Abstract
We advance the computation of physical modal expansions for unsteady incompressible flows. Point of departure is a linearization of the Navier–Stokes equations around its fixed point in a frequency domain formulation. While the most amplified stability eigenmode is readily identified by a power method, the technical challenge is the computation of more damped higher-order eigenmodes. This challenge is addressed by a novel method to compute unstable periodically forced solutions of the linearized Navier–Stokes solution. This method utilizes two key enablers. First, the linear dynamics is transformed by a complex shift of the eigenvalues amplifying the flow response at the given frequency of interest. Second, the growth rate is obtained from an iteration procedure. The method is demonstrated for several wake flows around a circular cylinder, a fluidic pinball, i.e. the wake behind a cluster of cylinders, a wall-mounted cylinder, a sphere and a delta wing. The example of flow control with periodic wake actuation and forced physical modes paves the way for applications of physical modal expansions. These results encourage Galerkin models of three-dimensional flows utilizing Navier–Stokes based modes.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Landau, L.D.: On the problem of turbulence. C.R. Acad. Sci. USSR 44, 311–314 (1944)
Hopf, E.: A mathematical example displaying features of turbulence. Commun. Pure Appl. Math. 1, 303–322 (1948)
Feigenbaum, M.J.: Quantitative universality for a class of nonlinear transformations. J. Stat. Phys. 19, 25–52 (1978)
Ruelle, D., Takens, F.: On the nature of turbulence. Commun. Math. Phys. 20, 167–192 (1971)
Newhouse, S., Ruelle, D., Takens, F.: Occurence of strange Axiom-A attractors near quasiperiodic flow on \(t^m\), \(m \le 3\). Commun. Math. Phys. 64, 35 (1978)
Pomeaou, Y., Manneville, P.: Intermittent transition to turbulence in dissipative dynamical systems. Commun. Math. Phys. 74, 189–197 (1980)
Åkervik, E., Hœpffner, J., Ehrenstein, U., Henningson, D.S.: Optimal growth, model reduction and control in separated boundary-layer flow using global eigenmodes. J. Fluid Mech. 579, 305–314 (2007)
Noack, B.R., Afanasiev, K., Morzyński, M., Tadmor, G., Thiele, F.: A hierarchy of low-dimensional models for the transient and post-transient cylinder wake. J. Fluid Mech. 497, 335–363 (2003)
Grosch, C.E., Salwen, H.: The continuous spectrum of the Orr–Sommerfeld equation. Part I. The spectrum and the eigenfunctions. J. Fluid Mech. 87, 33–54 (1978)
Salwen, H., Grosch, C.E.: The continuous spectrum of the Orr–Sommerfeld equation. Part 2. Eigenfunction expansions. J. Fluid Mech. 104, 445–465 (1981)
Taira, K., Brunton, S.L., Dawson, S., Rowley, C.W., Colonius, T., McKeon, B.J., Schmidt, O.T., Gordeyew, S., Theofilis, V., Ukeiley, L.S.: Modal analysis of fluid flows: an overview. AIAA. J. 55(12), 4013–4041 (2017)
Theofilis, V.: Advances in global linear instability analysis of nonparallel and three-dimensional flows. Prog. Aerosp. Sci. 39(4), 249–315 (2003)
Theofilis, V.: Global linear instability. Ann. Rev. Fluid Mech. 43, 319–352 (2011)
Wolter, D., Morzyński, M., Schütz, H., Thiele, F.: Numerische Untersuchungen zur Stabilität der Kreiszylinderumströmung. Z. Angew. Math. Mech. 69, T601–T604 (1989)
Morzyński, M., Afanasiev, K., Thiele, F.: Solution of the eigenvalue problems resulting from global non-parallel flow stability analysis. Comput. Methods Appl. Mech. Eng. 169(1), 161–176 (1999)
Morzyński, M., Thiele, F.: 3D FEM global stability analysis of viscous flow. Parallel Processing and Applied Mathematics. Lecture Notes in Computer Science, vol. 4967, pp. 1293–1302. Springer, Berlin (2008)
Gómez, F., Pérez, J.M., Blackburn, H.M., Theofilis, V.: On the use of matrix-free shift-invert strategies for global flow instability analysis. Aerosp. Sci. Technol. 44, 69–76 (2015)
Liu, Q., Gómez, F., Pérez, J.M., Theofilis, V.: Instability and sensitivity analysis of flows using openfoam. Chin. J. Aeronaut. 29(2), 316–325 (2016)
Semeraro, O., Bagheri, S., Brandt, L., Henningson, D.S.: Feedback control of three-dimensional optimal disturbances using reduced-order models. J. Fluid Mech. 677, 63–102 (2011)
Garnaud, X., Lesshafft, L., Schmid, P.J., Huerre, P.: The preferred mode of incompressible jets: linear frequency response analysis. J. Fluid Mech. 716, 189–202 (2013)
Åkervik, E., Hœpffner, J., Ehrenstein, U., Henningson, D.S.: Optimal growth, model reduction and control in a separated boundary-layer flow using global eigenmodes. J. Fluid Mech. 579, 305–314 (2007)
Chomaz, J.M.: Global instabilities in spatially developing flows: non-normality and nonlinearity. Ann. Rev. Fluid Mech. 37, 357–392 (2005)
Qu, Z.Q.: Model Order Reduction Techniques with Applications in Finite Element Analysis. Springer Science & Business Media, Berlin (2013)
Wilson, E.L., Yuan, M.W., Dickens, J.M.: Dynamic analysis by direct superposition of Ritz vectors. Earthq. Eng. Struct. Dyn. 10(6), 813–821 (1982)
Taylor, C., Hood, P.: A numerical solution of the Navier–Stokes equations using the finite element technique. Comput. Fluids 1(1), 73–100 (1973)
Noack, B.R., Morzyński, M.: The fluidic pinball — a toolkit for multiple-input multiple-output flow control (version 1.0). Technical report 02/2017, Chair of Virtual Engineering, Poznan University of Technology, Poland (2017)
Åkervik, E., Brandt, L., Henningson, D.S., Hœpffner, J., Marxen, O., Schlatter, P.: Steady solutions of the Navier–Stokes equations by selective frequency damping. Phys. Fluids 18(6), 068102 (2006)
Thiria, B., Goujon-Durand, S., Wesfreid, J.E.: The wake of a cylinder performing rotary oscillations. J. Fluid Mech. 560, 123–147 (2006)
Cornejo Maceda, G.Y.: Machine learning control applied to wake stabilization. M2 Master of Science Internship Report, LIMSI and ENSAM, Paris, France (2017)
Tadmor, G., Lehmann, O., Noack, B.R., Cordier, L., Delville, J., Bonnet, J.P., Morzyński, M.: Reduced-order models for closed-loop wake control. Philos. Trans. R. Soci. Lond. A Math. Phys. Eng. Sci. 369(1940), 1513–1524 (2011)
Brunton, S.L., Noack, B.R.: Closed-loop turbulence control: progress and challenges. Appl. Mech. Rev. 67(5), 050801:01–050801:48 (2015)
Luchtenburg, D.M., Günter, B., Noack, B.R., King, R., Tadmor, G.: A generalized mean-field model of the natural and actuated flows around a high-lift configuration. J. Fluid Mech. 623, 283–316 (2009)
Luchini, P., Bottaro, A.: Adjoint equations in stability analysis. Ann. Rev. Fluid Mech. 46(1), 493–517 (2014)
Michalke, A.: On spatially growing disturbances in an inviscid shear layer. J. Fluid Mech. 23, 521–561 (1965)
Barkley, D.: Linear analysis of the cylinder wake mean flow. Europhys. Lett. 75, 750–756 (2006)
Noack, B.R., Schlegel, M., Ahlborn, B., Mutschke, G., Morzyński, M., Comte, P., Tadmor, G.: A finite-time thermodynamics of unsteady fluid flows. J. Non-Equilibr. Thermodyn. 33, 103–148 (2008)
Acknowledgements
The authors acknowledge support by the Polish National Science Center (NCN) under the Grant No.: DEC-2011/01/B/ST8/07264 and by the Polish National Center for Research and Development under the Grant No. PBS3/B9/34/2015 and travel support of the Bernd Noack Cybernetics Foundation.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer International Publishing AG, part of Springer Nature
About this chapter
Cite this chapter
Morzyński, M., Szeliga, W., Noack, B.R. (2019). Unstable Periodically Forced Navier–Stokes Solutions–Towards Nonlinear First-Principle Reduced-Order Modeling of Actuator Performance. In: Gelfgat, A. (eds) Computational Modelling of Bifurcations and Instabilities in Fluid Dynamics. Computational Methods in Applied Sciences, vol 50. Springer, Cham. https://doi.org/10.1007/978-3-319-91494-7_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-91494-7_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-91493-0
Online ISBN: 978-3-319-91494-7
eBook Packages: EngineeringEngineering (R0)