**Claus Wagner**
*German Aerospace Center (DLR), Institute for Aerodynamics and Flow Technology, Bunsenstrasse 10, 37073 Göttingen, Germany; Ilmenau University of Technology, Institute of Thermodynamics and Fluid Mechanics, Germany*

**Rainer Friedrich**
*Lehrstuhl für Aerodynamik, Technische Universität München Boltzmannstrasse 15, D-85748 Garching, Germany *

### Abstract

Fully-developed low-Reynolds-number turbulent flow
through straight permeable pipes with circular cross-
section is investigated by means of direct numerical simulation. Three different cases of wall permeability are treated and compared with the case of a solid wall. In two of these
cases the wall satisfies the no-slip condition, but allows for
the wall normal velocity fluctuations in two different ways.
In the third case, the pipe wall has rectangular openings
of size 6 (Δz × *(RΔφ)*^{+}), regularly distributed over the
whole surface, similar to a chess-board where the black areas represent the openings and the white ones the solid wall. Velocity boundary conditions in the openings are such that the mean mass flux across the wall is zero and the flow in the openings is stress free. All flows are driven by the same mean pressure-gradient. Consequently, those flows which satisfy the no-slip condition have the same wall shear stress and hence the same turbulence Reynolds number *Re*_{τ} = 360. Pipe flow with wall openings exhibits a small, but finite Reynolds shear stress at the wall. If the friction velocity is defined via the total stress at this wall, the flow has nominally the same turbulence Reynolds number. The overall effect of a permeable wall with rectangular openings is a mean axial slip velocity at the wall and reduced viscous stress. In a thin near-wall layer of thickness *v/u*_{τ} the turbulence activity is increased compared to the flow cases where the velocity components satisfy the no-slip condition. All three rms-velocity fluctuations are non-zero. As a result the structure of the Reynolds stress tensor is modified in this region. This is also reflected in higher order central moments of the velocity fluctuations. A permeable wall with rectangular openings may be viewed as a model for a rough wall with a mean non-dimensional roughness height of 8.3 wall units. Close to such a wall the budgets of the Reynolds stress tensor differ strongly from those for flow along a smooth impermeable wall.