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Dispersion and reaction in random flows: Single realization versus ensemble average
Phys. Rev. Fluids 4, 124502 – Published 6 December, 2019
DOI: https://doi.org/10.1103/PhysRevFluids.4.124502
Abstract
We examine the dispersion of a passive scalar released in an incompressible fluid flow in an unbounded domain. The flow is assumed to be spatially periodic, with zero spatial average, and random in time, in the manner of the random-phase alternating sine flow which we use as an exemplar. In the long-time limit, the scalar concentration takes the same, predictable form for almost all realizations of the flow, with a Gaussian core characterized by an effective diffusivity, and large-deviation tails characterized by a rate function (which can be evaluated by computing the largest Lyapunov exponent of a family of random-in-time partial differential equations). We contrast this single-realization description with that which applies to the average of the concentration over an ensemble of flow realizations. We show that the single-realization and ensemble-average effective diffusivities are identical but that the corresponding rate functions are not, and that the ensemble-averaged description overestimates the concentration in the tails compared with that obtained for single-flow realizations. This difference has a marked impact for scalars reacting according to the Fisher-Kolmogorov-Petrovskii-Piskunov (FKPP) model. Such scalars form an expanding front whose shape is approximately independent of the flow realization and can be deduced from the single-realization large-deviation rate function. We test our predictions against numerical simulations of the alternating sine flow.
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References (26)
- G. I. Taylor, Diffusion by continuous movements, Proc. Lond. Math. Soc. s2-20, 196 (1922).
- R. H. Kraichnan, Dispersion of particle pairs in homogeneous turbulence, Phys. Fluids 9, 1937 (1966).
- A. J. Majda and P. R. Kramer, Simplified models for turbulent diffusion: Theory, numerical modeling, and physical phenomena, Phys. Rep. 314, 237 (1999).
- G. Falkovich, K. Gawędzki, and M. Vergassola, Particles and fields in fluid turbulence, Rev. Mod. Phys. 73, 913 (2001).
- G. A. Pavliotis and A. M. Stuart, Multiscale Methods: Averaging and Homogenization (Springer-Verlag, New-York, 2007).
- P. H. Haynes and J. Vanneste, Dispersion in the large-deviation regime. Part 1: Shear flows and periodic flows, J. Fluid Mech. 745, 321 (2014).
- R. T. Pierrehumbert, Tracer microstructure in the large-eddy dominated regime, Chaos Solitons Fractals 4, 1091 (1994).
- J. Gärtner and M. I. Freidlin, On the propagation of concentration waves in periodic and random media, Dokl. Akad. Nauk 249, 521 (1979).
- M. I. Freidlin, Functional Integration and Partial Differential Equations. (AM-109) (Princeton University Press, Princeton, 1985).
- P. H. Haynes and J. Vanneste, What controls the decay of passive scalars in smooth flows? Phys. Fluids 17, 097103 (2005).
- P. Meunier and E. Villermaux, The diffusive strip method for scalar mixing in two dimensions, J. Fluid Mech. 662, 134 (2010).
- A. J. Valocchi and M. Malmstead, Accuracy of operator splitting for advection-dispersion-reaction problems, Water Resour. Res. 28, 1471 (1992).
- 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).
- J.-P. and A. Georges, Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications, Phys. Rep. 195, 127 (1990).
- P. Le Doussal and J. Machta, Annealed versus quenched diffusion coefficient in random media, Phys. Rev. B 40, 9427 (1989).
- B. Øksendal, Stochastic Differential Equations (Springer-Verlag, Berlin, Heidelberg, 1998).
- V. I. Oseledec, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trans. Mosc. Math. Soc. 19, 197 (1968).
- D. Ruelle, Characteristic exponents and invariant manifolds in Hilbert space, Ann. Math. 115, 243 (1982).
- DLMF, NIST digital library of mathematical functions, http://dlmf.nist.gov/, release 1.0.22 of 2019-03-15, edited by W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, and B. V. Saunders.
- A. Tzella and J. Vanneste, Dispersion in Rectangular Networks: Effective Diffusivity and Large-Deviation Rate Function, Phys. Rev. Lett. 117, 114501 (2016).
- R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics 7, 355 (1937).
- A. N. Kolmogorov, I. G. Petrovsky, and N. S. Piskunov, A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem, Mosc. Univ. Bull. Math. 1, 1 (1937).
- J. Xin, An Introduction to Fronts in Random Media (Springer-Verlag, New York, 2000).
- A. Tzella and J. Vanneste, Front propagation in cellular flows for fast reaction and small diffusivity, Phys. Rev. E 90, 011001(R) (2014).
- A. Tzella and J. Vanneste, FKPP fronts in cellular flows: The large-Péclet regime, SIAM J. Appl. Math. 75, 1789 (2015).
- M. B. Isichenko, Percolation, statistical topography, and transport in random media, Rev. Mod. Phys. 64, 961 (1992).
