On Scaling Laws for the Transition to Turbulence in Uniform-Shear Flows

Numerical and experimental results indicate that the critical amplitude for non-linear growth of disturbances in a uniform shear flow follows two scaling laws, depending on the shape of the initial disturbance: i) a Re-1 law for disturbances with zero streamwise dependence (streaks) (Kreiss et al., submitted to J. Fluid Mech.); ii) a Re-0.4±0.1 law for disturbances with non-zero streamwise dependence (spots) (Dauchot and Daviaud, submitted to Phys. Fluid A). It is shown that these laws can be explained by the competition between non-linear growth and viscous dissipation and that the difference in exponent simply reflects the difference in viscous-decay mechanism. Lower bounds for conditions for non-linear growth are derived analytically. They follow a Re-1 and Re-1/3 law for streamwise and spotwise initial perturbations. The transition to a self-sustained turbulent state is also discussed. Upper bounds for this transition are derived. They follow, respectively, a Re-1/2 and Re-1/6 law for streak or spot initial perturbations.