Near wall coherence in wall-bounded flows and implications for flow control

Abstract: Opposition-control of the energetic cycle of near wall streaks in wall-bounded turbulence, using numerical approaches, has shown promise for drag reduction. For practical implementation, opposition control is only realizable if there is a degree of coherence between the sensor--actuator pairs of the control system (these sensors and actuators should typically be wall-based to avoid parasitic drag). As such, we here inspect the feasibility of real-time control of the near-wall cycle, by considering the coherence between a measurable wall-quantity, being the wall-shear stress fluctuations, and the streamwise and wall-normal velocity fluctuations in a turbulent boundary layer. Synchronized spatial and temporal velocity data from numerical simulations at $Re_\tau \approx 590$ and $ 2000$ are employed. It is shown that the spectral energy of the streamwise velocity fluctuations that is stochastically incoherent with wall signals is independent of Reynolds number in the near wall region. Consequently, the streamwise energy-fraction that is stochastically wall-coherent grows with Reynolds number due to the increasing range of energetic large scales. This thus implies that a wall-based control system has the ability to manipulate a larger portion of the total turbulence energy at off-wall locations, at higher Reynolds numbers. Coherence values of 0.55 and 0.4, which are considerably lower than the maximum possible coherence 1, were found between the streamwise and wall-normal velocity fluctuations at the near wall peak in the energy spectrogram, respectively, and the streamwise fluctuating friction velocity. This suggests that a closed-loop drag reduction scheme targeting near wall cycle of streaks alone will be of limited success in practice as the Reynolds number grows.