Disturbance structures that achieve maximum growth over a specified interval of time have recently been obtained for unbounded constant shear flow making use of closed-form solutions. Optimal perturbations have also been obtained for the canonical bounded shear flows, the Couette, and plane Poiseuille flows, using numerical solution of the linearized Navier-Stokes equations. In this note it is shown that these optimal perturbations have similar spectra and structure indicating an underlying universality of shear flow dynamics that is not revealed by traditional methods based on modal analysis. 

The three-dimensional perturbations to viscous constant shear flow that increase maximally in energy over a chosen time interval are obtained by optimizing over the complete set of analytic solutions. These optimal perturbations are intrinsically three dimensional, of restricted morphology, and exhibit large energy growth on the advective time scale, despite the absence of exponential normal modal instability in constant shear flow. The optimal structures can be interpreted as combinations of two fundamental types of motion associated with two distinguishable growth mechanisms: streamwise vortices growing by ‘ advection of mean streamwise velocity to form streamwise streaks, and upstream tilting waves growing by the down gradient Reynolds stress mechanism of two-dimensional shear instability. The optimal excitation over a chosen interval of time comprises a combination of these two mechanisms, characteristically giving rise to tilted roll vortices with greatly amplified perturbation energy. It is suggested that these disturbances provide the initial growth leading to transition to turbulence, in addition to providing an explanation for coherent structures in a wide variety of turbulent shear flows. 

The mean streak spacing of approximately 100 wall units that is observed in wall-bounded turbulent shear flow is shown to be consistent with near-wall streamwise vortices optimally configured to gain the most energy over an appropriate turbulent eddy turnover time. The streak spacing arising from the optimal perturbation increases with distance from the wall and is nearly independent of Reynolds number, in agreement with experiment. 

Transition to turbulence in plane channel flow occurs even for conditions under which modes of the linearized dynamical system associated with the flow are stable. In this paper an attempt is made to understand this phenomena by finding the linear three-dimensional perturbations that gain the most energy in a given time period. A complete set of perturbations, ordered by energy growth, is found using variational methods. The optimal perturbations are not of modal form, and those which grow the most resemble streamwise vortices, which divert the mean flow energy into streaks of streamwise velocity and enable the energy of the perturbation to grow by as much as three orders of magnitude. It is suggested that excitation of these perturbations facilitates transition from laminar to turbulent flow. The variational method used to find the optimal perturbations in a shear flow also allows construction of tight bounds on growth rate and determination of regions of absolute stability in which no perturbation growth is possible.

Evidence, both theoretical and experimental, is accumulating to support a mechanism for transition to turbulence in shear flow based on the 3-D secondary instability of finite 2-D departures from plane parallelism. It is of central importance for using this mechanism to understand how the finite amplitude 2-D disturbances arise. To be sure, it is possible that in many experiments the disturbance energy without calling on the store of kinetic energy using properly configured perturbations that develop into the required primary disturbance on time scales comparable to those associated with the secondary instabilities even though the shear slow is stable or supports, at most, weak exponential instability.