Turbulent skin-friction drag reduction by travelling waves induced by spanwise Lorentz force

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The streamwise and spanwise travelling waves induced by spanwise Lorentz force are studied for skin-friction drag reduction in a turbulent channel. The streamwise travelling wave by spanwise Lorentz force on drag reduction is compared to the with the spanwise wall motion. The drag reduction map shows a drag reduction region and a drag increase region, depending on a time scale $\mathscr{T}=\lambda/(\mathscr{U}_c-\omega/\kappa)$. For spanwise travelling wave, a large drag reduction appears at large oscillation frequencies and small spanwise wave numbers, while all stationary wave cases give a drag increase. When the wave travels at an oblique angle to the streamwise mean flow, the optimal drag reduction appears in backward travelling wave case. Generally, the backward streamwise travelling wave is found to be most efficient in drag reduction among all oblique travelling waves. Spanwise oscillation, forward streamwise travelling, spanwise travelling and backward streamwise travelling wave cases share a similar drag reduction mechanism: first, the spanwise motion directly breaks the near wall quasi-streamwise vortices structure array \cite{Jeong_etal1997}, which results in the shortening of streamwise streaks; second, the spanwise velocity layer maintains the asymmetry of the positive and negative quasi-streamwise vortices, which leads to a sustained drag reduction.