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Direct spectral domain decomposition method for 2D incompressible Navier-Stokes equations

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Abstract

An efficient direct spectral domain decomposition method is developed coupled with Chebyshev spectral approximation for the solution of 2D, unsteady and incompressible Navier-Stokes equations in complex geometries. In this numerical approach, the spatial domains of interest are decomposed into several non-overlapping rectangular sub-domains. In each sub-domain, an improved projection scheme with second-order accuracy is used to deal with the coupling of velocity and pressure, and the Chebyshev collocation spectral method (CSM) is adopted to execute the spatial discretization. The influence matrix technique is employed to enforce the continuities of both variables and their normal derivatives between the adjacent sub-domains. The imposing of the Neumann boundary conditions to the Poisson equations of pressure and intermediate variable will result in the indeterminate solution. A new strategy of assuming the Dirichlet boundary conditions on interface and using the first-order normal derivatives as transmission conditions to keep the continuities of variables is proposed to overcome this trouble. Three test cases are used to verify the accuracy and efficiency, and the detailed comparison between the numerical results and the available solutions is done. The results indicate that the present method is efficiency, stability, and accuracy.

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References

  1. Canuto, C., Hussaini, M. Y., Quarteroni, A., and Zang, T. A. Spectral Methods in Fluid Dynamics, Springer, New York (1988)

    Book  MATH  Google Scholar 

  2. Peyret, R. Spectral Methods for Incompressible Viscous Flow, Springer, New York (2002)

    Book  MATH  Google Scholar 

  3. Orszag, S. A. Spectral methods for problems in complex geometries. Journal of Computational Physics, 37(1), 70–92 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  4. Canuto, C. and Funaro, D. The Schwarz algorithm for spectral methods. SIAM Journal on Numerical Analysis, 25(1), 24–40 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  5. Pateraa, A. T. A spectral element method for fluid dynamics: laminar flow in a channel expansion. Journal of Computational Physics, 54(3), 468–488 (1984)

    Article  Google Scholar 

  6. Phillips, T. N. and Karageorghis, A. Efficient direct method for solving the spectral collocation equations for stokes flow in rectangularly decomposable domains. SIAM Journal on Scientific and Statistical Computing, 10(1), 89–103 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  7. Macaraeg, M. G. and Streett, C. L. Improvement in spectral collocation discretization through a multiple domain technique. Applied Numerical Mathematics, 2(2), 95–108 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  8. Zanolli, P. Domain decomposition algorithms for spectral methods. Calcolo, 24, 201–240 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  9. Funaro, D., Quarteroni, A., and Zanolli, P. An iterative procedure with interface relaxation for domain decomposition methods. SIAM Journal on Numerical Analysis, 25(6), 1213–1236 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  10. Louchart, O., Randriamampianina, A., and Leonardi, E. Spectral domain decomposition technique for the incompressible Navier-Stokes equations. Numerical Heat Transfer, Part A, 34(5), 495–518 (1998)

    Article  Google Scholar 

  11. Louchart, O. and Randriamampianina, A. A spectral iterative domain decomposition technique for the incompressible Navier-Stokes equations. Applied Numerical Mathematics, 33(1), 233–240 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  12. Pulicani, J. P. A spectral multi-domain method for the solution of 1-D-Helmholtz and Stokes-type equations. Computers & Fluids, 16(2), 207–215 (1988)

    Article  MATH  Google Scholar 

  13. Lacroix, J. M., Peyret, R., and Pulicani, J. P. A pseudospectral multi-domain method for the Navier-Stokes equations with application to double-diffusive convection. GAMM Conference on Numerical Methods in Fluid Mechanics, 167–174 (1988)

    Google Scholar 

  14. Raspo, I. and Bontoux, P. Direct multidomain spectral method for the computation of various fluid dynamic problems. Journal of Applied Mathematics and Mechanics, 79(1), 33–36 (1999)

    Google Scholar 

  15. Raspo, I., Ouazzani, J., and Peyret, R. A spectral multidomain technique for the computation of the Czochralskimelt configuration. International Journal of Numerical Methods for Heat & Fluid Flow, 6(1), 31–58 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  16. Bwemba, R. and Pasquetti, R. On the influence matrix used in the spectral solution of the 2D Stokes problem (vorticity-stream function formulation). Applied Numerical Mathematics, 16(3), 299–315 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  17. Boguslawski, A. and Kubacki, S. An influence matrix technique for multi-domain solution of the Navier-Stokes equations in a vorticity-streamfunction formulation. Journal of Theoretical and Applied Mechanics, 47(1), 17–40 (2009)

    Google Scholar 

  18. Raspo, I. A direct spectral domain decomposition method for the computation of rotating flows in a T-shape geometry. Computers & Fluids, 32, 431–456 (2003)

    Article  MATH  Google Scholar 

  19. Danabasoglu, G., Biringen, S., and Streett, C. L. Application of the spectral multidomain method to the Navier-Stokes equations. Journal of Computational Physics, 113(2), 155–164 (1994)

    Article  MATH  Google Scholar 

  20. Sabbah, C. and Pasquetti, R. A divergence-free multidomain spectral solver of the Navier-Stokes equations in geometries of high aspect ratio. Journal of Computational Physic, 139(2), 359–379 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  21. Droll, P. and Schäfer, M. A pseudospectral multi-domain method for the incompressible Navier-Stokes equations. Journal of Scientific Computing, 17, 365–374 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  22. Chen, S. S. and Li, B. W. Application of collocation spectral domain decomposition method to solve radiative heat transfer in 2D partitioned domains. Journal of Quantitative Spectroscopy & Radiative Transfer, 149, 275–284 (2014)

    Article  Google Scholar 

  23. Vay, J. L., Haber, I., and Godfrey, B. B. A domain decomposition method for pseudo-spectral electromagnetic simulations of plasmas. Journal of Computational Physic, 243, 260–268 (2013)

    Article  MathSciNet  Google Scholar 

  24. Grinberg, L. and Karniadakis, G. E. A new domain decomposition method with overlapping patches for ultrascale simulations: application to biological flows. Journal of Computational Physic, 229(15), 5541–5563 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Sidler, R., Carcione, J. M., and Holliger, K. A pseudo-spectral method for the simulation of poroelastic seismic wave propagation in 2D polar coordinates using domain decomposition. Journal of Computational Physic, 235, 846–864 (2013)

    Article  MathSciNet  Google Scholar 

  26. Golbabai, A. and Javidi, M. A spectral domain decomposition approach for the generalized Burger’s-Fisher equation. Chaos, Solitons and Fractals, 39(1), 385–392 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. Taleei, A. and Dehghan, M. Time-splitting pseudo-spectral domain decomposition method for the soliton solutions of the one- and multi-dimensional nonlinear Schrödinger equations. Computer Physics Communications, 185(6), 1515–1528 (2014)

    Article  MathSciNet  Google Scholar 

  28. Hameed, M. and Ellahi, R. Numerical and analytical solutions of an Oldroyd 8-constant MHD fluid with nonlinear slip conditions. International Journal for Numerical Methods in Fluids, 67(10), 1234–1246 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  29. Hameed, M. and Ellahi, R. Thin film flow of non-Newtonian MHD fluid on a vertically moving belt. International Journal for Numerical Methods in Fluids, 66(11), 1409–1419 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  30. Ellahi, R. Numerical analysis of steady non-Newtonian flows with heat transfer analysis, MHD and nonlinear slip effects. International Journal of Numerical Methods for Heat & Fluid Flow, 22(1), 24–28 (2012)

    Article  Google Scholar 

  31. Ellahi, R. The effects of MHD and temperature dependent viscosity on the flow of non-Newtonian nanofluid in a pipe: analytical solutions. Applied Mathematical Modelling, 37(3), 1451–1467 (2013)

    Article  MathSciNet  Google Scholar 

  32. Luo, X. H., Li, B. W., Zhang, J. K., and Hu, Z. M. Simulation of thermal radiation effects on MHD free convection in a square cavity using Chebyshev collocation spectral method. Numerical Heat Transfer, Part A, 66(7), 792–815 (2014)

    Article  Google Scholar 

  33. Hugues, S. and Randriamampianina, A. An improved projection scheme applied to pseudospectral methods for the incompressible Navier-Stokes equations. International Journal for Numerical Methods in Fluids, 28(3), 501–521 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  34. Tuckerman, L. S. Divergence-free velocity fields in nonperiodic geometries. Journal of Computational Physics, 80(2), 403–441 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  35. Chen, H. L., Su, Y. H., and Shizgal, B. D. A direct spectral collocation Poisson solver in polar and cylindrical coordinates. Journal of Computational Physics, 160(2), 453–469 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  36. Johnston, H. and Liu, J. G. Accurate, stable and efficient Navier-Stokes solvers based on explicit treatment of the pressure term. Journal of Computational Physics, 199(1), 221–259 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  37. Burggraf, O. R. Analytical and numerical studies of the structure of steady separated flows. Journal of Fluid Mechanics, 24(1), 113–151 (1966)

    Article  Google Scholar 

  38. Ghia, U., Ghia, K. N., and Shin, C. T. High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method. Journal of Computational Physics, 48(3), 387–411 (1982)

    Article  MATH  Google Scholar 

  39. Wahba, E. M. Steady flow simulations inside a driven cavity up to Reynolds number 35 000. Computers & Fluids, 66, 85–97 (2012)

    Article  MathSciNet  Google Scholar 

  40. Erturk, E., Corke, T. C., and Gokcol, C. Numerical solutions of 2D steady incompressible driven cavity flow at high Reynolds numbers. International Journal for Numerical Methods in Fluids, 48(7), 747–774 (2005)

    Article  MATH  Google Scholar 

  41. Moffat, H. K. Viscous and resistive eddies near a sharp corner. Journal of Fluid Mechanics, 18(1), 1–18 (1964)

    Article  MathSciNet  Google Scholar 

  42. Ehrenstein, U. and Peyret, R. A Chebyshev collocation method for the Navier-Stokes equations with application to double-diffusive convection. International Journal for Numerical Methods in Fluids, 9(4), 427–452 (1989)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Institute of Thermal Engineering, School of Energy and Power Engineering, Dalian University of Technology, Dalian, Liaoning Province, 116024, China

    Benwen Li

  2. Key Laboratory of Electromagnetic Processing of Materials (Ministry of Education), Northeastern University, Shenyang, 110819, China

    Shangshang Chen

Authors
  1. Benwen Li
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  2. Shangshang Chen
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Correspondence to Benwen Li.

Additional information

Project supported by the National Natural Science Foundation of China (No. 51176026) and the Fundamental Research Funds for the Central Universities (No.DUT14RC(3)029)

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Li, B., Chen, S. Direct spectral domain decomposition method for 2D incompressible Navier-Stokes equations. Appl. Math. Mech.-Engl. Ed. 36, 1073–1090 (2015). https://doi.org/10.1007/s10483-015-1964-7

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  • DOI: https://doi.org/10.1007/s10483-015-1964-7

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