU Darmstadt, Germany

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. The problem is solved by the method of matched asymptotic expansions at high values of the logarithm of the Reynolds number based on the friction 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 is found, which makes it possible to describe the relation between the wall shear stress, the Reynolds number, and the transpiration velocity by a single 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 DNS data.

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