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

  • O. Tammisola (a1), F. Giannetti (a2), V. Citro (a2) and M. P. Juniper (a1)

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 (first 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 second order in the perturbation, and show how the second-order eigenvalue changes can be computed numerically by overlapping the adjoint eigenfunction with the first-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 second order. Further analysis reveals that the second-order change in the eigenvalue arises through a resonance of the original (2-D) 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.

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Baumgärtel H. 1984 Perturbation Theory for Matrices and Operators, Licensed edition by Birkhäuser Verlag edn. Akademie Verlag.
Davis T. A. 2004 Algorithm 832: UMFPACK, an unsymmetric-pattern multifrontal method. Trans. Math. Software 30 (2), 196199.
Fischer P. F. 1997 An overlapping Schwarz method for spectral element solution of the incompressible Navier–Stokes equations. J. Comput. Phys. 133, 84101.
Giannetti F. & Luchini P. 2007 Structural sensitivity of the first instability of the cylinder wake. J. Fluid Mech. 581, 167197.
Del Guercio G., Cossu C. & Pujals G. 2014 Stabilizing effect of optimally amplified streaks in parallel wakes. J. Fluid Mech. 739, 3756.
Hinch E. J. 1991 Perturbation Methods. Cambridge University Press.
Hwang Y., Kim J. & Choi H. 2013 Stabilization of absolute instability in spanwise wavy two-dimensional wakes. J. Fluid Mech. 727, 346378.
Juniper M., Tammisola O. & Lundell F. 2011 The local and global stability of confined planar wakes at intermediate Reynolds number. J. Fluid Mech. 686, 218238.
Kim J. & Choi H. 2005 Distributed forcing of flow over a circular cylinder. Phys. Fluids 17, 033103.
Kolmogorov A. N. 1954 On the conservation of conditionally periodic motions under small perturbation of the Hamiltonian. Dokl. Akad. Nauk SSSR 74, 527530.
Lashgari I., Tammisola O., Citro V., Brandt L. & Juniper M. P. 2014 The planar ${X}$ -junction flow: stability and control. J. Fluid Mech. 753, 128.
Lehoucq R. B., Sorensen D. C. & Yang C. 1998 ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods. SIAM.
Lieu B. K., Moarref R. & Jovanovic M. R. 2010 Controlling the onset of turbulence by streamwise travelling waves. Part 2. Direct numerical simulation. J. Fluid Mech. 663, 100119.
Luchini P. & Bottaro A. 2014 Adjoint equations in stability analysis. Annu. Rev. Fluid Mech. 46, 493517.
Marquet O., Sipp D. & Jacquin L. 2008 Sensitivity analysis and passive control of cylinder flow. J. Fluid Mech. 615, 221252.
Maschhoff K. J. & Sorensen D. C. 1996 PARPACK: an efficient portable large scale eigenvalue package for distributed memory parallel architectures. In Proceedings of the Third International Workshop on Applied Parallel Computing, Industrial Computation and Optimization, pp. 478486. Springer.
Moarref R. & Jovanovic M. R. 2010 Controlling the onset of turbulence by streamwise travelling waves. Part 1. Receptivity analysis. J. Fluid Mech. 663, 7099.
Pralits J. O., Byström M., Hanifi A. & Henningson D. S. 2007 Optimal disturbances in three-dimensional boundary-layer flows. Ercoftac Bulletin 74, 2331.
Tammisola O., Lundell F., Schlatter P., Wehrfritz A. & Söderberg L. D. 2011 Global linear and nonlinear stability of viscous confined plane wakes with co-flow. J. Fluid Mech. 675, 397434.
Theofilis V. 2003 Advances in global linear instability analysis of nonparallel and three-dimensional flows. Prog. Aerosp. Sci. 39, 249315.
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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
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