# Aerodynamic Shape Optimization for Delaying Dynamic Stall of Airfoils by Regression Kriging

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## Abstract

The phenomenon of dynamic stall produce adverse aerodynamic loading which can adversely affect the structural strength and life of aerodynamic systems. Aerodynamic shape optimization (ASO) provides an effective approach for delaying and mitigating dynamic stall characteristics without the addition of auxiliary system. ASO, however, requires multiple evaluations time-consuming computational fluid dynamics models. Metamodel-based optimization (MBO) provides an efficient approach to alleviate the computational burden. In this study, the MBO approach is utilized for the mitigation of dynamic stall characteristics while delaying dynamic stall angle of the flow past wind turbine airfoils. The regression Kriging metamodeling technique is used to approximate the objective and constrained functions. The airfoil shape design variables are described with six PARSEC parameters. A total of 60 initial samples are used to construct the metamodel, which is further refined with 20 infill points using expected improvement. The metamodel is validated with the normalized root mean square error based on 20 test data samples. The refined metamodel is used to search for the optimal design using a multi-start gradient-based method. The results show that an optimal design with a \(3^\circ \) delay in dynamic stall angle as well a reduction in the severity of pitching moment coefficients can be obtained.

## Keywords

Dynamic stall Unsteady CFD Surrogate-based optimization Regression Kriging Expected improvement## 1 Introduction

The dynamic stall phenomenon was first observed on retreating blades of helicopter rotor [6]. Horizontal and vertical axis wind turbines are prone to dynamic stall. Wind turbines are subjected to dynamic loading from multiple sources, such as wind shear, turbulence, yaw angles, upwind turbine wake, and tower shadow, that cause unsteady inflow to the turbine rotor which results in dynamic stall. In vertical axis wind turbines (VAWT), dynamic stall arises from rapid changes in angle of attack on each blade in every rotation cycle [2, 25]. The dynamic loading in wind turbines generates adverse loading conditions, significantly impacting the blade, hub, tower structure, performance and turbine life.

Significant research has been conducted to mitigate or control dynamic stall via active and passive control systems [10, 15, 27, 28]. The addition of structures and control systems to the wind turbines increases their mass as well as their cost and complexity. Mitigating the adverse dynamic stall characteristics passively through aerodynamic shape optimization (ASO) has recently received interest from multiple researchers offering promising improvement in airfoil performance [12, 16, 23, 24, 26]. ASO studies for dynamic stall mitigation are typically done with adjoint-based computational fluid dynamics (CFD) simulations [4, 12, 16, 26] and have shown promising results for multiple dynamic stall optimization cases. Adjoint-based CFD simulations is a modern approach to solve ASO problems using gradient-based optimization (GBO) algorithms [8]. The advantage of the adjoint method is the ability to estimate gradient information cheaply. The GBO approach, however, can get easily get stuck in local minima, especially if the CFD data is noisy. Wang et al. [23, 24] used sequential quadratic programming (SQP) to alleviate aerodynamic loads during dynamic stall cycle on rotor airfoils.

Genetic algorithms have the ability to search the design space globally, but they require multiple design evaluations and can be impractical to use for high dimensional design problems. Ma et al. [11] used a multi-island genetic algorithm, which is a global search method, for VAWT performance improvement.

Metamodel-based optimization (MBO) (also called surrogate-based optimization) [22] is an approach to alleviate the computational burden of costly simulation-based design problems. In MBO, a metamodel (also called a surrogate) of the objective function is constructed using a limited number of the time-consuming simulations. The surrogate model is fast to evaluate and can be used within GBO or with genetica algorithms to search for the optimal design. To the best of our knowledge, MBO has not yet been utilized for ASO to mitigate dynamic stall characteristics of airfoils.

In this work, MBO is used for ASO of wind turbine airfoils to delay stall. The surrogate is constructed using regression Kriging [7] and is sequentially refinement using expected improvement infill criteria. The PARSEC airfoil parameterization technique [20] with six design variables is used for generating the airfoil shapes. The surrogate model is searched using a multi-start gradient-based optimizer.

The next section presents the problem statement for dynamic stall mitigation and the setup of the computational model. The following section describes the MBO approach. Results of numerical experiments are presented for the ASO. Conclusions and suggestions of future work are then described.

## 2 Problem Statement

This section describes the problem formulation and the airfoil parameterization method used for the current study, as well as the CFD modeling and validation.

### 2.1 Problem Formulation

*t*given as

*A*and \(\omega \) represent the mean angle of attack, amplitude of oscillation, and rotational rate, respectively. The reduced frequency,

*k*, is another important parameter and is defined as

*c*is the airfoil chord length, and

*U*is the free-stream speed. In this work, a deep dynamic stall case from Lee et al. [9] is used. The parameters defining the case are: \(\alpha _m = 10^{\circ }\), \(A=15^{\circ }\), \(k=0.05\), and a Reynolds number of \(Re=135,000\).

*N*denotes the number of time steps in each pitching cycle. For this study, we will only consider the upstroke part of the pitching cycle, which is predominantly affected by formation of dynamic stall vortex.

### 2.2 Design Variables

Design variables and their bounds for upper airfoil surface

Description | \(\mathbf {x}\) | \(\mathbf {x_u}\) | \(\mathbf {x_l}\) |
---|---|---|---|

Surface crest x coordinate |
| 0.5011 | 0.2733 |

Surface crest z coordinate |
| 0.09 | 0.054 |

Second order surface derivative | \(Z_{xx}\) | \(-\)0.4036 | \(-\)0.6726 |

Leading edge radius | \(R_{LE}\) | 0.0222 | 0.0104 |

Trailing edge directional angle | \(\theta _{TE}\) | \(-\)7.0294 | \(-\)11.7156 |

TE wedge angle | \(\beta _{TE}\) | 5.8803 | 3.52818 |

### 2.3 Computational Fluid Dynamics Modeling

The current study is performed with the Stanford University Unstructured (\(\mathrm {SU^2}\)) unsteady compressible Navier-Stokes (URANS) solver [17]. The dynamic stall simulations are performed using dual time stepping strategy, rigid grid motion and Menter’s shear stress transport (SST) turbulence model [14]. The convective fluxes calculated using second-order Jameson-Schmidt-Turkel (JST) scheme [17] and time discretization is done by the Euler implicit scheme [17] with maximum Courant-Friedrichs-Lewy (CFL) number selected as 4. The two-level multigrid W-cycle method [17] is also used for convergence acceleration. The Cauchy convergence criteria [1] is applied with Cauchy epsilon as \(10^{-6}\) over last 100 iterations. No-slip boundary condition is used on airfoil surface with farfield condition on external boundary with Reynolds number of 135,000 and Mach number of 0.1. The c-grid mesh is set up an with outer boundary at 55c from airfoil is generated using blockmesh utility provided by OpenFoam [3]. The mesh is refined near the airfoil surface with first layer thickness to obtain \(y^+ \le 0.5 \) and growth ratio of 1.05, which is necessary to accurately capture the onset of the dynamic stall vortex. Figure 1 show a coarse version of the mesh.

Grid convergence study at Re=135000, \(\alpha =4^\circ \)

Mesh | Number of cells \(\times \) \(10^3\) | \(c_l\) | \(c_d, counts\) | *Simulation time (min) |
---|---|---|---|---|

1 | 259 | 0.395 | 174.3 | 75 |

2 | 387 | 0.414 | 180.4 | 146 |

3 | 540 | 0.416 | 184.7 | 220 |

4 | 720 | 0.417 | 184.2 | 298 |

Time study at \(\alpha = 10^{\circ }+ 15^{\circ }sin(\omega t)\) with \(k=0.05\) at \(Re=135,000\)

[ | \(c_{d_{avg}}\) [ | **Simulation Time [ | \(Est_{err}= [c_{d_{avg}}- c_{d_{Est}}]\) [ |
---|---|---|---|

0.004 | 2,019 | 51 | 88.4 |

0.002 | 2,093 | 65 | 14.9 |

0.0015 | 2,103 | 69 | 4.8 |

0.0010 | 2,105 | 78 | 2.1 |

0.0005 | 2,107 | 99 | 0.52 |

After selecting the spatial resolution, a time independent study is conducted with multiple time steps of an airfoil in a sinusoidal pitching cycle in order to select the temporal resolution. This is done using the generalized Richardson extrapolation method (REM) [18] with the use of average drag coefficient per oscillation cycle \(c_{d_{avg}}\) as a lower order value to an estimation parameter. The REM estimate \(c_{d_{Est}}\) represents the average drag coefficient per cycle at a zero time step, which is calculated as \(c_{d_{Est}}=2,108\) counts. Table 3 summarizes the results. The simulation time and estimated error \(Est_{err}\) are then considered to select time step of 0.0015 for all further investigations.

## 3 Methods

This section describes the MBO algorithm and the mathematical details of the metamodeling. In particular, the details of the workflow, sampling plan, regression Kriging, infill criteria, and validation are described.

### 3.1 Workflow

### 3.2 Sampling Plan

The accurate construction of metamodel requires an appropriate sampling plan which captures the trend of objective function throughout design space. In this study, Latin hypercube sampling (LHS) [5, 13] is used to generate initial and test data samples. For this study, an initial sample size is considered as ten times the number of design variables.

### 3.3 Regression Kriging

*D*are correlated with each other through covariance matrix of function \(Z(\mathbf {x})\) given by

*r*denotes regression.

### 3.4 Infill Criteria

### 3.5 Validation

*NRMSE*) defined as

*y*could be an objective function \(f(\mathbf {x})\) or constraint function \(g_1(\mathbf {x})\) values for their respective error estimation. The \(n_T\) indicates the number of test data samples. The denominator of \((y_{max}- y_{min})_I\) represents maximum and minimum of response values of initial sample

*I*data. In this work, \(NRMSE \le 10\% \) and a fixed budget of 20 infill samples are considered as acceptable criteria for accurate global metamodel.

### 3.6 Optimization

Once an accurate metamodel is obtained it is used by the optimizer to find an optimal design for given problem. For this study, we use a multi-start gradient-based search algorithm to find the optimal design. The sequential least squares programming (SLSQP) algorithm offered by Scipy [21] python package is utilized in this work. A total 240 starting points are used in this study. These start points are distributed over the design space by using the LHS technique. The best obtained result is reported as optimal design.

## 4 Results

This section presents the results of the metamodel generation and the validation study for the dynamic stall mitigation problem. The optimization results are discussed.

### 4.1 Metamodel Construction

As discussed earlier, the optimization algorithm generates the metamodel and sequentially refines it. Initially, the design space is sampled using LHS. A total of 60 design samples (10\(\times \) number of design variables) are generated. Each design sample is then evaluated with the CFD module to generate the objective and constraint function values. Note that in this study we only simulated the upstroke of the pitching cycle where dynamic stall vortex formation occurs. The obtained observations are used to construct two separate metamodels, one for the objective and another for the constrained function. Both these metamodels are validated with 20 test data points (one third of initial samples). The test data points are also generated using LHS technique separately and evaluated with the CFD module. The global accuracy of the metamodel is tested using the *NRMSE* metric. If the accuracy of the model satisfies the termination criteria then it is passed to the optimizer, else an infill point is evaluated and added to the initial sampling plan to construct a new metamodel. This process is iterated until the metamodel satisfies the termination criteria of \(NRMSE \le 10\%\) and fixed budget of 20 infill points.

*NRMSE*for the objective and constraints functions every 5 infill points. It can be seen that both the metamodels satisfy global accuracy error criteria well before infill points reach the fixed budget criteria. The constraint function metamodel shows a higher accuracy than the objective function metamodel reaching \(2.4\%\) and \(8.8\%\), respectively, by total 80 sample points (60 initial samples plus 20 infill points).

### 4.2 Optimal Design

Figure 4 shows the baseline and optimum airfoil results. Table 4 gives the aerodynamic characteristics of the airfoils. There are major shape variations between the baseline (NACA0012) and the optimized airfoil. The optimized airfoil has a higher maximum thickness (\(t/c_{max}=0.146\)) with a maximum camber \((M)=1.89\%\) located at \(x/c=0.62\). The optimum design is able to delay the dynamic stall angle (\(\alpha _{ds}\)) by more than \(3^\circ \), whereas the moment stall angle \(\alpha _{ms}\) is delayed to \(20.26^\circ \). The \(\alpha _{ms}\) indicates formation of dynamic stall vortex which is responsible for sudden divergence in drag and pitching moment coefficients. The delay in dynamic stall vortex formation provides an increase in operational range without adverse loading on the airfoil. Moreover, optimum shape also shows the reduction in severity of pitching moment (Fig. 4d).

Aerodynamic and shape characteristics of baseline and optimized airfoil

Airfoil | \(\alpha _{ds}\) | \(\alpha _{ms}\) | \((t/c)_{max}\) | \(M(\%)\) |
---|---|---|---|---|

Baseline (NACA0012) | \(19.15^\circ \) | \(16.55^\circ \) | 0.12 | 0 |

Optimized | \(22.52^\circ \) | \(20.26^\circ \) | 0.146 | 1.89 |

## 5 Conclusion

In this work, efficient aerodynamic shape optimization using regression Kriging metamodeling is used for mitigating the adverse effects of dynamic stall on wind turbine airfoil shapes. The optimal airfoil shape shows a significant delay in the dynamic stall angle when compared to a baseline airfoil. It was found that the optimal shape has a higher maximum thickness and maximum camber compared to the baseline airfoil. Future work will consider global sensitivity analysis to provide the sensitivities of the individual variables with respect to objective and constraint functions, and to explore the interaction effects of variables. This will reveal how the airfoil aerodynamics affects dynamic stall response.

## Notes

### Acknowledgements

The second and third authors were supported in part by RANNIS grant number 174573.

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