Componentwise energy amplification in channel flows

Abstract

We study the linearized Navier-Stokes (LNS) equations in channel flows from an input-output point of view by analysing their spatio-temporal frequency responses. Spatially distributed and temporally varying body force fields are considered as inputs, and components of the resulting velocity fields are considered as outputs into these equations. We show how the roles of Tollmien-Schlichting (TS) waves, oblique waves, and streamwise vortices and streaks in subcritical transition can be explained as input-output resonances of the spatio-temporal frequency responses. On the one hand, we demonstrate the effectiveness of input field components, and on the other, the energy content of velocity perturbation components. We establish that wall-normal and spanwise forces have much stronger influence on the velocity field than streamwise force, and that the impact of these forces is most powerful on the streamwise velocity component. We show this using the relative scaling of the different input-output system components with the Reynolds number. We further demonstrate that for the streamwise constant perturbations, the spanwise force localized near the lower wall has, by far, the strongest effect on the evolution of the velocity field.

Stability analysis forn-periodic arrays of fluid systems

“…To this end, forcing is applied only to a specific subdomain of the flow or to only a particular state variable; likewise, only limited and user-specified output quantities are measured and assessed. In this manner, the strength and characteristics of a link between an input and output quantity can be probed, leading to more physical insight into the fluid system [22]. For example, the influence of fuel-mixture fluctuations in the feedpipe on the acoustic far-field output of a premixed flame could be quantified, to name a representative case from combustion dynamics for this type of analysis.…”

Closed-loop control of boundary layer streaks induced by free-stream turbulence

“…Substituting (47) into (44) and putting the resulting equation together with (45) in matrix form yields the following inhomogeneous (i.e., forced) Hamiltonian system:…”

“…This is known as the adjoint of the discretization [37]. Since it leads directly to the discrete form of the adjoint equations (45), it hides important physical properties of the full direct-adjoint system that can be revealed only by examining the differential form of the equations. In this section the differential form of the adjoint equations is derived, and it is proven that, under certain conditions, the adjoint and the controlled field have self-similar behavior for large κ.…”

“…Usually randomness appears additively as a stochastic body force on the right-hand side of an otherwise deterministic system and the independent variable is time (see, e.g., Refs. [44,45]). For the case examined in the paper we consider one particular frequency and the independent variable is x, so we would expect the form…”

Saturation of the response to stochastic forcing in two-dimensional backward-facing step flow: A self-consistent approximation

“…The optimal forcing and corresponding response are described by the singular vector of the resolvent operator R(ω) = (iω + L ) −1 [2,4]. Optimal forcing/response structures have been assessed in plane Couette [13] as well as in spatially developing open flows [8,10,11,14] and particularly in the backward-facing step [15][16][17][18][19]. A slightly different approach is undertaken by Garnaud, Lesshaftt, Schmid, and Huerre [14] where, in an attempt to describe more precisely the actual physics involved in the strong noise amplification exhibited in turbulent jets, they apply the optimal gain analysis on a model mean flow instead of the stable steady solution of the Navier-Stokes equations (NSEs) as in the previously mentioned studies.…”