Physics of Fluids
Feedback | Help | Exit
   
 
 
 
Previous Article
Sensitivity of two-dimensional spatially developing mixing layers with respect to uncertain inflow conditions
The fidelity of numerical simulations of transport phenomena is often compromised by the difficulty in modeling the inherent experimental uncertainties. In this study, we examine the sensitivity of di...
Next Article
Analysis and control of cavity flow
A flow above a cavity leads to an unsteady separated flow. This configuration exhibits an intense aeroacoustic coupling, where very intense aerodynamic noise can be emitted. Moreover, a majority of ta...

Constraints on scalar diffusion anomaly in three-dimensional flows having bounded velocity gradients

Phys. Fluids 20, 077103 (2008); doi:10.1063/1.2957022

Published 16 July 2008

You are not logged in to this journal. Log in

Chuong V. Tran
School of Mathematics and Statistics, University of St Andrews, St Andrews KY16 9SS, United Kingdom
This study is concerned with the decay behavior of a passive scalar theta in three-dimensional flows having bounded velocity gradients. Given an initially smooth scalar distribution, the decay rate d<theta2>/dt of the scalar variance <theta2> is found to be bounded in terms of controlled physical parameters. Furthermore, in the zero diffusivity limit, kappa-->0, this rate vanishes as kappaalpha0 if there exists an alpha0[is-an-element-of](0,1] independent of kappa such that <|(−Delta)alpha/2theta|2><[infinity] for alpha<=alpha0. This condition is satisfied if in the limit kappa-->0, the variance spectrum Theta(k) remains steeper than k−1 for large wave numbers k. When no such positive alpha0 exists, the scalar field may be said to become virtually singular. A plausible scenario consistent with Batchelor's theory is that Theta(k) becomes increasingly shallower for smaller kappa, approaching the Batchelor scaling k−1 in the limit kappa-->0. For this classical case, the decay rate also vanishes, albeit more slowly—like (ln  Pr)−1, where Pr is the Prandtl or Schmidt number. Hence, diffusion anomaly is ruled out for a broad range of scalar distribution, including power-law spectra no shallower than k−1. The implication is that in order to have a kappa-independent and nonvanishing decay rate, the variance at small scales must necessarily be greater than that allowed by the Batchelor spectrum. These results are discussed in the light of existing literature on the asymptotic exponential decay <theta2>~e−gammat, where gamma>0 is independent of kappa. ©2008 American Institute of Physics
History: Received 21 January 2008; accepted 10 June 2008; published 16 July 2008
Permalink: http://link.aip.org/link/?PHFLE6/20/077103/1
BUY THIS ARTICLE   (US$28)
Download PDF (134 kB) View Cart

KEYWORDS and PACS

Keywords
PACS
  • 47.10.A-
    Mathematical formulations in fluid dynamics
  • 05.60.-k
    Transport processes
  • YEAR: 2008

RELATED DATABASES


To view database links for this article,
you need to log in.
To view database links for this article,
you need to log in.

PUBLICATION DATA

ISSN:
1070-6631 (print)   1089-7666 (online)
Publisher:
AIP is a member of CrossRef AIP

REFERENCES (28)
View in Separate Window/Tab
For access to fully linked references, you need to log in. For access to fully linked references, you need to Log in.
  1. T. M. Antonsen, Jr., Z. Fan, E. Ott, and E. Garcia-Lopez, “The role of chaotic orbits in the determination of power spectra of passive scalars,” Phys. Fluids 8, 3094 (1996).
  2. E. Balkovsky and A. Fouxon, “Universal long-time properties of Lagrangian statistics in the Batchelor regime and their application to the passive scalar problem,” Phys. Rev. E 60, 4164 (1999).
  3. G. K. Batchelor, “Small-scale variation of convected quantities like temperature in turbulent fluid. Part 1. General discussion and the case of small conductivity,” J. Fluid Mech. 5, 113 (1959).
  4. G. K. Batchelor, I. A. Howell, and A. A. Townsend, “Small-scale variation of convected quantities like temperature in turbulent fluid. Part 2. The case of large conductivity,” J. Fluid Mech. 5, 134 (1959).
  5. M. Chertkov, G. Falkovich, and I. Kolokolov, “Intermittent dissipation of a passive scalar in turbulence,” Phys. Rev. Lett. 80, 2121 (1998).
  6. S. Corrsin, “On the spectrum of isotropic temperature fluctuation in isotropic turbulence,” J. Appl. Phys. 22, 469 (1951).
  7. D. R. Fereday and P. H. Haynes, “Scalar decay in two-dimensional chaotic advection and Batchelor-regime turbulence,” Phys. Fluids 16, 4359 (2004).
  8. P. H. Haynes and J. Vanneste, “What controls the decay rate of passive scalar in smooth random flows?” Phys. Fluids 17, 097103 (2005).
  9. R. H. Kraichnan, “Small-scale structure of a scalar field convected by turbulence,” Phys. Fluids 11, 945 (1968).
  10. W. Liu, “Does a fast mixer really exist?” Phys. Rev. E 72, 016312 (2005).
  11. A. D. Majda, “The random uniform shear layer: An explicit example of turbulent diffusion with broad tail probability distributions,” Phys. Fluids A 5, 1963 (1993).
  12. A. M. Obukhov, “The structure of the temperature field in a turbulent flow,” Izv. Akad. Nauk SSSR, Ser. Geogr. Geofiz. 13, 58 (1949).
  13. R. T. Pierrehumbert, “Tracer microstructure in the large-eddy dominated regime,” Chaos, Solitons Fractals 4, 1091 (1994).
  14. A. A. Schekochihin, P. H. Haynes, and S. C. Cowley, “Diffusion of passive scalar in a finite random flow,” Phys. Rev. E 70, 046304 (2004).
  15. K. R. Sreenivasan, “The passive scalar spectrum and the Obukhov–Corrsin constant,” Phys. Fluids 8, 189 (1996).
  16. K. R. Sreenivasan, “On local isotropy of passive scalars in turbulent shear flows,” Proc. R. Soc. London, Ser. A 434, 165 (1991).
  17. J. Sukhatme and R. T. Pierrehumbert, “Decay of passive scalars under the action of single scale smooth velocity fields in bounded two-dimensional domains: From non-self-similar probability distribution function to self-similar eigenmodes,” Phys. Rev. E 66, 056302 (2002).
  18. C. V. Tran, “An upper bound for passive scalar diffusion in shear flows,” Phys. Fluids 19, 068104 (2007).
  19. Y. K. Tsang, T. M. Antonsen, and E. Ott, “Exponential decay of chaotically advected passive scalars in the zero diffusivity limit,” Phys. Rev. E 71, 066301 (2005).
  20. A. Wonhas and J. C. Vassilicos, “Mixing in fully chaotic flows,” Phys. Rev. E 66, 051205 (2002).
  21. P. K. Yeung, S. Xu, and K. R. Sreenivasan, “Schmidt number effects on turbulent transport with uniform mean scalar gradient,” Phys. Fluids 14, 4178 (2002).
  22. P. A. Davidson, Turbulence: An Introduction for Scientists and Engineers (Oxford University Press, Oxford, 2004).
  23. C. V. Tran and D. G. Dritschel, “Vanishing enstrophy dissipation in two-dimensional Navier–Stokes turbulence in the inviscid limit,” J. Fluid Mech. 559, 107 (2006).
  24. C. V. Tran, “Nonlinear transfer and spectral distribution of energy in alpha turbulence,” Physica D 191, 137 (2004).
  25. C. V. Tran and T. G. Shepherd, “Constraints on the spectral distribution of energy and enstrophy dissipation in forced two-dimensional turbulence,” Physica D 165, 199 (2002).
  26. D. G. Dritschel, C. V. Tran, and R. K. Scott, “Revisiting Batchelor's theory of two-dimensional turbulence,” J. Fluid Mech. 591, 379 (2007).
  27. C. V. Tran, “Constraints on inertial range scaling laws in forced two-dimensional Navier–Stokes turbulence,” Phys. Fluids 19, 108109 (2007).
  28. J. Schumacher and K. R. Sreenivasan, “Geometric features of the mixing of passive scalars at high Schmidt numbers,” Phys. Rev. Lett. 91, 174501 (2003).

CITING ARTICLES
For access to citing articles, you need to log in.
For access to citing articles, you need to Log in.


Copyright © American Institute of Physics