Hostname: page-component-8448b6f56d-qsmjn Total loading time: 0 Render date: 2024-04-19T07:29:07.547Z Has data issue: false hasContentIssue false
  • English
  • Français

Localization of flow structures using $\infty $-norm optimization

Published online by Cambridge University Press:  24 July 2013

D. P. G. Foures ,
C. P. Caulfield  and
P. J. Schmid
D. P. G. Foures*
Affiliation:
DAMTP, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK
C. P. Caulfield
Affiliation:
DAMTP, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK BP Institute, University of Cambridge, Madingley Rise, Madingley Road, Cambridge CB3 0EZ, UK
P. J. Schmid
Affiliation:
Laboratoire d’Hydrodynamique (LadHyX), CNRS-Ecole Polytechnique, 91128 Palaiseau, France
*
†Email address for correspondence: dpgf2@cam.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

Stability theory based on a variational principle and finite-time direct-adjoint optimization commonly relies on the kinetic perturbation energy density ${E}_{1} (t)= (1/ {V}_{\Omega } )\int \nolimits _{\Omega } e(\boldsymbol{x}, t)\hspace{0.167em} \mathrm{d} \Omega $ (where $e(\boldsymbol{x}, t)= \vert \boldsymbol{u}{\vert }^{2} / 2$) as a measure of disturbance size. This type of optimization typically yields optimal perturbations that are global in the fluid domain $\Omega $ of volume ${V}_{\Omega } $. This paper explores the use of $p$-norms in determining optimal perturbations for ‘energy’ growth over prescribed time intervals of length $T$. For $p= 1$ the traditional energy-based stability analysis is recovered, while for large $p\gg 1$, localization of the optimal perturbations is observed which identifies confined regions, or ‘hotspots’, in the domain where significant energy growth can be expected. In addition, the $p$-norm optimization yields insight into the role and significance of various regions of the flow regarding the overall energy dynamics. As a canonical example, we choose to solve the $\infty $-norm optimal perturbation problem for the simple case of two-dimensional channel flow. For such a configuration, several solutions branches emerge, each of them identifying a different energy production zone in the flow: either the centre or the walls of the domain. We study several scenarios (involving centre or wall perturbations) leading to localized energy production for different optimization time intervals. Our investigation reveals that even for this simple two-dimensional channel flow, the mechanism for the production of a highly energetic and localized perturbation is not unique in time. We show that wall perturbations are optimal (with respect to the $\infty $-norm) for relatively short and long times, while the centre perturbations are preferred for very short and intermediate times. The developed $p$-norm framework is intended to facilitate worst-case analysis of shear flows and to identify localized regions supporting dominant energy growth.

JFM classification

Instability: Instability Mathematical Foundations: Mathematical Foundations Mathematical Foundations: Variational methods
Type
Papers
Information
Journal of Fluid Mechanics , Volume 729 , 25 August 2013 , pp. 672 - 701
Copyright
©2013 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arratia, C., Caulfield, C. P. & Chomaz, J.-M. 2013 Transient perturbation growth in time-dependent mixing layers. J. Fluid Mech. 717, 90–133.CrossRefGoogle Scholar
Blackburn, H. M., Barkley, D. & Sherwin, S. J. 2008 Convective instability and transient growth in flow over a backward-facing step. J. Fluid Mech. 603, 271–304.Google Scholar
Butler, K. M. & Farrell, B. F. 1992 Three-dimensional optimal perturbations in viscous shear flow. Phys. Fluids A 4, 1637–1650.Google Scholar
Cherubini, S., De Palma, P., Robinet, J.-Ch. & Bottaro, A. 2010 Rapid path to transition via nonlinear localized optimal perturbations in a boundary-layer flow. Phys. Rev. E 82, 066302.Google Scholar
Chorin, A. J. 1968 Numerical solution of the Navier–Stokes equations. Maths Comput. 22 (104), 745–762.CrossRefGoogle Scholar
Douglas, S. C., Amari, S. & Kung, S.-Y. 1998 Gradient adaptation under unit-norm constraints. In Proceedings of the Ninth IEEE SP Workshop on Statistical Signal and Array Processing, pp. 144–147.Google Scholar
Farrell, B. F. 1988 Optimal excitation of perturbations in viscous shear flow. Phys. Fluids 31, 2093–2102.Google Scholar
Farrell, B. F. & Ioannou, P. J. 1996 Generalized stability theory. Part II. Nonautonomous operators. J. Atmos. Sci. 53, 2041–2053.Google Scholar
Farrell, B. F. & Ioannou, P. J. 2003 Structural stability of turbulent jets. J. Atmos. Sci. 60, 2101–2118.Google Scholar
Foures, D. P. G., Caulfield, C. P. & Schmid, P. J. 2012 Variational framework for flow optimization using seminorm constraints. Phys. Rev. E 86, 026306.Google Scholar
Gustavsson, L. H. 1991 Energy growth of three-dimensional disturbances in plane Poiseuille flow. J. Fluid Mech. 224, 241–260.Google Scholar
Lindzen, R. S. 1988 Instability of plane parallel shear flow (toward a mechanistic picture of how it works). Pure Appl. Geophys. 126 (1), 103–121.Google Scholar
Marquet, O., Sipp, D., Chomaz, J.-M. & Jacquin, L. 2008 Amplifier and resonator dynamics of a low-Reynolds-number recirculation bubble in a global framework. J. Fluid Mech. 605, 429–443.CrossRefGoogle Scholar
Monokrousos, A., Bottaro, A., Brandt, L., Di Vita, A. & Henningson, D. S. 2011 Nonequilibrium thermodynamics and the optimal path to turbulence in shear flows. Phys. Rev. Lett. 106, 134502.Google Scholar
Orr, W. M’F. 1907 The stability or instability of the steady motions of a perfect liquid and of a viscous liquid. Part I. A perfect liquid. Part II. A viscous liquid. Proc. R. Irish Acad. Section A: Math. Phys. Sci. 27, 9–138.Google Scholar
Orszag, S. A. 1971 Accurate solution of the Orr–Sommerfeld stability equation. J. Fluid Mech. 50, 689–703.CrossRefGoogle Scholar
Polak, E. 1971 Computational Methods in Optimization: A Unified Approach, vol. 77. Academic.Google Scholar
Pringle, C. C. T. & Kerswell, R. R. 2010 Using nonlinear transient growth to construct the minimal seed for shear flow turbulence. Phys. Rev. Lett. 105, 154502.Google Scholar
Rabin, S. M. E., Caulfield, C. P. & Kerswell, R. R. 2012 Triggering turbulence efficiently in plane Couette flow. J. Fluid Mech. 712, 244–272.Google Scholar
Reddy, S. C. & Henningson, D. S. 1993 Energy growth in viscous channel flows. J. Fluid Mech. 252, 209–238.Google Scholar
Reddy, S. C., Schmid, P. J. & Henningson, D. S. 1993 Pseudospectra of the Orr–Sommerfeld operator. SIAM J. Appl. Maths. 53 (1), 15–47.Google Scholar
Schmid, P. J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129–162.Google Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.Google Scholar
Selle, A., Fedkiw, R., Kim, B., Liu, Y. & Rossignac, J. 2008 An unconditionally stable MacCormack method. J. Sci. Comput. 35 (2), 350–371.Google Scholar
Thomas, J. W. 1995 Numerical Partial Differential Equations: Finite Difference Methods, vol. 1. Springer.Google Scholar
Trefethen, L. N., Trefethen, A. E., Reddy, S. C. & Driscoll, T. A. 1993 Hydrodynamic stability without eigenvalues. Science 261, 578–584.CrossRefGoogle ScholarPubMed