# American Institute of Mathematical Sciences

January  2011, 15(1): 75-92. doi: 10.3934/dcdsb.2011.15.75

## Numerical simulations of diffusion in cellular flows at high Péclet numbers

 1 Department of Mathematics, University of Houston, Houston, TX 77204, United States 2 Department of Mathematics, Ajou University, Suwon 443-749, South Korea 3 Department of Mathematics, Pennsylvania State University, University Park, State College, PA 16802, United States

Received  November 2009 Revised  April 2010 Published  October 2010

We study numerically the solutions of the steady advection-diffu-sion problem in bounded domains with prescribed boundary conditions when the Péclet number Pe is large. We approximate the solution at high, but finite Péclet numbers by the solution to a certain asymptotic problem in the limit Pe $\to \infty$. The asymptotic problem is a system of coupled 1-dimensional heat equations on the graph of streamline-separatrices of the cellular flow, that was developed in [21]. This asymptotic model is implemented numerically using a finite volume scheme with exponential grids. We conclude that the asymptotic model provides for a good approximation of the solutions of the steady advection-diffusion problem at large Péclet numbers, and even when Pe is not too large.
Citation: Yuliya Gorb, Dukjin Nam, Alexei Novikov. Numerical simulations of diffusion in cellular flows at high Péclet numbers. Discrete and Continuous Dynamical Systems - B, 2011, 15 (1) : 75-92. doi: 10.3934/dcdsb.2011.15.75
##### References:
 [1] R. E. Bank, "PLTMG: A Software Package for Solving Elliptic Partial Differential Equations," SIAM Books, Philadelphia, 1994. [2] A. Bensoussan, J. L. Lions and G. Papanicoalou, "Asymptotic Analysis for Periodic Structures," North-Holland, 1978. [3] S. Childress, Alpha-effect in flux ropes and sheets, Phys. Earth Planet Int., 20 (1979), 172-180. doi: doi:10.1016/0031-9201(79)90039-6. [4] B. Cushman-Roisin, "Introduction to Geophysical Fluid Dynamics," Prentice-Hall, Englewood Cliffs, NJ, 1994. [5] C. Cuvelier, A. Segal and A. A. van Steenhoven, "Finite Element Methods and Navier-Stokes Equations," Mathematics and its Applications, 22, D. Reidel Publishing Co., Dordrecht, 1986. [6] J. Douglas, Jr. and T. F. Russell, Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures, SIAM J. Numer. Anal., 19 (1982), 871-885. doi: doi:10.1137/0719063. [7] R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, in "Handbook of Numerical Analysis," Vol. VII, North-Holland, Amsterdam, (2000), 713-1020. [8] A. Fannjiang and G. Papanicolaou, Convection enhanced diffusion for periodic flows, SIAM J. Appl. Math., 54 (1994), 333-408. doi: doi:10.1137/S0036139992236785. [9] A. Fannjiang and G. Papanicolaou, Convection-enhanced diffusion for random flows, J. Statist. Phys., 88 (1997), 1033-1076. doi: doi:10.1007/BF02732425. [10] J. H. Ferziger and M. Perić, "Computational Methods for Fluid Dynamics," 2nd edition, Springer-Verlag, Berlin, 1999. [11] D. Funaro and O. Kavian, Approximation of some diffusion evolution equations in unbounded domains by Hermite functions, Math. Comp., 57 (1991), 597-619. [12] P. H. Haynes and E. F. Shuckburgh, Effective diffusivity as a measure of atmospheric transport, Part I: Stratosphere, J. Geophys. Res., 105 (2000), 777-794. [13] P. H. Haynes and E. F. Shuckburgh, Effective diffusivity as a measure of atmospheric transport, Part II: Troposphere and lower stratosphere, J. Geophys. Res., 105 (2000), 795-810. doi: doi:10.1029/2000JD900092. [14] S. Heinze, Diffusion-advection in cellular flows with large Péclet numbers, Arch. Ration. Mech. Anal., 168 (2003), 329-342. doi: doi:10.1007/s00205-003-0256-7. [15] C. Johnson, "Numerical Solution of Partial Differential Equations by the Finite Element Method," Cambridge University Press, Cambridge, Great Britain, 1987. [16] R. B. Kellogg and A. Tsan, Analysis of some difference approximations for a singular perturbation problem without turning points, Mathematics of Computation, 32 (1978), 1025-1039. [17] L. Koralov, Random perturbations of 2-dimensional Hamiltonian flows, Probab. Theory Related Fileds, 129 (2004), 37-62. doi: doi:10.1007/s00440-003-0320-0. [18] J. J. H. Miller, E. O'Riordan and G. I. Shishkin, "Fitted Numerical Methods for Singular Perturbation Problems," World Scientific, Singapore, 1996. [19] K. W. Morton, Numerical solution of convection-diffusion problems, in "Applied Mathematics and Mathematical Computation," Vol. 12, Chapman & Hall, London, 1996. [20] N. Nakamura, Two-dimensional mixing, edge formation and permeability diagnosed in an area coordinate, J. Atmos. Sci., 53 (1996), 1524-1537. doi: doi:10.1175/1520-0469(1996)053<1524:TDMEFA>2.0.CO;2. [21] A. Novikov, G. Papanicolaou and L. Ryzhik, Boundary layers for cellular flows at high Péclet numbers, Comm. Pure Appl. Math., 58 (2005), 867-922. doi: doi:10.1002/cpa.20058. [22] P. B. Rhines and W. R. Young, How rapidly is passive scalar mixed within closed streamlines?, J. Fluid Mech., 133 (1983), 135-145. doi: doi:10.1017/S0022112083001822. [23] M. N. Rosenbluth, H. L. Berk, I. Doxas and W. Horton, Effective diffusion in laminar convective flows, Phys. Fluids, 30 (1987), 2636-2647. doi: doi:10.1063/1.866107. [24] T. A. Shaw, J.-L. Thiffeault and C. R. Doering, Stirring up trouble: Multi-scale mixing measures for steady scalar sources, Phys. D, 231 (2007), 143-164. doi: doi:10.1016/j.physd.2007.05.001. [25] J. Shen, Stable and efficient spectral methods in unbounded domains using Laguerre functions, SIAM J. Numer. Anal., 38 (2000), 1113-1133. doi: doi:10.1137/S0036142999362936. [26] B. I. Shraiman, Diffusive transport in a Rayleigh-Bénard convection cell, Phys. Rev. A, 36 (1987), 261-267. doi: doi:10.1103/PhysRevA.36.261. [27] E. Shuckburgh, H. Jones, J. Marshall and C. Hill, Quantifying the eddy diffusivity of the Southern Ocean I: Temporal variability I, J. Phys. Oceanogr., to appear (2010).

show all references

##### References:
 [1] R. E. Bank, "PLTMG: A Software Package for Solving Elliptic Partial Differential Equations," SIAM Books, Philadelphia, 1994. [2] A. Bensoussan, J. L. Lions and G. Papanicoalou, "Asymptotic Analysis for Periodic Structures," North-Holland, 1978. [3] S. Childress, Alpha-effect in flux ropes and sheets, Phys. Earth Planet Int., 20 (1979), 172-180. doi: doi:10.1016/0031-9201(79)90039-6. [4] B. Cushman-Roisin, "Introduction to Geophysical Fluid Dynamics," Prentice-Hall, Englewood Cliffs, NJ, 1994. [5] C. Cuvelier, A. Segal and A. A. van Steenhoven, "Finite Element Methods and Navier-Stokes Equations," Mathematics and its Applications, 22, D. Reidel Publishing Co., Dordrecht, 1986. [6] J. Douglas, Jr. and T. F. Russell, Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures, SIAM J. Numer. Anal., 19 (1982), 871-885. doi: doi:10.1137/0719063. [7] R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, in "Handbook of Numerical Analysis," Vol. VII, North-Holland, Amsterdam, (2000), 713-1020. [8] A. Fannjiang and G. Papanicolaou, Convection enhanced diffusion for periodic flows, SIAM J. Appl. Math., 54 (1994), 333-408. doi: doi:10.1137/S0036139992236785. [9] A. Fannjiang and G. Papanicolaou, Convection-enhanced diffusion for random flows, J. Statist. Phys., 88 (1997), 1033-1076. doi: doi:10.1007/BF02732425. [10] J. H. Ferziger and M. Perić, "Computational Methods for Fluid Dynamics," 2nd edition, Springer-Verlag, Berlin, 1999. [11] D. Funaro and O. Kavian, Approximation of some diffusion evolution equations in unbounded domains by Hermite functions, Math. Comp., 57 (1991), 597-619. [12] P. H. Haynes and E. F. Shuckburgh, Effective diffusivity as a measure of atmospheric transport, Part I: Stratosphere, J. Geophys. Res., 105 (2000), 777-794. [13] P. H. Haynes and E. F. Shuckburgh, Effective diffusivity as a measure of atmospheric transport, Part II: Troposphere and lower stratosphere, J. Geophys. Res., 105 (2000), 795-810. doi: doi:10.1029/2000JD900092. [14] S. Heinze, Diffusion-advection in cellular flows with large Péclet numbers, Arch. Ration. Mech. Anal., 168 (2003), 329-342. doi: doi:10.1007/s00205-003-0256-7. [15] C. Johnson, "Numerical Solution of Partial Differential Equations by the Finite Element Method," Cambridge University Press, Cambridge, Great Britain, 1987. [16] R. B. Kellogg and A. Tsan, Analysis of some difference approximations for a singular perturbation problem without turning points, Mathematics of Computation, 32 (1978), 1025-1039. [17] L. Koralov, Random perturbations of 2-dimensional Hamiltonian flows, Probab. Theory Related Fileds, 129 (2004), 37-62. doi: doi:10.1007/s00440-003-0320-0. [18] J. J. H. Miller, E. O'Riordan and G. I. Shishkin, "Fitted Numerical Methods for Singular Perturbation Problems," World Scientific, Singapore, 1996. [19] K. W. Morton, Numerical solution of convection-diffusion problems, in "Applied Mathematics and Mathematical Computation," Vol. 12, Chapman & Hall, London, 1996. [20] N. Nakamura, Two-dimensional mixing, edge formation and permeability diagnosed in an area coordinate, J. Atmos. Sci., 53 (1996), 1524-1537. doi: doi:10.1175/1520-0469(1996)053<1524:TDMEFA>2.0.CO;2. [21] A. Novikov, G. Papanicolaou and L. Ryzhik, Boundary layers for cellular flows at high Péclet numbers, Comm. Pure Appl. Math., 58 (2005), 867-922. doi: doi:10.1002/cpa.20058. [22] P. B. Rhines and W. R. Young, How rapidly is passive scalar mixed within closed streamlines?, J. Fluid Mech., 133 (1983), 135-145. doi: doi:10.1017/S0022112083001822. [23] M. N. Rosenbluth, H. L. Berk, I. Doxas and W. Horton, Effective diffusion in laminar convective flows, Phys. Fluids, 30 (1987), 2636-2647. doi: doi:10.1063/1.866107. [24] T. A. Shaw, J.-L. Thiffeault and C. R. Doering, Stirring up trouble: Multi-scale mixing measures for steady scalar sources, Phys. D, 231 (2007), 143-164. doi: doi:10.1016/j.physd.2007.05.001. [25] J. Shen, Stable and efficient spectral methods in unbounded domains using Laguerre functions, SIAM J. Numer. Anal., 38 (2000), 1113-1133. doi: doi:10.1137/S0036142999362936. [26] B. I. Shraiman, Diffusive transport in a Rayleigh-Bénard convection cell, Phys. Rev. A, 36 (1987), 261-267. doi: doi:10.1103/PhysRevA.36.261. [27] E. Shuckburgh, H. Jones, J. Marshall and C. Hill, Quantifying the eddy diffusivity of the Southern Ocean I: Temporal variability I, J. Phys. Oceanogr., to appear (2010).
 [1] Qing Tang. On an optimal control problem of time-fractional advection-diffusion equation. Discrete and Continuous Dynamical Systems - B, 2020, 25 (2) : 761-779. doi: 10.3934/dcdsb.2019266 [2] Michael Taylor. Random walks, random flows, and enhanced diffusivity in advection-diffusion equations. Discrete and Continuous Dynamical Systems - B, 2012, 17 (4) : 1261-1287. doi: 10.3934/dcdsb.2012.17.1261 [3] Assyr Abdulle. Multiscale methods for advection-diffusion problems. Conference Publications, 2005, 2005 (Special) : 11-21. doi: 10.3934/proc.2005.2005.11 [4] Lena-Susanne Hartmann, Ilya Pavlyukevich. Advection-diffusion equation on a half-line with boundary Lévy noise. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 637-655. doi: 10.3934/dcdsb.2018200 [5] S. Sadeghi, H. Jafari, S. Nemati. Solving fractional Advection-diffusion equation using Genocchi operational matrix based on Atangana-Baleanu derivative. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3747-3761. doi: 10.3934/dcdss.2020435 [6] Masahiro Yamamoto. Uniqueness for inverse problem of determining fractional orders for time-fractional advection-diffusion equations. Mathematical Control and Related Fields, 2022  doi: 10.3934/mcrf.2022017 [7] Shitao Liu, Roberto Triggiani. Determining damping and potential coefficients of an inverse problem for a system of two coupled hyperbolic equations. Part I: Global uniqueness. Conference Publications, 2011, 2011 (Special) : 1001-1014. doi: 10.3934/proc.2011.2011.1001 [8] Alexandre Caboussat, Roland Glowinski. A Numerical Method for a Non-Smooth Advection-Diffusion Problem Arising in Sand Mechanics. Communications on Pure and Applied Analysis, 2009, 8 (1) : 161-178. doi: 10.3934/cpaa.2009.8.161 [9] Patrick Henning, Mario Ohlberger. The heterogeneous multiscale finite element method for advection-diffusion problems with rapidly oscillating coefficients and large expected drift. Networks and Heterogeneous Media, 2010, 5 (4) : 711-744. doi: 10.3934/nhm.2010.5.711 [10] Patrick Henning, Mario Ohlberger. A-posteriori error estimate for a heterogeneous multiscale approximation of advection-diffusion problems with large expected drift. Discrete and Continuous Dynamical Systems - S, 2016, 9 (5) : 1393-1420. doi: 10.3934/dcdss.2016056 [11] Yuriy Golovaty, Anna Marciniak-Czochra, Mariya Ptashnyk. Stability of nonconstant stationary solutions in a reaction-diffusion equation coupled to the system of ordinary differential equations. Communications on Pure and Applied Analysis, 2012, 11 (1) : 229-241. doi: 10.3934/cpaa.2012.11.229 [12] Renhao Cui. Asymptotic profiles of the endemic equilibrium of a reaction-diffusion-advection SIS epidemic model with saturated incidence rate. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 2997-3022. doi: 10.3934/dcdsb.2020217 [13] Yichen Zhang, Meiqiang Feng. A coupled $p$-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075 [14] Yuki Kaneko, Hiroshi Matsuzawa, Yoshio Yamada. A free boundary problem of nonlinear diffusion equation with positive bistable nonlinearity in high space dimensions I : Classification of asymptotic behavior. Discrete and Continuous Dynamical Systems, 2022, 42 (6) : 2719-2745. doi: 10.3934/dcds.2021209 [15] Marco Di Francesco, Alexander Lorz, Peter A. Markowich. Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavior. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1437-1453. doi: 10.3934/dcds.2010.28.1437 [16] Abdallah Benabdallah, Mohsen Dlala. Rapid exponential stabilization by boundary state feedback for a class of coupled nonlinear ODE and $1-d$ heat diffusion equation. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 1085-1102. doi: 10.3934/dcdss.2021092 [17] Xinxin Jing, Yuanyuan Nie, Chunpeng Wang. Asymptotic behavior of solutions to coupled semilinear parabolic equations with general degenerate diffusion coefficients. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022107 [18] Cédric Bernardin, Valeria Ricci. A simple particle model for a system of coupled equations with absorbing collision term. Kinetic and Related Models, 2011, 4 (3) : 633-668. doi: 10.3934/krm.2011.4.633 [19] Dominika Pilarczyk. Asymptotic stability of singular solution to nonlinear heat equation. Discrete and Continuous Dynamical Systems, 2009, 25 (3) : 991-1001. doi: 10.3934/dcds.2009.25.991 [20] Dan Wei, Shangjiang Guo. Qualitative analysis of a Lotka-Volterra competition-diffusion-advection system. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2599-2623. doi: 10.3934/dcdsb.2020197

2021 Impact Factor: 1.497