Numerical simulations of diffusion in cellular flows at high Péclet numbers

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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

Abstract
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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 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).

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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).

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