Skip to main content
SpringerLink
Log in

Isogeometric Compatible Discretizations for Viscous Incompressible Flow

  • Chapter
  • First Online:
IsoGeometric Analysis: A New Paradigm in the Numerical Approximation of PDEs

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2161))

Abstract

In this chapter, isogeometric discretizations for viscous incompressible flow are presented that satisfy the incompressibility constraint in a pointwise manner. As incompressibility is satisfied pointwise, these discretizations replicate the geometric structure of the Navier-Stokes equations and properly balance energy, enstrophy, and helicity. The result is a method with enhanced accuracy and robustness as compared with classical finite element methods for incompressible flow. Within the chapter, we review the geometric structure of the Navier-Stokes equations, outline the construction of compatible B-spline spaces which allow for pointwise mass conservation, and present a suite of illustrative numerical results demonstrating the potential of compatible B-splines in computational fluid dynamics.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 49.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. J.C. Andre, M. Lesieur, Influence of helicity on the evolution of isotropic turbulence at high Reynolds number. J. Fluid Mech. 81, 187–207 (1977)

    Article  MATH  Google Scholar 

  2. V. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits. Annales de L’institut Fourier 16, 319–361 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  3. V. Arnold, The Hamiltonian nature of the Euler equations in the dynamics of a rigid body and of a perfect fluid. Uspekhi Matematicheskikh Nauk 24, 225–226 (1969)

    Google Scholar 

  4. D.N. Arnold, D. Boffi, R.S. Falk, Quadrilateral H(div) finite elements. SIAM J. Numer. Anal. 42, 2429–2451 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. D.N. Arnold, R.S. Falk, R. Winther, Mixed finite elements for linear elasticity with weakly imposed symmetry. Math. Comput. 76, 1699–1723 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. W.T. Ashurst, A.R. Kerstein, R.M. Kerr, G.H. Gibson, Alignment of vorticity and scalar gradient with strain rate in simulated Navier-Stokes turbulence. Phys. Fluids 30, 2343–2353 (1987)

    Article  Google Scholar 

  7. F. Bashforth, J.C. Adams, Theories of Capillary Action (Cambridge University Press, London, 1883)

    Google Scholar 

  8. J.T. Beale, T. Kato, A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Commun. Math. Phys. 94, 61–66 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  9. M. Behr, D. Hastreiter, S. Mittal, T.E. Tezduyar, Incompressible flow past a circular cylinder: dependence of the computed flow field on the location of the lateral boundaries. Comput. Methods Appl. Mech. Eng. 123, 309–316 (1995)

    Article  Google Scholar 

  10. L.C. Berselli, D. Cordoba, On the regularity of the solutions to the 3D Navier-Stokes equations: a remark on the role of helicity. C.R. Math. 347, 613–618 (2009)

    Google Scholar 

  11. D. Boffi, A note on the deRham complex and a discrete compactness property. Appl. Math. Lett. 14, 33–38 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  12. M.E. Brachet, Small-scale structure of the Taylor-Green vortex. J. Fluid Mech. 130, 411–452 (1983)

    Article  MATH  Google Scholar 

  13. A. Buffa, G. Sangalli, R. Vázquez, Isogeometric analysis in electromagnetics: B-splines approximation. Comput. Methods Appl. Mech. Eng. 199, 1143–1152 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. A. Buffa, J. Rivas, G. Sangalli, R. Vázquez, Isogeometric discrete differential forms in three dimensions. SIAM J. Numer. Anal. 49, 818–844 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. L. Caffarelli, R. Kohn, L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations. Commun. Pure Appl. Math. 35, 771–831 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  16. S. Caorsi, P. Fenrandes, M. Raffetto, On the convergence of Galerkin finite element approximations of electromagnetic eigenproblems. SIAM J. Numer. Anal. 38, 580–607 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  17. P. Constantin, C. Fefferman, A. Majda, Geometric constraints on potentially singular solutions for the 3-D Euler equations. Commun. Partial Differential Equations 21, 559–571 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  18. J.A. Cottrell, T.J.R. Hughes, Y. Bazilevs, Isogeometric Analysis: Toward Integration of CAD and FEA (Wiley, New York, 2009)

    Book  Google Scholar 

  19. J. Crank, P. Nicolson, A practical method for numerical evaluation of solutions of partial differential equations of the heat conduction type. Math. Proc. Camb. Philos. Soc. 43, 50–67 (1947)

    Article  MathSciNet  MATH  Google Scholar 

  20. R. de Vogelaere, Methods of integration which preserve the contact transformation property of Hamiltonian equations. Technical Report, Department of Mathematics, University of Notre Dame (1956)

    Google Scholar 

  21. J. Deng, T.Y. Hou, X. Yu, Geometric properties and the non-blow-up of the three-dimensional Euler equation. Commun. Partial Differential Equations 30, 225–243 (2005)

    Article  Google Scholar 

  22. J. Deng, T.Y. Hou, X. Yu, Improved geometric conditions for non-blowup of the 3D incompressible Euler equation. Commun. Partial Differential Equations 31, 293–306 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  23. C.R. Doering, C. Foias, Exponential decay rate of the power spectrum for solutions of the Navier-Stokes equations. Phys. Fluids 7, 1385–1390 (1995)

    Article  MathSciNet  Google Scholar 

  24. C.R. Doering, C. Foias, Energy dissipation in body-forced turbulence. J. Fluid Mech. 467, 289–306 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  25. C.R. Doering, J. Gibbon, Applied Analysis of the Navier-Stokes Equations (Cambridge University Press, Cambridge, 1995)

    Book  MATH  Google Scholar 

  26. J. Douglas, J. Roberts, Global estimates for mixed methods for second order elliptic equations. Math. Comput. 44, 39–52 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  27. D.G. Ebin, J.E. Marsden, Group of diffeomorphisms and the motion of an incompressible fluid. Ann. Math. 92, 102–163 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  28. J.A. Evans, T.J.R. Hughes, Discrete spectrum analyses for various mixed discretizations of the Stokes eigenproblem. Comput. Mech. 50, 667–674 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  29. J.A. Evans, T.J.R. Hughes, Isogeometric divergence-conforming B-splines for the Darcy-Stokes-Brinkman equations. Math. Models Methods Appl. Sci. 23, 671–741 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  30. J.A. Evans, T.J.R. Hughes, Isogeometric divergence-conforming B-splines for the steady Navier-Stokes equations. Math. Models Methods Appl. Sci. 23, 1421–1478 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  31. J.A. Evans, T.J.R. Hughes, Isogeometric divergence-conforming B-splines for the unsteady Navier-Stokes equations. J. Comput. Phys. 241, 141–167 (2013)

    Article  MATH  Google Scholar 

  32. J.A. Evans, T.J.R. Hughes, G. Sangalli, Enforcement of constraints and maximum principles in the variational multiscale method. Comput. Methods Appl. Mech. Eng. 199, 61–76 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  33. M. Fortin, An analysis of the convergence of mixed finite element methods. Revue Française d’Automatique Informatique et Recherche Operationnellle. Analyse Numérique 11, 341–354 (1977)

    MathSciNet  MATH  Google Scholar 

  34. U. Frisch, Turbulence: The Legacy of A. N. Kolmogorov (Cambridge University Press, Cambridge, 1995)

    Google Scholar 

  35. V. Girault, P.A. Raviart, Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms (Springer, Berlin, 1986)

    Google Scholar 

  36. S. Goto, J.C. Vassilicos, The dissipation rate coefficient of turbulence is not universal and depends on the internal stagnation point structure. Phys. Fluids 21, 035104–035104–8 (2009)

    Google Scholar 

  37. H. Gümral, Lagrangian description, symplectic structure, and invariants of 3D fluid flow. Phys. Lett. A 232, 416–424 (1997)

    Article  MATH  Google Scholar 

  38. J.G. Heywood, R. Rannacher, Finite element approximation of the nonstationary Navier-Stokes problem. I. Regularity of solutions and second-order error estimates for spatial discretization. SIAM J. Numer. Anal. 19, 275–311 (1982)

    MATH  Google Scholar 

  39. T.J.R. Hughes, L.P. Franca, M. Mallet, A new finite element formulation for computational fluid dynamics: I. Symmetric forms of the compressible Euler and Navier-Stokes equations and the second law of thermodynamics. Comput. Methods Appl. Mech. Eng. 54, 223–234 (1986)

    MATH  Google Scholar 

  40. T.J.R. Hughes, L.P. Franca, M. Mallet, A new finite element formulation for computational fluid dynamics: VI. Convergence analysis of the generalized SUPG formulation for linear time-dependent multi-dimensional advective-diffusive systems. Comput. Methods Appl. Mech. Eng. 63, 97–112 (1987)

    MATH  Google Scholar 

  41. T. Ishihara, T. Gotoh, Y. Kaneda, Study of high-Reynolds number isotropic turbulence by direct numerical simulation. Annu. Rev. Fluid Mech. 41, 165–180 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  42. T. Kato, Non-stationary flows of viscous and ideal fluids in \(\mathbb{R}^{3}\). J. Funct. Anal. 21, 296–309 (1972)

    Article  Google Scholar 

  43. L. Kelvin, On vortex motion. Transactions of the Royal Society of Edinburgh 25, 217–260 (1869)

    Google Scholar 

  44. R.M. Kerr, Evidence for a singularity of the three-dimensional incompressible Euler equations. Phys. Fluids A 5, 1725–1746 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  45. A.A. Kiselev, O.A. Ladyzhenskaya, On the existence and uniqueness of solutions of the non-stationary problems for flows of non-compressible fluids. Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya 9, 655–680 (1957)

    MATH  Google Scholar 

  46. H. Konzono, Y. Taniuchi, Limiting case of the Sobolev inequality in BMO with application to the Euler equations. Commun. Math. Phys. 214, 191–200 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  47. R.H. Kraichan, Helical turbulence and absolute equilibrium. J. Fluid Mech. 59, 745–752 (1973)

    Article  Google Scholar 

  48. P.D. Lax, A.N. Milgram, Parabolic equations, in Contributions to the Theory of Partial Differential Equations (AM-33) (Princeton University Press, Princeton, NJ, 1974), pp. 167–190

    Google Scholar 

  49. J. Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63, 193–248 (1934)

    Article  MathSciNet  Google Scholar 

  50. P.L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 1, Incompressible Models. Oxford Lecture Series in Mathematics and its Applications (Clarendon, Oxford, 1996)

    Google Scholar 

  51. J.E. Marsden, A. Weinstein, Coadjoint orbits, vortices and Clebsch variables for incompressible fluids. Phys. D 7, 305–323 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  52. C. Meneveau, K.R. Sreenivasan, The multifractal nature of turbulent energy dissipation. J. Fluid Mech. 224, 429–484 (1991)

    Article  MATH  Google Scholar 

  53. H.K. Moffatt, The degree of knottedness of tangled vortex lines. J. Fluid Mech. 36, 117–129 (1969)

    Article  MATH  Google Scholar 

  54. H.K. Moffatt, Simple topological aspects of turbulent vorticity dynamics, in Turbulence and Chaotic Phenomena in Fluids, ed. by T. Tatsumi (Elsevier, New York, 1984)

    Google Scholar 

  55. H.K. Moffatt, Fixed points of turbulent dynamical systems and suppression of nonlinearity, in Whither Turbulence? ed. by J.L. Lumley (Springer, Berlin, 1990), pp. 250–257

    Google Scholar 

  56. H.K. Moffatt, Spiral structures in turbulent flow, in Proceedings of the IMA Conference “Wavelets, Fractals and Fourier Transforms”, Cambridge (1991)

    Google Scholar 

  57. H.K. Moffatt, A. Tsoniber, Helicity in laminar and turbulent flow. Annu. Rev. Fluid Mech. 24, 281–312 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  58. H.K. Moffatt, S. Kida, K. Ohkitani, Stretched vortices - the sinews of turbulence; large-Reynolds-number asymptotics. J. Fluid Mech. 259, 241–264 (1994)

    Article  MathSciNet  Google Scholar 

  59. J.-J. Moreau, Constants d’un ilot tourbillionnaire en fluide parfait barotrope. R. Acad. Sci. (Paris) 252, 2810–2812 (1961)

    MathSciNet  MATH  Google Scholar 

  60. J.C. Nédélec, Mixed finite elements in \(\mathbb{R}^{3}\). Numer. Math. 35, 315–341 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  61. C. Norberg, An experimental investigation of the flow around a circular cylinder: influence of aspect ratio. J. Fluid Mech. 258, 287–316 (1994)

    Article  Google Scholar 

  62. K. Ohkitani, P. Constantin, Numerical study on the Eulerian-Lagrangian analysis of Navier-Stokes turbulence. Phys. Fluids 20, 75–102 (2008)

    Article  MATH  Google Scholar 

  63. P.A. Raviart, J.M. Thomas, A mixed finite element method for second order elliptic problems. Lect. Notes Math. 606, 292–315 (1977)

    Article  MATH  Google Scholar 

  64. R.L. Ricca, H.K. Moffatt, The helicity of a knotted vortex filament, in Topological Aspects of the Dynamics of Fluids and Plasmas, ed. by H.K. Moffatt (Kluwer, Dordrecht, 1987)

    Google Scholar 

  65. R. Stenberg, Analysis of mixed finite element methods for the Stokes problem: a unified approach. Math. Comput. 42, 9–23 (1984)

    MathSciNet  MATH  Google Scholar 

  66. H.S.G. Swann, Convergence with vanishing viscosity of nonstationary Navier-Stokes flow to ideal flow in \(\mathbb{R}^{3}\). Trans. Am. Math. Soc. 157, 373–397 (1971)

    MathSciNet  MATH  Google Scholar 

  67. T.E. Tezduyar, R. Shih, Numerical experiments on downstream boundary of flow past cylinder. J. Eng. Mech. 117, 854–871 (1991)

    Article  Google Scholar 

  68. M. Van Dyke, An Album of Fluid Motion (Parabolic Press, Stanford, CA, 1982)

    Google Scholar 

  69. R. Verürth, Error estimates for a mixed finite element approximation of the Stokes equation. Revue Française d’Automatique Informatique et Recherche Operationnellle. Analyse Numérique 18, 175–182 (1984)

    MathSciNet  Google Scholar 

  70. D. Vieru, W. Akhtar, C. Fetecau, C. Fetecau, Starting solutions for the oscillating motion of a Maxwell fluid in cylindrical domains. Meccanica 42, 573–583 (2007)

    Article  MATH  Google Scholar 

  71. W. Wolibner, Un théorème sur l’existence du mouvement plan d’un fluide parfait, homogène, incompressible, pendant un temps infiniment longue. Math. Z. 37, 698–726 (1933)

    Article  MathSciNet  MATH  Google Scholar 

  72. T.A. Zang, On the rotation and skew-symmetric forms for incompressible flow simulations. Appl. Numer. Math. 7, 27–40 (1991)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Colorado Boulder, Boulder, CO, 80309, USA

    John A. Evans

  2. Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, TX, 78712-0027, USA

    Thomas J. R. Hughes

Authors
  1. John A. Evans
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Thomas J. R. Hughes
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to John A. Evans .

Editor information

Editors and Affiliations

  1. IMATI - CNR , Pavia, Italy

    Annalisa Buffa

  2. Dipartimento di Matematica, Università di Pavia, Pavia, Italy

    Giancarlo Sangalli

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Evans, J.A., Hughes, T.J.R. (2016). Isogeometric Compatible Discretizations for Viscous Incompressible Flow. In: Buffa, A., Sangalli, G. (eds) IsoGeometric Analysis: A New Paradigm in the Numerical Approximation of PDEs. Lecture Notes in Mathematics(), vol 2161. Springer, Cham. https://doi.org/10.1007/978-3-319-42309-8_4

Download citation

Publish with us

Policies and ethics

Search

Navigation