Theoretical drag analyses of laminar channel and pipe flows with wall roughness

In this paper, an exact expression for the drag coefficient of a streamwise-periodic steady incompressible laminar channel and pipe flow with micro- or macro-scale wall roughness is derived, whereby the drag coefficient is decomposed into contributions from different components of the velocity gradient tensor in the flow field. It is shown through our theoretical analysis that drag reduction cannot be achieved by adding micro- or macro-scale spanwise-periodic/-symmetry wall roughness structures to the smooth inner walls of streamwise-periodic steady incompressible laminar channel/pipe flows while maintaining the same volumetric flow rate. It is also shown that wall roughness produces a higher drag due to two factors: (a) wall roughness induces other non-zero velocity gradient terms apart from the wall-normal/radial gradient of streamwise velocity that exists in a smooth channel/pipe flow; (b) the profile of streamwise velocity in the wall-normal/radial direction deviates from the parabolic profile that produces the minimum kinetic energy loss for a given volumetric flow rate. Finally, numerical simulations of laminar channel flow with longitudinal and transverse bars are conducted, and the numerical results confirm the theoretical finding.