Second-order perturbation of global modes and implications for spanwise wavy actuation

Abstract

Sensitivity analysis has successfully located the most efficient regions in which to apply passive control in many globally unstable flows. As is shown here and in previous studies, the standard sensitivity analysis, which is linear (1 st order) with respect to the actuation amplitude, predicts that steady spanwise wavy alternating actuation/modification has no effect on the stability of planar flows, because the eigenvalue change integrates to zero in the spanwise direction. In experiments however, spanwise wavy modification has been shown to stabilize the flow behind a cylinder quite efficiently.In this paper, we generalize sensitivity analysis by examining the eigenvalue drift (including stabilization/destabilization) up to 2 nd order in the perturbation, and show how the 2 nd order eigenvalue changes can be computed numerically by overlapping the adjoint eigenfunction with the 1 st order global eigenmode correction, shown here for the first time. We confirm the prediction against a direct computation, showing that the eigenvalue drift due to a spanwise wavy base flow modification is of 2 nd order. Further analysis reveals that the 2 nd order change in the eigenvalue arises through a resonance of the original (2D) eigenmode with other unperturbed eigenmodes that have the same spanwise wavelength as the base flow modification. The eigenvalue drift due to each mode interaction is inversely proportional to the distance between the eigenvalues of the modes (which is similar to resonance), but also depends on mutual overlap of direct and adjoint eigenfunctions (which is similar to pseudoresonance). By this argument, and by calculating the most sensitive regions identified by our analysis, we explain why an in-phase actuation/modification is better than an out-of-phase actuation for control of wake flows by spanwise wavy suction and blowing. We also explain why wavelengths several times longer than the wake thickness are more efficient than short wavelengths.

Breaking axi-symmetry in stenotic flow lowers the critical transition Reynolds number

“…The analysis shows that the growth rate of the leading eigenvalue increases with eccentricity. Further, the eigenvalue drift seems to be of second order, 17 demonstrating an application of the theory on second order sensitivity developed by Tammisola et al 16 Extrapolating from a quadratic least square fit at Re = 400 shows that the mode becomes unstable at around E = 0.17%, which matches the discontinuity in Figure 3(b). An identical study performed for Re = 350 predicts the leading mode to become unstable at E = 0.27%, which is also where the discontinuity occurs in Figure 3(b).…”

“…The full three-dimensional eigenpairs of the linearized Navier-Stokes operator are computed using the linearized DNS time-stepper (available in Nek5000) coupled with the implicit restart Arnoldi method implemented in PARPACK. 15 This matrix-free global stability solver implementation was also used in Tammisola et al 16 and was there verified against a FreeFem++ based solver.…”

Second-order sensitivity in the cylinder wake: Optimal spanwise-periodic wall actuation and wall deformation

“…The second-order sensitivity tensor has been explicitly computed in [26] for the Ginzburg-Landau equation, and in [27] and [1] for parallel flows. More specifically, in our previous study [1] we computed optimal spanwise-periodic base flow modifications for parallel flows by rewriting and manipulating the second-order perturbation system into a Hessian matrix form.…”

The linearized pressure Poisson equation for global instability analysis of incompressible flows

“…The latter is solved in FreeFEM++ by a matrix-forming approach, while in nektar++ and Semtex [7] the time-stepping algorithm discussed by Barkley et al [5] is used. The FreeFEM++ based code used for the present instability analysis has been presented by Tammisola et al [53] and was further validated by Lashgari et al [32]. It uses an unstructured mesh comprising a total of 29,132 triangles and 14,787 vertices has been used to spatially discretize both the base flow of the regularized cavity (14) and the corresponding global eigenvalue problem.…”

“…The analogous procedure of weak formulation of the equations of motion, followed in the finite-element package FreeFEM++ [25], ensures that no issues exist regarding appropriate boundary conditions for the pressure perturbation, when this package is adapted to perform global linear instability analysis [12,17,32,53].…”