A fictitious domain method for the numerical two-dimensional simulation of potential flows past sails

¹ Departamento de Matemática Aplicada, Universidad de Santiago de Compostela, 15706, Santiago de Compostela, Spain. alfredo.bermudez@usc.es; marialuisa.seoane@usc.es
² CI2MA, Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile. rodolfo@ing-mat.udec.cl

Received: 19 March 2010
Revised: 19 January 2011

Abstract

This paper deals with the mathematical and numerical analysis of a simplified two-dimensional model for the interaction between the wind and a sail. The wind is modeled as a steady irrotational plane flow past the sail, satisfying the Kutta-Joukowski condition. This condition guarantees that the flow is not singular at the trailing edge of the sail. Although for the present analysis the position of the sail is taken as data, the final aim of this research is to develop tools to compute the sail shape under the aerodynamic pressure exerted by the wind. This is the reason why we propose a fictitious domain formulation of the problem, involving the wind velocity stream function and a Lagrange multiplier; the latter allows computing the force density exerted by the wind on the sail. The Kutta-Joukowski condition is imposed in integral form as an additional constraint. The resulting problem is proved to be well posed under mild assumptions. For the numerical solution, we propose a finite element method based on piecewise linear continuous elements to approximate the stream function and piecewise constant ones for the Lagrange multiplier. Error estimates are proved for both quantities and a couple of numerical tests confirming the theoretical results are reported. Finally the method is used to determine the sail shape under the action of the wind.