Transient amplification of a particular set of favorably configured forcing functions in the stochastically driven Navier-Stokes equations linearized about a mean shear flow is shown to produce high levels of variance concentrated in a distinct set of response functions. The dominant forcing functions are found as solutions of a Lyapunov equation and the response functions are found as the distinct solutions of a related Lyapunov equation. Neither the forcing nor the response functions can be identified with the normal modes of the linearized dynamical operator. High variance levels are sustained in these systems under stochastic forcing, largely by transfer of energy from the mean flow to the perturbation field, despite the exponential stability of all normal modes of the system. From the perspective of modal analysis the explanation for this amplification of variance can be traced to the non-normality of the linearized dynamical operator. The great amplification of perturbation variance found for Couette and Poiseuille flow implies a mechanism for producing and sustaining high levels of variance in shear flows from relatively small intrinsic or extrinsic forcing disturbances. Transition to turbulence typically occurs in channel flow experiments at R = O( lOOO), where R is the flow Reynolds number based on the mean centerline velocity and the half-width of the channel. Careful control of the intensity of disturbances results in persistence of laminar flow to much higher Reynolds numbers, which for pipe Poiseuille flo~'~' can reach as high as Rz50 000. The dynamics of such small perturbations as are associated with high R transition can, at least initially, be described with accuracy t by the Navier-Stokes equations linearized about the background flow. Assuming that disturbances of whatever origin can be modeled as noise, it is of interest to address, making use of linearized perturbation theory, the level of variance sustained in the mean by stochastic forcing. Moreover, it is well known3 that even in fully turbulent flows the linear growth mechanism is solely responsible for the transfer of energy from the mean flow to perturbations, so that a complete characterization of this linear mechanism is a necessary step toward understanding turbulent flows. The familiar problem of the damped harmonic oscillator excited by random and uncorrelated impulses exemplifies the physical processes operating in the most familiar dynamical systems. It can be shown that the ensemble average variance of the displacement, x, of the oscillator is given by (IxI')=gp where 1 is the variance of the random forcing, w is the natural frequency of the oscillator, and y is the damping coefficient.4 When o=O there is no restoring force, and we obtain Brownian motion, which is not stationary. When y=O there is no damping, and random driving again leads to nonstationary statistics, while for nonvanishing restoring force and nonvanishing damping the variance reaches a tinite statistically stationary level, such that the input of energy from the driving balances the dissipation, and the variance level is inversely proportional to the damping coefficient. This behavior is characteristic of all dynamical systems with normal dynamical operators. The total variance in such systems is the sum of the variance of each of the normal modes taken separately, the same as if each mode were independently excited by stochastic forcing. Consider now a fluid with a background flow field having nonvanishing rate of strain, but with sufficient dissipation so that all small perturbations impressed on the tlow eventually decay. Linearization of this dynamical system about its background flow results in a non-normal dynamical operator and an associated set of modes that are not mutually orthogonal, either in the inner product associated with the L, or the energy norm. This lack of orthogonality corresponds to the potential for extraction of energy by the perturbations from the background flow, irrespectively of the existence of exponential instabilities, a result well known since the seminal work of Orr.5 The energy balance in such a system is between the stochastic forcing, together with the induced extraction of energy from the background flow, on the one hand, and the dissipation on the other. Without stochastic forcing the perturbation tleld would vanish. With stochastic forcing an enhanced level of variance can be maintained by the stochastically induced transfer of background flow energy to the perturbation field, and this level of variance may exceed that arising simply by accumulation of energy from the forcing. In this respect stochastically forced non-normal dynamical systems differ fundamentally from the classical problem of stochastically forced coupled harmonic oscillators.
To investigate this further, the full Navier-Stokes equations can be rewritten as linearised equations forced by nonlinear terms, and then these nonlinear terms can be modelled as statistically stationary stochastic forcing REF , so that the amplitude of the fluctuations remains time independent.
Stochastic forcing of the linearized Navier-Stokes equations
Physics of Fluids
Physics