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Abstract
Airfoil shape optimization for enhanced aerodynamic performance at a moderate Reynolds number is carried out using a novel technique for constrained optimization of expensive engineering functions. The proposed method is a co-kriging surrogate model using two levels of fidelity in conjunction with a gradient-free trust-region method to drive the model towards the global optimum. The methodology is first applied on a generic test function to demonstrate its efficiency. Later, the proposed optimization framework is applied on constrained airfoil shape optimization problems involving maximizing the lift-to-drag ratio, maximizing the endurance factor, and minimizing the drag coefficient of an Eppler E214 airfoil. The optimal lift-to-drag ratio and the endurance factor are found to be 13.25% and 16.05%, respectively, higher than the baseline airfoil. The optimal drag coefficient is 6.86% lower than the baseline airfoil. The flow transition is also delayed for the some of the optimal airfoils. The present results show that the proposed optimization methodology is successful in improving the aerodynamic characteristics in the sensitive low/moderate Reynolds number regime.
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References
- Priyanka, R. and Sivapragasam, M., "Multi-Fidelity Surrogate Model-Based Airfoil Optimization at a Transitional Low Reynolds Number", Sadhana,Vol.46, pp.1-19, 2021.
- Forrester, A. I. J., Sóbester, A. and Keane, A. J., "Multi-Fidelity Optimization via Surrogate Modelling", Proceedings of the Royal Society A, Vol.463, pp.3251-3269, 2007.
- Long, T., Li, X., Shi, R., Liu, J., Guo, X. and Liu, L., "Gradient-Free Trust-Region-Based Adaptive Response Surface Method for Expensive Aircraft Optimization", AIAA Journal, Vol.56, pp.862-873, 2018.
- Forrester, A. I. J., Sóbester, A. and Keane, A. J., "Engineering Design via Surrogate Modelling: A Practical Guide", John Wiley, Chichester, 2008.
- Toal, D., Bressloff, N. and Keane, A., "Kriging Hyperparameter Tuning Strategies", AIAA Journal, Vol.46, pp.1240-1252, 2008.
- Alexandrov, N. M., Dennis, J. E., Lewis, R. M. and Torczon, V., "A Trust-Region Framework for Managing the use of Approximation Models in Optimization, Structural Optimization", Vol.15, pp.16-23, 1998.
- Alexandrov, N. M., Lewis, R. M., Gumbert, C. R., Green, L. L. and Newman, P. A., "Approximation and Model Management in Aerodynamic Optimization with Variable-Fidelity Models", Journal of Aircraft, Vol.38, pp.1093-1101, 2001.
- Menter, F. R., "Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications", AIAA Journal, Vol.32, pp.1598-1605, 1994.
- Langtry, R. B. and Menter, F. R., "Correlation-Based Transition Modeling for Unstructured Parallelized Computational Fluid Dynamics Codes", AIAA Journal, Vol.47, pp.2894-2906, 2009.
- Celik, I. B., Ghia, U., Roache, P. J., Freitas, C. J., Coleman, H. and Raad, P. E., "Procedure for Estimation of Uncertainty Due to Discretization in CFD Applications", Journal of Fluids Engineering, Vol.130, 078001, 2008.
- Drela, M., XFOIL: An Analysis and Design System for Low Reynolds Number Airfoils, Low Reynolds Number Aerodynamics, Springer, 1989.
- Drela, M. and Giles, M. B., "Viscous-Inviscid Analysis of Transonic and Low Reynolds Number Airfoils", AIAA Journal, Vol. 25, pp.1347-1355, 1987.
- van Ingen, J. L., "The e N Method for Transition Prediction. Historical Review of Work at TU Delft", AIAA 2008-3830, 2008.
- Mack, L. M., "Transition and Laminar Instability", JPL Publication 77-15 (also NASA-CP-153203), 1977.
- McGhee, R. J., Walker, B. S. and Millard, B. F., "Experimental Results for the Eppler 387 Airfoil at Low Reynolds Number in the Langley Low-Turbulence Pressure Tunnel", NASA-TM-4062, 1988.
- Driver, J. and Zingg, D. W., "Numerical Aerodynamic Optimization Incorporating Laminar-Turbulent Transition Prediction", AIAA Journal, Vol.45, pp.1810-1818, 2007.
- Rashad, R. and Zingg, D. W., "Aerodynamic Shape Optimization for Natural Laminar Flow using a Discrete-Adjoint Approach", AIAA Journal, Vol.54, pp.3321-3337, 2016.
- Amoignon, O., Pralits, J., Hanifi, A., Berggren, M. and Henningson, D., "Shape Optimization for Delay of Laminar-Turbulent Transition", AIAA Journal, Vol.44, pp.1009-1024, 2006.
References
Priyanka, R. and Sivapragasam, M., "Multi-Fidelity Surrogate Model-Based Airfoil Optimization at a Transitional Low Reynolds Number", Sadhana,Vol.46, pp.1-19, 2021.
Forrester, A. I. J., Sóbester, A. and Keane, A. J., "Multi-Fidelity Optimization via Surrogate Modelling", Proceedings of the Royal Society A, Vol.463, pp.3251-3269, 2007.
Long, T., Li, X., Shi, R., Liu, J., Guo, X. and Liu, L., "Gradient-Free Trust-Region-Based Adaptive Response Surface Method for Expensive Aircraft Optimization", AIAA Journal, Vol.56, pp.862-873, 2018.
Forrester, A. I. J., Sóbester, A. and Keane, A. J., "Engineering Design via Surrogate Modelling: A Practical Guide", John Wiley, Chichester, 2008.
Toal, D., Bressloff, N. and Keane, A., "Kriging Hyperparameter Tuning Strategies", AIAA Journal, Vol.46, pp.1240-1252, 2008.
Alexandrov, N. M., Dennis, J. E., Lewis, R. M. and Torczon, V., "A Trust-Region Framework for Managing the use of Approximation Models in Optimization, Structural Optimization", Vol.15, pp.16-23, 1998.
Alexandrov, N. M., Lewis, R. M., Gumbert, C. R., Green, L. L. and Newman, P. A., "Approximation and Model Management in Aerodynamic Optimization with Variable-Fidelity Models", Journal of Aircraft, Vol.38, pp.1093-1101, 2001.
Menter, F. R., "Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications", AIAA Journal, Vol.32, pp.1598-1605, 1994.
Langtry, R. B. and Menter, F. R., "Correlation-Based Transition Modeling for Unstructured Parallelized Computational Fluid Dynamics Codes", AIAA Journal, Vol.47, pp.2894-2906, 2009.
Celik, I. B., Ghia, U., Roache, P. J., Freitas, C. J., Coleman, H. and Raad, P. E., "Procedure for Estimation of Uncertainty Due to Discretization in CFD Applications", Journal of Fluids Engineering, Vol.130, 078001, 2008.
Drela, M., XFOIL: An Analysis and Design System for Low Reynolds Number Airfoils, Low Reynolds Number Aerodynamics, Springer, 1989.
Drela, M. and Giles, M. B., "Viscous-Inviscid Analysis of Transonic and Low Reynolds Number Airfoils", AIAA Journal, Vol. 25, pp.1347-1355, 1987.
van Ingen, J. L., "The e N Method for Transition Prediction. Historical Review of Work at TU Delft", AIAA 2008-3830, 2008.
Mack, L. M., "Transition and Laminar Instability", JPL Publication 77-15 (also NASA-CP-153203), 1977.
McGhee, R. J., Walker, B. S. and Millard, B. F., "Experimental Results for the Eppler 387 Airfoil at Low Reynolds Number in the Langley Low-Turbulence Pressure Tunnel", NASA-TM-4062, 1988.
Driver, J. and Zingg, D. W., "Numerical Aerodynamic Optimization Incorporating Laminar-Turbulent Transition Prediction", AIAA Journal, Vol.45, pp.1810-1818, 2007.
Rashad, R. and Zingg, D. W., "Aerodynamic Shape Optimization for Natural Laminar Flow using a Discrete-Adjoint Approach", AIAA Journal, Vol.54, pp.3321-3337, 2016.
Amoignon, O., Pralits, J., Hanifi, A., Berggren, M. and Henningson, D., "Shape Optimization for Delay of Laminar-Turbulent Transition", AIAA Journal, Vol.44, pp.1009-1024, 2006.