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

• Volume: 45, Issue: 6, page 1033-1058
• ISSN: 0764-583X

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## Abstract

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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.

## How to cite

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Bermúdez, Alfredo, Rodríguez, Rodolfo, and Seoane, María Luisa. "A fictitious domain method for the numerical two-dimensional simulation of potential flows past sails." ESAIM: Mathematical Modelling and Numerical Analysis 45.6 (2011): 1033-1058. <http://eudml.org/doc/276353>.

@article{Bermúdez2011,
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. },
author = {Bermúdez, Alfredo, Rodríguez, Rodolfo, Seoane, María Luisa},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Finite element approximation; fluid-structure interaction; fictitious domain; flow past sails; finite element approximation; fictitious domain},
language = {eng},
month = {6},
number = {6},
pages = {1033-1058},
publisher = {EDP Sciences},
title = {A fictitious domain method for the numerical two-dimensional simulation of potential flows past sails},
url = {http://eudml.org/doc/276353},
volume = {45},
year = {2011},
}

TY - JOUR
AU - Bermúdez, Alfredo
AU - Rodríguez, Rodolfo
AU - Seoane, María Luisa
TI - A fictitious domain method for the numerical two-dimensional simulation of potential flows past sails
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2011/6//
PB - EDP Sciences
VL - 45
IS - 6
SP - 1033
EP - 1058
AB - 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.
LA - eng
KW - Finite element approximation; fluid-structure interaction; fictitious domain; flow past sails; finite element approximation; fictitious domain
UR - http://eudml.org/doc/276353
ER -

## References

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8. P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman (1985).  Zbl0695.35060
9. M.E. Gurtin, An Introduction to Continuum Mechanics. Academic Press (1981).  Zbl0559.73001
10. F. Muttin, A finite element for wrinkled curved elastic membranes, and its application to sails. Comm. Numer. Methods Engrg12 (1996) 775–785.  Zbl0863.73059
11. N. Parolini and A. Quarteroni, Mathematical models and numerical simulations for the America's Cup. Comput. Methods Appl. Mech. Engrg194 (2005) 1001–1026.  Zbl1091.76013
12. H. Schoop, Structural and aerodynamic theory for sails. Eur. J. Mech., A/SolidsIX (1990) 37–52.
13. L.R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp.54 (1990) 483–493.  Zbl0696.65007
14. B. Thwaites, The aerodynamic theory of sails. I. Two-dimensional sails. Proc. Roy. Soc. A261 (1961) 402–422.  Zbl0096.40803

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