We examine the efficacy of streamwise travelling waves generated by a zero-net-mass-flux surface blowing and suction for controlling the onset of turbulence in a channel flow. For small-amplitude actuation, we utilize a weakly nonlinear analysis to determine base-flow modifications and assess the resulting net power balance. Receptivity analysis of the velocity fluctuations around this base flow is then employed to design the travelling waves. Our simulation-free approach reveals that, relative to the flow with no control, the downstream travelling waves with properly designed speed and frequency can significantly reduce receptivity, which makes them well suited for controlling the onset of turbulence. In contrast, the velocity fluctuations around the upstream travelling waves exhibit larger receptivity to disturbances. Our theoretical predictions, obtained by perturbation analysis (in the wave amplitude) of the linearized Navier-Stokes equations with spatially periodic coefficients, are verified using full-scale simulations of the nonlinear flow dynamics in the companion paper (Lieu et al., J. Fluid Mech., 2010, doi:10.1017/S002211201000340X).
Bibliographical noteFunding Information:
Financial support from the National Science Foundation under CAREER Award CMMI-06-44793 and 3M Science and Technology Fellowship (to R.M.) is gratefully acknowledged. The University of Minnesota Supercomputing Institute is acknowledged for providing computing resources. This work was initiated during the 2006 Center for Turbulence Research Summer Program with financial support from Stanford University and NASA Ames Research Center. M.R.J. would like to thank Professor P. Moin for creating an inspiring intellectual atmosphere and Dr D. You for his hospitality during the stay.
- boundary-layer receptivity
- flow control
- transition to turbulence