Advertisement

A Predictor–Corrector Meshless Based Scheme for Incompressible Navier–Stokes Flows

  • 7 Accesses

Abstract

In this paper, a predictor–corrector scheme in meshless radial basis functions for solving incompressible Navier–Stokes equations in vorticity–streamfunction formulation is presented. The method is based on a local collocation formulation and does not require either a grid generation or evaluation of an integral. Predictor–corrector techniques are used to evaluate numerical fluxes in the forward–backward stencil parts and a relaxation scheme is investigated in order to reach high order accuracy. The main advantages of this approach are that no mesh generations nor Riemann problem solvers are required during the solution process. Numerical results are shown for several test examples including problems on driven cavity flows, backward-facing step flows and Rayleigh–Benard convection flows. The main focus is to examine the performance of the proposed meshless method for Navier–Stokes problems with high Reynolds number. The obtained results demonstrate its ability to capture the main solution features.

This is a preview of subscription content, log in to check access.

Access options

Buy single article

Instant unlimited access to the full article PDF.

US$ 39.95

Price includes VAT for USA

Subscribe to journal

Immediate online access to all issues from 2019. Subscription will auto renew annually.

US$ 99

This is the net price. Taxes to be calculated in checkout.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

References

  1. 1.

    Alhuri, Y., Benkhaldoun, F., Ouazar, D., Seaid, M., Taik, A.: A meshless method for numerical simulation of depth-averaged turbulence flows using a \(k{-}\epsilon \) model. Int. J. Numer. Methods Fluids 80(1), 3–22 (2016)

  2. 2.

    Atluri, S., Zhu, T.: A new meshless local Petrov–Galerkin (MLPG) approach in computational mechanics. Comput. Mech. 22, 117–127 (1998)

  3. 3.

    Batchelor, G.K.: An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge (2000)

  4. 4.

    Bell, J.B., Colella, P., Glaz, H.M.: A second-order projection method for the incompressible Navier–Stokes equations. J. Comput. Phys. 85(2), 257–283 (1989)

  5. 5.

    Belytschko, T., Krongauz, Y., Organ, D., Fleming, M., Krysl, P.: Meshless methods: an overview and recent developments. Comput. Methods Appl. Mech. Eng. 139, 3–47 (1996)

  6. 6.

    Benkhaldoun, F., Halassi, A., Ouazar, D., Seaid, M., Taik, A.: A stabilized meshless method for time-dependent convection-dominated flow problems. Math. Comput. Simul. 137, 159–176 (2017)

  7. 7.

    Brown, D.L., Cortez, R., Minion, M.L.: Accurate projection methods for the incompressible Navier–Stokes equations. J. Comput. Phys. 168(2), 464–499 (2001)

  8. 8.

    Bruneau, C.H., Jouron, C.: An efficient scheme for solving steady incompressible Navier–Stokes Equations. J. Comput. Phys. 89(2), 389–413 (1990)

  9. 9.

    Buhamman, M.: Radial Basis Functions: Theory and Implementations. Cambridge University Press, Cambridge (2003)

  10. 10.

    Chinchapatnam, P., Djidjeli, K., Nair, P., Tan, M.: A compact RBF-FD based meshless method for the incompressible Navier–Stokes equations. Proc. Inst. Mech. Eng. Part M J. Eng. Marit. Environ. 223(3), 275–290 (2009)

  11. 11.

    Chorin, A.: Numerical solution of the Navier–Stokes equations. Math. Comput. 22(104), 745–762 (1968)

  12. 12.

    El-Amrani, M., Seaïd, M.: Weakly compressible and advection approximations of incompressible viscous flows. Commun. Numer. Methods Eng. 22(7), 831–847 (2006)

  13. 13.

    Erturk, E.: Numerical solutions of 2-D steady incompressible flow over a backward-facing step, part I: high Reynolds number solutions. Comput. Fluids 37, 633–655 (2008)

  14. 14.

    Erturk, E., Gokcol, O.: Fourth order compact formulation of steady Navier–Stokes equations on non-uniform grids. Int. J. Mech. Eng. Technol. 9, 1379–1389 (2018)

  15. 15.

    Erturk, E., Corke, T.C., Gokcol, C.: Numerical solutions of 2-D steady incompressible driven cavity flow at high Reynolds numbers. Int. J. Numer. Methods Fluids 48, 747–774 (2005)

  16. 16.

    Gartling, D.K.: A test problem for outflow boundary conditions-flow over a backward-facing step. Int. J. Numer. Methods Fluids 11(7), 953–967 (1990)

  17. 17.

    Ghia, U., Ghia, K., Shin, C.: High-Re solutions for incompressible flow using the Navier–Stokes equations and a multigrid method. J. Comput. Phys. 48(3), 387–411 (1982)

  18. 18.

    Golberg, M., Chen, C.: The theory of radial basis function applied to the BEM for inhomogeneous partial differential equations. Bound. Elem. Commun. 5, 57–61 (1994)

  19. 19.

    Golub, G., Kahan, W.: Calculating the singular values and pseudo-inverse of a matrix. J. Soc. Ind. Appl. Math. Ser. B Numer. Anal. 2(2), 205–224 (1965)

  20. 20.

    Kadanoff, L.: Turbulent heat flow: structures and scaling. Phys. Today 54(8), 34–39 (2001)

  21. 21.

    Kansa, E.: Multiquadrics—a scattered data approximation scheme with applications to computational fluid-dynamics—I surface approximations and partial derivative estimates. Comput. Math. Appl. 19, 127–145 (1990)

  22. 22.

    Kansa, E., Power, H., Fasshauer, G., Ling, L.: A volumetric integral radial basis function method for time-dependent partial differential equations I. Formulation. Eng. Anal. Bound. Elem. 28, 1191–1206 (2004)

  23. 23.

    Kao, P.H., Yang, R.J.: Simulating oscillatory flows in Rayleigh–Bénard convection using the lattice Boltzmann method. Int. J. Heat Mass Transf. 50, 3315–3328 (2007)

  24. 24.

    Khoshfetrat, A., Abedini, M.: Numerical modeling of long waves in shallow water using LRBF-DQ and hybrid DQ/LRBF-DQ. Ocean Model. 65, 1–10 (2013)

  25. 25.

    Lashckarbolok, M., Jabbari, E.: Collocated discrete least squares (CDLS) meshless method for the stream function-vorticity formulation of 2D incompressible Navier–Stokes equations. Sci. Iran. 19(6), 1422–1430 (2012)

  26. 26.

    Lestandi, L., Bhaumik, S., Avatar, G., Azaiez, M., Sengupta, T.: Multiple Hopf bifurcations and flow dynamics inside a 2D singular lid driven cavity. Comput. Fluids 166, 86–103 (2018)

  27. 27.

    Liu, M., Ren, Y.X., Zhang, H.: A class of fully second order accurate projection methods for solving the incompressible Navier–Stokes equations. J. Comput. Phys. 200(1), 325–346 (2004)

  28. 28.

    MacCormack, R.: The effect of viscosity in hypervelocity impact cratering. J. Spacecr. Rockets 40(5), 757–763 (2003)

  29. 29.

    Micchelli, C.: Interpolation of scattered data: distance matrices and conditionally positive definite functions. Constr. Approx. 2, 11–22 (1986)

  30. 30.

    Moran, J.: An Introduction to Theoretical and Computational Aerodynamics. Dover Books on Aeronautical Engineering. Dover, New York (1984)

  31. 31.

    Morgan, K., Periaux, J., Thomasset, F. (eds.): Analysis of Laminar Flow Over a Backward Facing Step, vol. 9. Springer, Berlin (1984)

  32. 32.

    Powell, M.: The theory of radial basis function approximation in 1990, in advances in numerical analysis. In: Light, W. (ed.) Wavelets, Subdivision Algorithms and Radial Functions, vol. II, pp. 105–210. Oxford University Press, Oxford (1992)

  33. 33.

    Rizzo, E.: Nuclear Fusion, Current Lead, High Temperature Superconductor, Numerical simulation, Computational thermal Fluid Dynamics. KIT Scientific Publishing, Singapore (2014)

  34. 34.

    Shen, J.: Projection methods for time-dependent Navier–Stokes equations. Appl. Math. Lett. 5(1), 35–37 (1992)

  35. 35.

    Shu, C.: An upwind local RBF-DQ method for simulation of inviscid compressible flows. Comput. Methods Appl. Mech. Eng. 194, 2001–2017 (2005)

  36. 36.

    Stortkuhl, T., Zenger, C., Zimmer, S.: An asymptotic solution for the singularity at the angular point of the lid driven cavity. Int. J. Numer. Methods Heat Fluid Flow 4, 47–59 (1994)

  37. 37.

    Tabbakh, Z., Seaid, M., Ellaia, R., Ouazar, D., Benkhaldoun, F.: A local radial basis function projection method for incompressible flows in water eutrophication. Eng. Anal. Bound. Elem. 106, 528–540 (2019)

  38. 38.

    Velivelli, A.C., Keneth, M.B.: Domain decomposition based coupling between the lattice boltzmann method and traditional cfd methods—part II: numerical solution to the backward facing step flow. Adv. Eng. Softw. 82, 65–74 (2015)

  39. 39.

    Wei, Y., Wang, Z., Yang, J., Dou, H.S., Qian, Y.: A simple lattice Boltzmann model for turbulence Rayleigh–Bénard thermal convection. Comput. Fluids 118, 167–171 (2015)

  40. 40.

    Xiang, S., Wang, Km, ting Ai, Y., dong Sha, Y., Shi, H.: Trigonometric variable shape parameter and exponent strategy for generalized multiquadric radial basis function approximation. Appl. Math. Model. 36(5), 1931–1938 (2012)

  41. 41.

    Yang, X., Wang, Q.: A 2D numerical study of polar active liquid crystal flows in a cavity. Comput. Fluids 155, 33–49 (2017)

  42. 42.

    Young, D.M.: Iterative methods for solving partial difference equations of elliptical type. Ph.D. Thesis, Harvard University (1950)

Download references

Acknowledgements

Financial support provided by Centre National pour la Recherche Scientifique et Technique through the “Projet EuroMéditerranée 3+3/MedLagoon” is gratefully acknowledged. The authors are grateful to the anonymous reviewers for their valuable comments and helpful suggestions which greatly improved the paper’s quality.

Author information

Correspondence to Abdoul-hafar Halassi Bacar.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Halassi Bacar, A., Ouazar, D. & Taik, A. A Predictor–Corrector Meshless Based Scheme for Incompressible Navier–Stokes Flows. Int. J. Appl. Comput. Math 6, 18 (2020) doi:10.1007/s40819-020-0769-x

Download citation

Keywords

  • Navier–Stokes equations
  • Radial basis functions
  • Predictor–corrector scheme
  • Driven-cavity flow
  • Backward-facing step flows
  • Rayleigh–Benard convection