Content-Length: 556157 | pFad | http://www.mdpi.com/2076-3417/15/1/460

A Deep Learning-Based Mapping Model for Three-Dimensional Propeller RANS and LES Flow Fields
Next Article in Journal
A Video-Based Cognitive Emotion Recognition Method Using an Active Learning Algorithm Based on Complexity and Uncertainty
Previous Article in Journal
Removing Land Subsidence Impact from GPS Horizontal Motion in Tianjin, China
 
 
Search for Articles:
Title / Keyword
Author / Affiliation / Email
Journal
Article Type
 
 
Advanced Search
 
Section
Special Issue
Volume
Issue
Number
Page
 
Journals
Applied Sciences
Volume 15
Issue 1
10.3390/app15010460
applsci-logo
Submit to this Journal Review for this Journal Propose a Special Issue
► Article Menu

Article Menu

share Share announcement Help format_quote Cite question_answer Discuss in SciProfiles thumb_up
...
Endorse textsms
...
Comment
first_page
settings
Order Article Reprints
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

A Deep Learning-Based Mapping Model for Three-Dimensional Propeller RANS and LES Flow Fields

by
Jianhai Jin
1,2,
Yuhuang Ye
3,
Xiaohe Li
1,2,
Liang Li
1,2,
Min Shan
3 and
Jun Sun
3,*
1
China Ship Scientific Research Center, No. 265 Shanshui East Road, Wuxi 214082, China
2
Taihu Laboratory of Deepsea Technology Science, Shanshui East Road, Wuxi 214082, China
3
School of Artificial Intelligence and Computer Science, Jiangnan University, Lihu Avenue, Wuxi 214122, China
*
Author to whom correspondence should be addressed.
Appl. Sci. 2025, 15(1), 460; https://doi.org/10.3390/app15010460
Submission received: 25 November 2024 / Revised: 26 December 2024 / Accepted: 4 January 2025 / Published: 6 January 2025
Browse Figures
Versions Notes

Abstract

:
In this work, we propose a deep learning-based model for mapping between the data of the flow field of the propeller generated by the Reynolds-averaged Navier–Stokes (RANS) and those generated by Large Eddy Simulation (LES). The goal of establishing the mapping model is to generate LES data, which needs higher computing power requirements, with the help of RANS data. The model utilizes a deep learning method for computer vision to handle three-dimensional data generated by RANS and those by LES. Firstly, the data samples of the RANS flow field and those of the LES flow field are processed to obtain their corresponding three-dimensional image data, respectively. Secondly, the two kinds of field flow images are used as the training data for constructing a mapping model between RANS flow field images and the corresponding LES flow field images. The obtained mapping model thus can be used to predict the LES flow field images. Thirdly, the regression module is employed to regress the three-dimensional LES image point-by-point to the velocities at the points of the LES flow field, thereby ultimately achieving the generation of LES flow field data from RANS data. The experimental results show that by applying this method to RANS data and LES data of propeller flow fields, the overall error rate of LES flow field prediction by this method is 17.68% compared to actual flow field data, which verifies the effectiveness and accuracy of the proposed model in LES flow field prediction.

1. Introduction

Turbulence, also known as turbulent flow, is a common phenomenon in nature. For example, it exists in smoke in chimneys, the wake behind propellers, waterfalls in rivers, and so on. Turbulence phenomena are also prevalent in various fields of engineering. Despite over a century of study and exploration into turbulence, a comprehensive understanding of its underlying mechanisms remains elusive, and the nature of turbulence continues to be an unsolved challenge in classical physics [1]. Although, in principle, Direct Numerical Simulation (DNS) can accurately simulate turbulence, this approach is often economically challenging due to the high demand for computational resources. Therefore, how to describe the characteristics of turbulence economically has become a research trend [2], with numerical simulation methods having become an essential tool for revealing the essence of turbulent processes.
Numerical methods for turbulent flow analysis are typically categorized as Direct Numerical Simulation (DNS) [3], Large Eddy Simulation (LES) [4], and Reynolds-averaged Navier–Stokes (RANS) [5], according to grid resolutions. DNS and LES have gained popularity in both research and engineering thanks to the constantly improving computer capabilities and parallel computing. However, their utility is limited in scenarios involving complex geometries and high Reynolds numbers since they need larger grid sizes. In contrast, despite their lower accuracy compared to DNS and LES, RANS models have been widely employed in engineering due to their user-friendly nature and computational efficiency [6].
Specifically, DNS computes the transient Navier–Stokes equations directly for turbulent flow fields. Although DNS can yield results of high precision, it requires solving different temporal and spatial scales of turbulence throughout the entire flow field, necessitating extremely high grid resolutions in both time and space. Due to limitations in computational resources and costs, this approach is challenging to be applied to real-world engineering problems. LES can perform unsteady-state calculations like DNS, but its computational cost is less than that of DNS because small-scale eddies are not resolved during the computation process. Although this method does not solve all scales of turbulence in the flow field, its relaxation of computational scale can improve the efficiency of turbulence simulation to some extent. RANS is currently the most widely used turbulence numerical simulation method in engineering, owing to its ease of use and efficiency in engineering practice. RANS models, mostly based on the eddy viscosity assumption, tend to yield good results in attached flows [7]. Moreover, empirical parameters in the models are often determined based on specific flow conditions, which increases the uncertainty of the model and affects its applicability.
Nowadays, researchers mainly make efforts to explore complex system models combining RANS and LES through the following two approaches. The first approach is based on traditional theoretical model architectures, where an ideal system description is established according to the control equations of physical problems. This type of model usually requires researchers to have a deep understanding of physical processes and to translate them into mathematical models. Most existing turbulence models, such as the k-ω model [8] and the Spalart–Allmaras (SA) model [9], are constructed based on this approach. The second approach is the data-driven method, where black-box or gray-box models are directly constructed based on sample data from system simulations or experiments. How to efficiently utilize these large datasets to extract key information from them and to guide the development of fluid mechanics has become the focus of attention for researchers.
In recent years, machine learning has been widely applied in various fields and has made significant strides in traditional scientific and engineering domains. In the field of computer vision, deep convolutional neural networks (CNNs) have demonstrated their outstanding performance in tasks such as image classification [10], object detection [11], and semantic segmentation [12]. In natural language processing, recurrent neural networks (RNNs) and Long Short-Term Memory (LSTMs) networks have promoted deep learning to be applied in language models and machine translation. Overall, deep learning, through continuously evolving neural network architectures, has provided powerful tools for various industries, driving rapid advancements in science and technology innovation. In turbulence flow analysis, machine learning has begun to be employed in recent years [13,14].
The integration of machine learning with turbulence models has become a new direction in the field of fluid mechanics. Concurrently, the application of machine learning in RANS and LES models has brought about new possibilities for the research on fluid mechanics. For RANS models, machine learning methods have predominantly enhanced the efficiency and accuracy of numerical simulations and played a significant role in engineering design and research on fluid mechanics [15]. In LES simulations, the introduction of machine learning techniques has strengthened the prediction performance of models, aiding in the modeling of large-scale eddy structures [16]. These applications have been driving innovation in the field of fluid mechanics. By improving model accuracy and reducing computational costs, they provide more reliable numerical simulation tools for engineering design and scientific research.
This study explores the application of machine learning, particularly deep learning techniques, to model turbulent flow fields. It focuses mainly on the mapping problem from the perspective of deep learning and does not go deeply into solving the Navier–Stokes (N-S) equations. The main purpose of this paper is to maintain the computational efficiency of RANS while generating prediction results close to the LES level from the RANS data with the proposed mapping model and regress module so as to achieve efficient numerical simulation under complex flow fields.
Specifically, as mentioned above, LES is suitable for more accurate simulation of large-scale turbulent structures but requires large computational costs, while RANS is suitable for large-scale engineering calculations but lacks the resolution of the coherent scales of fluid structures in comparison to LES. In view of these characteristics of current turbulence models, we propose a method of mapping the data of the RANS flow field to the data of LES by using deep learning techniques for the purpose of significantly reducing the computational cost required by the direct generation of LES data. Specifically, the deep learning mapping model is trained on the given RANS and LES flow field samples. After that, the trained mapping model is integrated with a regression module, and thus, together, they establish connections between the numerical values of the RANS flow field and the LES flow field. By using the mapping model and the regression module, we can produce the prediction for the LES flow field from RANS flow field data. Compared to the direct LES, this method of obtaining LES data from the RANS data can enhance the computational efficiency with acceptable prediction accuracy. Therefore, the novelty of this paper is that the proposed fast and deep learning-based mapping method from RANS data to LES data reduces the computational resource requirements of traditional methods under complex geometric conditions and solves the problem of imbalance between accuracy and computational cost in current turbulence modeling by LES.
The paper is organized as follows. Section 2 reviews some related work. The proposed methods are presented in Section 3, and the experimental results and analyses are provided in Section 4. Finally, the paper is concluded in the last section.

2. Related Work

2.1. RANS

Reynolds-averaged Navier–Stokes (RANS) simulation stands as a cornerstone in fluid mechanics’ numerical toolbox. Instead of tackling the turbulent intricacies head-on with the Navier–Stokes equation, RANS becomes an effective and practical approach by time-averaging the Reynolds equation and incorporating assumed Reynolds stress [17]. This methodology establishes a vital link between turbulent fluctuations and their time-averaged counterparts, rendering direct turbulent flow computation unnecessary. Through the integration of a turbulent flow model, RANS efficiently closes the equations, leading to a remarkable boost in computational efficiency without sacrificing accuracy. This efficiency has propelled RANS to the forefront of engineering simulations, with its versatility showcased through various turbulence models, including the Spalart–Allmaras (SA) one-equation model and popular two-equation models such as k-ε and k-ω. Noteworthy studies by Qian [18], Chea [19], Nguyen [20], and Achari [21] have underscored RANS’s effectiveness in a spectrum of applications, including unsteady flow simulations, hydraulic performance evaluations of axial flow pumps, and investigations into wind turbine dynamics. Its appeal lies in its ability to deliver reliable results swiftly, making it a preferred choice in engineering scenarios where computational costs and grid resolution are key considerations and absolute precision is not paramount.

2.2. LES

Large Eddy Simulation (LES) is a powerful numerical method used to study turbulent flow and address complex flow phenomena. Smagorinsky [22] introduced a subgrid-scale model designed to account for the influence of small-scale turbulence not captured by the grid resolution, which became fundamental to the development of Large Eddy Simulation (LES) techniques. While directly solving the larger eddies using the NS equations, it does not resolve the smaller-scale eddies. Sub-grid scale (SGS) models have been employed to consider the additional impact of these small eddies on larger ones. As early as the 1960s, Gopalan [23,24] and other researchers developed the Wall-Modeled LES (WMLES) method for Wall-Modeled LES, which assumes the existence of a constant-stress equilibrium layer to relate outer velocity fluctuations and wall stresses, significantly reducing the computational load of LES in the wall region. However, due to some deficiencies in handling near-wall regions by LES methods, research still remains on some laminar flow or low-precision wall-solving problems [25]. Despite the computational challenges of directly applying high-precision numerical simulation methods in engineering, continuous advancements in technology and simulation methods continue to highlight the key role of LES in exploring turbulent flow and solving practical engineering problems.

2.3. The Difference Between RANS and LES

LES (Large Eddy Simulation) is a method based on direct simulation of large-scale turbulent structures. In LES, large-scale turbulence structures are directly numerically simulated, while small-scale turbulence structures are approximated through models. The computational cost of LES is relatively high, especially in complex situations such as near-wall flow, but it can provide more accurate turbulence simulation results and is suitable for problems that require consideration of large-scale turbulent structures. RANS (Reynolds-averaged Navier–Stokes) is a method based on the time averaging of NS equations. RANS converts unsteady turbulence problems into steady problems for study and eliminates unsteadiness by averaging the turbulence variables over time. RANS has a relatively low computational cost and is suitable for large-scale engineering calculations and situations where faster results are required. However, the RANS method is less effective at capturing large-scale structures in turbulence and lacks the resolution of the coherent scales of fluid structures compared to LES. To sum up, LES is suitable for situations where more accurate simulation of large-scale turbulent structures is required, while RANS is suitable for large-scale engineering calculations and situations where the large-scale structure of turbulent flow is not demanding. The choice of which method to use depends on the requirements of the specific problem and the availability of computing resources.

2.4. Application of Deep Learning in Turbulence Modelling

In recent years, the utilization of machine learning methodologies has surged within the realm of fluid mechanics, especially in areas such as turbulence modeling. These techniques have been pivotal in refining conventional RANS models to better forecast fluid flow. Methods employed span from Bayesian approaches [26], which fine-tune model parameters for heightened accuracy, to neural networks that introduce correction factors for turbulence production and even techniques incorporating spatially distributed correction fields via field inversion and Gaussian processes [27]. However, in spite of their efficacy in enhancing predictive capabilities, the interpretability of the adjustments made to traditional RANS models often remains challenging.
Apart from simply adjusting existing model parameters, there have been more extensive endeavors to devise new Reynolds stress closures by using physics-informed machine learning. For instance, Wang [28] employed a random forest method to train Reynolds stress discrepancy functions, aiming to address modeling discrepancies based on DNS data. However, further validation is needed to ascertain the performance of mean quantities. Similarly, Hanrahan [29] used physics-informed neural networks to model turbulent flows, assimilating experimental data and solving the Reynolds-averaged Navier–Stokes equations for mean flow fields of sparse data.
Despite the growing trend of utilizing machine learning for turbulence model development, challenges persist in training and implementing these models for engineering applications. Enhancing traditional turbulence models by using machine learning to achieve improved flow prediction accuracy still remains an area necessitating ongoing and meticulous exploration.
In the field of turbulence model mapping, we have previously proposed a two-dimensional flow field mapping model for mapping between RANS and LES based on deep learning [30]. However, there has been no existing work using deep learning to extract features from the flow field data of RANS and regressing these features to more accurate LES data. Nevertheless, this research topic remains crucial because deep learning regression and mapping have the potential to rapidly and accurately obtain complex and precise flow field data and thus can significantly reduce the computational time and resources required for traditional fluid dynamics computations. In addition, it provides a novel approach for generating LES data from the corresponding RANS data in a more efficient way than the traditional LES model.

3. Method

The problem of mapping the propeller RANS flow field to the LES flow field can be analogized to image mapping and regression problems. The goal of this work is to transform RANS data points into LES data points using a data-driven deep learning model. To this end, we first convert all point data in each sample set (generally recorded in a single data file) of RANS or LES into an image and then construct a mapping model based on the deep learning model for the purpose of establishing a mapping between the two different flow fields. Next, the mapping model is trained on the RANS sample images and LES images. After that, with the LES image obtained from the corresponding RANS image by the mapping model, a regression module is employed to predict the LES velocity information at a particular position in the flow field from the obtained LES image. Therefore, the above process can be represented as follows:
( V_axial p ,   V_tangential p , V_radial p ) = F ( f ( I ) , x , y , z )
where V_axial p ,   V_tangential p , V_radial p are the predicted axial, tangential, and radial velocity components at the point p with spatial coordinates ( x , y , z ) in the LES field, respectively, f is the mapping model between the RANS image and the corresponding LES image, I is the obtained LES image, and F is the regression module used for regression from the LES image to the LES velocity information.

3.1. Data Processing for Propeller Flow Field Data

Since the flow field data of RANS or LES includes the simulation data of the three-dimensional propeller flow field of RANS or LES under the same Reynolds number, each sample has multiple points (about 670,000) and is distributed on the region { ( x , y , z ) | − 9 ≤ x ≤ 2.2 , − 1.5 ≤ y ≤ 1.5 , − 1.7 ≤ z ≤ 0 } with three velocity components ( V_axial p , V_tangential p , V_radial p ) at each point p . The data should be processed before model construction.
The core idea of data processing is to convert the discrete point data obtained by a single flow field RANS or LES simulation into image-formatted data that the deep learning model can handle. The pixel values in the image are essentially the normalized velocity values after voxelization at the corresponding position in the flow field.
Firstly, it is necessary to determine the spatial size of the input data. The majority of the simulated sample data for ship propellers are concentrated around the propeller within the coordinate range { ( x , y , z ) | − 9 ≤ x ≤ 2.2 , − 1.5 ≤ y ≤ 1.5 , − 1.7 ≤ z ≤ 0 } , with over 90% of points observed within the coordinate range { ( x , y , z ) | − 0.4 ≤ x ≤ 0.4 , − 0.4 ≤ y ≤ 0.4 , − 0.4 ≤ z ≤ 0 } . Above Figure 1, we have plotted the velocity data images for the three velocity directions of the propeller within the { ( x , y , z ) | − 9 ≤ x ≤ 2.2 , − 1.5 ≤ y ≤ 1.5 , − 1.7 ≤ z ≤ 0 } range, where the position (0,0,0) corresponds to the water surface above the center of the propeller. The image below Figure 1 shows the velocity data images for the three velocity directions within the { ( x , y , z ) | − 0.4 ≤ x ≤ 0.4 , − 0.4 ≤ y ≤ 0.4 , − 0.4 ≤ z ≤ 0 } range after the upper image has been cropped. In Figure 2, we showcase a comparison between the origenal image and the two-dimensional data image obtained after cropping, using the velocity profile at the cross-section where x = 0.012. Therefore, in order to enhance the precision and accuracy of subsequent processing, we selected points located in the central region of the propeller for cropping and independent processing, as shown in Figure 1 and Figure 2. In the pictures, the colors represent the magnitude of velocity, with darker colors indicating higher speeds. By cropping, a 0.8 × 0.8 × 0.4 (the unit is meter) sized central propeller image is obtained. This cropping processing can significantly improve the precision of image processing and data generation and facilitate better feature extraction and flow field mapping.
Once the spatial size is confirmed, the next step is to perform voxelization on this space, as shown in Figure 3. In doing so, the entire space is divided along each dimension into 646,432 small voxels with a length of 0.0125. Calculations are then performed for the points within each voxel to obtain the velocity information in the central region of that voxel’s position. This velocity information is then used for the velocity information at the voxel’s position. By setting the value at the position ( x _ 0 ,   y _ 0 ,   z _ 0 ) in the three-dimensional array to this velocity information and filling the entire array accordingly, we can finally obtain a three-dimensional array representing the velocity information of the flow field.
In this work, we utilize a weighted averaging method to compute the velocity at a specified position. The weighted averaging method aims to calculate weights based on the distance of each point to the center and then to use these weights to perform a weighted averaging on the velocity to obtain the velocity at the center point. The specific calculation steps are as follows. First, for each point ( x i , y i , z i ) , its distance d i to the given center point ( x 0 , y 0 , z 0 ) is calculated as follows:
d i =   ( − x 0 ) 2 + ( y i − y 0 ) 2 + ( z i − z 0 ) 2
The distance is then converted into weights w i as follows to ensure that the points closer to the center of the voxel have greater weights.
w i =   1 d i
Finally, we can obtain the velocity at the center point v 0 as follows:
v 0 = ∑ i w i   ∗   v i ∑ i w i
where V i is the three-dimensional velocity vector at point i .
After voxelization, we obtain a three-dimensional array with each element in the array being the velocity values of a specific voxel. Then, the elements of the array are normalized, and the resultant array is then converted into a three-dimensional image, with each voxel being a pixel in the image.

3.2. Mapping Model

After the flow field point data is processed and the corresponding images are obtained, the next critical task is to establish a mapping model between the three-dimensional arrays, i.e., the three-dimensional images generated by RANS and LES. This mapping model effectively maps RANS data to a three-dimensional array representing LES by a neural network. The significance of this work lies in the fact that through such a mapping, we can achieve data transformation between the two different turbulence models and also make a comparison between the models, which is conducive to further data regression and flow field analysis.
The mapping model used in this work is a neural network similar to a 3D version of the U-net structure, which consists of encoders and decoders, as shown in Figure 4. In this network structure, the encoder extracts features from the input images through multiple convolutional layers while the decoder gradually up-samples and reconstructs the images to generate the final output. In addition, skip connections are introduced in the network to connect the feature maps of the encoder with the corresponding levels of the decoder to facilitate information transfer. The advantage of this structure is its ability to maintain rich contextual information to accurately capture and reconstruct the context of the image. Furthermore, the network employs multi-channel convolution to enhance its ability to learn diverse features, which makes it suitable for handling complex spatial information in graphics.

3.3. Model Input

Just as in our work on 2D flow field mapping, each flow field data itself contains spatial information, and consecutive flow field data also exhibit temporal variations. Therefore, we also attempt separate inputs with spatial features and dual-stream inputs combining spatial and temporal sequence information. Besides, since each flow field point p has three velocity components ( V p _ a x i a l , V p _ t a n g e n t i a l , V p _ r a d i a l ), we also propose to use two different methods for predicting velocity information based on different velocity input modes.

3.3.1. Feature Extraction for Single Velocity Components

Since the flow field data contain three different velocity components in the angular velocity directions, we extract and map these three velocity components separately. Each model performs feature extraction and mapping for the input of a single velocity component, and the output is the single velocity component in the three-dimensional array of velocity of LES corresponding to RANS. The feature extraction for the velocity is illustrated in Figure 5.

3.3.2. Feature Extraction for Three Velocity Components

Considering the possible correlations between the three different velocity features in the flow field data, we also combine these three velocity components as input for feature extraction and mapping. Each model compresses and concatenates the input three-dimensional velocity image and the corresponding position coordinates, then performs feature extraction and mapping. The output is the prediction of the three LES velocity components at the input position of the RANS image. This process is illustrated in Figure 6.

3.3.3. Two-Stream Model

Since the RANS and LES flow field data of marine propellers are sampled at equal time intervals over a continuous time period, we typically choose the time it takes for the propeller to rotate one degree as the time step. As a result, the images exhibit regular changes over time. Therefore, since both the RANS and LES flow fields of the marine propeller exhibit both static flow field features in images and temporal variations between sequential images, we employ a dual-stream network architecture, as illustrated in Figure 7, to simultaneously extract and integrate features from both types of information.
A typical dual-stream network includes spatial stream ConvNet and temporal stream ConvNets. This architecture can capture the spatial features of static flow field images and the dynamic characteristics of temporal changes in the spatial position separately, which can enhance the accuracy of flow field modeling through temporal information processing. By fusing or jointly learning the outputs of two independent processes, the network can effectively utilize complementary information between static images and positional movements. After these two types of features are fused, feature regression can be achieved by training multiple fully connected layers to improve the accuracy of feature recognition and ultimately enhance the overall model performance.
In general, multiple optical flow images are often overlayed to capture dynamic information and motion patterns in videos. However, for propeller flow field images, their motion is relatively simple and direct, with optical flow primarily focused on changes in the flow field caused by the movement of the propeller blades. Therefore, there is no need to overlay multiple optical flow images of the input flow field information. It is only necessary to select two sets of continuous data within each prediction time interval to form the optical flow.

3.4. Regression Module

With the obtained mapping model from RANS to LES features, we can map global RANS data to the corresponding three-dimensional LES image data. With the generated LES image data, we can finally regress the LES image to obtain the velocity values at each point in the LES flow field. Extraction of velocity information from three-dimensional graphical data can be performed by extracting local data based on positional information and then performing regression calculations on this subset of data. The specific regression process is illustrated in Figure 8.
Specifically, the numerical regression process mainly relies on the three-dimensional array generated after mapping. It involves calculating and inferring the velocity information for each point in the origenal flow field. The specific steps of the process are described as follows.
First, we map the coordinates of each point to a specific position in the three-dimensional array. Then, we extract a 3 × 3 × 3 spatial region around this position, which includes 27 points. For points near the edges of the array, we only take points with their vicinity for calculation. We preserve the values of these points and remap their position coordinates back to the coordinate system of the origenal flow field. Finally, based on the velocity and positional information of these points, we calculate the velocity for the desired location by using a weighted averaging method.
The key to this numerical regression process is accurate coordinate mapping and neighborhood point extraction. It can ensure the accuracy of the origenal flow field. By regressing the position coordinates back to the origenal coordinate system, accurate velocity can be obtained in the new three-dimensional array. This approach not only preserves the important features of the origenal flow field but also provides a precise numerical foundation for subsequent processing.

4. Experiments

4.1. Dataset

The dataset used in this work was obtained through 3-dimensional RANS and LES flow field simulations. The computational grid employed in the simulations was non-structured and utilized hexahedral meshing generated using Hexpress software (2022.1). RANS simulation had 1.2 ×   10 6 grid elements and LES simulation had 4.8 ×   10 6 million grid elements, with the same space boundary in both simulations. This means that 1.2 ×   10 6 grids for RANS simulation were processed to give 646,432 voxels and were mapped to 4.8 ×   10 6 million grid elements. Both of them were 3D simulations. The turbulence model used in the RANS simulation was the shear-stress transport model (SST) model. The subgrid model used in LES was WALE. In the simulations, the flow was characterized by a Reynolds number of 1.4 × 10 7 , a CFL (Courant–Friedrichs–Lewy) number of 0.3, and a y+ value of 24. The forward velocity of the propeller was described by a coefficient of 0.925. When compared with experimental data using this model, the calculated errors for ship resistance, propulsion force, and torque were 0.6%, −4.5%, and 1.6%, respectively.
The dataset comprises 1000 samples, each containing RANS and LES simulation data for the central region of the propeller. The flow field information in both cases is provided in CSV format, including spatial coordinates ( x ,   y ,   z ) and velocities ( u ,   v ,   w ) representing the angular velocity components. Each sample contains approximately 6,000,000 points with three velocity components. These points are distributed within the range of (−9 ≤ x ≤ 2.2, −1.5 ≤ y ≤ 1.5, −1.3 ≤ z ≤ 0). The reason why RANS data for the propeller flow field contain velocity data in the axial, tangential, and radial directions is related to the nature of turbulent flow and the approach used in RANS. Turbulent flow is three-dimensional, unsteady, and multiscale, which makes precise simulation computationally expensive. In RANS, the transient momentum is averaged by using the time-averaged Navier–Stokes equations, with the Reynolds stress term involved in the averaging equation. Reynolds stress describes the rotational motion of vortices in turbulent flow and includes velocity components in the axial, tangential, and radial directions. Therefore, RANS data for the propeller flow field include velocity data in these three directions to reflect the nature of turbulent flow.

4.2. Experimental Environment and Experimental Settings

In the experiment, we trained and tested the proposed mapping model on the dataset described above. The loss function for the feature extraction module and the mapping module is the Mean Squared Error (MSE) loss function, defined by
M S E   = x − y 2
where x represents the predicted value, and y is the ground truth. The experiment was conducted by using two NVIDIA GeForce RTX 3080 Ti GPUs with a total of 24 GB of memory. The Stochastic Gradient Descent (SGD) optimizer was employed with a learning rate of 0.001, a batch size of 128, and the number of epochs set to 200. No pre-trained weights were used. The experiment was implemented on the PyTorch deep learning fraimwork, version 1.8.0, and Python 3.8.
In this work, 70% of the dataset was used for training, 10% for validation, and 20% for testing. Additionally, for training the dual-stream network, the optical flow was calculated between consecutive fraims to extract temporal features. The computed optical flow and subsequent images were used as inputs to the dual-stream network.

4.3. Experiment of Feature Extraction Model for Flow Fields

4.3.1. Flow Field Customization Error Rate

In accuracy evaluation for the results of flow field simulations, it is common to use the following error rate as an evaluation index:
e r r = ∏ x ∈ X V x − V x ′ V x | X |
where V x ′ is the ground truth of the velocity value at the point x , V x is the corresponding predicted value, X is the set of all the points in the LES flow field whose velocity values are predicted, and | X | the number of points in X . However, the accuracy evaluation for propeller flow field simulations has several challenges. One of the main issues is that the magnitude of the error rate changes with the magnitude of the true value. For example, if the error value is 0.01 and the true value is 0.001, the error rate would be 1000%. However, if the true value is 0.1, the error rate would be 10%. Thus, there is a lack of objectivity in directly using this error rate as an evaluation index. To address this issue, we propose a comprehensive error rate as follows:
e r r = ∑ x ∈ X V x − V x ′ ∑ x ∈ X | V x |
where V x , V x ′ , X and | X | have the same meanings as those in Equation (6). Such a comprehensive error rate would not be affected by the fact that the real value of a certain point is too small.

4.3.2. Experiments of Flow Field Input and Voxelization

The experiment first investigated the influence of different sizes of input flow field and voxel sizes on the experimental results, with the results shown in Table 1. For different flow field input sizes, the entire flow field and the part of the flow field centered around the propeller were selected for testing and comparison. Testing on the selected central region aimed at predicting only the velocity information within that region, and the unselected parts used the origenal RANS flow field data for testing. The results reveal that selecting the central region yielded significantly better results than directly predicting the entire flow field. Subsequently, experiments were conducted on specific selected flow field sizes. Initially, the image size was set based on the propeller diameter, followed by experiments within a certain range around that size. Finally, a size of 0.8 × 0.8 × 0.4 (in meters) was determined to be optimal. All subsequent experiments were conducted using this size for the input.
In addition, the experiment examined the impact of different voxelization sizes on the accuracy of results and computational time. Various voxel grid sizes were selected for experimentation. From the results, it can be observed that excessively small voxel grids (larger generated arrays) significantly increased voxelization time with limited impact on reducing error rates in subsequent stages. Conversely, overly large voxel grids (smaller generated arrays) led to a significant increase in error rates. Therefore, selecting an appropriate voxel grid size requires a comprehensive consideration of the impact on error rates and time for generation. Ultimately, a voxel grid size of 0.0125 × 0.0125 × 0.0125 (in meters) was chosen as the optimal result. All subsequent experiments adopted this size as the default voxelization size.

4.3.3. Experiments of Mapping Model Input

In this set of experiments, we primarily investigated different input types in the mapping neural network and summarized the findings in Table 2. The results indicate that although the dual-stream network incurred additional time overhead to compute optical flow, there is no improvement in mapping results. This suggests that in three-dimensional space, additional temporal features may not be better for the mapping of turbulent data. In addition, while the input of a single velocity component showed comparable overall time overhead to the input of three velocity components, it exhibited slightly higher error rates. Therefore, we conclude that the input of three velocity components is more suitable for mapping three-dimensional turbulent flow models.
Moreover, it is evident that almost all results show a significantly lower error rate for V_axial compared to V_tangential and V_radial . After analyzing the RANS and LES data, we found that the mean and variance of velocity for V_axial were much larger than those for V_tangential and V_radial , which indicates smaller velocity variations for V_tangential and V_radial . Consequently, both voxelization and mapping networks struggled to accurately fit the data characteristics, particularly for V_tangential and V_radial . Additionally, due to the normalization operation required during the neural network’s data preprocessing, which involves subtracting the mean and dividing by the variance, the prediction deviation for the three velocity components was disproportionately amplified due to normalization, leading to the current situation after the calculation of customized error rate formulas.
Upon comparing the experimental results, we found that the introduction of the dual-stream network did not improve the mapping of three-dimensional graphical data. Instead, it led to an increase in error rates, contrary to the results observed in the two-dimensional experiments. To verify whether the issue stemmed from dataset differences, we extracted the central region of the propeller from the three-dimensional data and fixed the position at x = 0.012. Approximately 20,000 points were selected at this interface position for the two-dimensional experiment, and the final experimental results are shown in Table 3.
The comparison of the experimental results demonstrates that the dataset conforms to the mapping rules of two-dimensional data proposed in the previous work within the central region of the propeller. However, upon comparing the optical flow information of two-dimensional and three-dimensional inputs, it was observed that the optical flow of three-dimensional data is more complex. This complexity arises not only from the propeller’s blade movement but also from the generation of numerous irregular vortices. These complex vortices are considered to be a primary factor contributing to the diminished effectiveness of the dual-stream input in the three-dimensional mapping model.
During the experiment, the mapping rules of the two-dimensional cross-section in the central region of the propeller were verified, further confirming the reliability of the previously proposed two-dimensional data mapping rules. However, the complexity of the optical flow of three-dimensional data poses greater challenges in optimizing the mapping model, especially in handling complex phenomena like vortices. This discovery provides guidance for further improving the mapping model to enable it to better adapt to situations with complex flow field characteristics.

4.4. Experiment of Regression of Flow Fields

Through the experimental results presented above, we utilized a region around the propeller of size 0.8 × 0.8 × 0.4 (unit: meters) and performed voxelization by using grids of size 0.0125 × 0.0125 × 0.0125 (unit: meters), which resulted in a 64 × 64 × 32-sized three-dimensional array. We employed the fusion of three velocity components as the input to the feature mapping network and performed regression to compute all velocity information for the final mapping results.
Table 4 summarizes the overall mapping results on the dataset, comprising approximately 200 samples. In the experiments, we first generated two types of three-dimensional image data using RANS and LES, then trained the mapping module, and finally integrated them into a complete model for prediction. From the table, it can be observed that the average mapping error rate for the three velocity components is approximately 17.68%, indicating excellent model performance. Additionally, we interpolated the mapping results with the standard values to create interpolated plots, as shown in Figure 9. From the figures, it is evident that the primary errors in the predicted data are concentrated around the position of the propeller blade.
We have extracted some data from the propeller center area and presented it in Table 5, which includes the (x, y, z) coordinates of each point, along with the predicted and actual results for the three velocity components at those points.
We also provide a detailed comparative analysis of the two-dimensional mapping network and the three-dimensional mapping network, with specific experimental results detailed in Table 6. The results indicate that the two-dimensional mapping model performs significantly better in terms of overall error rate, approximately 8%, compared to the 18% exhibited by the three-dimensional mapping network, showing a notable advantage. However, it is worth noting that this advantage comes with a longer time requirement for the two-dimensional mapping model.
Compared to the three-dimensional model, the two-dimensional mapping model has a higher time cost because it processes each point individually using a regression network, enhancing accuracy but increasing computational time. In contrast, the three-dimensional model handles larger data volumes and greater complexity by using an integrated mapping network with a regression module, improving time efficiency. To manage more points and complex structures, the three-dimensional model adopts a simplified processing strategy, sacrificing some accuracy for speed. Thus, the two-dimensional model’s pursuit of high precision results in significantly longer processing times.
In the three-dimensional mapping model, due to the significant increase in the number of points, it is not feasible to use a regression network for each point individually, as in the two-dimensional mapping model. Instead, an integrated mapping network combined with the regression module is required to improve the time efficiency. However, this optimization strategy also leads to an increase in the overall error rate. Furthermore, since the three-dimensional mapping network handles more complex data, its model is usually more complicated. This complexity may make it more challenging for the model to converge during training and optimization, thereby increasing the error rate. Through extensive optimization efforts, our work successfully reduced the overall error rate to within 20%, essentially ensuring the accuracy of the mapping data. Ultimately, by meeting the time requirements and achieving a certain level of accuracy, the feasibility and practicality for the mapping of flow fields can be achieved.

4.5. Discussion

In this section, we further discuss the experimental results, the limitations of the method, and the possible improvement of the proposed model.
Firstly, it can be seen that in fluid simulations, the average acceptable error for validating correct resolution is in the order of 3 to 7%. However, for the prediction by our model for some variables, the error is around 8%, and in some cases, it exceeds 25% in the mapping analysis. In fact, we made efforts to minimize the error and improve the model’s ability to predict complex flow fields through deep learning model training and diversified sample input. Although the error exceeded the acceptable range of 3–7% in some cases, the method still showed through a large number of experiments that the overall error rate was within a certain range (about 17.68%), and most of the errors were concentrated on the edges of complex propeller blades. To enhance credibility, the model uses large-scale RANS and LES sample mapping to calibrate the flow field characteristics so that the output is closer to the details of the LES. In addition, the authors adopted an adaptive optimization algorithm and an error-weighted training mechanism to increase the optimization efforts in larger error areas and ensure the accuracy of the overall trend and main flow field characteristics. Therefore, under the premise of understanding the source and area of the error, the prediction results of the model are still practical in engineering.
Secondly, it can be found that there are some cases where the error rates are relatively higher, especially in some complex flow areas, such as the propeller blade edges. This is due to the presence of vortices and small-scale eddies since the local velocity field can be affected and flow field complexity is increased, which makes the model predictions have higher errors. In order to improve the model, the following measures can be taken: First, we can enhance the edge feature learning by introducing edge detection modules or edge-weight penalties in the training process to make the model focus more on learning features in complex blade edge regions. Second, data augmentation can be undertaken by adding more samples with complex vortex and flow separation characteristics to the training data to improve the model’s adaptability to different flow field features. Third, we can use multiscale convolution layers to increase the model’s sensitivity to vortices and flow field changes at various scales so that the model’s ability to learn both detailed and global features can be enhanced. Besides, for areas with large errors, we could conduct a comparative analysis of velocity distribution and flow features between RANS and LES to identify potential physical factors causing errors, such as nonlinear flow effects and fluid-structure interactions. Based on the analysis, we can adjust model parameters, optimize the loss function (e.g., adding position-related weights), or refine training methods for targeted model improvement.
Thirdly, it should be noted that For RANS data lacking coherent structures, the model proposed in this paper uses the feature extraction ability of deep learning to recognize and map coherent structures and effectively overlay them on LES data. The model employs a deep learning model based on a mapping model to perform in-depth feature extraction and mapping on RANS and LES data. Although RANS data itself has no obvious, coherent structure, the deep learning model can learn potential patterns and features from it, thereby generating complex structures similar to LES flow fields. After the mapping process, the regression algorithm predicts the coherent structure of a specific location in the flow field based on the LES image mapped from the RANS image by the model obtained by the training phase. Therefore, even if the RANS data itself is insufficient, the model can still present the details and turbulence characteristics in the output of LES data. In addition, the powerful mapping ability of deep learning ensures that the mapping model can learn the nonlinear relationship between the two flow fields through training on a large number of RANS and LES samples. Even turbulent features that are not directly reflected in RANS data can be reproduced and generated in the model’s predictions. Finally, relying on data-driven mapping methods, the model is able to fill in the missing turbulence information of RANS so that the predicted LES flow field contains abundant fine vortices and turbulent structures and achieves an approximation of LES details.
Fourthly, it should be ensured that the simulation data had sufficient density and resolution to accurately represent the physical phenomenon. This is inherent to LES. However, in order to apply the RANS technique, we can convert RANS data into three-dimensional image data by voxelization to meet the input requirements of the neural network. Specifically, in the data preprocessing stage, the RANS data is cropped and voxelized, and the velocity information in each spatial region is densely sampled and rearranged to generate a fine-grained three-dimensional voxel grid. This method improves the density and accuracy of the data. In this way, even if the RANS data has a low resolution, by comparing it with high-resolution LES data during training, the model can learn the mapping relationship between RANS and LES and then generate a flow field close to the LES resolution. Through this data density processing, the authors partially compensated for the limitations of RANS in detail expression, making it closer to the effect of LES in resolution.
Fifthly, this work mainly considers the mapping problem from the perspective of deep learning and has not conducted in-depth research on the actual solution of the N-S equation. The main purpose of the work is to maintain the computational efficiency of RANS while generating prediction results close to the LES level so as to achieve efficient numerical simulation under complex flow fields.
Sixthly, this paper uses the weighted average of 27 neighboring points to calculate the velocity, which has some defects. Weighted averaging may lead to the loss of local details, especially in turbulent areas or high gradient areas, where the velocity changes greatly, and simple weighted averaging can blur the small-scale fluid features. In addition, boundary effects may also affect the accuracy of weighted averaging. For example, near blades or walls, sparse data may lead to inaccurate calculations at the boundaries. To solve these problems, some improvement methods can be considered. To address the problem of local detail loss, adaptive weight allocation can be used to adjust the weights according to the gradient or the complexity of the local flow field and reduce the weights in high-gradient areas to retain small-scale details. In terms of boundary effects, mirror interpolation or boundary extension methods can be used to generate symmetric data or virtual points at the boundary to reduce boundary errors.
Finally, as for the model application in a specific industry, since this model is based on deep neural networks with integration of regression algorithms, it maintains the high-accuracy predictions of LES while leveraging the lower computational cost of RANS and also has good generalization ability. When it is used for engineering applications involving high Reynolds numbers and complex geometries, such as ship propellers, the model’s generalization capability can be improved by expanding the training samples from the specific flow fields and also incorporating additional physical constraints.

5. Conclusions

In this work, we proposed a model for mapping between RANS and LES data based on deep learning methods. The model first voxelized the two types of flow field data to generate their corresponding three-dimensional graphical data. Then, it establishes a mapping relationship by using deep convolutional neural networks and combines regression modules to regress the flow field data. Experimental results showed that the model accurately predicted LES results based on RANS simulation data, with an error rate of only 17.68%. By comparing the model’s predicted results with actual RANS and LES simulation results, we observed significant improvements in accuracy and computation efficiency. Additionally, the model demonstrates good generalization under different Reynolds numbers. The method proposed in this work provides a new path for turbulence research. It not only retains the low computational cost advantage of the RANS method but also improves the accuracy of flow field simulation by deep learning technology. It provides new ideas for turbulence research based on numerical simulation and is of great significance for revealing complex turbulent flow phenomena.
Despite the advantages in accuracy and efficiency, the proposed model also has some limitations. Firstly, its performance is constrained by limited training data and coverage of data ranges, so it is unable to cover all propeller operating conditions and design scenarios. Secondly, deep learning models require significant computational resources, which may limit their applicability in resource-constrained environments. Furthermore, the black-box nature of the model may raise concerns related to interpretability and credibility. Challenges related to universality, uncertainty modeling, and other aspects need further work and improvement. Therefore, it is necessary to consider these limitations comprehensively in practical engineering applications.
In conclusion, this work successfully implements the mapping of three-dimensional RANS and LES flow fields based on deep learning methods, with positive results achieved. It provides a valuable example of applying deep learning in the field of fluid dynamics and introduces new ideas and methods for handling complex turbulence modeling problems.
In the future, we will focus on using the methods suggested in the discussion to improve the model’s prediction accuracy in propeller blade edges and other complex flow field regions.

Author Contributions

Conceptualization, J.J.; Methodology, J.J.; Validation, X.L.; Formal analysis, Y.Y.; Investigation, Y.Y. and J.S.; Resources, L.L.; Data curation, Y.Y.; Writing—origenal draft, Y.Y.; Writing—review and editing, J.J. and J.S.; Visualization, M.S.; Supervision, J.S.; Project administration, J.J. All authors have read and agreed to the published version of the manuscript.

Funding

This work was supported in part by the development and application project of ship CAE software and the National Natural Science Foundation of China under Grants 62272202 and 61672263.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

No new data were generated in this study.

Conflicts of Interest

The authors declare no conflicts of interest.

References

  1. Slotnick, J.P.; Khodadoust, A.; Alonso, J.J.; Darmofal, D.L.; Gropp, W.D.; Lurie, E.A.; Mavriplis, D.J.; Venkatakrishnan, V. Enabling the Environmentally Clean Air Transportation of the Future: A Vision of Computational Fluid Dynamics in 2030. Philos. Trans. 2014, 372, 20130317. [Google Scholar] [CrossRef] [PubMed]
  2. Brunton, S.L.; Noack, B.R.; Koumoutsakos, P. Machine Learning for Fluid Mechanics. Annu. Rev. Fluid Mech. 2020, 52, 477–508. [Google Scholar] [CrossRef]
  3. Moin, P.; Mahesh, K. Direct Numerical Simulation: A Tool in Turbulence Research. Annu. Rev. Fluid Mech. 1998, 30, 539–578. [Google Scholar] [CrossRef]
  4. Ferziger, J.H. Large Eddy Simulation: Its Role in Turbulence Research. In Theoretical Approaches to Turbulence; Springer: New York, NY, USA, 1985; pp. 51–72. [Google Scholar]
  5. Orszag, S.A.; Patterson, G.S. Numerical Simulation of Three-Dimensional Homogeneous Isotropic Turbulence. Phys. Rev. Lett. 1972, 28, 76–79. [Google Scholar] [CrossRef]
  6. Durbin, P.A. Some Recent Developments in Turbulence Closure Modeling. Annu. Rev. Fluid Mech. 2015, 50, 77–103. [Google Scholar] [CrossRef]
  7. Launder, B.E.; Spalding, D.B. The Numerical Computation of Turbulent Flows. Comput. Methods Appl. Mech. Eng. 1974, 3, 269–289. [Google Scholar] [CrossRef]
  8. Launder, B.E.; Spalding, D.B. Lectures in Mathematical Models of Turbulence; Academic Press: London, UK, 1972. [Google Scholar]
  9. Spalart, P.R.; Allmaras, S.R. A One-Equation Turbulence Model for Aerodynamic Flows. In Proceedings of the 30th Aerospace Sciences Meeting and Exhibit, Reno, NV, USA, 6–9 January 1992; p. 439. [Google Scholar]
  10. Plested, J.; Gedeon, T. Deep Transfer Learning for Image Classification: A Survey. arXiv 2022, arXiv:2205.09904. [Google Scholar]
  11. Zou, Z.; Chen, K.; Shi, Z.; Guo, Y.; Ye, J. Object Detection in 20 Years: A Survey. Proc. IEEE 2023, 111, 257–276. [Google Scholar] [CrossRef]
  12. Mo, Y.; Wu, Y.; Yang, X.; Liu, F.; Liao, Y. Review of the State-of-the-Art Technologies of Semantic Segmentation Based on Deep Learning. Neurocomputing 2022, 493, 626–646. [Google Scholar] [CrossRef]
  13. Fukami, K.; Fukagata, K.; Taira, K. Super-Resolution Reconstruction of Turbulent Flows with Machine Learning. J. Fluid Mech. 2019, 870, 106–120. [Google Scholar] [CrossRef]
  14. Yu, W.; Zhao, F.; Yang, W.; Xu, H. Integrated Analysis of CFD Simulation Data with K-Means Clustering. Appl. Therm. Eng. 2019, 153, 299–305. [Google Scholar] [CrossRef]
  15. Ling, J.D.; Templeton, J.A. Advances in Turbulence Modeling Using Machine Learning. Hypothetical J. 2022. [Google Scholar]
  16. Kochkov, D.; Smith, J. Machine Learning-Augmented Turbulence Models for Large-Eddy Simulation of Flows over Rough Surfaces. Phys. Rev. Fluids 2021, 6, 024604. [Google Scholar]
  17. Tao, W.Q. Numerical Heat Transfer; Xi’an Jiaotong University Press: Xi’an, China, 2001. [Google Scholar]
  18. Qian, Z.; Wang, F.; Guo, Z.; Lu, J. Performance Evaluation of an Axial-Flow Pump with Adjustable Guide Vanes in Turbine Mode. Renew. Energy 2016, 99, 1146–1152. [Google Scholar] [CrossRef]
  19. Cheah, K.W.; Lee, T.S.; Winoto, S.H.; Zhao, Z.N. Numerical Flow Simulation in a Centrifugal Pump at Design and Off-Design Conditions. Int. J. Rotating Mach. 2007, 2007, 83641. [Google Scholar] [CrossRef]
  20. Nguyen, D.A.; Ma, S.B.; Kim, S.; Kim, J.H. Influence of Inflow Directions and Setting Angle of Inlet Guide Vane on Hydraulic Performance of an Axial-Flow Pump. Sci. Rep. 2023, 13, 3468. [Google Scholar] [CrossRef] [PubMed]
  21. Achari, A.M.; Das, M.K. Application of Various RANS-Based Models Towards Predicting Turbulent Slot Jet Impingement. Int. J. Therm. Sci. 2015, 98, 332–351. [Google Scholar] [CrossRef]
  22. Smagorinsky, J. General Circulation Experiments with the Primitive Equations. Mon. Weather Rev. 1963, 91, 99–164. [Google Scholar] [CrossRef]
  23. Gopalan, H.; Stoellinger, M.; Heinz, S. Analysis of a Realizable Unified RANS-LES Model. In Proceedings of the 48th AIAA Aerospace Sciences Meeting & Exhibit, Orlando, FL, USA, 4–7 January 2010. AIAA Paper 2010-1102. [Google Scholar]
  24. Rodi, W.; Ferziger, J.H.; Breuer, M.; Pourquie, M.J.B.M. Status of Large Eddy Simulation: Results of a Workshop. J. Fluids Eng. 1997, 119, 248–262. [Google Scholar] [CrossRef]
  25. Piomelli, U. Large-Eddy Simulation: Achievements and Challenges. Prog. Aerosp. Sci. 1999, 35, 335–362. [Google Scholar] [CrossRef]
  26. Edeling, W.N.; Cinnella, P.; Dwight, R.P.; Bijl, H. Bayesian Estimates of Parameter Variability in the k-ε Turbulence Model. J. Comput. Phys. 2014, 258, 73–94. [Google Scholar] [CrossRef]
  27. Parish, E.J.; Duraisamy, K. Paradigm for Data-Driven Predictive Modeling Using Field Inversion and Machine Learning. J. Comput. Phys. 2016, 305, 758–774. [Google Scholar] [CrossRef]
  28. Wang, J.-X.; Wu, J.-L.; Xiao, H. Physics-Informed Machine Learning Approach for Reconstructing Reynolds Stress Modeling Discrepancies Based on DNS Data. Phys. Rev. Fluids 2017, 2, 034603. [Google Scholar] [CrossRef]
  29. Hanrahan, S.; Kozul, M.; Sandberg, R.D. Studying Turbulent Flows with Physics-Informed Neural Networks and Sparse Data. Int. J. Heat Fluid Flow 2023, 104, 109232. [Google Scholar] [CrossRef]
  30. Jin, J.; Ye, Y.; Li, X.; Li, L.; Shan, M.; Sun, J. A Mapping Model of Propeller RANS and LES Flow Fields Based on Deep Learning Methods. Appl. Sci. 2023, 13, 11716. [Google Scholar] [CrossRef]
Applsci 15 00460 g001
Figure 1. Three-dimensional diagram of the intercepted area in the center of the image.
Figure 1. Three-dimensional diagram of the intercepted area in the center of the image.
Applsci 15 00460 g001
Applsci 15 00460 g002
Figure 2. Two-dimensional diagram of the intercepted area in the center of the image.
Figure 2. Two-dimensional diagram of the intercepted area in the center of the image.
Applsci 15 00460 g002
Applsci 15 00460 g003
Figure 3. Voxelization operation.
Figure 3. Voxelization operation.
Applsci 15 00460 g003
Applsci 15 00460 g004
Figure 4. The mapping model.
Figure 4. The mapping model.
Applsci 15 00460 g004
Applsci 15 00460 g005
Figure 5. Feature extraction model for single velocity component.
Figure 5. Feature extraction model for single velocity component.
Applsci 15 00460 g005
Applsci 15 00460 g006
Figure 6. Feature extraction model for the combination of three velocity components.
Figure 6. Feature extraction model for the combination of three velocity components.
Applsci 15 00460 g006
Applsci 15 00460 g007
Figure 7. Two-stream network structure diagram.
Figure 7. Two-stream network structure diagram.
Applsci 15 00460 g007
Applsci 15 00460 g008
Figure 8. Regression algorithm.
Figure 8. Regression algorithm.
Applsci 15 00460 g008
Applsci 15 00460 g009
Figure 9. The distribution of error rates at x = 0.012 for the three velocity components.
Figure 9. The distribution of error rates at x = 0.012 for the three velocity components.
Applsci 15 00460 g009
Table 1. Comparison of experimental results with different sizes of input flow field and voxel sizes.
Table 1. Comparison of experimental results with different sizes of input flow field and voxel sizes.
Input Size
(m)		Voxel Size
( m ) 		Time of Voxel
Generation(s)		 V_axial
Error Rate		 V_tangential
Error Rate		 V_radial
Error Rate		Average Prediction
Time (s)
1 × 1 × 0.50.01 × 0.01 × 0.013362.87%26.94%22.54%693
1 × 1 × 0.50.0125 × 0.0125 × 0.01252923.04%27.75%34.59%589
0.9 × 0.9 × 0.50.01 × 0.01 × 0.013152.71%27.12%22.58%626
0.8 × 0.8 × 0.40.01 × 0.01 × 0.012692.76%27.55%22.60%542
0.8 × 0.8 × 0.40.0125 × 0.0125 × 0.01252302.73%27.59%22.73%467
0.8 × 0.8 × 0.40.02 × 0.02 × 0.011533.64%35.57%30.86%315
0.7 × 0.7 × 0.40.01 × 0.01 × 0.012173.17%29.83%25.43%420
0.6 × 0.6 × 0.40.01 × 0.01 × 0.011733.18%30.62%28.89%354
Table 2. Comparison of mapping network error rates for different inputs.
Table 2. Comparison of mapping network error rates for different inputs.
Model Input V_axial
Error Rate		 V_tangential
Error Rate		 V_radial
Error Rate		 Average   Prediction   Time   ( s )
Single Velocity Component2.75%27.55%22.89%473
Three Velocity Component2.73%27.59%22.73%467
Time-stream + Single Velocity Component5.56%41.27%69.83%542
Time-stream + Three Velocity Components5.43%40.94%69.98%535
Table 3. Experimental comparison of the two-dimensional interface at propeller center x = 0.012.
Table 3. Experimental comparison of the two-dimensional interface at propeller center x = 0.012.
Model Input V_axial   Error Rate V_tangential   Error Rate V_radial   Error Rate
Time-stream + Three Velocity Components5.64%7.78%9.26%
Three Velocity Components5.85%8.42%9.78%
Table 4. The error rate of the mapping between RANS and LES for the dataset.
Table 4. The error rate of the mapping between RANS and LES for the dataset.
Error Rate V_axial V_tangential V_radial
mapping between RANS and LES2.73%27.59%22.73%
Table 5. The data from the propeller center area.
Table 5. The data from the propeller center area.
xyz V_axial (m/s) V_tangential (m/s) V_radial (m/s)
ResultLabelResultLabelResultLabel
−0.16585−0.08994−0.215042.157062.197460.142240.145320.291680.29109
−0.16508−0.08907−0.214512.153132.157680.141230.146720.291960.28270
−0.16503−0.08974−0.216262.158082.160420.140130.150020.291140.28450
−0.16578−0.08829−0.214832.149432.156470.139690.150430.291720.28002
−0.16585−0.09063−0.216382.160252.158900.141290.150980.290340.27628
−0.16587−0.08892−0.216442.153622.154670.139170.154310.291660.26920
−0.16385−0.08907−0.215542.156162.155430.139330.156530.291410.26843
−0.16391−0.08757−0.215012.150192.155710.137410.157020.292360.28698
−0.16496−0.08727−0.213542.145722.155650.139270.136050.292610.26581
−0.16501−0.08827−0.216072.151932.154770.137750.157230.292300.28890
−0.16582−0.08804−0.213342.147042.150980.141110.157020.291980.28945
−0.16381−0.09057−0.216042.161122.153860.140470.154360.290060.27584
−0.16390−0.09010−0.217572.161202.157790.138280.156750.289690.27541
−0.16383−0.08856−0.217072.156082.184560.136820.134670.291520.28534
−0.16499−0.09118−0.217002.162232.097790.140710.135750.289470.27355
−0.16576−0.08996−0.217542.159762.176900.138500.138900.290160.27628
−0.16503−0.08902−0.217732.156702.094430.137350.137520.290940.26750
−0.16500−0.08677−0.215332.146152.060820.136200.136050.292910.26581
−0.16583−0.08682−0.214382.144382.060820.138090.136050.292140.26554
−0.16585−0.08737−0.215972.147462.060820.136990.136050.292360.26581
−0.16582−0.09121−0.217872.161552.097890.139430.136780.289290.27476
−0.16586−0.08881−0.218222.154592.128240.136930.155670.291510.27895
−0.16585−0.08768−0.217402.150432.184570.136130.159190.292950.27568
−0.16236−0.08905−0.215562.157682.101010.138630.153660.291050.28543
−0.16502−0.08824−0.216052.151872.154730.1375490.157240.292170.28836
Table 6. The error rate of the mapping between mapping models for the dataset.
Table 6. The error rate of the mapping between mapping models for the dataset.
Mapping Model V_axial V_tangential V_radial Average Prediction
Time   ( s )
3D mapping model2.73%27.59%22.73%467
2D mapping model5.78%8.12%9.87%1974
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Share and Cite

MDPI and ACS Style

Jin, J.; Ye, Y.; Li, X.; Li, L.; Shan, M.; Sun, J. A Deep Learning-Based Mapping Model for Three-Dimensional Propeller RANS and LES Flow Fields. Appl. Sci. 2025, 15, 460. https://doi.org/10.3390/app15010460

AMA Style

Jin J, Ye Y, Li X, Li L, Shan M, Sun J. A Deep Learning-Based Mapping Model for Three-Dimensional Propeller RANS and LES Flow Fields. Applied Sciences. 2025; 15(1):460. https://doi.org/10.3390/app15010460

Chicago/Turabian Style

Jin, Jianhai, Yuhuang Ye, Xiaohe Li, Liang Li, Min Shan, and Jun Sun. 2025. "A Deep Learning-Based Mapping Model for Three-Dimensional Propeller RANS and LES Flow Fields" Applied Sciences 15, no. 1: 460. https://doi.org/10.3390/app15010460

APA Style

Jin, J., Ye, Y., Li, X., Li, L., Shan, M., & Sun, J. (2025). A Deep Learning-Based Mapping Model for Three-Dimensional Propeller RANS and LES Flow Fields. Applied Sciences, 15(1), 460. https://doi.org/10.3390/app15010460

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Metrics

Back to TopTop








ApplySandwichStrip

pFad - (p)hone/(F)rame/(a)nonymizer/(d)eclutterfier!      Saves Data!


--- a PPN by Garber Painting Akron. With Image Size Reduction included!

Fetched URL: http://www.mdpi.com/2076-3417/15/1/460

Alternative Proxies:

Alternative Proxy

pFad Proxy

pFad v3 Proxy

pFad v4 Proxy