Hostname: page-component-7c8c6479df-7qhmt Total loading time: 0 Render date: 2024-03-28T14:32:12.926Z Has data issue: false hasContentIssue false

Low-dimensional description of free-shear-flow coherent structures and their dynamical behaviour

Published online by Cambridge University Press:  26 April 2006

Mojtaba Rajaee
Affiliation:
Center for Fluid Mechanics and the Division of Engineering, Brown University, Providence, RI 02912, USA
Sture K. F. Karlsson
Affiliation:
Center for Fluid Mechanics and the Division of Engineering, Brown University, Providence, RI 02912, USA
Lawrence Sirovich
Affiliation:
Center for Fluid Mechanics and the Division of Engineering, Brown University, Providence, RI 02912, USA

Abstract

The snapshot form of the Karhunen-Loéve (K–L) expansion has been applied to twodimensional, two-component hot-wire data from the region of a weakly pertubed free shear layer that includes the first pairing process. Low-level external perturbation was provided by a loudspeaker driven by a computer-generated signal composed of two sine waves of equal amplitude at the frequencies of the naturally developing fundamental instability wave and its first subharmonic, separated by a controllable initial phase angle difference. It was found that a large fraction of the fluctuation energy is carried by the first few modes. A low-dimensional empirical eigenfunction space is obtained which describes the shear-flow coherent structures well. Galerkin projection of the Navier-Stokes equations onto this basis set of principal eigenfunction modes results in a low-order system of dynamical equations, and solution of this system of equations describes the dynamics of the coherent structures associated with eigenfunctions. Finally the simulation, as obtained from the system of dynamical equations, is shown to compare reasonably well with the experiments.

Type
Research Article
Copyright
© 1994 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

Ahmed, N. & Goldstein, M. H. 1975 Orthogonal Transforms for Digital Signal Processing. Springer.
Ash, R. B. & Gardner, M. F. 1975 Topics in Stochastic Processes. Academic.
Aubry, N., Holmes, P., Lumley, J. L. & Stone, E. 1988 The dynamics of coherent structures in the wall region of a turbulent boundary layer. J. Fluid Mech. 192, 115.Google Scholar
Bakewell, P. & Lumley, J. L. 1967 Viscous sublayer and adjacent wall region in turbulent pipe flow. Phys. Fluids 10, 1880.Google Scholar
Browand, F. K. & Weidman, P. D. 1976 Large scales in the developing mixing layer. J. Fluid Mech. 76, 127.Google Scholar
Brown, G. L. & Roshko, A. 1974 On density effects and large structure in turbulent mixing layers. J. Fluid Mech. 64, 775.Google Scholar
Dimotakis, P. & Brown, G. 1976 The mixing layer at high Reynolds number: large-structure dynamics and entrainment. J. Fluid Mech. 78, 535.Google Scholar
Fukunaga, K. 1972 Introduction to Statistical Pattern Recognition. Academic.
Fukuoka, A. A. 1951 A study on 10-day forecast (A synthetic report). Geophys. Mag. 177Google Scholar
Glauser, A., Lieb, S. J. & George, N. K. 1987 Coherent structures in the axisymmetric turbulent jet mixing layer. In Turbulent Shear Flows 5 (ed. F. Durst, B. E. Launder, J. L. Lumley, et al.), p. 134. Springer.
Glezer, A., Kadioglu, Z. & Pearlstein, A. J. 1989 Development of an extended proper orthogonal decomposition and its application to a time periodically forced plane mixing layer. Phys. Fluids A 1, 1363.Google Scholar
Herzog, S. 1986 The large scale structures in the near wall region of turbulent pipe flow PhD Thesis, Cornell University.
Ho, C.-M. & Huang, L. S. 1982 Subharmonics and vortex merging in mixing layers. J. Fluid Mech. 119, 443.Google Scholar
Ho, C.-M. & Huerre, P. 1984 Perturbed free shear layers. Ann. Rev. Fluid Mech. 16, 365.Google Scholar
Huang, L. S. & Ho, C.-M. 1990 Small-scale transition in a plane mixing layer. J. Fluid Mech. 210, 475.Google Scholar
Kawakubo, T. 1990 Characteristic-eddy decomposition of turbulent mixing layer. Proc. 17th Intl Symp. on Space Technology and Science, p. 2337. Tokyo: Japan Publications Trading Co.
Kutzbach, J. E. 1967 Empirical eigenvectors of sea-level pressure, surface temperature and precipitation complexes over North America. J. Appl. Met. 6, 791.Google Scholar
Loéave, M. 1955 Probability Theory. Van Nostrand.
Lorenz, E. N. 1956 Empirical orthogonal functions and statistical weather prediction. Sci. Rep. No. I, Statistical Forecasting Project, Department of Metrology, MIT.
Lumley, J. L. 1967 The structure of inhomogeneous turbulent flows. In Atmospheric Turbulence and Radio Wave Propagation (ed. A. M. Yaglom V. I. Tatarski), p. 166. Moscow: Nauka.
Lumley, J. L. 1981 Coherent structures in turbulence. In Transition and Turbulence (ed. R. E. Meyer), p. 215. Academic.
Michalke, A. & Hermann, J. 1982 On the inviscid instability of a circular jet with external flow. J. Fluid Mech. 114, 343.Google Scholar
Moin, P. 1984 Probing turbulence via large eddy simulation, AIAA 22nd Aerospace Sciences Meeting.
Mollo-Christensen, E. 1971 Physics of turbulent flow. AIAA J. 9, 1217.Google Scholar
Monkewitz, P. A. & Huerre, P. 1982 Influence of the velocity ratio on the spatial instability of mixing layers. Phys. Fluids 25, 1137.Google Scholar
Payne, F. R. & Lumley, J. L. 1967 Large eddy structure of the turbulent wake behind a circular cylinder. Phys. Fluids 10, S194.Google Scholar
Preisendofer, R. 1988 Principal Component Analysis in Meteorology and Oceanography. Elsevier.
Rajaee, M. 1991 Measurement and mathematical analysis of free shear flow PhD Thesis, Brown University.
Rajaee, M. & Karlsson, S. K. F. 1990 Shear flow coherent structures via Karhunen-Loéve expansion. Phys. Fluids A 2, 2249.Google Scholar
Rajaee, M. & Karlsson, S. K. F. 1992 On the Fourier space decomposition of free shear flow measurements and mode degeneration in the pairing process. Phys. Fluids A 4, 321.Google Scholar
Sirovich, L. 1987a Turbulence and the dynamics of coherent structures, Part 1: Coherent structures. Q. Appl. Maths 45/3, 561.Google Scholar
Sirovich, L. 1987b Turbulence and the dynamics of coherent structures, Part 2: Symmetries and transformations. Q. Appl. Maths 45/3, 573.Google Scholar
Sirovich, L. 1987c Turbulence and the dynamics of coherent structures, Part 3: Dynamics and scaling. Q. Appl. Maths 45/3, 583.Google Scholar
Sirovich, L. 1989 Chaotic dynamics of coherent structures. Physica D 37, 126.Google Scholar
Sirovich, L. 1991 Analysis of turbulent flows by means of the empirical eigenfunctions. Fluid Dyn. Res. 8, 85.Google Scholar
Sirovich, L. & Everson, R. 1992 Management and analysis of large scientific datasets. Int J. Supercomput. Applics. 6, 50.Google Scholar
Sirovich, L., Kirby, M. & Winter, M. 1990 An eigenfunction approach to large scale transitional structures in jet flow. Phys. Fluids A 2, 127.Google Scholar
Sirovich, L., Maxey, M. & Tarman, H. 1989 An eigenfunction analysis of turbulent thermal convection. In Turbulent Shear Flows 5 (ed. J.-C. André, J. Cousteix, F. Durst et al.), p. 68. Springer.
Sirovich, L. & Rodriguez, J. D. 1987 Coherent structures and chaos: A model problem. Phys. Lett. A 120, 211.Google Scholar
Sirovich, L. & Sirovich, H. 1989 Low dimensional description of complicated phenomena. Contemp. Maths 99, 277.Google Scholar
White, R. M., Cooley, D. S., Derby, R. C. & Seaver, F. A. 1958 The development of efficient linear statistical operators for the prediction of sea-level pressure. J. Met. 15, 426.Google Scholar
Winant, C. D. & Browand, F. K. 1974 Vortex pairing the mechanism of turbulent mixing layer growth at moderate Reynolds number. J. Fluid Mech. 63, 237.Google Scholar
Yang, Z. & Karlsson, S. K. F. 1991 Evolution of coherent structures in a plane shear layer. Phys. Fluids A 3, 2207.Google Scholar
Zhou, X. & Sirovich, L. 1992 Coherence and chaos in a model of turbulent boundary layer. Phys. Fluids A 4, 28552874.Google Scholar