Interaction and multiscale mechanics

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Multiscale method and pseudospectral simulations for linear viscoelastic incompressible flows

  • Zhang, Ling (Department of Applied Mathematics, Northwestern Polytechnical University) ;
  • Ouyang, Jie (Department of Applied Mathematics, Northwestern Polytechnical University)
  • Received : 2011.05.23
  • Accepted : 2012.02.12
  • Published : 2012.03.25
⟨ Previous Next ⟩

Abstract

The two-dimensional incompressible flow of a linear viscoelastic fluid we considered in this research has rapidly oscillating initial conditions which contain both the large scale and small scale information. In order to grasp this double-scale phenomenon of the complex flow, a multiscale analysis method is developed based on the mathematical homogenization theory. For the incompressible flow of a linear viscoelastic Maxwell fluid, a well-posed multiscale system, including averaged equations and cell problems, is derived by employing the appropriate multiple scale asymptotic expansions to approximate the velocity, pressure and stress fields. And then, this multiscale system is solved numerically using the pseudospectral algorithm based on a time-splitting semi-implicit influence matrix method. The comparisons between the multiscale solutions and the direct numerical simulations demonstrate that the multiscale model not only captures large scale features accurately, but also reflects kinetic interactions between the large and small scale of the incompressible flow of a linear viscoelastic fluid.

Keywords

References

  1. Bensoussan, A., Lions, J.L. and Papanicolaou, G. (1978), Asymptotic analysis for periodic structures, North-Holland Publishing Company, Netherlands.
  2. Beris, A.N. and Dimitropoulos, C.D. (1999), "Pseudospectral simulation of turbulent viscoelastic channel flow", Comput. Method. Appl. M., 180(3-4), 365-392. https://doi.org/10.1016/S0045-7825(99)00174-7
  3. Beris, A.N. and Sureshkumar, R. (1996), "Simulation of time-dependent viscoelastic channel poiseuille flow at high reynolds numbers", Chem. Eng. Sci., 51(9), 1451-1471. https://doi.org/10.1016/0009-2509(95)00313-4
  4. Canuto, C., Hussaini, M.Y., Quarteroni, A. and Zang, T.A. (1988), Spectral methods in fluid dynamics, Springer-Verlag, New York.
  5. Cioranescu, D. and Donato, P. (1999), An introduction to homogenization, Oxford University Press, Oxford.
  6. Hou, T.Y. and Li, R. (2006), "Dynamic depletion of vortex stretching and non-blowup of the 3-D incompressible Euler equations", J. Nonlinear Sci., 16, 639-664. https://doi.org/10.1007/s00332-006-0800-3
  7. Hou, T.Y., Yang, D.P. and Ran, H.Y. (2005), "Multiscale analysis in Lagrangian formulation for the 2D incompressible Euler equation", Discrete Cont. Dyn. S., 13(5), 1153-1186. https://doi.org/10.3934/dcds.2005.13.1153
  8. Hou, T.Y., Yang, D.P. and Ran, H.Y. (2008), "Multi-scale analysis and computation for the 3D incompressible Navier-Stokes equations", Multiscale Model Sim., 4, 1317-1346.
  9. Hou, T.Y., Yang, D.P. and Wang, K. (2004), "Homogenization of incompressible Euler equation", J. Comput. Math, 22(2), 220-229.
  10. Jikov, V., Kozlov, S.M. and Oleinik, O.A. (1994), Homogenization of differential operators and integral functionals, Springer-Verlag, Berlin.
  11. Li, A., Li, R. and Fish, J. (2008), "Generalized mathematical homogenization: from theory to practice", Comput. Method. Appl. M., 197(41-42), 3225-3248. https://doi.org/10.1016/j.cma.2007.12.002
  12. McLaughlin, D.W., Papanicolaou, G.C. and Pironneau, O.R. (1985), "Convection of microstructure and related problems", SIAM J. Appl. Math., 45(5), 780-797. https://doi.org/10.1137/0145046
  13. Oleinik, O.A., Shamaev, A.S. and Yosifian, G.A. (1992), Mathematical problems in elasticity and homogenization, North-Holland Publishing Company, Netherlands.
  14. Pavliotis, G.A. and Stuart, A.M. (2008), Multiscale methods: averaging and homogenization, Texts in Applied Mathematics, 53, Springer, New York.
  15. Phillips, T.N. and Soliman, I.M. (1991), "Influence matrix technique for the numerical spectral simulation of viscous incompressible flows", Numer. Meth. Part. D. E., 7(1), 9-24. https://doi.org/10.1002/num.1690070103
  16. Shen, J. and Tang, T. (2006), Spectral and high-order methods with applications, Science Press, Beijing.