Fast Prediction of Flow Field around Airfoils Based on Deep Convolutional Neural Network

Round 1

Reviewer 1 Report

Review comments

Title: Fast prediction of flow field around airfoils based on deep convolutional neural network

Manuscript Number: applsci-2014584

General comments:

This manuscript presents a data-driven method to predict the aerodynamics of NACA 0012 series. A consecutive framework of a convolutional neural network and a deconvolutional neural network is employed in constructing the training model. The effects of the training parameters on the training model are evaluated. It is found that  The prediction process using the proposed CNN-DCNN model is much faster than the traditional CFD simulations. The results indicate that the deep learning approach is promising in terms of prediction and optimization. The paper's quality is good and can be considered for publication after a minor revision.

Detailed comments:

1.       The introduction of the paper is very informative. The authors provided sufficient background about the machine learning and deep learning methods in airfoil/aerodynamic optimization studies. However, the motivation and uniqueness of this study were not well formulated. The authors should clearly state why this study is significant and unique among the others.

2.       The authors mentioned that the validation dataset is used to avoid overfitting. Please clarify.

3.       The authors should clearly state what the operating parameters mean and include. It also appears in Step 1 (Data generation) in Fig. 1, but was not well explained.

4.       The authors should explain what the weight parameter matrix and bias parameter matrix mean. They are the critical parameters in the training function.

5.       In equation 1, P and U are function of (S, Re; W, b). However, in the training function in Step II in Fig. 1, P and U are only function of (S; W, b). Please clarify. Why is the Re left out in the training function?

6.       The authors should clearly explain how the parameters (W, b) are optimized, as it is an essential part of constructing the trained CNN-DCNN.

7.       In equation 3, F is the features mined from the input S using CNN encoder. Please clarify what features are included in F.

8.       The authors claimed that “the model works as a nonlinear function that maps the low dimensional input variables to the high dimensional output variables”, but then mentioned that “the CNN encoder compresses the high-dimensional input states to the feature maps…”. Please clarify.

9.       Please clearly explain what type of filters (besides the size and number) are implemented in the model.

10.   Please double-check the equation numbers cited in the paper. “Behavior os Equation (3) makes sure that the output of ELU…”. Should it be Equation (4)? The same issue for all the following equations.

 

11.   The equations (Eq. 19) to calculate the lift and drag coefficients are not correct. Please double-check.

Author Response

Manuscript ID: applsci-2014584

Dear reviewer 1:

We wish to thank for your useful comments. In the revised manuscript, the response to you are all highlighted in red in detail.

Detailed comments:

■ Comment 1. The introduction of the paper is very informative. The authors provided sufficient background about the machine learning and deep learning methods in airfoil/aerodynamic optimization studies. However, the motivation and uniqueness of this study were not well formulated. The authors should clearly state why this study is significant and unique among the others.

▲ Response: Thanks for your suggestion. We have supplied additional statement to emphasize our motivation (at lines 110-112) and uniqueness (at lines 124-136) of this study in the Introduction.

■ Comment 2. The authors mentioned that the validation dataset is used to avoid overfitting. Please clarify.

▲ Response: Sorry for the confusion. Actually, what we want to express is that we are able to determine whether the model is overfitting by the performance of the validation dataset. With the validation dataset, each training iteration of the CNN-DCNN model will consist two steps. The training accuracy is first computed using the training dataset and the network parameters are updated, subsequently, the validation accuracy is calculated using the validation dataset while the network parameters keeping constant. Therefore, the learning curves will cover the training and validation accuracy, which can be seen in Figure 9. Thus, we can make judgments that the model is overfitting provided that the training accuracy is high and validation accuracy is much lower since it indicates that the learned mapping function is only applicable to the training dataset and differs greatly from the actual one. Moreover, we can conclude that the model is underfitting when both the training and validation accuracy are much lower since it suggests that the network cannot conform to the actual logic due to the data itself, or the bad characteristics of the network. To deal with this problem, we have modified the inappropriate expressions to “Validation dataset is used to judge whether the model is overfitting.” (at lines 180-181).

■ Comment 3. The authors should clearly state what the operating parameters mean and include. It also appears in Step 1 (Data generation) in Fig. 1, but was not well explained.

▲ Response: Sorry for the unclear description. We have provided a detailed explanation about the operating parameters, i.e. the Reynolds number at lines 175-176. Moreover, we also re-draw Figure 1 to correct the error.

■ Comment 4. The authors should explain what the weight parameter matrix and bias parameter matrix mean. They are the critical parameters in the training function.

▲ Response: Thanks for your suggestion. We have further explained the weight parameter matrix and bias parameter matrix: “Subsequently, the ELU activation function (see Equation (4)) will be applied to obtain the final output . Actually, the weight parameter refers to the actual values associated with each element and indicate the importance of that element in predicting the final value, which can be understood like the coefficients of each term of a nonlinear function. As described in Equation (7), the bias parameter is a threshold value to shift the activation function to the left or right, which can be called the intercept distance along y direction in the linear equation.” (at lines 255-261).

■ Comment 5. In equation 1, P and U are function of (S, Re; W, b). However, in the training function in Step II in Fig. 1, P and U are only function of (S; W, b). Please clarify. Why is the Re left out in the training function?

▲ Response: Sorry for the omission of the Re. As stated in the response to comment 3, we have checked the same issue and added the Re to the training function.

■ Comment 6. The authors should clearly explain how the parameters (W, b) are optimized, as it is an essential part of constructing the trained CNN-DCNN.

▲ Response: Thanks for the suggestion. We have provided a clear explanation about how the parameters (W, b) are optimized. “ and  are the network parameters to learn, whose updating rules will be introduced in Section 2.2.” (at lines 161-162) and “For ease of description,  and  in Equation (1) are collectively referred to as network parameters and represented by  since their updating rules are the same.” (at lines 312-313). In fact, we have mentioned this point at lines 316-317, i.e. “where  represents the network’s parameters set including the weight  and the bias ,  refers to the gradients of the loss function with respect to ”. Therefore, the optimization process of the parameters (W, b) can be known according to Equation (12)-(16).

■ Comment 7. In equation 3, F is the features mined from the input S using CNN encoder. Please clarify what features are included in F.

▲ Response: From the characteristics of the input (S, Re), the features F mined from the input include the information of airfoil profile and the flow conditions (i.e. Reynolds number). Furthermore, since the airfoil profile is the actual coordinates after rotational transformation, F also includes the information of the angle of attack (AOA). We have supplied the explanation at lines 205-208.

■ Comment 8. The authors claimed that “the model works as a nonlinear function that maps the low dimensional input variables to the high dimensional output variables”, but then mentioned that “the CNN encoder compresses the high-dimensional input states to the feature maps…”. Please clarify.

▲ Response: We apologize for the inappropriate expression. Since the CNN-DCNN model mainly consists two modules, namely, the CNN and DCNN, we said “the model works as a nonlinear function that maps the low dimensional input variables to the high dimensional output variables”, which is actually what DCNN decoder does. It is the fundamental function of the CNN-DCNN model that it can find out the reduced-order representation of the input matrices using CNN encoder and then map them to output using DCNN decoder. To avoid unnecessary confusion, we have corrected it to “the model works as a nonlinear function that maps the low dimensional features extracted by CNN encoder to the high dimensional output” at lines 190-191.

■ Comment 9. Please clearly explain what type of filters (besides the size and number) are implemented in the model.

▲ Response: To clarify the concept of the filter and the kernel, we added the explanation “where following the descriptive convention of convolutional networks, we refer to a 3D cube stacked with multiple 2D convolutional kernels as a filter in this work.” at lines 221-222. As stated at lines 219-220,  is the size of the filter (: width, :height, : channel),  is the number of the filters (), where  is the size of the convolutional kernel.

■ Comment 10. Please double-check the equation numbers cited in the paper. “Behavior os Equation (3) makes sure that the output of ELU…”. Should it be Equation (4)? The same issue for all the following equations.

▲ Response: Thanks for your reminder. Yes, it should be Equation (4). We have checked the same issue for all the following equations and corrected them.

■ Comment 11. The equations (Eq. 19) to calculate the lift and drag coefficients are not correct. Please double-check.

▲ Response: Thanks for your reminder. For convenience, we unified the description of the lift and drag coefficient in one equation, perhaps this is the reason for the ambiguity. Therefore, we have corrected it as follows:

 

 

Sincerely yours,

Wei-Tao Wu, PhD,

Professor, School of Mechanical Engineering,

Nanjing University of Science and Technology, Nanjing 210094, China.

Email: [email protected]

 

Nadine Aubry, PhD

Department of Mechanical Engineering

Tufts University, Medford, MA, 02155, USA

Email: [email protected]

 

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments for author File: Comments.pdf

Author Response

Manuscript ID: applsci-2014584

Dear reviewer 2:

We wish to thank for your useful comments. In the revised manuscript, the response to you are all highlighted in blue in detail.

Detailed comments:

■ Comment 1. Page 1: Is there any other than Signed Distance Function (SDF) available to parameterize the geometric and flow condition setups? Kindly mention.

▲ Response: Traditionally, the widely used methods to parameterize the geometric are basic profiles or geometric parameterized variables, which are all derived from numerical equations like B-splines, Bezier curve, Parsec (Parametric Section), Hicks-Henne, CST (class shape function transformation), and so on. However, they are applicable to conventional aerodynamic surrogate models like polynomial regression, Kriging and RBF (Radial Basis Function) regression network but not to CNN since convolutional operations on parametric vectors only move along one direction. The binary images to parameterize airfoils can work with CNN but they contain less information, i.e. 0 and 1, leading the neural network model for point-to-point flow field prediction not being trained efficiently. Adopting the convolutional approach with the geometric representation by the SDF has been one of the popular strategies since the Guo et al.’s publication in 2016 [1]. Therefore, our work applied SDF to parameterize the airfoil geometries. We have mentioned the above methods in Section 2.3.2 (at lines 372-384). As for the flow condition, there is no need to parameterize it as it is usually a single value. It can be merged directly with the airfoil representation or it can enter the network in other forms, such as concatenating it to the input of intermediate layer.

■ Comment 2. Page 4, line148, Include the formulation of nonlinear function ? in the manuscript. Though its given in Eq1, it's unclear. Mention explicitly in the manuscript.

▲ Response: Thanks for the suggestion. Actually, the formulation of nonlinear function ? is the dataflow transfer function with regard to the weight parameter matrix, bias parameter matrix and activation function of the CNN-DCNN model. Combining Equation (4) and (7), the  in Equation (1) can be summarized as follows provided that l indicates the last layer:

                                                                                                           (8)

where  is the output (l-1)th layer,  denotes the final output after using activation function, and  and  represent the weight and bias matrix of the lth layer, respectively. We have appended Equation (8) to complement the description of the nonlinear function ? at lines 262-266.

■ Comment 3. Page 7, line 242, Kindly include and describe the activation function used in this work.

▲ Response: Thanks for the suggestion. In fact, the activation function has been given at lines 225-234.

■ Comment 4. Page 14, Why is the data sampled every 200 epochs in Figure 9? Will the increase or decrease of it affect the learning rate?

▲ Response: As a matter of fact, the model parameters are updated every epoch. The sampling frequency of 200 just denotes the frequency of recording model accuracy, which is set to save time. The increase or decrease of it will not affect the learning rate at all since it does not affect the training of the model. To clarify this point, we have added the explanation “It should be noted that the sampling frequency of 200 just denotes the frequency of recording model accuracy, which is set to save time merely and will not affect the training of model.” at lines 445-447.

■ Comment 5. Page 15, 16, and 17, What are the possible reasons for error in Figs.10, 11, and 12.

▲ Response: As we can see form Figs.10, 11, and 12, the maximum relative errors are found around the leading and trailing edges of the airfoil, due to the flow separation induced by the circulation and incoming flow. Specifically, the error band of velocity on the trailing edge of the airfoil almost coincide with the wake caused by flow separation. Therefore, the reasons for error may be the sharp numerical changes, but it should be noted that the errors appear much smaller. We have supplied the explanation at lines 503-508.

 

Finally, we have included the following references in the revised manuscripts, i.e. https://doi.org/10.3390/en12163185, https://doi.org/10.3390/en14071808 as references [2-3].

 

 

 

 

Sincerely yours,

Wei-Tao Wu, PhD,

Professor, School of Mechanical Engineering,

Nanjing University of Science and Technology, Nanjing 210094, China.

Email: [email protected]

 

Nadine Aubry, PhD

Department of Mechanical Engineering

Tufts University, Medford, MA, 02155, USA

Email: [email protected]

[1]      X. Guo, W. Li, and F. Iorio, “Convolutional neural networks for steady flow approximation,” in Proceedings of the 22nd ACM SIGKDD international conference on knowledge discovery and data mining, 2016, pp. 481–490.

 

Author Response File: Author Response.pdf

Reviewer 3 Report

I.m.h.o. the submitted article is characterized by a low level of novelty and significance, thus, resulting of little interest to readers.

It appears to me, that the key innovative element is supposedly the proposed CNN-DCNN method applied to pressure, velocity fields.

However, it is not clear from the text, if this method a genuine contribution of this work is, or rather somehow already existing in literature. If the latter would be true, it is clear to me the challenge constituted by the application being two colored diagrams in place of a more routine random image.

Moreover, the usage of CFD for the described operating conditions remains questionable, due to their extreme simplicity, for which a simpler panel-code solver would have been sufficient. Nevertheless, the description of the CFD methodology used is poor, lacking of any details concerning turbulence modeling or grid independence results.

It is not clear what programming language has been used for the method development, neither a reference to a git package is provided. Having been submitted to an open-access journal, using open-source code like OpenFOAM and publicly available reference database, I would have expected from the authors a more transparent dissemination of their approach to the research community.

I have included within the attached PDF file some further comments.

For these reasons I do not recommend the submitted article for publication.

Comments for author File: Comments.pdf

Author Response

Manuscript ID: applsci-2014584

Dear reviewer 3:

We wish to thank for your useful comments. In the revised manuscript, the response to you are all highlighted in green in detail. Moreover, following your general recommendations, we have uploaded the codes related to this work to github.com (links are at the end).

Detailed comments:

■ Comment 1. Evaluation of what is?

▲ Response: Sorry for the confusion. The aerodynamic evaluation includes flow fields around an airfoil and aerodynamic performance parameters such as pressure distribution coefficient, lift coefficient, and drag coefficient. We have supplemented it at lines 41-42.

■ Comment 2. It is not clear what is the novelty of the proposed approach w.r.t. the literature survey provided. Moreover, what is the expected added value of such a proposal, differentiating it from what presented by other researchers in their work?

▲ Response: Sorry for the inadequate description about the novelty of our work. The main novelty of this work is that we combined the flow parameter (i.e. Reynolds number) with the popular parametric strategy (Signed Distance Function) to make the prediction model not only adaptive to different geometries but also different flow conditions, thus the designed data-driven reduced-order model (CNN-DCNN model) can predict the pressure and velocity field simultaneously with a mean relative error lower than 1%, costing only 25ms, three orders of magnitudes faster than CFD. We have emphasized the novelty of the proposed approach at the penultimate paragraph of Introduction (at lines 124-136).

■ Comment 3. What is the AoA? Is the flow fully turbulent or transitional from laminar to turbulent status? What turbulence model has been used for both simulations? Is there any grid convergence study available to guarantee the independence of the solution from the grid?

▲ Response: AOA is 10°, which was stated at line 347. The flow is fully turbulent since the Reynolds number in our study case is millions of orders of magnitude, which can be found at lines 349 and 413-414. Spalart-Allmaras (SA) turbulence model has been used for both simulations, which has been stated at line 349. Besides, for avoiding the tediousness of current work, we did not present the grid convergence study in this paper. To response this comment, we give the corresponding results as follows:

Fig. 1. Grid independence study.

The variations of  and  are shown in Fig. 1. When the grid number exceeds 5×10⁴, the sensitivity if lift and drag coefficients to the grid number is rather low. Moreover, with the grid number of 5×10⁴ under the validation condition, the calculated  and  are 1.1283 and 0.0129, respectively. Slight deviations are achieved compared with the experimental values of 1.0809 and 0.0117. Therefore, the grid scale about 5×10⁴ is selected for subsequent calculations. We have appended the above content at lines 368-374 and re-drawn Figure 6.

■ Comment 4. Based on this statement, it is necessary to argument, why the usage of the proposed CNN-DCNN model require a check on its prediction and training behavior. In other words, if the CNN-DCNN has been already proposed in the literature and verified/validated by means of image recognition problems, where is the need for further verification/validation when one simply substitute images with pressure and velocity field images, where colors represent values?

▲ Response: Though the CNN-DCNN network proposed in this work is similar to U-Net architecture [1]-[2], this work is unique as the detailed methods (e.g., specific architecture of the networks and the adaptability of flow condition) differ from existing methods. Moreover, the specific applied scenario is different in that the flow condition under high Reynolds numbers, different turbulence model, the generation method of deformed airfoil geometries, the predicted objects covering pressure and velocity field. As well known, the data-driven model mainly depends on the dataset itself and its intrinsic correlation. As such, due to the aforementioned differences, the CNN-DCNN model may behavior quite differently and then it is necessary to evaluate its prediction and training behavior. Actually, it is not as simple as just substituting images with pressure and velocity field images.

■ Comment 5. Can you provide a justification for the choice of this range of AoA? What is it representative for? I.m.h.o. there is no advantage of using CFD method over much simpler/faster panel code methods, as r.g. XFOIL et.al., in these operating conditions. Can you help me understand your approach on this?

▲ Response: First, the settings of the range of AOA is to enrich the airfoil geometry library for training the CNN-DCNN model. Secondly, it is to make the CNN-DCNN model not only adaptive to airfoil shapes but also to different AOAs. Finally, this range of AOA was chosen with reference to the relevant paper [3]. As for the advantage of using CFD method, the reasons lie in that one important session of this work is to obtain the flow field data and not just aerodynamic performance parameters, while XFOIL does well in analyzing the latter but cannot achieve our requirements. Therefore, we select the CFD method (i.e. open-source CFD software, OpenFOAM) to obtain the wanted flow field data and aerodynamic performance parameters. The main steps include grid generation, boundary conditions setting, selection of turbulence model, selection of the solver, and data postprocessing.

■ Comment 6. Can you please explain why this is confirmed from the Figure 14 for the pressure field, but not for the velocity field. I assume, the plotted line for the velocity field has a reversed x-axis, where the first mode actually on the right-most point located is.

Please comment on this.

▲ Response: Yes, as you said, the plotted line for the velocity field has a reversed x-axis since Figure 14 is a graph shared by y-axis. Therefore, the first mode for the pressure field is located on the left-most point while that for the velocity field is located on the right-most point.

Git url: https://github.com/njustwulab/CNN_DCNN_PU

 

 

Sincerely yours,

Wei-Tao Wu, PhD,

Professor, School of Mechanical Engineering,

Nanjing University of Science and Technology, Nanjing 210094, China.

Email: [email protected]

 

Nadine Aubry, PhD

Department of Mechanical Engineering

Tufts University, Medford, MA, 02155, USA

Email: [email protected]

[1]      J. Chen, J. Viquerat, and E. Hachem, “U-net architectures for fast prediction of incompressible laminar flows,” arXiv Prepr. arXiv1910.13532, 2019.

[2]      N. Thuerey, K. Weißenow, L. Prantl, and X. Hu, “Deep learning methods for Reynolds-averaged Navier–Stokes simulations of airfoil flows,” AIAA J., vol. 58, no. 1, pp. 25–36, 2020.

[3]      J.-Z. Peng, S. Chen, N. Aubry, Z.-H. Chen, and W.-T. Wu, “Time-variant prediction of flow over an airfoil using deep neural network,” Phys. Fluids, vol. 32, no. 12, p. 123602, 2020.

 

Author Response File: Author Response.pdf

Round 2

Reviewer 3 Report

Thanks for your responses, integrations and the link to the github repository.

A few remarks to your responses:

"The flow is fully turbulent since the Reynolds number in our study case is millions of orders of magnitude, ...": One can still have laminar to turbulent transition at Reynolds number three times higher than that. See: https://iopscience.iop.org/article/10.1088/1742-6596/749/1/012014.

"As for the advantage of using CFD method, the reasons lie in that one important session of this work is to obtain the flow field data and not just aerodynamic performance parameters, while XFOIL does well in analyzing the latter but cannot achieve our requirements. ": Being XFOIL based on singularity distributions is indeed capable of providing velocity and pressure fields in output, however with a bit of post-processing. CFD is with this regards ready-to-go.

 

Author Response

Manuscript ID: applsci-2014584

Dear reviewer 3:

We wish to thank for your useful comments again. We have read and considered your remarks carefully. In the revised manuscript, we have made appropriate revisions in response to the useful remarks, which are highlighted in green in detail.

A few remarks to your responses:

 

■ Comment 1. "The flow is fully turbulent since the Reynolds number in our study case is millions of orders of magnitude, ...": One can still have laminar to turbulent transition at Reynolds number three times higher than that. See: https://iopscience.iop.org/article/10.1088/1742-6596/749/1/012014.

▲ Response: We apologize for our previous inappropriate explanation and we agree with your remarks. We have carefully read the suggested references and cited it as reference [49] in the revised manuscript. Indeed, the flow can still have laminar to turbulent transition at Reynolds number three times higher than millions of orders of magnitude, which is also true for our cases. Nevertheless, we want to emphasize that the final collected flow field data is under steady flow status. At this time, the flow has transformed to fully turbulent status. We have added the above explanation at lines 377-380.

 

■ Comment 2. "As for the advantage of using CFD method, the reasons lie in that one important session of this work is to obtain the flow field data and not just aerodynamic performance parameters, while XFOIL does well in analyzing the latter but cannot achieve our requirements. ": Being XFOIL based on singularity distributions is indeed capable of providing velocity and pressure fields in output, however with a bit of post-processing. CFD is with this regards ready-to-go.

▲ Response: We apologize for the inadequate explanation. Yes, we agree with your comments. It is for the reasons you stated that we have used the CFD method instead of XFOIL for convenience, as such, we can collect more training data, which will help to render the CNN-DCNN model to its maximal effectiveness.  

 

 

Sincerely yours,

Wei-Tao Wu, PhD,

Professor, School of Mechanical Engineering,

Nanjing University of Science and Technology, Nanjing 210094, China.

Email: [email protected]

 

Nadine Aubry, PhD

Department of Mechanical Engineering

Tufts University, Medford, MA, 02155, USA

Email: [email protected]

 

Author Response File: Author Response.pdf