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Finiteness of corner vortices

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A Commentary to this article was published on 14 May 2018

Abstract

Till date, the sequence of vortices present in the solid corners of steady internal viscous incompressible flows was thought to be infinite. However, the already existing and most recent geometric theories on incompressible viscous flows that express vortical structures in terms of critical points in bounded domains indicate a strong opposition to this notion of infiniteness. In this study, we endeavor to bridge the gap between the two opposing stream of thoughts by diagnosing the assumptions of the existing theorems on such vortices. We provide our own set of proofs for establishing the finiteness of the sequence of corner vortices by making use of the continuum hypothesis and Kolmogorov scale, which guarantee a nonzero scale for the smallest vortex structure possible in incompressible viscous flows. We point out that the notion of infiniteness resulting from discrete self-similarity of the vortex structures is not physically feasible. Making use of some elementary concepts of mathematical analysis and our own construction of diametric disks, we conclude that the sequence of corner vortices is finite.

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References

  1. Albensoeder, S., Kuhlmann, H.C.: Accurate three-dimensional lid-driven cavity flow. J. Comput. Phys. 206, 536–558 (2005)

    Article  MATH  Google Scholar 

  2. Anderson, D.M., Davis, S.H.: Two-fluid viscous flow in a corner. J. Fluid Mech. 18, 1–31 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  3. Andersson, B., Andersson, R., Hakansson, L., Mortensen, M., Sudiyo, R., Wachem, B.V.: Computational Fluid Dynamics for Engineers. Cambridge University Press, New York (2012)

    MATH  Google Scholar 

  4. Bakker, P. G.: Bifurcations in flow patterns. Ph.D. thesis, Technical University of Delft, Netherlands (1989)

  5. Banks, D.C., Singer, B.A.: Vortex tubes in turbulent flows: identification, representation, reconstruction. In: Proceedings of the Conference of Visualization, pp. 132–139. IEEE Computer Society Press (1994)

  6. Batchelor, G.K.: An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge (1993)

    MATH  Google Scholar 

  7. Biswas, S., Kalita, J.C.: Moffatt vortices in the lid-driven cavity flow. J. Phys. Conf. Ser. 759, 012081 (2016)

    Article  Google Scholar 

  8. Biswas, G., Breuer, M., Drust, F.: Backward-facing step flows for various expansion ratios at low and moderate Reynolds numbers. J. Fluids Eng. 126, 362–374 (2004)

    Article  Google Scholar 

  9. Branicki, M., Moffatt, H.K.: Evolving eddy structures in oscillatory Stokes flows in domains with sharp corners. J. Fluid Mech. 551, 63–92 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bretherton, F.P.: Resonant interactions between waves. The case of discrete oscillations. J. Fluid Mech. 20(3), 457–479 (1964)

    Article  MathSciNet  Google Scholar 

  11. Collins, W.M., Dennis, S.C.R.: Viscous eddies near a 90\(^\circ \) and a 45\(^\circ \) corner in flow through a curved tube of triangular cross-section. J. Fluid Mech. 76, 417–432 (1976)

    Article  MATH  Google Scholar 

  12. Dean, W.R., Montagnon, P.E.: On the steady motion of viscous liquid in a corner. Math. Proc. Camb. Philos. Soc. 45(3), 389–395 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  13. Délery, J.: Three-Dimensional Separated Flow Topology. ISTE Ltd and Wiley, London, Hoboken (2013)

    Book  MATH  Google Scholar 

  14. Deshpande, M.D., Milton, S.G.: Kolmogorov scales in a driven cavity flow. Fluid Dyn. Res. 22, 359–381 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  15. Do Carmo, M.P.: Differential Geometry of Curves and Surfaces. Prentice-Hall, Englewood Cliffs (1976)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  17. Ghil, M., Liu, J.-G., Wang, C., Wang, S.: Boundary-layer separation and adverse pressure gradient for 2-D viscous incompressible flow. Phys. D 197, 149–173 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  18. Glowinski, R., Guidoboni, G., Pan, T.-W.: Wall-driven incompressible viscous flow in a two-dimensional semi-circular cavity. J. Comput. Phys. 216, 76–91 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. Guillemin, V., Pollack, A.: Differential Topology. Prentice-Hall, Englewood Cliffs (1974)

    MATH  Google Scholar 

  20. Gustafson, K., Leben. R.: Vortex subdomains. In 1st International symposium on domain decomposition methods for partial differential equations, Paris, France, 7–9 January (1987)

  21. Gustafson, K., Leben, R.: Mutigrid calculations of subvortices. Appl. Math. Comput. 19, 89–102 (1986)

    MATH  Google Scholar 

  22. Gustafson, K., Halasi, K., Leben, R.: Controversies concerning finite/infinite sequences of fluid corner vortices. Contemp. Math. Am. Math. Soc. 99, 351–357 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hall, O., Hills, C.P., Gilbert, A.D.: Slow flow between concentric cones. Q. J. Mech. Appl. Mech. 60(1), 27–48 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hellou, M., Coutanceau, M.: Cellular Stokes flow induced by rotation of a cylinder in a closed channel. J. Fluid Mech. 236, 557–577 (1992)

    Article  Google Scholar 

  25. Hirschel, E.H., Cousteix, J., Kordulla, W.: Three-Dimensional Attached Viscous Flow. Springer, Berlin (2014)

    Book  MATH  Google Scholar 

  26. Jiménez, J.: The contributions of A. N. Kolmogorov to the theory of turbulence. Arbor 178(704), 589–606 (2004)

    Article  Google Scholar 

  27. Kalita, J.C., Sen, S.: Unsteady separation leading to secondary and tertiary vortex dynamics: the sub-\(\alpha \)- and sub-\(\beta \)-phenomena. J. Fluid Mech. 730, 19–51 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kirkinis, E., Davis, S.H.: Moffatt vortices induced by the motion of a contact line. J. Fluid Mech. 746, R3 (2014)

    Article  Google Scholar 

  29. Kolmogorov, A.N.: The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. C. R. Acad. Sci. URSS 30, 301–305 (1941)

    MathSciNet  Google Scholar 

  30. Koseff, J.R., Street, R.L.: On end wall effects in a lid-driven cavity flow. J. Fluids Eng. 106(4), 385–389 (1984)

    Article  Google Scholar 

  31. Krasnopolskaya, T.S.: Two-dimensional Stokes flow near a corner in a right angle wedge and Moffatt’s eddies. Mech. Res. Commun. 22(1), 9–14 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  32. Krasnopolskaya, T.S., Meleshko, V.V., Peters, G.W.M., Meijer, H.E.H.: Steady Stokes flow in an annular cavity. Q. J. Mech. Appl. Mech. 49(4), 593–619 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  33. Lesieur, M.: Turbulence in Fluids, 4th edn. Springer, Netherlands (2001)

    MATH  Google Scholar 

  34. Liron, N., Blake, J.R.: Existence of viscous eddies near boundaries. J. Fluid Mech. 107, 109–129 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  35. Lohse, D., Xia, K.-Q.: Small-scale properties of turbulent Rayleigh–Bénard convection. Annu. Rev. Fluid Mech. 42, 335–364 (2010)

    Article  MATH  Google Scholar 

  36. Ma, T., Wang, S.: Geometric theory of incompressible flows with applications to fluid dynamics. In Mathematical Surveys and Monographs, Vol.-119, American Mathematical Society, Rhode Island, USA (2005)

  37. Malhotra, C.P., Weidman, P.D., Davis, A.M.J.: Nested toroidal vortices between concentric cones. J. Fluid Mech. 522, 117–139 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  38. Malyuga, V.S.: Viscous eddies in a circular cone. J. Fluid Mech. 522, 101–116 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  39. McLachlan, R.: A steady separated viscous corner flow. J. Fluid Mech. 231, 1–34 (1991)

    Article  MATH  Google Scholar 

  40. Mishra, P.K., Verma, M.K.: Energy spectra and fluxes for Rayleigh–Bénard convection. Phys. Rev. E 81, 056316 (2010)

    Article  MathSciNet  Google Scholar 

  41. Mishra, P.K., De, A.K., Verma, M.K., Eswaran, V.: Dynamics of reorientations and reversals of large-scale flow in Rayleigh–Bénard convection. J. Fluid Mech. 668, 480–499 (2011)

    Article  MATH  Google Scholar 

  42. Moffatt, H.K.: The Asymptotic Behaviour of Solutions of the Navier–Stokes Equations Near Sharp Corners. Lecture Notes in Mathematics, vol. 771, pp. 371–380. Springer (1979)

  43. Moffatt, H.K.: Viscous and resistive eddies near a sharp corner. J. Fluid Mech. 18, 1–18 (1964)

    Article  MATH  Google Scholar 

  44. Moffatt, H.K.: Viscous eddies near a sharp corner. Arch. Mech. Stosow. 2, 365–372 (1964)

    MathSciNet  MATH  Google Scholar 

  45. Moffatt, H.K., Duffy, B.R.: Local similarity solutions and their limitations. J. Fluid Mech. 96(2), 299–313 (1980)

    Article  MATH  Google Scholar 

  46. Neufeld, Z., Hernandez-García, E.: Chemical and Biological Processes in Fluid Flows: A Dynamical Systems Approach. Imperial College Press, London (2010)

    MATH  Google Scholar 

  47. Obabko, A.V., Cassel, K.W.: Navier–Stokes solutions of unsteady separation induced by a vortex. J. Fluid Mech. 465, 99–130 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  48. O’Neill, M.E.: On angles of separation in Stokes flows. J. Fluid Mech. 133, 427–442 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  49. Papanastasiou, T.C., Georgiou, G.C., Alexandrou, A.N.: Viscous Fluid Flow. CRC Press, Boca Raton (2000)

    MATH  Google Scholar 

  50. Perko, L.: Differential Equations and Dynamical Systems, 3rd edn. Springer, New York (2001)

    Book  MATH  Google Scholar 

  51. Pope, S.B.: Turbulent Flows. Cambridge University Press, Cambridge (2000)

    Book  MATH  Google Scholar 

  52. Pozrikidis, C.: Creeping flow in two-dimensional channels. J. Fluid Mech. 180, 495–514 (1987)

    Article  Google Scholar 

  53. Ricca, R.L.: An Introduction to the Geometry and Topology of Fluid Flows. Springer, New York (2001)

    Book  MATH  Google Scholar 

  54. Rudin, W.: Principles of Mathematical Analysis, 3rd edn. McGraw-hill, New York (1976)

    MATH  Google Scholar 

  55. Sahin, M., Owens, R.G.: A novel fully implicit finite volume method applied to the lid-driven cavity problem—part I: high Reynolds number flow calculations. Int. J. Numer. Methods Fluids 42, 57–77 (2003)

    Article  MATH  Google Scholar 

  56. Shankar, P.N.: The eddy structure in Stokes flow in a cavity. J. Fluid Mech. 250, 371–383 (1993)

    Article  MATH  Google Scholar 

  57. Shankar, P.N.: Moffatt eddies in the cone. J. Fluid Mech. 539, 113–135 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  58. Shankar, P.N., Deshpande, M.D.: Fluid mechanics in the driven cavity. Annu. Rev. Fluid Mech. 32, 93–136 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  59. Shtern, V.: Moffatt eddies at an interface. Theoret. Comput. Fluid Dyn. 28, 651–656 (2014)

    Article  Google Scholar 

  60. Sousa, R.G., Poole, R.J., Afonso, A.M., Pinho, F.T., Oliveira, P.J., Morozov, A., Alves, M.A.: Lid-driven cavity flow of viscoelastic liquids. J. Nonnewton Fluid Mech. 234, 129–138 (2016)

    Article  MathSciNet  Google Scholar 

  61. Taneda, S.: Visualization of separating Stokes flows. J. Phys. Soc. Jpn. 46(6), 1935–1942 (1979)

    Article  Google Scholar 

  62. Tyson, J.J., Strogatz, S.H.: The differential geometry of scroll waves. Int. J. Bifurc. Chaos 4, 723–744 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  63. Walstra, P.: Physical Chemistry of Foods, pp. 436–437. Marcel Dekker, New York (2003)

    Google Scholar 

  64. White, F.M.: Fluid Mechanics. Mcgraw-Hill, New York (2003)

    Google Scholar 

  65. Wu, J.-Z., Ma, H.-Y., Zhou, M.-D.: Vortical Flows. Springer, Berlin (2015)

    Book  Google Scholar 

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  1. Department of Mathematics, Indian Institute of Technology, Guwahati, 781039, India

    Jiten C. Kalita, Sougata Biswas & Swapnendu Panda

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  1. Jiten C. Kalita
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  2. Sougata Biswas
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Kalita, J.C., Biswas, S. & Panda, S. Finiteness of corner vortices. Z. Angew. Math. Phys. 69, 37 (2018). https://doi.org/10.1007/s00033-018-0933-x

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