Linear mechanisms and pressure fluctuations in wall turbulence with passivity-based linear feedback control

Abstract

Full-domain, linear feedback control of turbulent channel flow at Re{_\tau} < 400 is an effective method to attenuate turbulent fluctuations such that it is relaminarised. The passivity-based control approach is adopted and motivated by the conservative characteristics of the nonlinear terms in the Navier-Stokes equations with respect to the disturbance energy. The control acts on the wall-normal velocity fluctuations at low wavenumbers. The maximum spanwise wavelength that can be used without losing control is constant with Reynolds number at {\lambda_z^+} = 125. In the minimal flow unit at Re_{cl} = 5000, the maximum streamwise wavelength is {\lambda_x^+} = 1000. The effect of control on the pressure components is investigated via the Green's function approach. Only the spanwise spectra of p_{r} up to the designated controlled spanwise wavenumber k_{z} are effectively suppressed by the linear control. This indicates that the linear control operates via vdU/dy and thus acts on the pressure field via the linear ("rapid") source term of the Poisson equation for pressure fluctuations, 2(dU/dy)(dv/dx). The effectiveness of the linear control to suppress inherently nonlinear wall turbulence is explained by Landahl's theory of timescales, in that the linear control proceeds via the linear shear-interaction timescale which is significantly shorter than both the nonlinear and viscous timescales for turbulence. The linear shear-interaction timescale is effective as a result of the linear ("rapid") source term. The maximum control forcing is located at y^+ = 20, corresponding to the location of the maximum in the mean-square pressure gradient. The existence of Landahl's timescales in the near-wall region of the minimal flow unit at Re_{cl} = 5000 is confirmed. The dynamic mode decomposition (DMD) indicates that the linear operator is stable via the linear control. The application of DMD to nonlinear systems should be used with caution.