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On the Dynamics of the Boundary Vorticity for Incompressible Viscous Flows

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Abstract

The dynamical equation of the boundary vorticity has been obtained, which shows that the viscosity at a solid wall is doubled as if the fluid became more viscous at the boundary. For certain viscous flows the boundary vorticity can be determined via the dynamical equation up to bounded errors for all time, without the need of knowing the details of the main stream flows. We then validate the dynamical equation by carrying out stochastic direct numerical simulations (i.e. the random vortex method for wall-bounded incompressible viscous flows) by two different means of updating the boundary vorticity, one using mollifiers of the Biot–Savart singular integral kernel, another using the dynamical equations.

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The data that support the findings of this study are available from the corresponding author upon reasonable request.

References

  1. Anderson, C., Greengard, C.: On vortex methods. SIAM J. Numer. Anal. 22(3), 413–440 (1985)

    Article  MathSciNet  Google Scholar 

  2. Cherepanov, V., Qian, Z.: Monte–Carlo method for incompressible fluid flows past obstacles. arXiv:2304.09152 (2023)

  3. Chorin, A.J.: Numerical study of slightly viscous flow. J. Fluid Mech. 57, 785–796 (1973)

    Article  MathSciNet  Google Scholar 

  4. Chorin, A.J.: Vorticity and Turbulence. Springer, Berlin (1994)

    Book  Google Scholar 

  5. Chorin, A.J., Marsden, J.E.: A Mathematical Introduction to Fluid Mechanics. Springer, Berlin (1993)

    Book  Google Scholar 

  6. Cottet, G.-H., Koumoutsakos, P.D.: Vortex Methods: Theory and Practice. Cambridge University Press, Cambridge (2000)

    Book  Google Scholar 

  7. Goodman, J.: Convergence of the random vortex method. Commun. Pure Appl. Math. 40(2), 189–220 (1987)

    Article  MathSciNet  Google Scholar 

  8. Leonard, A.: 1980 Vortex methods for flow simulation. J. Comput. Phys. 289–335

  9. Li, J., Qian, Z., Xu, M.: Twin Brownian particle method for the study of Oberbeck–Boussinesq fluid flows. arXiv:2303.17260 (2023)

  10. Liu, J.-G., Weinan, E.: Simple finite element method in vorticity formulation for incompressible flow. Math. Comput. 69, 1385–1407 (2001)

    MathSciNet  Google Scholar 

  11. Majda, A.J., Bertozzi, A.L.: Vorticity and Incompressible Flow. Cambridge University Press, Cambridge (2002)

    Google Scholar 

  12. Marchioro, C., Pulvirenti, M.: Vortex Methods in Two-dimensional Fluid Dynamics. Springer, Berlin (1984)

    Google Scholar 

  13. Qian, Z.: Stochastic formulation of incompressible fluid flows in wall-bounded regions. arXiv:2206.05198 (2022)

  14. Qian, Z., Qiu, Y., Zhao, L., Wu, J.: Monte-Carlo simulations for wall-bounded fluid flows via random vortex method. (2022) arXiv:2208.13233

  15. Schlichting, H., Gersten, K.: Boundary-Layer Theory, 9th edn. Springer, Berlin (2017)

    Book  Google Scholar 

  16. Weinan, E., Liu, J.G.: Vorticity boundary condition and related issues for finite difference schemes. J. Comput. Phys. 124, 368–382 (1996)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank Oxford Suzhou Centre for Advanced Research for providing the excellent computing facility. JGL is partially supported by NSF under award DMS-2106988. VC and ZQ are is supported (fully and partially, respectively) by the EPSRC Centre for Doctoral Training in Mathematics of Random Systems: Analysis, Modelling and Simulation (EP/S023925/1).

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Authors and Affiliations

  1. Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK

    Vladislav Cherepanov & Zhongmin Qian

  2. Department of Mathematics and Department of Physics, Duke University, Durham, NC, 27708, US

    Jian-Guo Liu

  3. Oxford Suzhou Centre for Advanced Research, Suzhou, China

    Zhongmin Qian

Authors
  1. Vladislav Cherepanov
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  2. Jian-Guo Liu
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  3. Zhongmin Qian
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Correspondence to Jian-Guo Liu.

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Cherepanov, V., Liu, JG. & Qian, Z. On the Dynamics of the Boundary Vorticity for Incompressible Viscous Flows. J Sci Comput 99, 42 (2024). https://doi.org/10.1007/s10915-024-02498-1

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  • DOI: https://doi.org/10.1007/s10915-024-02498-1

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