A Perturbative Model for Predicting the High-Reynolds-Number Behaviour of the Streamwise Travelling Waves Technique in Turbulent Drag Reduction

The background of this work is the problem of reducing the aerodynamic turbulent friction drag, which is an important source of energy waste in innumerable technological fields (transportation being probably the most important). We develop a theoretical framework aimed at predicting the behaviour of existing drag reduction techniques when used at the large values of the Reynolds numbers Re which are typical of applications. We focus on one recently proposed and very promising technique, which consists in creating at the wall streamwise-travelling waves of spanwise velocity (M. Quadrio, P. Ricco, and C. Viotti, J. Fluid Mech. 627, 161-178, 2009). A perturbation analysis of the Navier-Stokes equations that govern the fluid motion is carried out, for the simplest wall-bounded flow geometry, i.e. the plane channel flow. The streamwise base flow is perturbed by the spanwise time-varying base flow induced by the travelling waves. An asymptotic expansion is then carried out with respect to the velocity amplitude of the travelling wave. The analysis, although based on several assumptions, leads to predictions of drag reduction that agree well with the measurements available in literature and mostly computed through Direct Numerical Simulations (DNS) of the full Navier-Stokes equations. New DNS data are produced on purpose in this work to validate our method further. The method is then applied to predict the drag-reducing performance of the streamwise-travelling waves at increasing Re, where comparison data are not available. The current belief, based on a Re-range of about one decade only above the transitional value, that drag reduction obtained at low Re is deemed to decrease as Re is increased is fully confirmed by our results. From a quantitative standpoint, however, our outlook based on several decades of increase in Re is much less pessimistic than other existing estimates, and motivates further, more accurate studies on the present subject.