^{1,2}

^{1,2}

^{1,2}

^{2,3}

^{1}

^{2}

^{3}

A Computational Fluid Dynamics (CFD) and response surface-based multiobjective design optimization were performed for six different 2D airfoil profiles, and the Pareto optimal front of each airfoil is presented. FLUENT, which is a commercial CFD simulation code, was used to determine the relevant aerodynamic loads. The Lift Coefficient (

According to the US Department of energy, the combustion of fossil fuels results a net increase of 10.65 billion tones of atmospheric carbon dioxide every year [

The BEM theory is based on the assumption that the flow at a given annulus does not affect the flow at adjacent annuli [

Over the last several years, the wind turbine community has started to look at CFD methods to complement wind tunnel [

The overall goal of this study is to perform a response surface-based multiobjective optimization of selected 2D airfoil profiles using Elitist Nondominated Sorting Genetic Algorithm (NSGA). In order to achieve this overall goal, several specific objectives were determined. The first specific objective was to identify several airfoil profiles with their geometric coordinates. The second objective was to perform CFD simulations around the airfoils. Simulations for each airfoil were performed for several values of Re and

In this work, we consider the flow around six different airfoil shapes (NACA 63-218, E387, FX63-137, NACA 63-421, NACA 64-421, and NACA 65-421) at 3 different Reynolds number (

The CFD data of the 15 simulated cases for each airfoil were used to generate a response surface. The response surfaces were fit using standard least-square regression with quadratic polynomial using JMP. These response surfaces are obtained between design variable (Re and

Grid generation is the most important step in the CFD simulations. The quality of the grid plays a direct role on the quality of the analysis, regardless of the flow solver used. Additionally, the solver will be more robust and efficient when using a well-constructed mesh. In this work, structured grids were generated using the commercial code GAMBIT. Figure

2D mesh of the entire domain using a map scheme with around 50,000 quadrilateral elements.

Mesh generated around the airfoil.

In order to have a stable solution, the generated grids had the least number of elements with high aspect ratios. To be able to resolve adequately the boundary layer along the airfoil wall, grid points were clustered near the wall. The grids were also clustered near the trailing edge in order to catch the flow separation.

Boundary conditions specify the flow and thermal variables on the boundaries of the physical model. They are, therefore, a critical component of the CFD simulations, and it is important that they are specified appropriately. In this work, 3 different types of boundary conditions were used: no-slip boundary condition over the airfoil surface, inlet boundary condition for free stream flow, and pressure outlet. The outlet boundary of the domain was set to a constant pressure value. It was set to be atmospheric pressure. The object in the computational domain (i.e., the airfoil surface), around which the flow was simulated, was set to be no-slip boundary (wall). The no-slip boundary condition sets the stream wise velocity to zero. The velocity Inlet boundary condition was used to define the flow velocity at the flow inlet. Figure

The flow domain with the boundary conditions.

In Gambit, the boundary conditions were declared (i.e., wall, velocity inlet, and pressure outlet), but actual values for these boundaries were defined in fluent. For velocity inlet, we used 3 different velocities for each airfoil at every angle of attack. We set

The response surface method fits an approximate function to a set of experimentally or numerically evaluated design data points [

Let

The methodology used for generating Pareto optimal front is a multiobjective evolutionary algorithm (MOEA). The specific algorithm used is the Elitist Nondominated Sorting Genetic Algorithm (NSGA-II) [

Population size: 100,

Generations: 250,

Crossover probability (P_{cross}): 1.00,

Distribution parameter (for crossover): 20,

Mutation probability (P_{mut}): 0.250,

Distribution parameter (for mutation): 200.

Where the population size is the size of the non-dominated solutions and the generations are equivalent to the number of iterations. Crossover probability, mutation probability, distribution parameter for crossover, and distribution parameter for mutation are used to create the offspring population from the parent population. The crossover probability is mainly responsible for the search aspect of the genetic algorithm while mutation probability keeps the diversity in the population. The distribution parameter for crossover controls the diversity of the children solutions obtained after crossover while distribution parameter for mutation controls the spread of the solutions after mutation.

In NSGA-II algorithm, the code first creates a parent population of

The CFD validation is a very important part of computational fluid dynamics. It is used to evaluate the accuracy of CFD results. We compared the CFD results of Lift Coefficient of E387 airfoil for 5 different angles of attack with National Renewable Energy Laboratory (NREL) experimental data [

Comparison of

Colour contours of static pressure and velocity vectors by velocity magnitude of NACA 64-421 airfoil at an angle of attack of 6° and at

Colour contour of static pressure around NACA 64-421 airfoil at

Colour velocity vectors by velocity magnitude around NACA 64-421 airfoil at

Figures

(a)

Figure

Integrated pressure coefficient of NACA 63-421 airfoil at different Re.

Figure

The variations of

The differences are more significant at higher

Both FX 63-137 and E 387 indicate reaching stall condition at around

Both FX 63-137 and E 387 show smaller variation with Re and reach higher values of

CFD simulation results of

It is obvious from the previous observations that different airfoils behave differently with angle of attack and Reynolds numbers.

Drag Coefficient as a function of angle of attack of NACA 63-421 at the three different Reynolds numbers is shown in Figure

CFD simulation results of

The CFD simulation results for the 15 cases are shown in Table

Normalized design variables and objective function values from CFD simulations of NACA 63-421.

CFD normalized results | |||

0.499999 | 0 | 0.208173299 | 0.152746 |

0.499999 | 0.25 | 0.467045629 | 0.167201 |

0.499999 | 0.5 | 0.682900599 | 0.204313 |

0.499999 | 0.75 | 0.868028883 | 0.25002 |

0.499999 | 1 | 0.967890613 | 0.335964 |

1 | 0 | 0.227377477 | 0.098445 |

1 | 0.25 | 0.450145952 | 0.120713 |

1 | 0.5 | 0.657935167 | 0.16095 |

1 | 0.75 | 0.828468275 | 0.217986 |

1 | 1 | 0.937932094 | 0.279319 |

0.071428 | 0 | 0.145183592 | 0.597703 |

0.071428 | 0.25 | 0.372561069 | 0.630909 |

0.071428 | 0.5 | 0.718397603 | 0.688374 |

0.071428 | 0.75 | 0.877093255 | 0.822994 |

0.071428 | 1 | 1 | 1 |

The CFD data of 15 cases were used to generate a response surface for each of the two objective functions for each airfoil shape. The response surfaces were fit using standard least-square regression with quadratic polynomial using JMP [

Lift Coefficient

Drag Coefficient

The quality of the response surface of this airfoil is shown in Table

Quality parameters of response surface of NACA 63-421 airfoil.

Observations | ||
---|---|---|

0.994164 | 0.992416 | |

0.990921 | 0.988203 | |

Root Mean Square Error | 0.028065 | 0.031312 |

Mean of Response | 0.627276 | 0.381842 |

In order to obtain the Pareto optimal solutions, the two response surface equations were incorporated in the NSGA-II code with the input parameters as mentioned in the previous section. After the simulation, the code generates a file containing 100 non-dominated solutions created during the final iteration. Non-dominated values are the best values according to the desired maximization of the objective functions among the entire population. For better understanding 2D plots of

2D presentation of non-dominated solutions of

2D presentation of non-dominated solutions of

Similar optimizations were performed on all the airfoil shapes. The results obtained are as follows: for NACA 63-218 airfoil, the optimum

Optimum

The Pareto optimal front in multiobjective optimization problem is useful to visualize the tradeoffs among different objectives. In addition to identify compromise solutions, this also helps the designer set realistic design goals. The goal of the current research is focused on the determination and optimization of wind turbine airfoil performance. For this purpose, six different two-dimensional airfoil profiles were studied over two important design variables. The NSGA-II approach of optimization and response surface methodology has been used to generate Pareto optimal front. The optimum

This work is supported by the National Science Foundation (NSF) through the Center for Energy and Environmental Sustainability (CEES), a CREST Center, award no. 1036593.