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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References
Anderson, C., Greengard, C.: On vortex methods. SIAM J. Numer. Anal. 22(3), 413–440 (1985)
Cherepanov, V., Qian, Z.: Monte–Carlo method for incompressible fluid flows past obstacles. arXiv:2304.09152 (2023)
Chorin, A.J.: Numerical study of slightly viscous flow. J. Fluid Mech. 57, 785–796 (1973)
Chorin, A.J.: Vorticity and Turbulence. Springer, Berlin (1994)
Chorin, A.J., Marsden, J.E.: A Mathematical Introduction to Fluid Mechanics. Springer, Berlin (1993)
Cottet, G.-H., Koumoutsakos, P.D.: Vortex Methods: Theory and Practice. Cambridge University Press, Cambridge (2000)
Goodman, J.: Convergence of the random vortex method. Commun. Pure Appl. Math. 40(2), 189–220 (1987)
Leonard, A.: 1980 Vortex methods for flow simulation. J. Comput. Phys. 289–335
Li, J., Qian, Z., Xu, M.: Twin Brownian particle method for the study of Oberbeck–Boussinesq fluid flows. arXiv:2303.17260 (2023)
Liu, J.-G., Weinan, E.: Simple finite element method in vorticity formulation for incompressible flow. Math. Comput. 69, 1385–1407 (2001)
Majda, A.J., Bertozzi, A.L.: Vorticity and Incompressible Flow. Cambridge University Press, Cambridge (2002)
Marchioro, C., Pulvirenti, M.: Vortex Methods in Two-dimensional Fluid Dynamics. Springer, Berlin (1984)
Qian, Z.: Stochastic formulation of incompressible fluid flows in wall-bounded regions. arXiv:2206.05198 (2022)
Qian, Z., Qiu, Y., Zhao, L., Wu, J.: Monte-Carlo simulations for wall-bounded fluid flows via random vortex method. (2022) arXiv:2208.13233
Schlichting, H., Gersten, K.: Boundary-Layer Theory, 9th edn. Springer, Berlin (2017)
Weinan, E., Liu, J.G.: Vorticity boundary condition and related issues for finite difference schemes. J. Comput. Phys. 124, 368–382 (1996)
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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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
Keywords
- Boundary vorticity
- Dynamical equation
- Incompressible fluid flow
- Stochastic integral representation
- Random vortex method