Onset of turbulence in Couette and Poiseuille flows

The differences between the onset of turbulence in Couette flow in a plane channel and that of Poiseuille flow are investigated analytically using a quasi-linear approximation and a simplified Orr-Sommerfeld equation with an effective dissipation coefficient. The Couette-flow onset is characterized as soft, with a channel-center inflection point which coincides with the maximum Reynolds shear stress. Steady-state turbulence in Poiseuille flow requires a finite positive value of the difference between the Reynolds number (R) and the critical Reynolds number (Rcr), making the onset hard; there is no inflection point, and the point of maximum Reynolds shear stress moves toward the wall as (R - Rcr) increases.