Direct numerical simulations of a Couette-Poiseuille flow were performed near the transition to turbulence to investigate the nonlinear relationship between streak waviness and rolls. This relationship is a key step in Waleffe's model for a self-sustaining process. Simulations were conducted for Reynolds numbers ranging from 500 to 940, and a range of initial perturbation amplitudes was used. In these simulations, the streaks, rolls, and streak waviness initially grow. The optimal time for this growth closely matches the linear transient growth period for small perturbations but is much shorter when the initial perturbations are large and highly nonlinear. For higher Reynolds numbers and large initial perturbations, the velocity field reaches a turbulent steady state, while in the remaining cases, the flow relaminarizes. The main result is that the waviness of the streaks is a quadratic function of the rolls, provided that the roll amplitude is sufficiently large.

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• Shear flows
• Turbulence