Hostname: page-component-77c89778f8-swr86 Total loading time: 0 Render date: 2024-07-19T02:28:12.618Z Has data issue: false hasContentIssue false
  • English
  • Français

Two-point similarity in temporally evolving plane wakes

Published online by Cambridge University Press:  19 April 2007

D. EWING ,
W. K. GEORGE ,
M. M. ROGERS  and
R. D. MOSER
D. EWING
Affiliation:
Department of Mechanical Engineering, McMaster University, Hamilton, ON, Canada
W. K. GEORGE
Affiliation:
Department of Applied Mechanics, Chalmers University of Technology, Gothenburg, Sweden
M. M. ROGERS
Affiliation:
NASA Ames Research Center, Moffett Field, CA, USA
R. D. MOSER
Affiliation:
Department of Mechanical Engineering and Institute for Computational Engineering and Sciences, University of Texas at Austin, 1 University Station, Stop C22, Austin, TX, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

The governing equations for the two-point correlations of the turbulent fluctuating velocity in the temporally evolving wake were analysed to determine whether they could have equilibrium similarity solutions. It was found that these equations could have such solutions for a finite-Reynolds-number wake, where the two-point velocity correlations could be written as a product of a time-dependent scale and a function dependent only on similarity variables. It is therefore possible to collapse the two-point measures of all the scales of motions in the temporally evolving wake using a single set of similarity variables. As in an earlier single-point analysis, it was found that the governing equations for the equilibrium similarity solutions could not be reduced to a form that was independent of a growth-rate dependent parameter. Thus, there is not a single ‘universal’ solution that describes the state of the large-scale structures, so that the large-scale structures in the far field may depend on how the flow is generated.

The predictions of the similarity analysis were compared to the data from two direct numerical simulations of the temporally evolving wakes examined previously. It was found that the two-point velocity spectra of these temporally evolving wakes collapsed reasonably well over the entire range of scales when they were scaled in the manner deduced from the equilibrium similarity analysis. Thus, actual flows do seem to evolve in a manner consistent with the equilibrium similarity solutions.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 577 , 25 April 2007 , pp. 287 - 307
Copyright
Copyright © Cambridge University Press 2007

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

REFERENCES

Antonia, R. A., Browne, L. W. B., Bisset, D. K. & Fulachier, L. 1987 A description of the organized motion in the turbulent far wake of a cylinder at low Reynolds number. J. Fluid Mech. 184, 423–444.CrossRefGoogle Scholar
Batchelor, G. K. 1948 Energy decay and self-preserving correlation functions in isotropic turbulence. Q. Appl. Maths 6, 97–116.CrossRefGoogle Scholar
Bonnet, J. P., Delville, J. & Garem, H. 1986 Space and space-time longitudinal velocity correlations in the turbulent far wake of a flat plate in incompressible flow. Exps. Fluids 4, 189–196.Google Scholar
Brede, M., Leder, A. & Westergaard, C. 2003 Time-resolved PIV investigation of the separated shear layer in the transitional cylinder wake. In Proc. 5th Intl Symp. on Particle Image Velocimetry PIV 03 (ed. Kim, K. C.), p. 3215. The Korean Society of Visualization, Pusan, Korea.Google Scholar
Durbin, P. A. & Reif, B. A. P. 2001 Statistical Theory and Modeling for Turbulent Flows. John Wiley.Google Scholar
Ewing, D. 1995 On multi-point similarity solutions in turbulent free-shear flows. PhD thesis, State University of New York at Buffalo.Google Scholar
Ewing, D. 2001 Evolution of the large-scale structures in the far-field of turbulent shear flows. In Coherent Structures in Complex System (ed. Regura, D., Bonilla, L. & Rubi, J.), Lecture Notes in Physics vol. 567, pp. 92–102. Springer.Google Scholar
Ewing, D. & George, W. K. 1995 Similarity analysis of the two-point velocity correlation tensor in a turbulent axisymmetric jet. In Turbulence, Heat, and Mass Transfer I (ed. Hanjalic, K. & Pereira, J. C. F.), pp. 49–56. Begell House.Google Scholar
Ewing, D. W., George, W. K., Moser, R. D. & Rogers, M. M. 1995 A similarity hypothesis for the two-point correlation tensor in a temporally evolving plane wake. In Annual Research Brief 1995, pp. 163–174. Stanford University: Center for Turbulence Research.Google Scholar
Ewing, D., George, W. K., Moser, R. D. & Rogers, M. M. 1996 A similarity hypothesis for the two-point velocity correlations in a temporally evolving wake. In Advances in Turbulence VI (ed. Gavrilakis, S., Machiels, L. & Monkewitz, P.), pp. 223–226. Kluwer.CrossRefGoogle Scholar
Gamard, S., Jung, D. & George, W. K. 2004 Downstream evolution of the most energetic modes in a turbulent axisymmetric jet at high Reynolds number. Part 2. The far-field region. J. Fluid Mech. 514, 205–230.Google Scholar
George, W. K. 1989 The self-preservation of turbulent flows and its relation to initial conditions and coherent structures. In Advances in Turbulence (ed. George, W. K. & Arndt, R. E.), pp. 39–73. Hemisphere.Google Scholar
George, W. K. 1992 The decay of homogeneous isotropic turbulence. Phys. Fluids A 4, 1492–1507.CrossRefGoogle Scholar
George, W. K. 1995 Some new ideas for similarity of turbulent shear flows. In Turbulence, Heat, and Mass Transfer I (ed. Hanjalic, K. & Pereira, J. C. F.), pp. 13–24. New York: Begell House.Google Scholar
George, W. K. & Gibson, M. M. 1992 The self preservation of homogeneous shear flow turbulence. Exps. Fluids 13, 229–238.CrossRefGoogle Scholar
George, W. K., Beuther, P. D. & Lumley, J. L. 1978 Processing of random signals. In Proc. of the Dynamic Flow Conference, Skovlunde, Denmark, pp. 757–800.Google Scholar
Ghosal, S. & Rogers, M. M. 1997 A numerical study of self-similarity in a turbulent plane wake using large-eddy simulation. Phys. Fluids 9, 1729–1739.CrossRefGoogle Scholar
Gordeyev, S. V. & Thomas, F. O. 2000 Coherent structure in the turbulent plane jet. Part 1. Extraction of proper orthogonal eigenmodes and their self-similarity. J. Fluid Mech. 414, 145–194.Google Scholar
Grant, H. L. 1958 The large eddies of turbulent motion. J. Fluid Mech. 4, 149–190.Google Scholar
Hayakawa, M. & Hussain, F. 1989 Three-dimensionality of organized structures in a plane turbulent wake. J. Fluid Mech. 206, 375–404.Google Scholar
Hinze, J. O. 1975 Turbulence. McGraw-Hill.Google Scholar
Hussein, H. J., Capp, S. P. & George, W. K. 1994 Velocity measurements in a high-Reynolds-number, momentum-conserving, axisymmetric, turbulent jet. J. Fluid Mech. 258, 31–75.CrossRefGoogle Scholar
Johansson, P. B. V. & George, W. K. 2006 The far downstream evolution of the high Reynolds number axisymmetric wake behind a disk. Part 2. Slice proper orthogonal decomposition. J. Fluid Mech. 555, 363–385.CrossRefGoogle Scholar
Johansson, P. B. V., George, W. K. & Gourlay, M. J. 2003 Equilibrium similarity, effects of initial conditions and local Reynolds number on the axisymmetric wake. Phys. Fluids 15, 603–617.Google Scholar
von Kármán, T. & Howarth, L. 1938 On the statistical theory of isotropic turbulence. Proc. R. Soc. Lond. (A) 164, 476.Google Scholar
Keffer, J. F. 1965 The uniform distortion of a turbulent wake. J. Fluid Mech. 22, 135–159.Google Scholar
Lumley, J. L. 1970 Stochastic Tools. Academic.Google Scholar
McIlwain, S., Ewing, D. & Pollard, A. 1997 Evolution of large- and small-scale coherent stuctures in non-homogeneous shear flows. Bull. Am. Phys. Soc. 41, 1816.Google Scholar
Maekawa, H., Mansour, N. N. & Buell, J. C. 1992 Instability mode interactions in a spatially developing plane wake. J. Fluid Mech. 235, 223–254.Google Scholar
Mi, J., Nobes, D. S. & Nathan, G. S. 2001 Influence of jet exit conditions on the passive scalar field of an axisymmetric free jet. J. Fluid Mech. 432, 91–125.Google Scholar
Monin, A. S. & Yaglom, A. M. 1975 Statistical Fluid Mechanics, vol. 2. MIT Press.Google Scholar
Moser, R. D. & Rogers, M. M. 1994 Direct simulation of a self-similar plane wake. Tech. Rep. TM 108815. NASA Tech. Memo.Google Scholar
Moser, R. D., Rogers, M. M. & Ewing, D. 1998 Self-similarity of time-evolving plane wakes. J. Fluid Mech. 367, 255–289.Google Scholar
Mumford, J. C. 1983 The structure of the large eddies in fully developed turbulent shear flows. Part 2. The plane wake. J. Fluid Mech. 137, 447–456.CrossRefGoogle Scholar
Narasimha, R. 1992 The utilities and drawbacks of traditional approaches. In Whither Turbulence? Turbulence at the Crossroads (ed. Lumley, J. L.), pp. 13–49. Springer.Google Scholar
Pope, S. 2000 Turbulent Flows. Cambridge University Press.Google Scholar
Slessor, M. D., Bond, C. L. & Dimotakis, P. E. 1998 Turbulent shear-layer mixing at high Reynolds numbers: effects of initial conditions. J. Fluid Mech. 376, 115–138.CrossRefGoogle Scholar
Spalart, P. R. 1988 Direct simulation of a turbulent boundary layer up to. J. Fluid Mech. 187, 61–98.Google Scholar
Speziale, C. G. & Bernard, P. S. 1992 The energy decay in self-preserving isotropic turbulence revisited. J. Fluid Mech. 241, 645–667.Google Scholar
Tennekes, H. & Lumley, J. L. 1972 A First Course in Turbulence. MIT Press.CrossRefGoogle Scholar
Townsend, A. A. 1956 The Structure of Turbulent Shear Flow. Cambridge University Press.Google Scholar
Townsend, A. A. 1979 Flow patterns of large eddies in a wake and in a boundary layer. J. Fluid Mech. 95, 515–537.Google Scholar
Wang, H. & George, W. K. 2002 The integral scale in homogeneous isotropic turbulence. J. Fluid Mech. 459, 429–443.Google Scholar
Weygandt, J. H. & Mehta, R. D. 1993 Three-dimensional structure of straight and curved plane wakes. Tech. Rep. JIAA TR-110. Stanford University.Google Scholar
Williamson, C. H. K. & Prasas, A. 1993 A new mechanism for oblique wave resonance in the ‘natural’ far wake. J. Fluid Mech. 256, 269–313.Google Scholar
Wygnanski, I., Champagne, F. & Marasli, B. 1986 On the large-scale structures in two-dimensional small-deficit, turbulent wakes. J. Fluid Mech. 168, 31–71.Google Scholar