  Despite the nonlinear nature of wall turbulence, there is evidence that the
energy-injection mechanisms sustaining wall turbulence can be ascribed to
linear processes. The different scenarios stem from linear stability theory and
comprise exponential instabilities from mean-flow inflection points, transient
growth from non-normal operators, and parametric instabilities from temporal
mean-flow variations, among others. These mechanisms, each potentially capable
of leading to the observed turbulence structure, are rooted in simplified
theories and conceptual arguments. Whether the flow follows any or a
combination of them remains unclear. In the present study, we devise a
collection of numerical experiments in which the Navier-Stokes equations are
sensibly modified to quantify the role of the different linear mechanisms. This
is achieved by direct numerical simulation of turbulent channel flows with
constrained energy extraction from the streamwise-averaged mean-flow. We
demonstrate that (i) transient growth alone is not sufficient to sustain wall
turbulence and (ii) the flow remains turbulent when the exponential
instabilities are suppressed. On the other hand, we show that (iii) transient
growth combined with the parametric instability of the time-varying mean-flow
is able to sustain turbulence.
