J. Fluid Mech. (2006), vol. 561, pp. 279–305. c  2006 Cambridge University Press doi:10.1017/S0022112006000723 Printed in the United Kingdom 279 Turbulent channel ﬂow with either transverse or longitudinal roughness elements on one wall By P. ORLANDI 1 , S. LEONARDI 1 AND R. A. ANTONIA 2 1 Dipartimento di Meccanica e Aeronautica Universit` a La Sapienza, Via Eudossiana 18, 00184, Roma, Italy 2 Discipline of Mechanical Engineering, University of Newcastle, NSW 2308 Australia (Received 7 January 2005 and in revised form 18 January 2006) Direct numerical simulation results are presented for turbulent channel ﬂows with two-dimensional roughness elements of diﬀerent shapes. The focus is mainly on a geometry where the separation between consecutive roughness elements is small and for which the rate of change of the roughness function with respect to the separation between consecutive elements is large. Roughness elements are placed either along the ﬂow direction or orthogonally to it. In the latter case, the drag is increased. For the former case, the possibility of drag reduction reﬂects the diﬀerent relative contributions from viscous and Reynolds shear stresses. The Reynolds shear stress depends on the shape of the surface more than the viscous stress and is closely related to the near-wall structures. For orthogonal elements, there is no satisfactory correlation between the roughness function and parameters describing the roughness geometry. On the other hand, a satisfactory collapse of the data is achieved when the roughness function is plotted against the root mean square wall-normal velocity averaged over the plane of the roughness crests. Relative to a smooth wall surface, the Reynolds stress tensor near the wall tends to become more isotropic when the elements are orthogonal to the ﬂow and less isotropic when the elements are aligned with the ﬂow. The interdependencies between the departure from isotropy in the wall region, the organization of the wall structures, and the magnitude of the drag are assessed by examining the rotational component of the turbulent kinetic energy production and the probability density function of the helicity density. 1. Introduction The eﬀect of roughness may be thought to cause, in a broad sense, either an increase or a reduction in drag. The former case is usually associated with either three-dimensional or transverse two-dimensional roughness elements, while the latter is generally achieved with the use of riblets (elements aligned to the ﬂow). Nikuradse (1933) was the ﬁrst to investigate the Reynolds-number dependence of the drag for ﬂows over a uniform sand grain roughness. Clauser (1954) showed that the eﬀect of the roughness was to shift the velocity distribution in the log-region according to U + = κ -1 ln(y + )+ B - U + , (1.1) where κ is the K´ arm ´ an constant, B is a constant (equal to about 5.5 for channel ﬂows) and U + is the so-called roughness function (+ denotes normalization by wall variables, i.e. the frictional velocity U τ and the kinematic viscosity ν ) which