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

Numerical simulations of diffusion in cellular flows at high Péclet numbers

Yuliya Gorb 1, , Dukjin Nam 2, and  Alexei Novikov 3,

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.
Keywords: asymptotic model, high Péclet number, a system of coupled heat equations., Advection-diffusion equation.
Mathematics Subject Classification: Primary: 65-05, 35Q35, 35J25, 65N0.
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, Philadelphia, 1994.  Google Scholar

[2]

A. Bensoussan, J. L. Lions and G. Papanicoalou, "Asymptotic Analysis for Periodic Structures," North-Holland, 1978.  Google Scholar

[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.  Google Scholar

[4]

B. Cushman-Roisin, "Introduction to Geophysical Fluid Dynamics," Prentice-Hall, Englewood Cliffs, NJ, 1994. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, in "Handbook of Numerical Analysis," Vol. VII, North-Holland, Amsterdam, (2000), 713-1020.  Google Scholar

[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.  Google Scholar

[9]

A. Fannjiang and G. Papanicolaou, Convection-enhanced diffusion for random flows, J. Statist. Phys., 88 (1997), 1033-1076. doi: doi:10.1007/BF02732425.  Google Scholar

[10]

J. H. Ferziger and M. Perić, "Computational Methods for Fluid Dynamics," 2nd edition, Springer-Verlag, Berlin, 1999.  Google Scholar

[11]

D. Funaro and O. Kavian, Approximation of some diffusion evolution equations in unbounded domains by Hermite functions, Math. Comp., 57 (1991), 597-619.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

C. Johnson, "Numerical Solution of Partial Differential Equations by the Finite Element Method," Cambridge University Press, Cambridge, Great Britain, 1987.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

J. J. H. Miller, E. O'Riordan and G. I. Shishkin, "Fitted Numerical Methods for Singular Perturbation Problems," World Scientific, Singapore, 1996.  Google Scholar

[19]

K. W. Morton, Numerical solution of convection-diffusion problems, in "Applied Mathematics and Mathematical Computation," Vol. 12, Chapman & Hall, London, 1996.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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). Google Scholar

show all references

References:
[1]

R. E. Bank, "PLTMG: A Software Package for Solving Elliptic Partial Differential Equations," SIAM Books, Philadelphia, 1994.  Google Scholar

[2]

A. Bensoussan, J. L. Lions and G. Papanicoalou, "Asymptotic Analysis for Periodic Structures," North-Holland, 1978.  Google Scholar

[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.  Google Scholar

[4]

B. Cushman-Roisin, "Introduction to Geophysical Fluid Dynamics," Prentice-Hall, Englewood Cliffs, NJ, 1994. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, in "Handbook of Numerical Analysis," Vol. VII, North-Holland, Amsterdam, (2000), 713-1020.  Google Scholar

[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.  Google Scholar

[9]

A. Fannjiang and G. Papanicolaou, Convection-enhanced diffusion for random flows, J. Statist. Phys., 88 (1997), 1033-1076. doi: doi:10.1007/BF02732425.  Google Scholar

[10]

J. H. Ferziger and M. Perić, "Computational Methods for Fluid Dynamics," 2nd edition, Springer-Verlag, Berlin, 1999.  Google Scholar

[11]

D. Funaro and O. Kavian, Approximation of some diffusion evolution equations in unbounded domains by Hermite functions, Math. Comp., 57 (1991), 597-619.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

C. Johnson, "Numerical Solution of Partial Differential Equations by the Finite Element Method," Cambridge University Press, Cambridge, Great Britain, 1987.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

J. J. H. Miller, E. O'Riordan and G. I. Shishkin, "Fitted Numerical Methods for Singular Perturbation Problems," World Scientific, Singapore, 1996.  Google Scholar

[19]

K. W. Morton, Numerical solution of convection-diffusion problems, in "Applied Mathematics and Mathematical Computation," Vol. 12, Chapman & Hall, London, 1996.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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). Google Scholar

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