Advertisement

Two Decades of Wave-Like Equation for the Numerical Simulation of Incompressible Viscous Flow: A Review

  • Roland GlowinskiEmail author
  • Tsorng-Whay Pan
Chapter
Part of the Computational Methods in Applied Sciences book series (COMPUTMETHODS, volume 47)

Abstract

A wave-like equation based method for the numerical solution of the Navier-Stokes equations modeling incompressible viscous flow was introduced nearly twenty years ago. From its inception to nowadays it has been applied successfully to the numerical solution of two and three dimensional flow problems for incompressible Newtonian and non-Newtonian viscous fluids, in flow regions with fixed or moving boundaries. The main goals of this article are: (i) To recall the foundations of the wave-like equation methodology, and (ii) to review some typical viscous flow problems where it has been applied successfully.

Keywords

Incompressible viscous flow Operator splitting time discretization schemes Wave-like equation method for the numerical treatment of the advection step Finite element approximations 

Notes

Acknowledgements

The authors of this article are thanking the editors of this volume, a very special thank being due to Marja-Leena Rantalainen, from the University of Jyväskylä in Finland, for her beautiful processing of this contribution. The support of the US National Science Foundation via grant DMS-1418308 is also acknowledged.

References

  1. 1.
    Anupindi K, Lai W, Frankel S (2014) Characterization of oscillatory instability in lid driven cavity flows using lattice Boltzmann method. Comput Fluids 92:7–21MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Baaijens FPT (1998) Mixed finite element methods for viscoelastic flow analysis: a review. J Non-Newton Fluid Mech 79(2–3):361–385zbMATHCrossRefGoogle Scholar
  3. 3.
    Bristeau MO, Glowinski R, Périaux J (1987) Numerical methods for the Navier-Stokes equations. Application to the simulation of compressible and incompressible viscous flow. Comput Phys Rep 6:73–187MathSciNetCrossRefGoogle Scholar
  4. 4.
    Chhabra RP (1993) Bubbles, drops, and particles in non-Newtonian fluids. CRC Press, Boca Raton, FLGoogle Scholar
  5. 5.
    Chiang TP, Sheu WH, Hwang RR (1998) Effect of Reynolds number on the eddy structure in a lid-driven cavity. Int J Numer Meth Fluids 26(5):557–579zbMATHCrossRefGoogle Scholar
  6. 6.
    Chilcott MD, Rallison JM (1988) Creeping flow of dilute polymer solutions past cylinders and spheres. J Non-Newton Fluid Mech 29:381–432zbMATHCrossRefGoogle Scholar
  7. 7.
    Chippada S, Dawson CN, Martinez ML, Wheeler MF (1998) Finite element approximations to the system of shallow water equations I: continuous-time a priori error estimates. SIAM J Numer Anal 35(2):692–711MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Chippada S, Dawson CN, Martínez-Canales ML, Wheeler MF (1998) Finite element approximations to the system of shallow water equations, Part II: discrete-time a priori error estimates. SIAM J Numer Anal 36(1):226–250zbMATHCrossRefGoogle Scholar
  9. 9.
    Chorin AJ (1967) A numerical method for solving incompressible viscous flow problems. J Comput Phys 2(1):12–26MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Dean EJ, Glowinski R (1997) A wave equation approach to the numerical solution of the Navier-Stokes equations for incompressible viscous flow. CR Acad Sci Paris Sér I Math 325(7):783–791MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Dean EJ, Glowinski R, Guidoboni G (2007) On the numerical simulation of Bingham visco-plastic flow: old and new results. J Non-Newton Fluid Mech 142(1–3):36–62zbMATHCrossRefGoogle Scholar
  12. 12.
    Dean EJ, Glowinski R, Pan T-W (1998) A wave equation approach to the numerical simulation of incompressible viscous fluid flow modelled by the Navier-Stokes equations. In: De Santo JA (ed) Mathematical and numerical aspects of wave propagation (Golden. CO, 1998). Philadelphia, PA, pp 65–74Google Scholar
  13. 13.
    Economides MJ, Nolte KG (1989) Reservoir stimulation. Prentice Hall, Englewood Cliffs, NJGoogle Scholar
  14. 14.
    Fattal R, Kupferman R (2004) Constitutive laws for the matrix-logarithm of the conformation tensor. J Non-Newton Fluid Mech 123(2–3):281–285zbMATHCrossRefGoogle Scholar
  15. 15.
    Fattal R, Kupferman R (2005) Time-dependent simulation of viscoelastic flows at high Weissenberg number using the log-conformation representation. J Non-Newton Fluid Mech 126(1):23–37zbMATHCrossRefGoogle Scholar
  16. 16.
    Feldman Y, Gelfgat AY (2010) Oscillatory instability of a three-dimensional lid-driven flow in a cube. Phys Fluids 22:093602CrossRefGoogle Scholar
  17. 17.
    Fujima S, Tabata M, Fukasawa Y (1994) Extension to three-dimensional problems of the upwind finite element scheme based on the choice of up- and downwind points. Comput Methods Appl Mech Eng. 112:109–131MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Ghia UK, Ghia KN, Shin CT (1982) High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method. J Comput Phys 48(3):387–411zbMATHCrossRefGoogle Scholar
  19. 19.
    Giannetti R, Luchini P, Marino L (2009) Linear stability analysis of three-dimensional lid-driven cavity flow. In: Atti del XIX Congresso AIMETA di Meccanica Teorica e Applicata (Ancona, 2009), Aras Edizioni, pp 738.1–738.10Google Scholar
  20. 20.
    Glowinski R (2003) Finite element methods for incompressible viscous flow. In: Ciarlet PG, Lions JL (eds) Handbook of numerical analysis, vol IX. North-Holland, Amsterdam, pp 3–1176Google Scholar
  21. 21.
    Glowinski R (2015) Variational methods for the numerical solution of nonlinear elliptic problems. SIAM, Philadelphia, PAzbMATHCrossRefGoogle Scholar
  22. 22.
    Glowinski R, Dean EJ, Guidoboni G, Juárez LH, Pan T-W (2008) Applications of operator-splitting methods to the direct numerical simulation of particulate and free-surface flows and to the numerical solution of the two-dimensional elliptic Monge-Ampère equation. Jap J Indust Appl Math 25(1):1–63zbMATHCrossRefGoogle Scholar
  23. 23.
    Glowinski R, Guidoboni G, Pan T-W (2006) Wall-driven incompressible viscous flow in a two-dimensional semi-circular cavity. J Comput Phys 216(1):76–91MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Glowinski R, Lawton W, Ravachol M, Tenenbaum E (1990) Wavelet solution of linear and nonlinear elliptic, parabolic and hyperbolic problems in one space dimension. In: Glowinski R, Lichnewsky A (eds) Computing methods in applied sciences and engineering (Paris, 1990). SIAM, Philadelphia, PA, pp 55–120zbMATHGoogle Scholar
  25. 25.
    Glowinski R, Osher S, Yin W (eds) (2016) Splitting methods in communication and imaging, science and engineering. Springer, New YorkzbMATHGoogle Scholar
  26. 26.
    Glowinski R, Pan T-W, Hesla TI, Joseph DD, Périaux J (2001) A fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies: application to particulate flow. J Comput Phys 169(2):363–426MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Glowinski R, Wachs A (2011) On the numerical simulation of viscoplastic fluid flow. In: Ciarlet PG, Glowinski R, Xu J (eds) Handbook of numerical analysis, vol XVI. North-Holland, Amsterdam, pp 483–718zbMATHGoogle Scholar
  28. 28.
    Gustafsson B, Kreiss H-O, Oliger J (1995) Time dependent problems and difference methods. Wiley, New YorkzbMATHGoogle Scholar
  29. 29.
    Hao J, Pan T-W, Glowinski R, Joseph DD (2009) A fictitious domain/distributed Lagrange multiplier method for the particulate flow of Oldroyd-B fluids: a positive definiteness preserving approach. J Non-Newton Fluid Mech 156(1–2):95–111zbMATHCrossRefGoogle Scholar
  30. 30.
    He Q, Glowinski R, Wang XP (2011) A least-squares/finite element method for the numerical solution of the Navier-Stokes-Cahn-Hilliard system modeling the motion of the contact line. J. Comput. Phys. 230(12):4991–5009MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Hou S, Pan T-W, Glowinski R (2014) Circular band formation for incompressible viscous fluid-rigid-particle mixtures in a rotating cylinder. Phys Rev E 89(2):023013CrossRefGoogle Scholar
  32. 32.
    Huang PY, Hu HH, Joseph DD (1998) Direct simulation of the sedimentation of elliptic particles in Oldroyd-B fluids. J Fluid Mech 362:297–326MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Hur SC, Choi SE, Kwon S, Di Carlo D (2011) Inertial focusing of non-spherical microparticles. Appl Phys Lett 99(4):044101CrossRefGoogle Scholar
  34. 34.
    Iwatsu R, Hyun JM, Kuwahara K (1990) Analyses of three-dimensional flow calculations in a driven cavity. Fluid Dyn Res 6(2):91–102CrossRefGoogle Scholar
  35. 35.
    Jeffery GB (1922) The motion of ellipsoidal particles immersed in a viscous fluid. Proc R Soc London A 102:161–179zbMATHCrossRefGoogle Scholar
  36. 36.
    Joseph DD (1990) Fluid dynamics of viscoelastic liquids. Springer, New YorkzbMATHCrossRefGoogle Scholar
  37. 37.
    Karnis A, Goldsmith HL, Mason SG (1966) The flow of suspensions through tubes: V. Inertial effects. Can J Chem Eng 44(4):181–193CrossRefGoogle Scholar
  38. 38.
    Keunings R (2000) A survey of computational rheology. In: Binding DM et al (eds) Proceedings of the 13th international congress on rheology, vol 1. British Society of Rheology, Glasgow, pp 7–14Google Scholar
  39. 39.
    Ku HC, Hirsh RS, Taylor TD (1987) A pseudospectral method for solution of the three-dimensional incompressible Navier-Stokes equations. J Comput Phys 70(2):439–462zbMATHCrossRefGoogle Scholar
  40. 40.
    Lee Y-J, Xu J (2006) New formulations, positivity preserving discretizations and stability analysis for non-Newtonian flow models. Comput Methods Appl Mech Eng 195(9–12):1180–1206MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Liberzon A, Feldman Y, Gelfgat A (2011) Experimental observation of the steady-oscillatory transition in a cubic lid-driven cavity. Phys Fluids 23:084106CrossRefGoogle Scholar
  42. 42.
    Liu YJ, Joseph DD (1993) Sedimentation of particles in polymer solutions. J Fluid Mech 255:565–595CrossRefGoogle Scholar
  43. 43.
    Lozinski A, Owens RG (2003) An energy estimate for the Oldroyd-B model: theory and applications. J Non-Newton Fluid Mech 112(2–3):161–176zbMATHCrossRefGoogle Scholar
  44. 44.
    Lynch DR, Gray WG (1979) A wave equation model for finite element tidal computations. Comput Fluids 7(3):207–228zbMATHCrossRefGoogle Scholar
  45. 45.
    McKinley GH (2002) Steady and transient motion of spherical particles in viscoelastic liquids. In: De Kee D, Chhabra RP (eds) Transport processes in bubbles, drops, and particles, 2nd edn. Taylor & Francis, New York, pp 338–375Google Scholar
  46. 46.
    Pan T-W, Chang C-C, Glowinski R (2008) On the motion of a neutrally buoyant ellipsoid in a three-dimensional Poiseuille flow. Comput Methods Appl Mech Eng 197(25–28):2198–2209MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Pan T-W, Glowinski R (2000) A projection/wave-like equation method for the numerical simulation of incompressible viscous fluid flow modeled by the Navier-Stokes equations. Comput Fluid Dyn J 9(2):28–42Google Scholar
  48. 48.
    Pan T-W, Hao J, Glowinski R (2009) On the simulation of a time-dependent cavity flow of an Oldroyd-B fluid. Int J Numer Meth Fluids 60(7):791–808MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Pan T-W, Hao J, Glowinski R (2011) Positive definiteness preserving approaches for viscoelastic flow of Oldroyd-B fluids: applications to a lid-driven cavity flow and a particulate flow. In: Ciarlet PG, Glowinski R, Xu J (eds) Handbook of numerical analysis, vol XVI. North-Holland, Amsterdam, pp 433–481zbMATHGoogle Scholar
  50. 50.
    Pan T-W, Joseph DD, Glowinski R (2005) Simulating the dynamics of fluid-ellipsoid interactions. Comput Struct. 83(6–7):463–478CrossRefGoogle Scholar
  51. 51.
    Patankar NA, Hu HH (2000) A numerical investigation of the detachment of the trailing particle from a chain sedimenting in Newtonian and viscoelastic fluids. J Fluids Eng 122(3):517–521CrossRefGoogle Scholar
  52. 52.
    Pironneau O (1989) Finite element methods for fluids. Wiley, ChichesterzbMATHGoogle Scholar
  53. 53.
    Rallison JM, Hinch EJ (1988) Do we understand the physics in the constitutive equation? J Non-Newton Fluid Mech 29:37–55CrossRefGoogle Scholar
  54. 54.
    Segré G, Silberberg A (1961) Radial particle displacements in Poiseuille flow of suspensions. Nature 189:209–210CrossRefGoogle Scholar
  55. 55.
    Segré G, Silberberg A (1962) Behaviour of macroscopic rigid spheres in Poiseuille flow Part 1. Determination of local concentration by statistical analysis of particle passages through crossed light beams. J Fluid Mech 14(1):115–135zbMATHCrossRefGoogle Scholar
  56. 56.
    Singh P, Joseph DD, Hesla TI, Glowinski R, Pan T-W (2000) A distributed Lagrange multiplier/fictitious domain method for viscoelastic particulate flows. J Non-Newton Fluid Mech 91(2–3):165–188zbMATHCrossRefGoogle Scholar
  57. 57.
    Süli E (1988) Convergence and nonlinear stability of the Lagrange-Galerkin method for the Navier-Stokes equations. Numer Math 53(4):459–483MathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    Süli E (1988) Stability and convergence of the Lagrange-Galerkin method with nonexact integration. In: Whiteman JR (ed) The mathematics of finite elements and applications, VI (Uxbridge, 1987). Academic Press, London, pp 435–442Google Scholar
  59. 59.
    Tagliabue A, Dedè L, Quarteroni A (2014) Isogeometric analysis and error estimates for high order partial differential equations in fluid dynamics. Comput Fluids 102:277–303MathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    Wu J (1994) Wave equation models for solving advection-diffusion equation. Int J Numer Methods Eng 37(16):2717–2733zbMATHCrossRefGoogle Scholar
  61. 61.
    Wu J (1997) A wave equation model to solve the multidimensional transport equation. Int J Numer Meth Fluids 24(5):423–439MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of HoustonHoustonUSA
  2. 2.Department of MathematicsHong-Kong Baptist UniversityKowloon TongHong Kong

Personalised recommendations