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Relationship between temporal and spatial averages in grid turbulence

Published online by Cambridge University Press:  02 August 2013

L. Djenidi ,
S. F. Tardu  and
R. A. Antonia
L. Djenidi*
Affiliation:
Discipline of Mechanical Engineering, School of Engineering, University of Newcastle, Newcastle, 2308 NSW, Australia
S. F. Tardu
Affiliation:
Laboratoires des Ecoulements Geophysiques et Industriels, LEGI BP 53 X Grenoble, France
R. A. Antonia
Affiliation:
Discipline of Mechanical Engineering, School of Engineering, University of Newcastle, Newcastle, 2308 NSW, Australia
*
†Email address for correspondence: [email protected]
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Abstract

A long-time direct numerical simulation (DNS) based on the lattice Boltzmann method is carried out for grid turbulence with the view to compare spatially averaged statistical properties in planes perpendicular to the mean flow with their temporal counterparts. The results show that the two averages become equal a short distance downstream of the grid. This equality indicates that the flow has become homogeneous in a plane perpendicular to the mean flow. This is an important result, since it confirms that hot-wire measurements are appropriate for testing theoretical results based on spatially averaged statistics. It is equally important in the context of DNS of grid turbulence, since it justifies the use of spatial averaging along a lateral direction and over several realizations for determining various statistical properties. Finally, the very good agreement between temporal and spatial averages validates the comparison between temporal (experiments) and spatial (DNS) statistical properties. The results are also interesting because, since the flow is stationary in time and spatially homogeneous along lateral directions, the equality between the two types of averaging provides strong support for the ergodic hypothesis in grid turbulence in planes perpendicular to the mean flow.

JFM classification

Turbulent Flows: Isotropic turbulence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 730 , 10 September 2013 , pp. 593 - 606
Copyright
©2013 Cambridge University Press 

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References

Androulakis, G. & Dostoglu, S. 2004 Space averages and homogeneous fluid flows. Math. Phys. Electron. J. 10, 1–12.Google Scholar
Ashok, A., Hultmark, M., Bailey, S. C. C. & Smits, A. J. 2012 Hot wire spatial resolution effects in measurements of grid generated turbulence. Exp. Fluids 53, 1713–1722.CrossRefGoogle Scholar
Batchelor, G. K. 1953 The Theory of Homogeneous Turbulence. Cambridge University Press.Google Scholar
Batchelor, G. K & Townsend, A. A. 1947 Decay of vorticity in isotropic turbulence. Proc. R. Soc. Lond. A 190, 534–550.Google Scholar
Birkhoff, G. D. 1931 Proof of the ergodic problem. Proc. Natl Acad. Sci. USA 17, 600–656.Google Scholar
Boltzmann, L. 1884 Uber die Eigenschaften Monozyklischer und Amderer damit ver vandter Systeme. Crelle’s J. 98, 68–94.Google Scholar
Burattini, P., Lavoie, P., Agrawal, A., Djenidi, L. & Antonia, R. A. 2006 On the power law of decaying homogeneous isotropic turbulence at low ${R}_{\lambda } $ . Phys. Rev. E 73, 066304.CrossRefGoogle Scholar
Carver, H. B., Nash, R. W., Bernabeu, M. O., Hetherington, J., Groen, D., Krger, T. & Coveney, P. V. 2012 Choice of boundary condition and collision operator for lattice-Boltzmann simulation of moderate Reynolds number flow in complex domains. Phys. Rev. E (arXiv preprint, arXiv:1211.0205).Google Scholar
Chen, S. & Doolen, G. D. 1998 Lattice Boltzmann method for fluid flows. Annu. Rev. Fluid Mech. 30, 329–364.Google Scholar
Chikatamarla, S. S., Ansumali, S. & Karlin, I. V. 2006 Grad’s approximation for missing data in lattice Boltzmann simulations. Europhys. Lett. 74, 215–221.CrossRefGoogle Scholar
Corrsin, S. 1963 Turbulence: Experimental Methods, vol. 8. Springer.Google Scholar
Cramer, H. & Leadbetter, M. R. 2004 Stationary and related stochastic processes. In Sample Function Properties and their Applications. Dover.Google Scholar
Djenidi, L. 2006 Lattice Boltzmann simulation of grid-generated turbulence. J. Fluid Mech. 552, 13–35.Google Scholar
Djenidi, L. 2008 Study of the structure of a turbulent crossbar near-wake by means of Lattice Boltzmann. Phys. Rev. E 77, 036310.CrossRefGoogle Scholar
Djenidi, L. & Tardu, S. F. 2012 On the anisotropy of a low-Reynolds-number grid generated turbulence. J. Fluid Mech. 702, 332–353.Google Scholar
Ertunc, L., Ozyilmaz, N., Lienhart, H., Durst, F. & Beronov, K. 2010 Homogeneity of turbulence generated by static-grid structures. J. Fluid Mech. 654, 473–500.Google Scholar
Frisch, U., Hasslacher, B. & Pomeau, Y. 1986 Lattice gas automata for the Navier–Stokes equations. Phys. Rev. Lett. 56, 1505–1508.Google Scholar
Galanti, B. & Tsinober, A. 2004 Is turbulence ergodic? Phys. Lett. A, 173–180.Google Scholar
Grant, H. L. & Nisbet, I. C. T. 1957 The inhomogeneity of grid turbulence. J. Fluid Mech. 2, 263–272.CrossRefGoogle Scholar
Hinze, J. O. 1975 Turbulence. McGraw-Hill Classic Textbook Reissues Series.Google Scholar
Hou, S., Sterlin, J., Chen, S. & Doolen, G. D. 1996 A lattice Boltzmann subgrid model for high Reynolds number flows. In Field Institute Communications. Pattern Fornmation and Lattice Gas Automata (ed. Lawniczak, A. T. & Kapral, R.), vol. 6, pp. 151–166. American Mathematical Society. Also arXiv:comp-gas/9401004v1.Google Scholar
Monin, A. S. & Yaglom, A. M. 1971 Statistical Fluid Mechanics, vol. 1. MIT.Google Scholar
Nillsen, R. 2010 Randomness and Recurrence in Dynamical Systems. The Carus Mathematical Monographs, vol. 31. Mathematical Association of America.CrossRefGoogle Scholar
Papoulis, A. 1984 Probability, Random Variables, and Stochastic Processes, 2nd edn. McGraw Hill.Google Scholar
Succi, S. 2001 The lattice Boltzmann equation for fluid dynamics and beyond. In Numerical Mathematics and Scientific Computation (ed. Golub, G. H., Jeltsch, R., Light, W. A. & Süli, E.). Oxford University Press.Google Scholar
Taylor, G. I. 1935 Statistical theory of turbulence. Proc. R. Soc. Lond. A 151, 412–478.Google Scholar
Taylor, G. I. 1938 The spectrum of turbulence. Proc. R. Soc. Lond. A 164, 476–490.Google Scholar
Walter, P. 1982 An Introduction to Ergodic Theory. Graduate Texts in Mathematics, vol. 79. Springer.Google Scholar