Objectivity in Turbulence under Change of Reference Frame and Superposed Rigid Body Motion
Publication: Journal of Engineering Mechanics
Volume 137, Issue 10
Abstract
The validity of objectivity or frame-indifference in turbulence is examined simultaneously from the dual perspective of change of reference frame and superposed rigid-body motion. Similarities and differences arising from the two approaches and their effect on modeling in turbulence are discussed in detail. The use of the absolute spin for noninertial frames and of the net spin for superposed rigid-body motions in closure representations for the pressure-strain-rate correlations tensor are critically discussed. The enigmatic statement made by Speziale that the frame-indifference or objectivity of Reynolds stress does not establish the validity of material frame-indifference in turbulence and the subsequent case-study test of objectivity under superposed rigid-body mean motion within a rotating frame presented to prove it are investigated and fully explained. It is shown analytically that such apparently contrasting conclusions on the validity of objectivity result from the application of the objectivity test to the mean rather than the instantaneous motion, a case not encountered in classical continuum mechanics, which considers only one motion, the instantaneous.
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Acknowledgments
The writer would like to acknowledge the very important influence the articles of the late Professor Speziale and collaborators had on his understanding of the subject matter on objectivity in turbulence theory. Recommendations by the reviewers made this work more complete. Discussions with the author’s colleague, Bassam Younis, are gratefully acknowledged.
References
Ahmadi, G. (1987). “On material frame-indifference of turbulence closure models.” Geophys. Astrophys. Fluid Dyn., 38, 131–144.
Astarita, G. (1979). “Objective and generally applicable criteria for flow classification.” J. Non-Newtonian Fluid Mech., 6, 69–76.
Dafalias, Y. F., and Younis, B. A. (2007). “Objective tensorial representation of the pressure-strain correlations of turbulence.” Mech. Res. Commun., 34, 319–324.
Dafalias, Y. F., and Younis, B. A. (2009). “Objective model for the fluctuating pressure-strain-rate correlations.” J. Eng. Mech., 135(9), 1006–1014.
Daly, B., and Harlow, F. (1970). “Transport equations in turbulence.” Phys. Fluids, 13, 2634–2649.
Drouot, R., and Lucius, M. (1976). “Approximation du second ordre de la loi de comportement des fluides simples. Lois classiques deduites de l’introduction d’un nouveau tenseur objectif.” Archives de Mécanique Appliquée, 28, 189–199 (in French).
Frewer, M. (2008). “More clarity on the concept of material frame-indifference in classical continuum mechanics.” Acta Mech., 202(1–4), 213–246.
Gatski, T. B., and Speziale, C. G. (1993). “On explicit algebraic stress models for complex turbulent flows.” J. Fluid Mech., 254, 59–78.
Gatski, T. B., and Wallin, S. (2004). “Extending the weak-equilibrium condition for algebraic Reynolds stress models to rotating and curved flows.” J. Fluid Mech., 518, 147–155.
Hamba, F. (2006). “Euclidean invariance and weak-equilibrium condition for the algebraic Reynolds stress model.” J. Fluid Mech., 569, 399–408.
Liu, I. S. (1982). “On representation of anisotropic invariants.” Int. J. Eng. Sci., 20(10), 1099–1109.
Luca, I., and Sadiki, A. (2008). “New insight into the functional dependence rules in turbulence modeling.” Int. J. Eng. Sci., 46(11), 1053–1062.
Lumley, J. L. (1970). “Toward a turbulent constitutive relation.” J. Fluid Mech., 41, 413–434.
Muller, I. (1972). “On the frame dependence of stress and heat flux.” Arch. Ration. Mech. Anal., 45(4), 241–250.
Murdoch, A. I. (1983). “On material frame-indifference, intrinsic spin, and certain constitutive relations motivated by the kinetic theory of gases.” Arch. Ration. Mech. Anal., 83(2), 185–194.
Murdoch, A. I. (2003). “Objectivity in classical continuum physics: A rationale for discarding the ‘principle of invariance under superposed rigid body motions’ in favor of purely objective considerations.” Continuum Mech. Thermodyn., 15, 309–320.
Pope, S. B. (1975). “A more general effective-viscocity hypothesis.” J. Fluid Mech., 72(2), 331–340.
Rivlin, R. S. (2006). “Some thoughts on frame indifference.” Math. Mech. Solids, 11(2), 113–122.
Smith, G. F. (1971). “On isotropic functions of symmetric tensors, skew-symmetric tensors, and vectors.” Int. J. Eng. Sci., 9(10), 899–916.
Spencer, A. J. M., and Rivlin, R. S. (1957). “The theory of matrix polynomials and its application to the mechanics of isotropic continua.” Arch. Ration. Mech. Anal., 2(1), 309–336.
Speziale, C. G. (1979). “Invariance of turbulent closure models.” Phys. Fluids, 22(6), 1033–1037.
Speziale, C. G. (1998). “A review of material frame-indifference in mechanics.” Appl. Mech. Rev., 51(8), 489–504.
Speziale, C. G., Sarkar, S., and Gatski, T. B. (1991). “Modeling the pressure-strain correlation of turbulence: An invariant dynamical systems approach.” J. Fluid Mech., 227, 245–272.
Truesdell, C., and Noll, W. (1965). “The nonlinear field theories of mechanics.” Encyclopedia of Physics, 3rd Ed., S. Flugge, ed., Vol. 3, Springer-Verlag, Berlin.
Wallin, S., and Johansson, A. V. (2002). “Modeling steamline curvature effects in explicit algebraic Reynolds stress turbulence models.” Int. J. Heat Fluid Flow, 23(5), 721–730.
Wang, C. C. (1970). “A new representation theorem for isotropic functions, Part 2.” Arch. Ration. Mech. Anal., 36, 198–223.
Wang, L. (1997). “Frame-indifferent and positive-definite Reynolds stress-strain relation.” J. Fluid Mech., 352, 341–348.
Weis, J., and Hutter, K. (2003). “On Euclidean invariance of algebraic Reynolds stress models in turbulence.” J. Fluid Mech., 476, 63–68.
Younis, B. A., Speziale, C. G., and Berger, S. A. (1998). “Accounting for effects of a system rotation on the pressure-strain correlation.” AIAA J., 36(9), 1746–1748.
Younis, B. A., Weigand, B., and Vogler, A. D. (2009). “Prediction of momentum and scalar transport in turbulent swirling flows with an objective Reynolds-stress transport closure.” Heat Mass Transfer, 45(10), 1271–1283.
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Information
Published In
Journal of Engineering Mechanics
Volume 137 • Issue 10 • October 2011
Pages: 699 - 707
Copyright
© 2011 American Society of Civil Engineers.
History
Received: Sep 15, 2010
Accepted: Mar 24, 2011
Published online: Mar 26, 2011
Published in print: Oct 1, 2011
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