Drag reduction in near-critical turbulent open-channel flow over undular bottoms

Reduced drag is often desirable in hydraulic engineering, e.g. to increase the mass flow in an open channel. Steady turbulent open-channel flow with very large Reynolds numbers is considered. The Froude number in the reference state is assumed to be close to the critical value 1, i.e., Fr = 1 + (3/2)ε with ε « 1. By allowing for a slightly uneven bottom with bottom elevations of the order O(ε^5/2), a steady-state version of an extended Korteweg–de Vries (KdV) equation, describing the non-dimensional free-surface elevation, is derived by an asymptotic analysis. The amplitudes of the free-surface elevation are of O(ε). Choosing the bottom elevation of a particular undular shape, i.e., a superposition of a linear and a periodic part, leads to stationary periodic cnoidal waves as solutions of the extended KdV equation, as if the flow were inviscid. The analysis of the wall friction distribution over a wave period in comparison to a plane bottom shows that bottom elevations as small as O(ε^5/2) lead to drag changes of order O(ε). It turns out that the vast majority of feasible parameter configurations results in drag reduction. Thus, the present analysis shows a suitable way for passively lowering the effective friction coefficient in near-critical turbulent open-channel flows.