Abstract
This work reports numerical explorations in the advection of one passive tracer by point vortices living in the unbounded plane. The main objective is to find the energy-optimal displacement of one passive particle (point vortex with zero circulation) surrounded by N point vortices. The direct formulation of the corresponding control problems is presented for the case of \(N=1\), \(N=2\), \(N=3\) and \(N=4\) vortices. The restrictions are due to (i) the ordinary differential equations that govern the displacement of the passive particle around the point vortices, (ii) the available time T to go from the initial position \(z_{0}\) to the final destination \(z_{f}\,,\) and (iii) the maximum absolute value \(u_{\max }\) that is imposed on the control variables. The resulting optimization problems are solved numerically. The numerical results show the existence of nearly/quasi-optimal controls.
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References
Aref, H.: Integrable, chaotic, and turbulent vortex motion in two-dimensional flows. Ann. Rev. Fluid Mech. 15(1), 345–389 (1983)
Vainchtein, D., Mezić, I.: Vortex-based control algorithms. Control of Fluid Flow, pp. 189–212. Springer, Berlin (2006)
Babiano, A., Boffetta, G., Provenzale, A., Vulpiani, A.: Chaotic advection in point vortex models and two-dimensional turbulence. Phys. Fluids 6(7), 2465–2474 (1994)
Pereira, F.L., Grilo, T., Gama, S.: Optimal multi-process control of a two vortex driven particle in the plane. IFAC-PapersOnLine 50(1), 2193–2198 (2017)
Chertovskih, R., Karamzin, D., Khalil, N.T., Pereira, F.L.: Regular path-constrained time-optimal control problems in three-dimensional flow fields. Eur. J. Control (2020). https://doi.org/10.1016/j.ejcon.2020.02.003
Protas, B.: Vortex dynamics models in flow control problems. Nonlinearity 21(9), R203 (2008)
Betts, J.T., Kolmanovsky, I.: Practical methods for optimal control using nonlinear programming. Appl. Mech. Rev. 55, B68 (2002)
Biegler, L.T.: Nonlinear Programming: Concepts, Algorithms, and Applications to Chemical Processes. Cambridge University PR, New York (2010)
Pontryagin, L.S.: Mathematical Theory of Optimal Processes (Classics of Soviet Mathematics), vol. 4, Gordon and Breach Science Publishers (1986)
Balsa, C., Gama, S.M.A., Braz César, M.: Control problem in passive tracer advection by point vortex flow: a case study. In: Papadrakakis, M., Fragiadakis, M. (eds.) COMPDYN 2019 - 7th ECCOMAS Thematic Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (2019)
Pereira, F.L., Grilo, T., Gama, S.: Optimal power consumption motion control of a fish-like vehicle in a vortices vector field. In: OCEANS 2017 - Aberdeen. IEEE (2017)
MathWorks: Matlab Optimization Toolbox: User’s Guide (R2020a). The MathWorks, Inc., Natick, Massachusetts, United State (2020). https://www.mathworks.com/help/pdf_doc/optim/
Chorin, A.J.: Vorticity and Turbulence. Springer, New York (1994)
Newton, P.K.: The N-vortex Problem: Analytical Techniques. Springer Science & Business Media, New York (2013)
Gama, S., Milheiro-Oliveira, P.: Statistical properties of passive tracers in a positive four-point vortex model. Phys. Rev. E 62(1), 1424 (2000)
Waltz, R.A., Morales, J.L., Nocedal, J., Orban, D.: An interior algorithm for nonlinear optimization that combines line search and trust region steps. Math. Program. 107(3), 391–408 (2006)
Batchelor, G.K.: An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge (2000)
Acknowledgements
SG was partially supported by CMUP, which is financed by national funds through FCT – Fundação para a Ciência e a Tecnologia, I.P., under the project with reference UIDB/00144/2020; by Project STRIDE NORTE-01-0145-FEDER-000033, funded by ERDF NORTE 2020; and by project MAGIC POCI-01-0145- FEDER-032485, funded by FEDER via COMPETE 2020 - POCI and by FCT/MCTES via PIDDAC.
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Balsa, C., Gama, S. (2021). The Numerical Control of the Motion of a Passive Particle in a Point Vortex Flow. In: Gonçalves, J.A., Braz-César, M., Coelho, J.P. (eds) CONTROLO 2020. CONTROLO 2020. Lecture Notes in Electrical Engineering, vol 695. Springer, Cham. https://doi.org/10.1007/978-3-030-58653-9_14
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