# 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 & 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, (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. doi: doi:10.1016/0031-9201(79)90039-6. [4] B. Cushman-Roisin, "Introduction to Geophysical Fluid Dynamics,", Prentice-Hall, (1994). [5] C. Cuvelier, A. Segal and A. A. van Steenhoven, "Finite Element Methods and Navier-Stokes Equations,", Mathematics and its Applications, 22 (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. doi: doi:10.1137/0719063. [7] R. Eymard, T. Gallouët and R. Herbin, Finite volume methods,, in, VII (2000), 713. [8] A. Fannjiang and G. Papanicolaou, Convection enhanced diffusion for periodic flows,, SIAM J. Appl. Math., 54 (1994), 333. doi: doi:10.1137/S0036139992236785. [9] A. Fannjiang and G. Papanicolaou, Convection-enhanced diffusion for random flows,, J. Statist. Phys., 88 (1997), 1033. doi: doi:10.1007/BF02732425. [10] J. H. Ferziger and M. Perić, "Computational Methods for Fluid Dynamics," 2nd edition,, Springer-Verlag, (1999). [11] D. Funaro and O. Kavian, Approximation of some diffusion evolution equations in unbounded domains by Hermite functions,, Math. Comp., 57 (1991), 597. [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. [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. doi: doi:10.1029/2000JD900092. [14] S. Heinze, Diffusion-advection in cellular flows with large Péclet numbers,, Arch. Ration. Mech. Anal., 168 (2003), 329. 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, (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. [17] L. Koralov, Random perturbations of 2-dimensional Hamiltonian flows,, Probab. Theory Related Fileds, 129 (2004), 37. 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, (1996). [19] K. W. Morton, Numerical solution of convection-diffusion problems,, in, 12 (1996). [20] N. Nakamura, Two-dimensional mixing, edge formation and permeability diagnosed in an area coordinate,, J. Atmos. Sci., 53 (1996), 1524. 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. 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. 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. 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. 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. doi: doi:10.1137/S0036142999362936. [26] B. I. Shraiman, Diffusive transport in a Rayleigh-Bénard convection cell,, Phys. Rev. A, 36 (1987), 261. 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., (2010).

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##### References:
 [1] R. E. Bank, "PLTMG: A Software Package for Solving Elliptic Partial Differential Equations,", SIAM Books, (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. doi: doi:10.1016/0031-9201(79)90039-6. [4] B. Cushman-Roisin, "Introduction to Geophysical Fluid Dynamics,", Prentice-Hall, (1994). [5] C. Cuvelier, A. Segal and A. A. van Steenhoven, "Finite Element Methods and Navier-Stokes Equations,", Mathematics and its Applications, 22 (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. doi: doi:10.1137/0719063. [7] R. Eymard, T. Gallouët and R. Herbin, Finite volume methods,, in, VII (2000), 713. [8] A. Fannjiang and G. Papanicolaou, Convection enhanced diffusion for periodic flows,, SIAM J. Appl. Math., 54 (1994), 333. doi: doi:10.1137/S0036139992236785. [9] A. Fannjiang and G. Papanicolaou, Convection-enhanced diffusion for random flows,, J. Statist. Phys., 88 (1997), 1033. doi: doi:10.1007/BF02732425. [10] J. H. Ferziger and M. Perić, "Computational Methods for Fluid Dynamics," 2nd edition,, Springer-Verlag, (1999). [11] D. Funaro and O. Kavian, Approximation of some diffusion evolution equations in unbounded domains by Hermite functions,, Math. Comp., 57 (1991), 597. [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. [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. doi: doi:10.1029/2000JD900092. [14] S. Heinze, Diffusion-advection in cellular flows with large Péclet numbers,, Arch. Ration. Mech. Anal., 168 (2003), 329. 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, (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. [17] L. Koralov, Random perturbations of 2-dimensional Hamiltonian flows,, Probab. Theory Related Fileds, 129 (2004), 37. 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, (1996). [19] K. W. Morton, Numerical solution of convection-diffusion problems,, in, 12 (1996). [20] N. Nakamura, Two-dimensional mixing, edge formation and permeability diagnosed in an area coordinate,, J. Atmos. Sci., 53 (1996), 1524. 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. 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. 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. 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. 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. doi: doi:10.1137/S0036142999362936. [26] B. I. Shraiman, Diffusive transport in a Rayleigh-Bénard convection cell,, Phys. Rev. A, 36 (1987), 261. 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., (2010).
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