Role of parasitic modes in nonlinear closure via the resolvent feedback loop

We use the feedback formulation of McKeon and Sharma [J. Fluid Mech. 658, 336 (2010)], where the nonlinear term in the Navier-Stokes equations is treated as an intrinsic forcing of the linear resolvent operator, to educe the structure of fluctuations in the range of scales (wave numbers) where linear mechanisms are not active. In this region, the absence of dominant linear mechanisms is reflected in the lack of low-rank characteristics of the resolvent and in the disagreement between the structure of resolvent modes and actual flow features. To demonstrate the procedure, we choose low Reynolds number cylinder flow and the Couette equilibrium solution EQ1, which are representative of very low-rank flows dominated by one linear mechanism. The former is evolving in time, allowing us to compare resolvent modes with dynamic mode decomposition (DMD) modes at the first and second harmonics of the shedding frequency. There is a match between the modes at the first harmonic but not at the second harmonic where there is no separation of the resolvent operator's singular values. We compute the self-interaction of the resolvent mode at the shedding frequency and illustrate its similarity to the nonlinear forcing of the second harmonic. When it is run through the resolvent operator, the “forced” resolvent mode shows better agreement with the DMD mode. A similar phenomenon is observed for the fundamental streamwise wave number of the EQ1 solution and its second harmonic. The importance of parasitic modes, labeled as such since they are driven by the amplified frequencies, is their contribution to the nonlinear forcing of the main amplification mechanisms as shown for the shedding mode, which has subtle discrepancies with its DMD counterpart.

Figure 1

Rows 1–3: Cylinder results. (a) The leading singular values computed for ${\mathit{k}}_h$, (b) the $u$ component of the optimal response mode ${\widehat{\mathbf{\psi }}}_1\left({\mathit{k}}_h\right)$, and (c) the corresponding DMD mode. (d) The leading singular values computed for $2{\mathit{k}}_h$, (e) the $u$ component of the optimal response mode ${\widehat{\mathbf{\psi }}}_1\left(2{\mathit{k}}_h\right)$, and (f) the corresponding DMD mode. (g) The self-interaction of ${\widehat{\mathbf{\psi }}}_1\left({\mathit{k}}_h\right)$ compared to the true nonlinear forcing $\widehat{\mathit{f}}\left(2{\mathit{k}}_h\right)$ in (h), and (i) the forced resolvent mode ($u$ component) for $2{\mathit{k}}_h$. Rows 4–6: EQ1 results. As in (a)–(i) but for EQ1.

Figure 2

The top of the schematic illustrates a variation of the self-sustaining process as viewed through the lens of resolvent analysis. The mean profile, fed into the resolvent operator, generates a highly amplified mode (obtained via the SVD) which self-interacts to generate the Reynolds stresses needed to sustain the mean and close the cycle. The saturated amplitude of the mode is obtained by considering the nonlinear interactions which give rise to higher harmonics and their subsequent feedback on the system, as shown in the bottom of the schematic.