A novel thermal turbulence reconstruction method using proper orthogonal decomposition and compressed sensing coupled based on improved particle swarm optimization for sensor arrangement

ANN

Artificial neural network

a

Weight power

b

Modal coefficient

{b_i}^m_i=1

Modal coefficient of {ψ_i}^l_i=1

c₁, c₂

Learning factors

c_end

Dynamic termination values of c

CFD

Computational fluid dynamics

CNN

Convolutional neural network

CS

Compressive sensing

c_start

Dynamic initial values of c

DNN

Deep neural network

DNSs

Direct numerical simulations

GAN

Generative adversarial network

g

Local acceleration due to gravity

k

Number of iterations

LESs

Large eddy simulations

N

Number of observation points

n

Mask vector

P

Air pressure

PIV

Particle image velocimetry

POD

Proper orthogonal decomposition

PSO

Particle swarm optimization

RANS

Reynolds-average Navier–Stokes

r₁, r₂

Random numbers

T

Average temperature during gas flow

T′

Fluctuating temperature

T_a

Air reference temperature

T_m

Maximum number of iterations

u_i, u_j

Mean velocities

u_i′, u_j′

Fluctuating velocity components

x_i, x_j

Cartesian coordinates

Y_pre

Reconstructed dataset

Z_i

Data at the observation point

α

Air thermal diffusivity coefficient

β

Air thermal expansion coefficient

δ_ij

Kronecker delta symbol

λ_i

Weight coefficient

ν

Air kinematic viscosity

ρ

Air density under specific conditions

Φ

Measurement matrix

ψ

Sparse basis

{ψ_i}^l_i=1

POD_basis

ω

Inertia coefficient

ω_end

Dynamic termination values of ω

ω_start

Dynamic initial values of ω

The global trend toward carbon neutrality has accelerated the transformation of energy systems, with the use of wind energy accounting for approximately 30% of the renewable energy.^1,2 Offshore wind turbines are developing in the direction of high power with a standalone capacity of 10 MW or higher. However, the consequent increase in heat loss from components hinders the balance of temperature and power in electromechanical devices (gearboxes, generators, etc.).^3,4 Therefore, it is crucial to accurately reconstruct the turbulent thermal environment of the nacelle in real time and control the heat flow by, for example, controlling the start and stop of the cooling system.

Influenced by the nacelle structure and temperature difference, the nacelle is subjected to a complex turbulent thermal environment, i.e., triple coupling of a high-turbulence-intensity (Ra > 10⁹) airflow, heat conduction, and convective heat transfer. The higher turbulence intensity makes the airflow intermittent and fluctuating, while the coupling of turbulence and heat transfer makes the turbulent thermal environment highly nonlinear. Existing methods for the reconstruction of the above-mentioned complex turbulent thermal environments suffer from large reconstruction errors and low efficiency of sensor application.^5,6

The reconstruction of the turbulent thermal environment is mainly carried out by experimental methods and numerical calculations.

Regarding the experimental methods, the traditional monitoring method is ground testing using a wind tunnel, particle image velocimetry (PIV), and sensors.^7,8 PIV can provide an accuracy of 1%–2%, but the integrated in-nacelle design hinders PIV in the small available space. Sensors can be used to detect the turbulent thermal environment but can obtain only a physical information of a single node. The current sensor application is inefficient and susceptible to damage due to temperature changes in the wind power environment. The above-mentioned problems hinder the implementation of the experiment.

Numerical computations are mainly divided into computational fluid dynamics (CFD) simulations and data-driven predictions including machine learning.

With the development of high-performance computers, data from direct numerical simulations (DNSs) or large eddy simulations (LESs) in CFD can be used to analyze specific physical phenomena. However, DNS and LES have strict requirements on grid division and computational equipment, and the computational speed is low. Thus, it is very challenging for precalculation and poststorage.^9–11 Nowadays, the Reynolds-average Navier–Stokes (RANS) method has been used for time averaging of the control equations. By simplifying the traditional nacelle model through an ideal hot and cold plate model, thermal behaviors including laminar flow¹² and coupling of turbulent and convective heat transfers^13–15 have been computed separately. Other researchers improved the meshing technique for hot and cold plate models and obtained a lower average relative error of approximately 10%.¹⁶ However, the error of 10% is still significant, and most of the current CFD studies are based on the turbulent thermal environment of low-power wind turbine generators. For example, in Ref. 16, only the turbulent thermal environment in a nacelle of 8 kW has been simulated, which has a lower turbulence intensity and simpler turbulent thermal environment, which is easier to implement in CFD. Therefore, most of the current CFDs cannot meet the accuracy requirement for nacelle reconstruction above 10 MW.

In recent years, data-driven techniques based on machine learning and deep learning have been widely used for turbulence modeling in fluid dynamics. The flow is often coupled with heat transfer phenomena such as natural convection. Artificial neural network (ANN) and deep neural network (DNN), among others, have been used to predict or quantify the uncertainty in the RANS method. António et al.¹⁷ reconstructed the temperature field of a domestic refrigerator using ANN and dataset of actual detected temperature points, compared it to RANS simulation results, and obtained better results for ANN than for the RANS reconstruction. Milano et al.¹⁸ estimated the flow field above the wall using an ANN with 26 000 inputs and reported that too many inputs can lead to a slow model training. Other researchers have conducted similar studies using 3000–7000 data points.^19–22 

To provide an accurate description of the thermal environment in strongly nonlinear fluid domains, researchers have used DNN [mainly convolutional neural network (CNN) and generative adversarial network (GAN)] to more naturally consider the spatial structure of the input data and accurately describe the thermal environments of fluid domains in two dimensions and three dimensions. Wang et al.²³ used a ten-layer DNN for a turbulent thermal environment synthesis and obtained thermal environments with relative errors within 6% using 1.48% of the computational nodes. Yilmaz and German²⁴ used CNN to transform the regression problem of pressure coefficient prediction on an airfoil into a classification problem, resulting in an accuracy above 80%. Li et al.²⁵ combined CNN and GAN for a 3D thermal environment reconstruction using a fin 2D thermal environment and reported that the relative error of the reconstruction results was below 1.728%. CNN and GAN are widely used in turbulence synthesis and thermal environment reconstruction.^26–31 

The aforementioned results show that the shallow network of ANN cannot accurately describe the nonlinear or high-fidelity characteristics of fluid flow and heat transfer but is more accurate than the RANS method. DNN can synthesize turbulence more accurately. However, limited by the thermal environment measurements in the nacelle, it is not possible to provide a complete high-resolution thermal environment dataset for DNN training. In addition, the unexplainable nature of the black-box principle in ANNs and DNNs, as well as the problems of model training overfitting and predictive instability, remain unresolved.

With the development of multimodal machine learning, researchers have adopted methods such as proper orthogonal decomposition (POD) to improve the reconstruction accuracy, stability, and model interpretability.^32,33 POD is widely used for dynamic flow field reconstruction in fluid dynamics, ocean engineering, etc.^34,35 Based on the POD model, sensor measurements, and compressive sensing (CS) theory, researchers developed a coupled POD-CS reconstruction model (also called Gappy proper orthogonal decomposition). The characteristics of the flow field are obtained by calculating the flow field distribution using CFD, which enables a real-time reconstruction of the flow field. The method overcomes the shortcomings of the uncertainty of statistical methods and computationally intensive physical forecasting methods. The method has been applied to a single physical field,^36–40 and in this paper, the method is applied to a complex physical field with turbulent natural convection coupling.

Combining the above-mentioned methods, we obtained a POD-CS coupling reconstruction method based on improved particle swarm optimization (PSO) by improving the existing methods. First, the sensor positions are optimally calculated using the improved PSO. This ensures the stability of the reconstruction and saves sensor resources. Second, the improved PSO-POD-CS uses CFD simulation data as a priori data. In order to evaluate the applicability of highly turbulent convective heat transfer in various nacelles, this paper restricts the sensor arrangement to internal and boundary arrangements and evaluates the effects of the dimensionality, location, and number of sensor information on the reconstruction accuracy of different numbers of snapshots of the thermal environment. Finally, the generality of the method is validated and compared with the reconstruction results from a physical information neural network (PINN). The method is instructive for sparse reconstruction of complex thermally coupled environments with high turbulence intensity.

A physical model of a 10-MW offshore wind turbine is constructed. The wind turbine nacelle is a 15 × 6 × 6 m³ hexahedral nacelle, as shown in Fig. 1(a). The unit consists mainly of a low-speed rotor shaft 1 connected to the fan blades outside the nacelle, acceleration gearbox 2, semidirect-drive medium-speed rotor shaft 3, permanent-magnet synchronous generator 4, and control cabinet 5, as shown in Fig. 1(b).

According to the geometry of the computational domain, Ra ≈ 6 × 10¹² (Ra > 10⁹, the flow state is turbulent). The chamber exhibits a high-intensity turbulence. CFD is used as a standard for the reconstruction. To reduce the computational difficulty, the 3D computational domain of the model is simplified to that in Fig. 1(c), while the 2D computational domain to that in Fig. 1(d) by analyzing the influence of each component on the velocity and temperature fields according to Ref. 14.

The measurement matrix is the initial missing dataset for reconstructing the turbulent thermal environment field. It can be defined as a dot product of a mask vector $Φ$ (the mask vector $Φ$ is a matrix containing only 0 and 1). The element of mask vector $Φ$ with 1 is the location of the sensor's turbulent thermal environment measurement arrangement, which is determined using Improved PSO for optimization as in Sec. II B 2. Ψ and α are parameters in the POD matrix decomposition as in Sec. II B 3.

In the field of fluid dynamics, the POD method allows to extract the spatiotemporal information and the principal modes of the flow. Therefore, we use the principal modes of the flow obtained by POD as the required sparse basis functions, and use POD to extract features of the turbulent thermal environment distribution data to construct the basis functions of the turbulent thermal environment. POD extracts the basic features of the turbulent thermal environment from a priori data, i.e., obtains low-dimensional pictures to prevent a too large amount of data leading to dimensional explosion.

In CS reconstruction, the initial vectors are some sparse vectors, which need to be reconstructed from “incomplete” vectors to “complete” vectors. The missing vectors can be characterized in the form of a product of the observed matrix with the product of the complete matrix. The sparsity coefficients can be calculated by the least squares method.

The transient turbulent thermal environment dataset consists of 600 snapshots. Split according to different temporal resolutions, 100, 200, and 300 snapshot datasets were obtained as Data_100, Data_200, and Data_300, respectively, for the reconstruction of the generic validation.

As shown in Fig. 3, in terms of spatial resolution, these snapshots consider 1823 grid points as the allowable distribution locations of the sensor. The [P, U, V, T] physical field of the transient dataset is matrix-decomposed and used as the reconstructed POD_basis vectors by summing the energy ratios of the first n basis vectors.

POD_basis vectors with larger energy ratios capture the main features of transient datasets. However, using more modes leads to slower reconstruction and dimensional catastrophe, while using fewer modes leads to lower reconstruction accuracy. Therefore, this paper analyzes the relationship between POD_basis dimensionality and reconstruction accuracy as shown in Fig. 4.

Figure 4 shows the average relative error in reconstructing the transient physical field using internal sensors, boundary sensors, and POD_basis with different numbers of truncations. We found that the trend of reconstruction results using different types and numbers of sensors and different truncation numbers of POD_basis is approximately the same. First, as the number of POD_basis truncations increases, the reconstruction relative error decreases substantially, which is the same as the theory in Fig. 3, because the first few orders of modes of the POD_basis take up most of the features in the dataset (the features are quantified by the Energy_ratio), and the feature information with a larger energy share will quickly improve the reconstruction accuracy. The relative error then changes slightly or remains stable as the number of truncations increases. Finally, the reconstruction error increases rapidly when the number of truncations reaches a certain value, which is due to the redundancy of feature information. Boundary sensors and internal sensors show the same trend, but the reconstruction error is larger than that of internal sensors. Therefore, for different physical field reconstruction tasks, it is necessary to determine the optimal range of POD_basis truncation numbers before reconstruction.

We quantitatively analyzed the relationship between the truncation number and the energy ratio of POD_basis. We used a range of cutoffs that did not exceed 0.5% of the optimal reconstruction error as the optimal range of cutoffs for POD_basis. The optimal Energy_ratio for the internal sensor reconstruction {P, U, V, T} ranges from {97. 7%–99.3%, 92.6%–95.5%, 93.3%–96.8%, 93.9%–98.3%} and that of the boundary sensor reconstruction {P, U, V, T} ranges from {98.4%–99.6%, 93.6%–95.5%, 93.3%–96.1%, 92.6%–97.5%}.

Therefore, in order to balance the POD basis vector dimensions and the reconstruction accuracy, this paper is used to reconstruct the final modes of the {P, U, V, T} physical field for both interior and boundary sensors as shown in Fig. 5.

Since the location of the sensors has a great influence on the reconstruction results (more feature physical information is obtained at locations with larger gradients in the turbulent thermal environment), in order to ensure the stability of the reconstruction and to save the sensor resources in the reconstruction, this paper adopts an improved PSO to optimize the sensor locations.

Linearizing the parameters of PSO can well regulate the step size of the population search, which can converge faster and avoid local optimums compared to the traditional PSO, as shown in Fig. 6. The physical information obtained from 50 snapshots and the three sensors are reconstructed using conventional PSO and improved PSO, respectively, and convergence is reached at 646 and 88 steps of the iteration, respectively.

The optimization parameters are different for different number of computing nodes and different number of transient datasets. In this paper, the population size and number of iterations of the optimization algorithm are obtained through several iterative tests as shown in Table I.

By optimizing the internal sensors, the results of reconstructing Data_100, Data_200, and Data_300 are shown in Fig. 7.

We found that the use of three internal sensors can only capture some of the important internal features, which can also lead to excessive average relative errors in the reconstructed results of the transient turbulent thermal environments for some time steps. Using five internal sensors essentially captures important features of the reconstructed turbulent thermal environment parameters, and the average relative error of the reconstructed turbulent thermal environments for Data_100 to Data_300 essentially does not exceed 10%.

As the number of transient turbulent thermal environments increases from Data_100 to Data_300, the reconstruction error of the turbulent thermal environments using three and five sensors increases due to the fact that the Data_300 transient turbulent thermal environments need to capture more physical information features. The use of seven and nine internal sensors still performs better in the reconstruction of transient turbulent thermal environments for Data_200 and Data_300, with nine sensors providing better reconstruction stability relative to seven sensors. Since too many internal sensors will destroy the heat flow direction and lead to poorer reconstruction results, this paper limits the number of sensors to no more than 9. It is necessary to ensure the sensor resources and reconstruction stability by optimizing the sensor locations.

In some models of nacelle turbulent thermal environment reconstruction, the integrated nacelle arrangement results in limited sensor locations in the nacelle. Therefore, such turbine turbulent thermal environment reconstruction will choose to arrange the sensors at the boundary.

The physical information points of the model boundary in this paper are 352, relative to the 1823 in Sec. III B 1, which can be used to achieve the optimization convergence with a smaller number of particle swarm populations and iteration steps, and the boundary optimization parameters are shown in Table II.

The result of the reconstruction through the boundary sensor is shown in Fig. 8. The use of boundary sensors makes it difficult to capture physical features that carry a large amount of information in the nacelle as opposed to internal sensors. In the reconstruction of transient snapshots, the stability of the reconstruction accuracy of boundary sensors is poor. However, with the increase in the number of boundary sensors, reconstruction results with better results can still be obtained. When nine boundary sensors are used to reconstruct the turbulent thermal environment, the maximum relative error of [P, V] can be controlled within 5%, and the maximum relative error of [U, T] can be controlled within 8%. Compared with the internal sensors, the boundary sensors cause less damage to the flow field when reconstructing the turbulent thermal environment, and the limitation of the number of boundary sensors is smaller.

The reconstruction results for the two sensor arrangements are shown in Tables III and IV. The reconstruction accuracy of the boundary sensors is closer to the reconstruction results of the internal sensors due to more information of the main features distributed at the pressure field boundary. For the reconstruction of velocity and temperature fields, the reconstruction accuracy is still unstable using 3 and 5 data points, but the reconstruction results using 7 and 9 sensors can narrow the gap with the reconstruction results from the internal sensors to some extent.

Therefore, using the coupled POD-CS reconstruction method, accurate turbulent thermal environment results can be obtained with a sufficient number of boundary sensors arranged, but the number of sensor weights required increases with the number of snapshots and parameter physical information features. However, the accuracy of the reconstructed snapshots with the same number of boundary sensors is poorer than that of the internal sensors.

For the method proposed in this paper, it can be used for both individual physical field reconstruction and multiple physical field reconstruction. For scalar physical fields such as pressure and temperature, it is recommended to reconstruct them individually, so as to avoid insufficient extraction of eigenmodes due to the difference in magnitude, resulting in less than ideal accuracy of the final physical field reconstruction. For vector fields such as velocity, it is recommended to use different directional velocities for collective reconstruction, which can enhance the computational efficiency of the reconstruction to a certain extent.

The optimization of the finite sensor locations by the improved PSO, using the optimized sensors to provide physical information for the coupled POD-CS reconstruction method can improve the stability of the reconstructed transient turbulent thermal environment to some extent. However, during the design process of the wind turbine structure, as the wind turbine is also moving toward space-saving direction, this imposes a great limitation on the distribution of sensor locations. Therefore, this paper considers the internal space of different wind motors and discusses the optimal location distribution for two cases, with limitations (boundary sensors) and without limitations (internal sensors), respectively.

The positional distribution of the [P, U, V, T] physical field for the three transient datasets Data_100, Data_200, and Data_300 were reconstructed using 3, 5, 7, and 9 internal and boundary sensors as shown in Figs. 9 and 10.

We find that most of the optimized internal sensors are distributed near the boundary layer, which also proves that the gradient of physical quantities near the boundary layer is larger, and it is more beneficial to use the sensors for physical information extraction to improve the reconstruction accuracy of the turbulent thermal environment.

The boundary sensor can be applied to most of the physical field reconstruction inside the wind turbine, and the physical information points of the boundary sensor are 352, compared with the 1823 of the internal sensors, the distribution optimization process of the Improved PSO with the boundary sensor will have a higher computational efficiency, and the convergence can be completed in a shorter number of iteration steps. However, the ability to capture physical information using boundary sensors is not as good as internal sensors.

To verify the generalization and applicability of the improved PSO-POD-CS, we use the data from Ref. 43 for reconstruction. As shown in Fig. 11(a), Exp is the experimental data. S1 and S2 are data from two simulations.

S1 and S2 are used as the a priori information for POD-CS. The reconstruction results of S1 and S2 obtained using the first 1 POD_basis and first 2 POD_basis with 1–12 computational nodes are discussed. Figure 11(b) shows that the accuracy of the improved PSO-POD-CS is related to the accuracy of the a priori information. The reconstruction accuracy of S1 is higher than that of S2. The average absolute reconstruction errors of S1 and S2 (0.023) even with one computational node are better than the simulated average absolute errors of S1 (0.0724) and S2 (0.132), which also indicates the applicability and reconstruction accuracy of the improved PSO-POD-CS.

As shown in Fig. 11(c), the improved PSO-POD-CS is advantageous over not using a priori reconstruction in terms of the number of computational nodes and reconstruction accuracy, because methods such as Cubic use at least four points in a one-dimensional reconstruction before reconstruction, while other reconstruction methods have larger reconstruction errors with less than four points. As shown in Fig. 11(c), the reconstruction error of POD-CS using one computational node is smaller than those of other methods using 12 computational nodes. The reconstruction accuracy of POD-CS is higher.

Figure 11(d) shows the reconstruction results of the improved PSO-POD-CS in a steady state turbulent thermal environment in comparison to the PINN reconstruction results of Ref. 23. The relative errors of [P, U, V, T] reconstructed by the method in Ref. 23 using 0.74% of data points at Ra = 10³ are [7.7%, 18.11%, 18.76%, 3.02%]. A larger Ra corresponds to a larger turbulence intensity in the chamber, larger parameter gradient, and more computational nodes required for reconstruction.²³ In this study, the improved PSO-POD-CS is used to reconstruct the turbulent thermal environment with a larger Ra using fewer (0.49%) computational nodes, and smaller reconstruction errors are obtained.

The relative errors of the steady-state turbulent thermal environment were [0.88%,1.47%,1.34%,1.75%] for the reconstruction [P, U, V, T] using the internal sensors and [1.24%, 2.62%, 2.20%, 2.19%] for the steady-state reconstruction using the boundary sensors. The improved PSO-POD-CS has a better accuracy in the reconstruction of conjugate heat transfer in high-Ra turbulent thermal environments, and is instructive in reconstructing coupled heat transfer problems at high turbulence intensities.

This study reconstructs the transient turbulent thermal environment in the nacelle of a wind turbine with a single power of 10 MW. The optimized sensor locations are optimized using a modified PSO, which accelerates the optimized convergence of a conventional PSO. This also ensures reconstruction stability and saves sensor resources. In order to investigate the sensor distribution for the reconstruction of the nacelle turbulent thermal environment for various models, the accuracy of the reconstruction of three transient turbulent thermal environments (Data_100, Data_200, and Data_300) with two types of sensor arrangements, namely, the internal arrangement and the boundary arrangement, is discussed. In addition, the effect of POD_basis with different feature dimensions on the reconstruction accuracy is also investigated. Finally, to ensure the universality and accuracy of this paper's method, the results of Refs. 23 and 43 are used for comparison, respectively, and this paper's method obtains good reconstruction results.

Based on the calculations by the above-mentioned methods, we can summarize the following conclusions:

• Linearizing the parameters such as population update rate and position in the traditional PSO can speed up the optimization convergence speed to some extent. In this paper, with the same parameters before and after the improvement, and taking the relative reconstruction error of the same 50 × 1823 snapshots as the objective function for optimization, the improved PSO completes the convergence 558 steps earlier than the traditional PSO.

• We calculated the reconstruction accuracy of POD_basis and internal sensors with different number of truncations and boundary sensors, respectively, and obtained the optimal Energy_ratio range for different turbulent thermal environments, and the Energy_ratio control of POD_basis is optimal at 92.6%–99.6%.

• In a study of two arrangements, boundary sensors and internal sensing, the improved PSO-POD-CS method and a sufficient number (7 and 9) of sensors were used, and both methods were able to reconstruct the transient turbulent thermal environment well. However, the relative error of the reconstruction using internal sensors is smaller in most cases. As the number of snapshots increases, the number of sensors needs to be increased to capture the physical information features, otherwise large local reconstruction errors will occur.

The focus of this study was on the theoretical approach. It would be helpful to validate this approach with more experimental data. The method alleviates the problem of high surface temperatures of key components, such as high-power nacelle generators to some extent, and could be valuable for the structural design of wind turbines, arrangement of sensor positions, and reconstruction of highly turbulent heat transfer coupled physical fields.

This study was supported by the National Natural Science Foundation of China (Grant No. 51776051).

We thank the editors and referees who provided valuable comments to improve this paper.

The authors have no conflicts to disclose.

Zhenhuan Zhang: Data curation (equal); Formal analysis (equal); Methodology (equal); Project administration (equal). Xiuyan Gao: Methodology (equal); Resources (equal); Software (equal). Qixiang Chen: Project administration (equal); Software (equal). Yuan Yuan: Formal analysis (equal); Funding acquisition (equal); Methodology (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.