Skip to main content
SpringerLink
Log in

Conservation of acceleration and dynamic entanglement in mechanics

  • Original Paper
  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

Within a flow of a viscous fluid the effects of compression and viscosity are closely coupled with inertia. Discrete mechanics gives a physical sense of these interactions by showing that they exist only in a dynamic vision where the variation of the velocity in time generates the entanglement of the effects of compression and shearing. These two phenomena are described in the form of a Helmholtz–Hodge decomposition by two orthogonal terms within the discrete equation of motion, the first curl-free and the second divergence-free. They can only exchange mechanical energy if the acceleration is nonzero. This entanglement, which occurs at all spatial scales, is a function of longitudinal and transverse celerities. After a presentation of the formal framework, simple examples allow to understand the temporal shifts of direct and induced flows in accordance with the causality principle.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

References

  1. Bhatia, H., Norgard, G., Pascucci, V., Bremer, P.: The Helmholtz–Hodge decomposition–a survey. IEEE Trans. Visual Comput. Graph. 19(8), 1386–1404 (2013). https://doi.org/10.1109/TVCG.2012.316

    Article  Google Scholar 

  2. Brachet, M., Meiron, D., Orszag, S., Nickel, B., Morf, R., Frish, U.: The Taylor-Green vortex and fully developed turbulence. J. Stat. Phys. 34, 1049–1063 (1984). https://doi.org/10.1007/BF01009458

    Article  MathSciNet  Google Scholar 

  3. Caltagirone, J.P.: Discrete Mechanics, Concepts and Applications. ISTE, John Wiley & Sons, London (2019). https://doi.org/10.1002/9781119482826

    Book  Google Scholar 

  4. Caltagirone, J.P.: On Helmholtz–Hodge decomposition of inertia on a discrete local frame of reference. Phys. Fluids 32, 083604 (2020). https://doi.org/10.1063/5.0015837

    Article  Google Scholar 

  5. Caltagirone, J.P.: Application of discrete mechanics model to jump conditions in two-phase flows. J. Comput. Phys. 432, 110151 (2021). https://doi.org/10.1016/j.jcp.2021.110151

    Article  MathSciNet  MATH  Google Scholar 

  6. Caltagirone, J.P.: On a reformulation of Navier–Stokes equations based on Helmholtz–Hodge decomposition. Phys. Fluids 33, 063605 (2021). https://doi.org/10.1063/5.0053412

    Article  Google Scholar 

  7. Caltagirone, J.P.: The role of inertia on the onset of turbulence in a vortex-filament. Fluids 8(16), 1–19 (2023). https://doi.org/10.3390/fluids8010016

    Article  Google Scholar 

  8. Caltagirone, J.P., Vincent, S.: On primitive formulation in fluid mechanics and fluid-structure interaction with constant piecewise properties in velocity-potentials of acceleration. Acta Mech. 231(6), 2155–2171 (2020). https://doi.org/10.1007/s00707-020-02630-w

    Article  MathSciNet  MATH  Google Scholar 

  9. deBonis, J.: Solutions of the Taylor-Green Vortex problem using high-resolution explicit finite difference methods. NASA/TM-2013-217850 (2013)

  10. Diosady, L., Murman, S.: Case 3.3: Taylor-green vortex evolution. Case summary for 3rd International Workshop on Higher-Order CFD Methods, Jan 3–4, 2015, Kissimmee (2015)

  11. Gad-El-Hak: Stokes hypothesis for a Newtonian, isotropic fluid. J. Fluids Eng. 117, 3–5 (1995)

    Article  Google Scholar 

  12. Gottlieb, D., Orszag, S.S.A.: Numerical analysis of spectral methods: theory and applications. The Society for Industrial and Applied Mathematics, Philadelphia (1977)

    Book  MATH  Google Scholar 

  13. Guermond, J., Minev, P., Shen, J.: An overview of projection methods for incompressible flows. Comput. Methods Appl. Mech. Eng. 195, 6011–6045 (2006). https://doi.org/10.1016/j.cma.2005.10.010

    Article  MathSciNet  MATH  Google Scholar 

  14. Kosmann-Schwarzbach, Y.: Noether Theorems. Invariance and Conservations Laws. Springer-Verlag, New York (2011). https://doi.org/10.1007/978-0-387-87868-3

    Book  MATH  Google Scholar 

  15. Liénard, A.: Champ électrique et magnétique produit par une charge électrique concentrée en un point et animée d’un mouvement quelconque. G. Carré et C. Naud (1898)

  16. Lima, F.: Simple ‘log formulae’ for pendulum motion valid for any amplitude. Eur. J. Phys. 29, 1091–1098 (2008)

    Article  MATH  Google Scholar 

  17. Maxwell, J.: A dynamical theory of the electromagnetic field. Philos. Trans. R. Soc. Lond. 155, 459–512 (1865)

    Google Scholar 

  18. Moet, H., Laporte, F., Chevalier, G., Poinsot, T.: Wave propagation in vortices and vortex bursting. Phys. Fluids 17(5), 054109 (2005). https://doi.org/10.1063/1.1896937

    Article  MathSciNet  MATH  Google Scholar 

  19. Newton, I.: Principes Mathématiques de la Philosophie Naturelle. traduit en francais moderne d’après l’œuvre de la marquise Du Châtelet sur les Principia, fac-similé de l’édition de 1759 publié aux Editions Jacques Gabay en 1990, Paris (1990)

  20. Orszag, S.A.: Analytical theories of turbulence. J. Fluid Mech. 41(2), 363–386 (1970). https://doi.org/10.1017/S0022112070000642

    Article  MATH  Google Scholar 

  21. Rajagopal, K.: A new development and interpretation of the Navier–Stokes fluid which reveals why the “Stokes assumption’’ is inapt. Int. J. Non-Linear Mech. 50, 141–151 (2013). https://doi.org/10.1016/j.ijnonlinmec.2012.10.007

    Article  Google Scholar 

  22. Ranocha, H.: Entropy conserving and kinetic energy preserving numerical methods for the euler equations using summation-by-parts operators. In: Sherwin, S.J., et al. (eds.) Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018, Lecture Notes in Computational Science and Engineering, vol. 134. Springer, Berlin (2020). https://doi.org/10.1007/978-3-030-39647-3-42

    Chapter  Google Scholar 

  23. Ranocha, H., Ostaszewski, K.P.H.: Discrete vector calculus and Helmholtz Hodge decomposition for classical finite difference summation by parts operators. arXiv:1908.08732v1 [math.NA] 23 Aug 2019 pp. 1–33 (2019)

  24. van Rees, W.M.: Vortex bursting. Phys. Rev. Fluids 5, 110504 (2020). https://doi.org/10.1103/PhysRevFluids.5.110504

    Article  Google Scholar 

  25. van Rees, W.M., Leonard, A., Pullin, D., Koumoutsakos, P.: A comparison of vortex and pseudo-spectral methods for the simulation of periodic vortical fows at high Reynolds numbers. J. Comput. Phys. 230, 2794–2805 (2011). https://doi.org/10.1016/j.jcp.2010.11.031

    Article  MathSciNet  MATH  Google Scholar 

  26. Sabelnikov, V.A., Lipatnikov, A.N., Nikitin, N., Nishiki, S., Hasegawa, T.: Solenoidal and potential velocity fields in weakly turbulent premixed flames. Proc. Combust. Inst. 38(2), 3087–3095 (2021). https://doi.org/10.1016/j.proci.2020.09.016

    Article  Google Scholar 

  27. Spalart, P.R.: Airplane trailing vortices. Annu. Rev. Fluid Mech. 30(1), 107–138 (1998). https://doi.org/10.1146/annurev.fluid.30.1.107

    Article  MathSciNet  MATH  Google Scholar 

  28. Wang, Z., Fidkowski, K., Abgrall, R., Bassi, F., Caraeni, D., Cary, A., Deconinck, H., Hartmann, R., Hillewaert, K., Huynh, H., Kroll, N., May, G., Persson, P.O., van Leer, B., Visbal, M.: High-order CFDmethods: current status and perspective. Int. J. Numer. Meth. Fluids 72, 811–845 (2013). https://doi.org/10.1002/fld.3767

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Bordeaux INP, Arts et Métiers Institute of Technology, University of Bordeaux, CNRS UMR no 5295, INRAE, I2M Bordeaux, 33405, Talence, France

    Jean-Paul Caltagirone

  2. Department of Engineering and Architecture, Via delle scienze 208, 33100, Udine, Italy

    Cristian Marchioli

  3. Université Paris-Est Marne-La-Vallée, Laboratoire Modélisation et Simulation Multi Echelle (MSME), UMR CNRS no 8208, 5 boulevard Descartes, 77454, Marne-la-Vallée Cedex, France

    Stéphane Vincent

Authors
  1. Jean-Paul Caltagirone
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Cristian Marchioli
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Stéphane Vincent
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jean-Paul Caltagirone.

Ethics declarations

Conflict of interest

There is no conflict of interest in this work.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Caltagirone, JP., Marchioli, C. & Vincent, S. Conservation of acceleration and dynamic entanglement in mechanics. Acta Mech 234, 5511–5541 (2023). https://doi.org/10.1007/s00707-023-03682-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-023-03682-4

Mathematics Subject Classification

Search

Navigation