## Abstract

In this work we couple an IGA-BEM solver for 2D lifting flows with the viscous model in X-Foil [1], a vintage but still widely used software tool for the design and analysis of subsonic airfoils, towards deeper integration of the isogeometric concept in 2D flow models that incorporate boundary-layer corrections.

The formulation of the exterior potential-ﬂow problem reduces to a Boundary Integral Equation (BIE) for the associated velocity potential. Adopting the approach presented in [3], the resulting BIE is handled by an IGA-BEM method, combining:

(i) A generic B-splines parametric modeler for generating hydrofoil shapes, using a set of 8 design-oriented parameters;

(ii) The very same basis of the geometric representation for representing the velocity potential, and

(iii) Collocation at the Greville abscissas of the knot vector of the hydrofoil’s B-splines representation, appropriately enhanced to accommodate the null-pressure jump Kutta condition at the trailing edge.

For the viscous part of the solution, the two-equation model of X-Foil [2] is employed. X-Foil’s inviscid solver is “circumvented” and inviscid isogeometric parameters are sent to its viscous component, namely the integral momentum and kinetic energy shape parameter equations presented in [2]. The derived coupled system is tested for NACA4412 and NACA0012 airfoils and the output lift and drag coefficients for different angle of attacks are compared to experimental data, uncoupled X-Foil results and one-way coupling results obtained [4] via the software tool PABLO [5].

The so-resulting coupled system can be used in airfoil/hydrofoil shape optimisation algorithms with a variety of optimisation criteria such as maximum lift coeﬃcient, maximum lift-over-drag-ratio, minimum deviation of the airfoil/hydrofoil area from a reference area, etc.

REFERENCES

[1] Drela, M. (1989) “XFOIL: An analysis and design system for low Reynolds number airfoils”, MIT, Massachusetts, USA.

[2] Drela, M., Giles, M. (1987) “Viscous – inviscid analysis of transonic and low Reynolds number airfoils”, AIAA Journal, vol. 25(10), pp. 1347 – 1355.

[3] Kostas, K.V., Ginnis, A.I., Politis, C.G., Kaklis, P.D. (2017) “Shape-optimization of 2D hydrofoils using an Isogeometric BEM solver”, Computer Aided Design, vol. 82, pp. 79-87.

[4] Kostas, K.V., Ginnis, A.-A.I, Politis, C.G., Kaklis P.D. (2017) “Shape-optimization of 2D hydrofoils using one-way coupling of an IGA-BEM solver with a boundary-layer model”, Coupled Problems 2017, VII International Conference on Coupled Problems in Science and Engineering, June 12-14, 2017, Rhodes (GR).

The formulation of the exterior potential-ﬂow problem reduces to a Boundary Integral Equation (BIE) for the associated velocity potential. Adopting the approach presented in [3], the resulting BIE is handled by an IGA-BEM method, combining:

(i) A generic B-splines parametric modeler for generating hydrofoil shapes, using a set of 8 design-oriented parameters;

(ii) The very same basis of the geometric representation for representing the velocity potential, and

(iii) Collocation at the Greville abscissas of the knot vector of the hydrofoil’s B-splines representation, appropriately enhanced to accommodate the null-pressure jump Kutta condition at the trailing edge.

For the viscous part of the solution, the two-equation model of X-Foil [2] is employed. X-Foil’s inviscid solver is “circumvented” and inviscid isogeometric parameters are sent to its viscous component, namely the integral momentum and kinetic energy shape parameter equations presented in [2]. The derived coupled system is tested for NACA4412 and NACA0012 airfoils and the output lift and drag coefficients for different angle of attacks are compared to experimental data, uncoupled X-Foil results and one-way coupling results obtained [4] via the software tool PABLO [5].

The so-resulting coupled system can be used in airfoil/hydrofoil shape optimisation algorithms with a variety of optimisation criteria such as maximum lift coeﬃcient, maximum lift-over-drag-ratio, minimum deviation of the airfoil/hydrofoil area from a reference area, etc.

REFERENCES

[1] Drela, M. (1989) “XFOIL: An analysis and design system for low Reynolds number airfoils”, MIT, Massachusetts, USA.

[2] Drela, M., Giles, M. (1987) “Viscous – inviscid analysis of transonic and low Reynolds number airfoils”, AIAA Journal, vol. 25(10), pp. 1347 – 1355.

[3] Kostas, K.V., Ginnis, A.I., Politis, C.G., Kaklis, P.D. (2017) “Shape-optimization of 2D hydrofoils using an Isogeometric BEM solver”, Computer Aided Design, vol. 82, pp. 79-87.

[4] Kostas, K.V., Ginnis, A.-A.I, Politis, C.G., Kaklis P.D. (2017) “Shape-optimization of 2D hydrofoils using one-way coupling of an IGA-BEM solver with a boundary-layer model”, Coupled Problems 2017, VII International Conference on Coupled Problems in Science and Engineering, June 12-14, 2017, Rhodes (GR).

Original language | English |
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Number of pages | 16 |

Publication status | Published - 19 Sept 2019 |

Event | IGA 2019 – VII International Conference on Isogeometric Analysis - Technische Universität München, Munich, Germany Duration: 18 Sept 2019 → 20 Sept 2019 http://congress.cimne.com/iga2019/frontal/default.asp |

### Conference

Conference | IGA 2019 – VII International Conference on Isogeometric Analysis |
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Abbreviated title | IGA 2019 |

Country/Territory | Germany |

City | Munich |

Period | 18/09/19 → 20/09/19 |

Internet address |

## Keywords

- coupling
- inviscid
- IGA-BEM solver
- X-Foil
- boundary layer model
- 2D flows
- parametric modeller