Turbulent Drag Reduction by Travelling Waves of Spanwise Lorentz Force

Travelling waves induced by spanwise Lorentz force for skin-friction drag reduction are studied in the present work using direct numerical simulations, with particular focus on the streamwise and spanwise travelling waves. The overall picture of drag reduction in the frequency-wavenumber parameter space, ? - ?(x)( )- ?(z) is uncovered. It is found that both drag reduction maps for the streamwise and spanwise travelling waves are featured with a drag reduction (DR) region and a drag increase (DI) region. For the streamwise travelling wave of spanwise Lorentz force, the DR and DI regions are located in the same parameter regime compared to that of the streamwise travelling wave of spanwise wall velocity, while for the spanwise travelling wave, the DR variation at any fixed ?(z) is similar to that of span wise oscillating Lorentz force. An exploration of the oblique travelling wave with an angle to the mean flow shows that the optimal drag reduction appears when the wave travels backward relative to the flow direction, and the "ribbon" structure is a general phenomenon appearing in all oblique travelling wave cases. The ensemble averaged positive and negative quasi-streamwise vortices become asymmetric when the travelling wave is imposed, and the near-wall high-and low-speed streaks are significantly tilted in the spanwise direction. Spanwise oscillation, streamwise and spanwise travelling waves share strong similarity in the statistics, energy spectra as well as turbulent structure modulation.