Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-23T15:05:38.841Z Has data issue: false hasContentIssue false
  • English
  • Français

Hyperbolicity, shadowing directions and sensitivity analysis of a turbulent three-dimensional flow

Published online by Cambridge University Press:  28 January 2019

Angxiu Ni Open the ORCID record for Angxiu Ni [Opens in a new window]
Angxiu Ni*
Affiliation:
Department of Mathematics, University of California, Berkeley, Berkeley, CA 94720, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper uses compressible flow simulation to analyse the hyperbolicity, shadowing directions and sensitivities of a weakly turbulent three-dimensional cylinder flow at Reynolds number 525 and Mach number 0.1. By computing the first 40 covariant Lyapunov vectors (CLVs), we find that unstable CLVs are active in the near-wake region, whereas stable CLVs are active in the far-wake region. This phenomenon is related to hyperbolicity since it shows that CLVs point to different directions; it also suggests that for open flows there is a large fraction of CLVs that are stable. However, due to the extra neutral CLV and the occasional tangencies between CLVs, our system is not uniform hyperbolic. By the non-intrusive least-squares shadowing (NILSS) algorithm, we compute shadowing directions and sensitivities of long-time-averaged objectives. Our results suggest that shadowing methods may be valid for general chaotic fluid problems.

JFM classification

Nonlinear Dynamical Systems: Chaos Mixing: Chaotic advection Wakes/Jets: Wakes
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 863 , 25 March 2019 , pp. 644 - 669
Copyright
© 2019 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

Albers, D. J. & Sprott, J. C. 2006 Structural stability and hyperbolicity violation in high-dimensional dynamical systems. Nonlinearity 19 (8), 1801–1847.10.1088/0951-7715/19/8/005Google Scholar
Antali, M. & Horvath, Z. 2018 Computation of sensitivity of periodically excited dynamical systems. J. Engng Maths 113 (1), 123–142.10.1007/s10665-018-9977-3Google Scholar
Becker, R. & Rannacher, R. 2001 An optimal control approach to a posteriori error estimation in finite element methods. Acta Numerica 10, 1–102.10.1017/S0962492901000010Google Scholar
Benettin, G., Galgani, L., Giorgilli, A. & Strelcyn, J.-M. 1980 Lyapunov characteristic exponents for smooth dynamical systems and for hamiltonian systems; a method for computing all of them. Part 2. Numerical application. Meccanica 15 (1), 21–30.10.1007/BF02128237Google Scholar
Bewley, T. R. 2001 Flow control: new challenges for a new renaissance. Prog. Aerosp. Sci. 37 (1), 21–58.10.1016/S0376-0421(00)00016-6Google Scholar
Blonigan, P. J. 2017 Adjoint sensitivity analysis of chaotic dynamical systems with non-intrusive least squares shadowing. J. Comput. Phys. 348, 803–826.10.1016/j.jcp.2017.08.002Google Scholar
Blonigan, P. J., Fernandez, P., Murman, S. M. M., Wang, Q., Rigas, G. & Magri, L. 2016a Towards a Chaotic Adjoint for LES, pp. 385–394. Center for Turbulence Research, Summer Program.Google Scholar
Blonigan, P. J., Wang, Q., Nielsen, E. J. & Diskin, B. 2016b Least squares shadowing sensitivity analysis of chaotic flow around a two-dimensional airfoil. AIAA J. 56, 658–672.10.2514/1.J055389Google Scholar
Bowen, R. 1970 Markov partitions for Axiom A diffeomorphisms. Amer. J. Math. 92 (3), 725–747.10.2307/2373370Google Scholar
Brès, G. A., Ham, F. E., Nichols, J. W. & Lele, S. K. 2017 Unstructured large-eddy simulations of supersonic jets. AIAA J. 55 (4), 1164–1184.10.2514/1.J055084Google Scholar
Chorin, A. J., Hald, O. H. & Kupferman, R. 2002 Optimal prediction with memory. Physica D 166 (3–4), 239–257.Google Scholar
Colonius, T., Lele, S. K. & Moin, P. 1993 Boundary conditions for direct computation of aerodynamic sound generation. AIAA J. 31 (9), 1574–1582.10.2514/3.11817Google Scholar
Fernandez, P. & Wang, Q.2017a. Lyapunov spectrum of chaotic flow simulations. Application to chaotic adjoints. AIAA Paper 2017–3797.Google Scholar
Fernandez, P. & Wang, Q. 2017b Lyapunov spectrum of the separated flow around the NACA 0012 airfoil and its dependence on numerical discretization. J. Comput. Phys. 350, 453–469.10.1016/j.jcp.2017.08.056Google Scholar
Fidkowski, K. J. & Darmofal, D. L. 2011 Review of output-based error estimation and mesh adaptation in computational fluid dynamics. AIAA J. 49 (4), 673–694.10.2514/1.J050073Google Scholar
Frederickson, P., Kaplan, J. L., Yorke, E. D. & Yorke, J. A. 1983 The Liapunov dimension of strange attractors. J. Differ. Equ. 49 (2), 185–207.10.1016/0022-0396(83)90011-6Google Scholar
Gallavotti, G. 2006 Entropy, thermostats, and chaotic hypothesis. Chaos 16 (4), 043114.10.1063/1.2372713Google Scholar
Gallavotti, G. 2008 Chaotic hypothesis. Scholarpedia 3(1):5906 3 (1), 8–11.Google Scholar
Gallavotti, G. & Cohen, E. G. D. 1995 Dynamical ensembles in stationary states. J. Stat. Phys. 80 (5–6), 931–970.10.1007/BF02179860Google Scholar
Giles, M. B. & Süli, E. 2002 Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality. Acta Numerica 11, 145–236.10.1017/S096249290200003XGoogle Scholar
Ginelli, F., Chaté, H., Livi, R. & Politi, A. 2013 Covariant Lyapunov vectors. J. Phys. A 46 (25), 254005.Google Scholar
Ginelli, F., Poggi, P., Turchi, A., Chaté, H., Livi, R. & Politi, A. 2007 Characterizing dynamics with covariant Lyapunov vectors. Phys. Rev. Lett. 99 (13), 1–4.10.1103/PhysRevLett.99.130601Google Scholar
Inubushi, M., Kobayashi, M. U., Takehiro, S. I. & Yamada, M. 2012 Covariant Lyapunov analysis of chaotic Kolmogorov flows and time-correlation function. Proc. IUTAM 5, 244–248; Published by Elsevier.10.1016/j.piutam.2012.06.033Google Scholar
Jameson, A. 1988 Aerodynamic design via control theory. J. Sci. Comput. 3 (3), 233–260.10.1007/BF01061285Google Scholar
Katok, A. B. & Hasselblatt, B. A. 1997 Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, vol. 54. Cambridge University Press.Google Scholar
Keefe, L., Moin, P. & Kim, J. 1992 The dimension of attractors underlying periodic turbulent Poiseuille flow. J. Fluid Mech. 242 (29), 1–29.10.1017/S0022112092002258Google Scholar
Mittal, R. & Balachandar, S. 1996 Direct numerical simulation of flow past elliptic cylinders. J. Comput. Phys. 367 (124), 351–367.10.1006/jcph.1996.0065Google Scholar
Ni, A.2018a Adjoint sensitivity analysis on chaotic dynamical systems by non-intrusive least squares adjoint shadowing (NILSAS), arXiv:1801.08674, pp. 1–44.Google Scholar
Ni, A.2018b Adjoint shadowing directions in hyperbolic systems for sensitivity analysis. arXiv:1807.05568, pp. 1–23.Google Scholar
Ni, A., Blonigan, P. J., Chater, M., Wang, Q. & Zhang, Z.2016 Sensitivity analysis on chaotic dynamical system by non-intrusive least square shadowing (NI-LSS). AIAA Paper 2016–4399.Google Scholar
Ni, A. & Wang, Q. 2017 Sensitivity analysis on chaotic dynamical systems by non-intrusive least squares shadowing (NILSS). J. Comput. Phys. 347, 56–77.10.1016/j.jcp.2017.06.033Google Scholar
Ni, A., Wang, Q., Fernandez, P. & Talnikar, C.2018 Sensitivity analysis on chaotic dynamical systems by finite difference non-intrusive least squares shadowing (FD-NILSS), arXiv:1711.06633, pp. 1–21.Google Scholar
Oseledets, V. I. 1968 A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems. Tr. Moskov. Mat. Obshchestva 19, 179–210.Google Scholar
Oseledets, V. I. 2008 Oseledets theorem. Scholarpedia 3 (1), 1846.10.4249/scholarpedia.1846Google Scholar
Parish, E. J. & Duraisamy, K. 2017 A dynamic subgrid scale model for large eddy simulations based on the Mori–Zwanzig formalism. J. Comput. Phys. 349, 154–175.10.1016/j.jcp.2017.07.053Google Scholar
Pilyugin, S. Y. 1997 Shadowing in structurally stable flows. J. Differ. Equ. 140 (2), 238–265.10.1006/jdeq.1997.3295Google Scholar
Pilyugin, S. Y. 1999 Shadowing in Dynamical Systems, Lecture Notes in Mathematics, vol. 1706. Springer.Google Scholar
Poinsot, T. J. & Lelef, S. K. 1992 Boundary conditions for direct simulations of compressible viscous flows. J. Comput. Phys. 101 (1), 104–129.10.1016/0021-9991(92)90046-2Google Scholar
Reick, C. H. 2002 Linear response of the Lorenz system. Phys. Rev. E 66 (3), 1–11.Google Scholar
Rezakhanlou, F. 2018 Lectures on Dynamical Systems. Department of Mathematics, University of California, Berkeley.Google Scholar
Ruelle, D. 1980 Measures describing a turbulent flow. Ann. N.Y. Acad. Sci. 357 (1), 1–9.10.1111/j.1749-6632.1980.tb29669.xGoogle Scholar
Ruelle, D. 1997 Differentiation of SRB states. Commun. Math. Phys. 187, 227–241.10.1007/s002200050134Google Scholar
Ruelle, D. 2008 Differentiation of SRB states for hyperbolic flows. Ergod. Theory Dyn. Sys. 28 (02), 613–631.Google Scholar
Sieber, M. 1987 Experiments on the attractor-dimension for turbulent pipe flow. Phys. Lett. A 122 (9), 467–470.10.1016/0375-9601(87)90868-1Google Scholar
Thepaut, J.-N. & Courtier, P. 1991 Four-dimensional variational data assimilation using the adjoint of a multilevel primitive-equation model. Q. J. R. Meteorol. Soc. 117 (502), 1225–1254.10.1002/qj.49711750206Google Scholar
Tromp, J., Tape, C. & Liu, Q. 2005 Seismic tomography, adjoint methods, time reversal and banana-doughnut kernels. Geophys. J. Intl 160, 195–216.10.1111/j.1365-246X.2004.02453.xGoogle Scholar
Wang, Q. 2014 Convergence of the least squares shadowing method for computing derivative of ergodic averages. SIAM J. Numer. Anal. 52 (1), 156–170.10.1137/130917065Google Scholar
Wang, Q., Hu, R. & Blonigan, P. 2014 Least squares shadowing sensitivity analysis of chaotic limit cycle oscillations. J. Comput. Phys. 267, 210–224.10.1016/j.jcp.2014.03.002Google Scholar
Williamson, C. H. K. & Roshko, A. 1990 Measurements of base pressure in the wake of a cylinder at low Reynolds numbers. Z. Flugwiss. Weltraumforsch. 14, 38–46.Google Scholar
Xu, M. & Paul, M. R. 2016 Covariant Lyapunov vectors of chaotic Rayleigh–Bénard convection. Phys. Rev. E 93 (6), 1–12.Google Scholar
Young, L.-S. 2002 What are SRB measures, and which dynamical systems have them? J. Stat. Phys. 108 (5), 733–754.10.1023/A:1019762724717Google Scholar
Young, L.-S. 2003 Entropy in dynamical systems. In Entropy, (ed. Greven, Keller & Warnecke), pp. 313–328. Princeton University Press.Google Scholar
Supplementary material: File

Ni supplementary material

Ni supplementary material 1

Download Ni supplementary material(File)
File 5.1 MB