Accurate modeling of the dynamic stall remains a challenge for the design and construction of turbine blades and helicopter rotors. At the same time, wind turbines, for instance, are becoming steadily larger, further increasing the demands on their structure and necessitating even more detailed modeling of the forces at hand. The primarily used (semi-)empirical models today have a long research history and are invariably based on phase-averaged data from oscillating blade pitch experiments. However, much potential for more accurate modeling of uncertainties and force peaks is wasted here, since averaging blurs many features of the response signals. Even computational fluid dynamics can help little in this regard, since the Reynolds-averaged Navier–Stokes equations used in practice cannot account for cycle variations, and scale-resolving models require extremely large amounts of computational resources. This paper presents an approach for a fully stochastic machine learning model that can nevertheless simulate these critical properties. Aerodynamic coefficients are compared with experimental data for different test cases. It is shown that synthetic force profiles which cannot be distinguished from the experimental data visually and are very close to them in the frequency spectrum can be generated. Additionally, attention is drawn to the difficulty of evaluating such a model, as traditional error metrics are of little use. A combination of dynamic time warping and the Earth mover's distance provides a robust solution for this problem.

As the trend towards increased performance of wind turbines continues, the calculation of the dynamic forces acting on the blades is becoming increasingly important. For classical horizontal axis turbines, a number of factors such as atmospheric turbulence, tower shadow, and yaw misalignment lead to highly unsteady and nonlinear aerodynamic conditions. In vertical axis turbines, the periodic change of the angle of attack is even an inherent part of the working principle. For all turbine types, however, there is a desire to predict loads and fatigue stresses as accurately as possible in order to build cost-effective and robust structures. Since the same phenomenon also occurs in rotary-wing aircraft such as helicopters, the common interest in these industries led to extensive research out of a desire to learn more details about the mechanisms behind it

The aerodynamic forces present on a wing show exceptionally strong fluctuations if the flow dynamically detaches from the suction surface. This unsteady phenomenon, called dynamic stall, is typically caused by a rapid change of the inflow conditions, such as a sudden increase in the angle of attack often in conjunction with a change in the inflow velocity. In his very well-known publication,

Modeling this phenomenon has always been a challenging task. Even the use of computational fluid dynamics (CFD) has not produced particularly satisfactory results.

Another approach is to use empirical (Gormont,

Problematically, other researchers have found that blade pitch experiments required up to 50 cycles to converge to a mean

This is where data-driven models come into play, especially machine learning, which has become very popular in recent years. Machine learning promises to solve the aforementioned problems, as it can detect the underlying probabilistic process of the problem and draw the right conclusions in the form of resulting distributions of statistics, estimators or metrics. This can be advantageous when the complexity of the data is such that physics-based models have difficulty fitting the data or fit them poorly. First attempts in the past with surrogate models deal with the fitting of kriging models

The work is organized as follows. First, Sect. 2 demonstrates the architecture and mathematical foundations of the neural network. Building on this, Sect. 3 shows how the experimental raw data are processed so that they can be fed to the model. Section 4 describes the challenges of evaluating such a model and describes how the best learning parameter combination was found. Then, in Sect. 5, the dynamic stall results of the model for three different test cases are presented. The results of one test case are further clustered in Sect. 6, followed by a brief discussion of the method and an outlook in Sect. 7, ending with a conclusion in Sect. 8.

The neural network architecture used here is based on a convolutional neural network called WaveNet

The most important features of such a model are that it is autoregressive; i.e., it generates new samples based on values that it itself has previously generated. And it is fully probabilistic; i.e., there is a probability distribution prediction from which the final value needs to be sampled each time.

The dilated causal convolutions shown in the network architecture diagram in Fig.

WaveNet-based model with

Visualization of stacked dilated causal convolutional layers. Each node in the input vector represents a single time step, and the time flow is from left to right.

The original model uses quantization of the output for 8-bit integers, resulting in 256 discrete possible values. This is too imprecise for our model and introduces additional difficulties for learning performance, since the cross-entropy loss used cannot differentiate the spatial distance between the categorical buckets. Therefore, the same loss can be assigned to a close hit as to one far-off target. For our purposes it makes more sense to use a mixed density output where we stack a user-defined amount

The loss is described by the average negative log likelihood of the probability density functions. First, the posterior probability is calculated by using the true solution

Then, all posterior probabilities are multiplied with their associated weights

The hyperparameter batch size, number of feature maps, number of stacked dilation filters and number of mixed Gaussians are fine-tuned by a grid search for the best score. For this, the data are split into a training and test set. The test set is ignored until the final predictions are made. The training set is further divided into five parts, and a

The data in this paper were originally prepared for an extensive series of experiments conducted by Hanns Müller-Vahl as part of his doctoral thesis

View of the test section

The wing model is made of Obomodulan^{®} and is pitched about the quarter-chord position. It has 40 surface pressure ports located in the mid-span area that are staggered to avoid interference. Piezoresistive pressure transducers are placed inside the wing to improve transient response. The experimental study relies solely on surface pressure measurements from which the instantaneous aerodynamic coefficients can be derived by integration. This means that drag forces due to friction are not considered. However, it is assumed that at the high incidence levels encountered in the experiments, the pressure-induced forces far outweigh the viscous drag forces. Overall, this is considered acceptable in the context of this work, since we are mainly interested in the corresponding lift forces. It should be noted that in the experiments presented in this section, both blowing slots were sealed with tape (75

Available experimental data sets for the S809 airfoil sorted by the mean angle of attack

Since in this paper we want to examine standard airfoils that do not actively blow, a large part of the test data was omitted. Also, Müller-Vahl's focus was mainly on the averaged data. In the end, however, there are still 91 data sets available for the observed S809 airfoil that meet our requirements. During the experiments, a large number of different frequencies and Reynolds numbers were recorded for a few combinations of angle of attack and amplitude. Therefore, a manual division into training and test set is necessary and makes it more difficult for the neural network. Otherwise the neural network would often practically already know the case at hand if, for example, only the Reynolds number is slightly different. An overview of the data used for training and testing can be found in Table

The experimental data are sampled at a high rate of 500 Hz with the pitch oscillation frequency

Example of raw results from the Technion Wind Tunnel Complex tests compared to the Beddoes–Leishman model from the QBlade package for the S809 airfoil (

Reviewing several parameter sets, it is revealed that not many interesting features are visible in the frequency spectrum beyond 30 Hz (see Fig.

Fast Fourier transform (FFT) of the S809 airfoil coefficient signals subtracted by their mean for

Each sample used as input is a small slice of specific length from the various experimental files. The length of each slice is kept relatively short with

Since there are not enough past time steps available during the beginning of the prediction phase, the startup slice is first filled with synthetic data. Here it has been shown to be sufficient to select a constant Reynolds number at a static 0

The final loss is calculated by applying Eq. (

The accuracy metric is more challenging than in the usual case for neural networks as we cannot use scores like the mean squared error, classification accuracy or the coefficient of determination,

The Earth mover's distance (EMD,

Applied to a 2D histogram, this means that the weights correspond to the value of the buckets and the distance corresponds to the spatial distance. While the metric works considerably better than the total variation, it still has some disadvantages. For example, it provides a usable metric for the global distribution similarity but ignores the frequency and appearance of the individual time series. Using a combination of Earth mover's distance and dynamic time warping (DTW,

DTW on its own is a distance measure, which is suitable to compare one-dimensional time series with each other. The algorithm searches for the “shortest” path from the beginning to the end of both signals over an array of the pairwise distance of all points of both signals (see Fig.

DTW distance between two representative time series. The sum of all local distances is the result.

For the full scoring, each set (predictions and experimental data) is treated as part of a bipartite graph. To prepare the EMD calculation that is essentially an optimal transport problem, the same weight

Two-dimensional visualization of optimal transport, where each point represents a time series that is part of either the predicted or experimental distribution. Instead of the Euclidean distance as shown here, the DTW distance is used in practice.

The usefulness of the WaveNet-based approach is illustrated in this section using an oscillating pitch S809 airfoil and comparing it to experimental data for unsteady lift, drag and moment coefficients. The pitching motion can be described by the equation

To make the different amplitudes of the coefficients comparable, all results are normalized to the range of [0, 1] by min–max scaling relative to the corresponding experimental data before calculating the scores. Otherwise, it would be difficult to compare different parameter ranges, because even if their relative differences are the same, the sum of all absolute differences can still vary significantly.

Unsteady aerodynamic coefficients under dynamic stall conditions in comparison with the experimental results for the parameters

Figure

Unsteady aerodynamic coefficients with gusts under dynamic stall conditions in comparison with the experimental results for the parameters

Figure

Unsteady aerodynamic coefficients under dynamic stall conditions in comparison with the experimental results for the parameters

The third case

Frequency response analysis for the S809 airfoil with the parameters

Another important feature that distinguishes this model from traditional methods is that the returned frequency spectrum is close to the real spectrum as well. This opens the possibility for a more accurate analysis of blade flutter and realistic aeroelastic responses. To illustrate this, Fig.

As far as the computational cost of applying this model is concerned, the following can be stated. The time needed to train the final model amounts to about 4 h on an NVIDIA GeForce GTX 1060 6 GB. Since only one-dimensional time series are considered here, the computational time and memory consumption is certainly low compared to hardware-intensive problems, such as image recognition. Prediction requires about 0.046 s per time step, or about 4.6 s wall-clock time per second of simulated time. The prediction is thus relatively slow, since while the model can process hundreds of parameter sets in parallel, it can only predict all sets step by step into the future. This serial mode of operation during evaluation probably has its bottleneck in the communication between CPU and GPU. However, it is still orders of magnitude faster than simulation using CFD.

Experimental data cluster analysis for the S809 airfoil with the parameters

Clustering of raw airfoil measurement data is a topic recently investigated by several authors. It is now consensus that the statistical mean and standard deviation used to represent cycle-to-cycle variations is inaccurate

Predicted synthetic data cluster analysis for the S809 airfoil with the parameters

The model discussed here is able to learn and predict multimodal distributions without the need for active switching between data groups. However, the data available do not show obvious furcation in the coefficient data. Nevertheless, we can cluster the time series with the method used by

While the model's capabilities are promising, its practical use is of course still limited in so far as only one blade profile can be used within a wide but still restricted parameter range. However, robust and fully functional models can be obtained by designing experiments specifically tailored to this machine learning problem. The additional information about the frequency response and possible load spectra represent a clear added value for the engineer. To be able to map a wider parameter range without gaps, more angle and oscillation frequency combinations should be used. In addition to the pitch motion, plunging could be added as a further parameter to allow the simulation of more demanding aeroelastic problems. If the necessary computing capacities are available, a comprehensive database of LES simulations could also extend or possibly replace the experiments.

With more data for different airfoils available global conditioning could be added to introduce geometry parameters that do not vary in time. In such a way, a very powerful and flexible model could be created to describe all types of airfoils, or even to discover novel airfoil shapes with desired characteristics through optimization.
Another complex model could be created by relying on the data from the pressure ports on the airfoils surface. The derived coefficients

One has to be very sure about the quality of the training data. Since there are no subsequent plausibility checks, some major errors in the experiments would also remain in the data and predictions. Nevertheless, the model could already be incorporated into existing turbine design tools that utilize blade element theory or lifting-line theory to describe dynamic stall for an S809 airfoil.

In this paper, a WaveNet-based neural network is established as a reduced-order model for the relationship between the motion parameters of an airfoil under dynamic stall and the aerodynamic loads on it. In contrast to existing (semi-)empirical models it is fully probabilistic and working with raw wind tunnel time series. The neural network is autoregressive and predicts one time step at a time by generating a probability distribution from which a sample is drawn. Thus, it can predict realistic frequency responses and the local variance of the aerodynamic coefficients. This opens up new possibilities in the study of blade flutter and other aeroelastic problems.

The presented model improves the prediction for the aerodynamic forces and their higher harmonic effects due to vortex shedding and introduces a new level of detail, which has not been possible with traditional modeling methods. Details on the model architecture, implementation and challenges have been summarized in the present work. Three test cases were shown with different mean angles of attack, amplitude, and oscillation frequencies. The results of one case were examined in more detail for its frequency response and decomposed into clusters for comparison with the experimental data.
The technique is currently limited to the S809 airfoil due to the small amount of data available but may be expanded through further studies. Finally, this work serves as a proof of concept for further elaboration of the method to apply stochastic machine learning models into the field of aerodynamics. The main conclusions can be summarized as follows.

Autoregressive machine learning models provide a promising base for future complex and accurate dynamic stall models.

Fully stochastic models can present a physically realistic frequency response of the aerodynamic coefficients.

Recovery of more raw data from old wind tunnel tests or new experiments at high sampling rates tailored to machine learning is necessary to create truly flexible models.

The phenomena detected by clustering wind tunnel data, such as furcations and bimodal distributions of forces, can be learned by the model.

The evaluation code and neural network model can be shared by contacting the corresponding author of the paper. The machine learning framework can be found under

The raw wind tunnel data and prediction results can be shared by contacting the corresponding author of the paper.

JPK conceptualized the original idea; developed the methodology; wrote the code for the data curation, the machine learning model and the evaluation; and wrote the original draft manuscript. TR supervised the work, was responsible for project administration, and reviewed and edited the manuscript.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The wind tunnel data were recorded by Hanns-Müller Vahl and David Greenblatt at the Technion – Israel Institute of Technology in cooperation with the Technical University of Berlin and were kindly provided for this project.

This paper was edited by Jens Nørkær Sørensen and reviewed by Galih Bangga and Helge Aagaard Madsen.