STOCHASTIC FORCING OF THE LINEARIZED NAVIER-STOKES EQUATIONS

Transient amplification of a particular set of favorably configured fo rcing functions in the stochastically driven Navier-Stokes equations l inearized about a mean shear flow is shown to produce high levels of v ariance concentrated in a distinct set of response functions. The domi nant forcing functions are found as solutions of a Lyapunov equation a nd the response functions are found as the distinct solutions of a rel ated Lyapunov equation. Neither the forcing nor the response functions can be identified with the normal modes of the linearized dynamical o perator. High variance levels are sustained in these systems under sto chastic forcing, largely by transfer of energy from the mean flow to t he perturbation field, despite the exponential stability of all normal modes of the system. From the perspective of modal analysis the expla nation for this amplification of variance can be traced to the non-nor mality of the linearized dynamical operator. The great amplification o f perturbation variance found for Couette and Poiseuille flow implies a mechanism for producing and sustaining high levels of variance in sh ear flows from relatively small intrinsic or extrinsic forcing disturb ances.