Transition to turbulence

Certain laminar flows are known to be linearly stable at all Reynolds numbers, R, although in practice they always become turbulent for sufficiently large R. Other flows typically become turbulent well before the critical Reynolds number of linear instability. One resolution of these paradoxes is that the domain of attraction for the laminar state shrinks for large R (as R^g say, with g<0) so that small but finite perturbations lead to transition. This small domain of attraction is due to the nonnormality of the Orr-Sommerfeld operator, which leads to initial conditions which give large transient growth before the solution eventually decays. A formal asymptotic analysis of the Navier-Stokes equations can quantify this growth, and lead to asymptotic estimates of the exponent g in the limit of large Reynolds number.