Boundary shape control of the Navier-Stokes equations and applications

Abstract

In this paper, the geometrical design for the blade’s surface $\Im $ in an impeller or for the profile of an aircraft, is modeled from the mathematical point of view by a boundary shape control problem for the Navier-Stokes equations. The objective function is the sum of a global dissipative function and the power of the fluid. The control variables are the geometry of the boundary and the state equations are the Navier-Stokes equations. The Euler-Lagrange equations of the optimal control problem are derived, which are an elliptic boundary value system of fourth order, coupled with the Navier-Stokes equations. The authors also prove the existence of the solution of the optimal control problem, the existence of the solution of the Navier-Stokes equations with mixed boundary conditions, the weak continuity of the solution of the Navier-Stokes equations with respect to the geometry shape of the blade’s surface and the existence of solutions of the equations for the Gäteaux derivative of the solution of the Navier-Stokes equations with respect to the geometry of the boundary.

Keywords

Blade / Boundary shape control / General minimal surface / Navier-Stokes equations / Euler-Lagrange equations

References

[1] Abergel F., Temam R.. On some control problems in fluid mechanics. Theoret. Comput. Fluid Dyn., 1990, 1: 303-325

[2] Barbu V.. H_∞ Boundary control with state feedback: the hyperbolic case. SIAM J. Control Option., 1995, 33: 648-701

[3] Barbu V., Sritharan S. S.. H_∞ Control theory of fluid dynamics. Proc. R. Soc. London, 1998, 454A: 3009-3033

[4] Bewley T. R., Temam R., Ziane M.. A general framework for robust control in fluid mechanics. Physica D, 2000, 138: 360-392

[5] Bewley T. R., Moin P., Temam R.. DNS-based predictive control of turbulence: an optimal benchmark for feedback algorithms. J. Fluid Mech., 1999, 8: 90-99

[6] Mohammadi B., Pironneau O.. Shape optimization in fluid mechanics. Aunu. Re. Fluid Mech., 2004, 36: 255-279

[7] Ciarlet P. G.. Mathematical Elasticity, Theory of Shell, 2000, Amsterdam: North-Holland

[8] Ciarlet P. G.. An Introduction to Differential Geometry with Applications to Elasticity, 2005, Netherland: Springer-Verlag

[9] Constantin P., Foias C.. Navier-Stokes Equations, 1988, Chicago, London: The University of Chicago Press

[10] Ekeland I., Temam R.. Convex Analysis and Variational Problems, 1974, Amsterdam: North-Holland

[11] Girault V., Raviart P. A.. Finite Element Methods for Navier-Stokes Equations, Theory and Algortithms, 1986, Berlin, Heidorberg, NewYork: Springer-Verleg

[12] Glowinski R.. Ciarlet P. G., Lions J. L.. Finite Element Methods for Incompressible Viscous Flow. Handbook of Numerical Analysis, 2003, Amsterdam: Elsevier Science

[13] Gunzburger M. D.. Flow Control, The Institute for Mathematics and Its Applications, 1995, Berlin: Springer-Verlag

[14] Heywood J. G., Rannacher R., Turek S.. Artificial boundaries and flux and pressure conditions for the incompressible Navier-Stokes equations. Inter. J. Numer. Meth. in Fluids, 1996, 22: 325-352

[15] Kracmar S., Neustupa J.. A weak solvability of a steady variational inequality of the Navier-Stokes type with mixed boundary conditions. Nonlinear Analysis, 2001, 47: 4169-4180

[16] Kucera P., Skalak Z.. Local solutions to the Navier-Stokes equations with mixed boundary conditions. Acta Applicandae Mathematicae, 1998, 54: 275-288

[17] Ladyzhenskaya O. A.. The Mathematical Theory of Viscous Incompressible Flow, 1969, London: Gordon and Breach

[18] Lagnese J. E., Russell D. L., White L. W.. Control and Optimal Design of Distributed Parameter Systems, 1995, Berlin: Springer-Verlag

[19] Li K. T., Ma Y. C.. Hilbert Space Methods for Mathematical Physical Equations (in Chinese), 1992, Xi’an: Xian Jiaotong University Press

[20] Li K. T., Huang A. X.. Tensor Analysis and Its Applications (in Chinese), 2004, Beijing: Chinese Academic Press

[21] Lions P. L.. Mathematical Topics in Fluid Mechanics, 1998, Oxford: Oxford Science Publications

[22] Lumley J. L., Blossey P. N.. Control of turbulence. Ann. Rev. Fluid Mech., 1998, 30: 311-327

[23] Temam R.. Navier-Stokes Equations, Theory and Numerical Analysis, 2001, A. M. S., Providence, RI: AMS Chelsea Publishing