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Dependence of advection-diffusion-reaction on flow coherent structures
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1.
1. G. I. Taylor,“Diffusion by continuous movements,” Proc. London Math. Soc. s2-20, 196212 (1922).
http://dx.doi.org/10.1112/plms/s2-20.1.196
2.
2. A. J. Majda and P. R. Kramer, “Simplified models of turbulent diffusion: Theory, numerical modeling, and physical phenomena,” Phys. Rep. 314, 237574 (1999).
http://dx.doi.org/10.1016/S0370-1573(98)00083-0
3.
3. Z. Warhaft,“Passive scalars in turbulent flows,” Annu. Rev. Fluid Mech. 32, 203240 (2000).
http://dx.doi.org/10.1146/annurev.fluid.32.1.203
4.
4. P. H. Haynes and J. Vanneste, “What controls the decay of passive scalars in smooth flows?,” Phys. Fluids 17, 097103 (2005).
http://dx.doi.org/10.1063/1.2033908
5.
5. J.-L. Thiffeault, “Stretching and curvature of material lines in chaotic flows,” Physica D 198(3–4), 169181 (2004).
http://dx.doi.org/10.1016/j.physd.2004.04.009
6.
6. C. R. Doering and J.-L. Thiffeault, “Multiscale mixing efficiencies for steady sources,” Phys. Rev. E 74, 025301 (2006).
http://dx.doi.org/10.1103/PhysRevE.74.025301
7.
7. G. I. Taylor, “The formation of emulsions in definable fields of flow,” Proc. R. Soc. London, Ser. A 146, 501523 (1934).
http://dx.doi.org/10.1098/rspa.1934.0169
8.
8. P. Welander,“Studies on the general development of motion in a two-dimensional, ideal fluid,” Tellus 7, 141156 (1955).
http://dx.doi.org/10.1111/j.2153-3490.1955.tb01147.x
9.
9. J. M. Ottino, The Kinematics of Mixing: Stretching, Chaos and Transport (Cambridge University Press, Cambridge, 1989), p. 364.
10.
10. G. Haller, “Distinguished material surfaces and coherent structures in 3D fluid flows,” Physica D 149, 248277 (2001).
http://dx.doi.org/10.1016/S0167-2789(00)00199-8
11.
11. G. Haller, “A variational theory of hyperbolic Lagrangian coherent structures,” Physica D 240, 574598 (2011).
http://dx.doi.org/10.1016/j.physd.2010.11.010
12.
12. S. C. Shadden, F. Lekien, and J. E. Marsden, “Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows,” Physica D 212, 271304 (2005).
http://dx.doi.org/10.1016/j.physd.2005.10.007
13.
13. G. Haller, “An objective definition of a vortex,” J. Fluid Mech. 525, 126 (2005).
http://dx.doi.org/10.1017/S0022112004002526
14.
14. W. Tang and P. Walker, “Finite-time statistics of scalar diffusion in Lagrangian coherent structures,” Phys. Rev. E 86, 045201 (2012).
http://dx.doi.org/10.1103/PhysRevE.86.045201
15.
15. I. Mezić, J. F. Brady, and S. Wiggins, “Maximal effective diffusivity for time-periodic incompressible fluid flows,” SIAM J. Appl. Math. 56(1), 4056 (1996).
http://dx.doi.org/10.1137/S0036139994270449
16.
16. M. Sandulescu, C. López, E. Hernández-García, and U. Feudel, “Plankton blooms in vortices: The role of biological and hydrodynamic time scales,” Nonlinear Processes Geophys. 14, 443454 (2007).
http://dx.doi.org/10.5194/npg-14-443-2007
17.
17. V. Artale, G. Boffetta, M. Celani, M. Cencini, and A. Vulpiani, “Dispersion of passive tracers in closed basins: Beyond the diffusion coefficient,” Phys. Fluids 9, 31623171 (1997).
http://dx.doi.org/10.1063/1.869433
18.
18. A. Tzella and P. H. Haynes, “Smooth and filamental structures in chaotically advected chemical fields,” Phys. Rev. E 81, 016322 (2010).
http://dx.doi.org/10.1103/PhysRevE.81.016322
19.
19. D. A. Birch, Y. K. Tsang, and W. R. Young, “Bounding biomass in the Fisher equation,” Phys. Rev. E 75, 066304 (2007).
http://dx.doi.org/10.1103/PhysRevE.75.066304
20.
20. W. J. McKiver and Z. Neufeld, “Influence of turbulent advection on a phytoplankton ecosystem with nonuniform carrying capacity,” Phys. Rev. E 79, 061902 (2009).
http://dx.doi.org/10.1103/PhysRevE.79.061902
21.
21. D. Bargteila and T. Solomon, “Barriers to front propagation in ordered and disordered vortex flows,” Chaos 22, 037103 (2012).
http://dx.doi.org/10.1063/1.4746764
22.
22. K. A. Mitchell and J. R. Mahoney, “Invariant manifolds and the geometry of front propagation in fluid flows,” Chaos 22, 037104 (2012).
http://dx.doi.org/10.1063/1.4746039
23.
23. A. Brandenburg, N. E. L. Haugen, and N. Babkovskaia, “Turbulent front speed in the Fisher equation: Dependence on Damköhler number,” Phys. Rev. E 83, 016304 (2011).
http://dx.doi.org/10.1103/PhysRevE.83.016304
24.
24. J. P. Crimaldi, J. R. Cadwell, and J. B. Weiss, “Reaction enhancement of isolated scalars by vortex stirring,” Phys. Fluids 20, 073605 (2008).
http://dx.doi.org/10.1063/1.2963139
25.
25. T. Tél, A. de Moura, C. Grebogi, and G. Károlyi, “Chemical and biological activity in open flows: A dynamical system approach,” Phys. Rep. 413, 91196 (2005).
http://dx.doi.org/10.1016/j.physrep.2005.01.005
26.
26. Z. Neufeld and E. Hernández-García, Chemical and Biological Activity in Open Flows: A Dynamical System Approach (Imperial College Press, London, 2009), p. 304.
27.
27. C. L. Winter, D. M. Tartakovsky, and A. Guadagnini, “Numerical solutions of moment equations for flow in heterogeneous composite aquifers,” Water Resour. Res. 38(5), 131138, doi:10.1029/2001WR000222 (2002).
http://dx.doi.org/10.1029/2001WR000222
28.
28. C. L. Winter, D. M. Tartakovsky, and A. Guadagnini, “Moment differential equations for flow in highly heterogeneous porous media,” Surv. Geophys. 24(1), 81106 (2003).
http://dx.doi.org/10.1023/A:1022277418570
29.
29. C. L. Winter, A. Guadagnini, D. Nychka, and D. M. Tartakovsky, “Multivariate sensitivity analysis of saturated flow through simulated highly heterogeneous groundwater aquifers,” J. Comput. Phys. 217(1), 166175 (2006).
http://dx.doi.org/10.1016/j.jcp.2006.01.047
30.
30. D. M. Tartakovsky, M. Dentz, and P. C. Lichtner, “Probability density functions for advective-reactive transport with uncertain reaction rates,” Water Resour. Res. 45, W07414, doi:10.1029/2008WR007383 (2009).
http://dx.doi.org/10.1029/2008WR007383
31.
31. M. Dentz and D. M. Tartakovsky, “Probability density functions for passive scalars dispersed in random velocity fields,” Geophys. Res. Lett. 37, L24406, doi:10.1029/2010GL045748 (2010).
http://dx.doi.org/10.1029/2010GL045748
32.
32. I. Mezić, S. Loire, V. A. Fonoberov, and P. Hogan, “A new mixing diagnostic and Gulf oil spill movement,” Science 330, 486489 (2010).
http://dx.doi.org/10.1126/science.1194607
33.
33. G. Haller and F. J. Beron-Vera, “Geodesic theory of transport barriers in two-dimensional flows,” Physica D 241, 16801702 (2012).
http://dx.doi.org/10.1016/j.physd.2012.06.012
34.
34. R. T. Pierrehumbert and H. Yang, “Global chaotic mixing on isentropic surfaces,” J. Atmos. Sci. 50, 24622480 (1993).
http://dx.doi.org/10.1175/1520-0469(1993)050<2462:GCMOIS>2.0.CO;2
35.
35. T. Peacock and J. Dabiri, “Introduction to focus issue: Lagrangian coherent structures,” Chaos 20, 017501 (2010).
http://dx.doi.org/10.1063/1.3278173
36.
36. F. Lekien, C. Coulliette, A. J. Mariano, E. H. Ryan, L. K. Shay, G. Haller, and J. Marsden, “Pollution release tied to invariant manifolds: A case study for the coast of Florida,” Physica D 210, 120 (2005).
http://dx.doi.org/10.1016/j.physd.2005.06.023
37.
37. S. C. Shadden, J. O. Dabiri, and J. E. Marsden, “Lagrangian analysis of fluid transport in empirical vortex ring flows,” Phys. Fluids 18(4), 047105 (2006).
http://dx.doi.org/10.1063/1.2189885
38.
38. W. Tang, M. Mathur, G. Haller, D. C. Hahn, and F. H. Ruggiero, “Lagrangian coherent structures near a subtropical jet stream,” J. Atmos. Sci. 67(7), 23072319 (2010).
http://dx.doi.org/10.1175/2010JAS3176.1
39.
39. J. Finn and D. del-Castillo-Negrete, “Lagrangian chaos and Eulerian chaos in shear flow dynamics,” Chaos 11(4), 816832 (2001).
http://dx.doi.org/10.1063/1.1418762
40.
40. T. Peacock and G. Haller, “Lagrangian coherent structures: The hidden skeleton of fluid flows,” Phys. Today 66(2), 4147 (2013).
http://dx.doi.org/10.1063/PT.3.1886
41.
41. M. Farazmand and G. Haller, “Computing Lagrangian coherent structures from variational LCS theory,” Chaos 22, 013128 (2012).
http://dx.doi.org/10.1063/1.3690153
42.
42. T. H. Solomon and J. P. Gollub, “Passive transport in steady Rayleigh Bénard convection,” Phys. Fluids 31, 13721379 (1988).
http://dx.doi.org/10.1063/1.866729
43.
43. W. R. Young, A. Pumir, and Y. Pomeau, “Anomalous diffusion of tracer in convection rolls,” Phys. Fluids A 1, 462469 (1989).
http://dx.doi.org/10.1063/1.857415
44.
44. F. Lekien and S. D. Ross, “The computation of finite-time Lyapunov exponents on unstructured meshes and for non-Euclidean manifolds,” Chaos 20, 017505 (2010).
http://dx.doi.org/10.1063/1.3278516
45.
45. R. A. Fisher, “The wave of advance of advantageous genes,” Annu. Eugen. 7, 355369 (1937).
http://dx.doi.org/10.1111/j.1469-1809.1937.tb02153.x
46.
46. A. Kolmogorov, I. Petrovskii, and N. Piscounov, “A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem,” in Selected Works of A. N. Kolmogorov I (Kluwer, 1991), pp. 248270, edited by V. M. Tikhomirov
46.[translated by V. M. Volosov, Bull. Moscow Univ., Math. Mech. 1, 125 (1937)].
http://dx.doi.org/10.1007/978-94-011-3030-1_38
47.
47. P. Moin and K. Mahesh, “Direct numerical simulation: A tool in turbulence research,” Annu. Rev. Fluid Mech. 30, 539578 (1998).
http://dx.doi.org/10.1146/annurev.fluid.30.1.539
48.
48. J. R. Taylor, “Numerical simulations of the stratified oceanic bottom boundary layer,” Ph.D. thesis (University of California, San Diego, 2008).
49.
49. E. Ott and T. M. Antonsen, Jr., “Chaotic fluid convection and the fractal nature of passive scalar gradients,” Phys. Rev. Lett. 61(25), 28392842 (1988).
http://dx.doi.org/10.1103/PhysRevLett.61.2839
50.
50. E. Ott and T. M. Antonsen, Jr., “Fractal measures of passively convected vector fields and scalar gradients in chaotic fluid flows,” Phys. Rev. A 39(7), 36603671 (1989).
http://dx.doi.org/10.1103/PhysRevA.39.3660
51.
51. A. Mahadevan and J. W. Campbell, “Biogeochemical patchiness at the sea surface,” Geophys. Res. Lett. 29(19), 1926, doi:10.1029/2001GL014116 (2002).
http://dx.doi.org/10.1029/2001GL014116
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Figures

Image of FIG. 1.

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FIG. 1.

Range of behaviors of the reaction process. (a) Variation of the average concentration ⟨⟩ in the periodic domain. The two black curves are the bounding cases of the fastest and slowest reaction behaviors among the realizations. (b) Probability density of ⟨⟩ at = 7.5. The range of behavior lies on the vertical dashed line in (a). (c) Probability density of the reaction time when ⟨⟩ = 0.5. The range of behavior lies on the horizontal solid line in (a).

Image of FIG. 2.

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FIG. 2.

Comparison between the variability in ⟨⟩ and LCS. Each point in (a) marks the center of a concentration blob initially released in the flow (a realization). The color value is the associated ⟨⟩ of this realization at = 7.5. (b) Repelling LCS marked by the forward-time FTLE field. The four colored dots mark the initial conditions of four representative realizations discussed later. Note the range of ⟨⟩ at this time and the strong correlation between the two panels.

Image of FIG. 3.

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FIG. 3.

Range of behaviors of the reaction process. (a) Placement of the reaction processes in the chaotic flow with other flow types. The inset plot shows the early-time behavior of these cases. (b) Difference in reaction rates between the actual realizations and the nominal logistic equation case red line). The inset plot shows the early time behavior in log scale. The vertical line is a reference line at ⟨⟩ = 0.0035. This line is also shown in the inset plot in (a). The blue line is the realization with no flow. (c) Correlation between the structures in Fig. 2 as a function of time. The inset plot shows the comparison between probability density of FTLE (black) and ⟨⟩ (blue). The black horizontal bar marks the range of FTLE between 2 and 4. The blue horizontal bar marks the range of ⟨⟩ between 0.52 and 0.55.

Image of FIG. 4.

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FIG. 4.

Chemical concentration at = 1. The four realizations are initially released at (a) black dot, (b) white dot, (c) red dot, and (d) magenta dot in Fig. 2 . The black (white) curves are the iso-contours for forward (backward)-time FTLE indicating repelling (attracting) structures. The titles indicate the average chemical content ⟨⟩.

Image of FIG. 5.

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FIG. 5.

Same as Fig. 4 but at = 7.5. Note the major difference between (b) and the other cases (a, c, and d).

Image of FIG. 6.

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FIG. 6.

Evolution of individual contributions from Eqs. (6) and (7) to the scalar variance ⟨2, , and ⟨ 2⟩, as functions of ⟨⟩. (a) Terms relevant to Eq. (6) . The black curve is ⟨2, which is a straight line in log scale as a function of ⟨⟩. The blue curve is 2η⟨2 − 2η⟨3. The green curves are for the four cases, plotted in thick solid (hyperbolic core), thick dashed (interior), dashed-dotted (repeller), and dotted (attractor) lines. The red vertical line references ⟨⟩ = 0.002, where all cases are around = 1. (b) Terms relevant to Eq. (7) . The black curves are . The blue curves are and the green curves are 2⟨| |2⟩/. Line styles are the same as those in panel (a). The red vertical line again marks ⟨⟩ = 0.002. (c) Terms relevant to evolution of ⟨ 2⟩. The black curves are ⟨ 2⟩, the blue curves are 2η⟨ 2⟩ − 2η⟨ 3⟩ and the green curves are 2⟨| |2⟩/. Line styles are the same as those in panel (a). The red solid vertical line marks ⟨⟩ = 0.002, the three red dashed vertical lines mark ⟨⟩ = 0.001, 0.01, and 0.1, respectively. All realizations reach ⟨⟩ = 0.1 between = 5 and 5.5.

Image of FIG. 7.

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FIG. 7.

Phases of the entire process for two realizations. Top row: release at hyperbolic core. Bottom row: release at interior. Left column: end of the first phase dominant by scalar decay. Center column: end of the second phase dominant by filament stretching. Right column: end of the third phase dominant by homogenization among neighboring filaments. End of the fourth phase is uniform concentration and is not shown.

Image of FIG. 8.

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FIG. 8.

(a) Integral scale of the scalar. The line styles are the same as those in Fig. 6 . The red vertical line indicates ⟨⟩ = 0.1. (b) Information dimension of the scalar. Line styles and the vertical line are the same as (a).

Image of FIG. 9.

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FIG. 9.

Comparison between simulations and models based on FTLE. Line styles for panels (a) and (b): solid curves: fastest reacting case; dashed curves: slowest reacting case; black curves: DNS results; red curves: modeled from FTLE. (a) ⟨⟩ against time. (b) ⟨ 2⟩ against ⟨⟩. (c) Modeled ⟨⟩ for all realizations at = 7.5.

Image of FIG. 10.

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FIG. 10.

(a) Correlation between and ⟨⟩ as a function of time for different . Black: = 10 000; blue: = 1000: green: = 100; and red: = 10. The scattered points are at = 7.5, marking the time the largest variation appears in the ⟨⟩ field. The top inset plot shows the largest gap of ⟨⟩ between extreme realizations at any given time as a function of . The bottom inset plot shows ⟨⟩ for = 10 at = 7.5. (b) Correlation between and ⟨⟩ as a function of time for different η. Black: η = 4; green: η = 2; blue: η = 1; red: η = 1/2: and magenta: η = 1/4. The scattered points are at the respective times when the variability within each ⟨⟩ field is the largest. This time scale is inversely proportional to the reaction rate η. The simulations for different η are only up to when ⟨⟩ gets close to 1 everywhere, hence the correlation beyond this time is extrapolated as the last value of each case (where all curves have started to show asymptotic behavior). The top inset plot shows the largest gap of ⟨⟩ between extreme realizations at any given time as a function of η. The bottom inset plot shows ⟨⟩ for η = 4 at = 1.875.

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/content/aip/journal/pof2/25/10/10.1063/1.4823991
2013-10-15
2014-04-20

Abstract

A study on an advection-diffusion-reaction system is presented. Variability of the reaction process in such a system triggered by a highly localized source is quantified. It is found, for geophysically motivated parameter regimes, that the difference in bulk concentration subject to realizations of different source locations is highly correlated with the local flow topology of the source. Such flow topologies can be highlighted by Lagrangian coherent structures. Reaction is relatively enhanced in regions of strong stretching, and relatively suppressed in regions where vortices are present. In any case, the presence of a divergence-free background flow helps speed up the reaction process, especially when the flow is time-dependent. Probability density of various quantities characterizing the reaction processes is also obtained. This reveals the inherent complexity of the reaction-diffusion process subject to nonlinear background stirring.

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Scitation: Dependence of advection-diffusion-reaction on flow coherent structures
http://aip.metastore.ingenta.com/content/aip/journal/pof2/25/10/10.1063/1.4823991
10.1063/1.4823991
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