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Le, T.; Kang, H.; Kim, H. Computational Fluid Dynamics Simulations. Encyclopedia. Available online: https://encyclopedia.pub/entry/31129 (accessed on 10 December 2023).

Le T, Kang H, Kim H. Computational Fluid Dynamics Simulations. Encyclopedia. Available at: https://encyclopedia.pub/entry/31129. Accessed December 10, 2023.

Le, Thi-Thu-Huong, Hyoeun Kang, Howon Kim. "Computational Fluid Dynamics Simulations" *Encyclopedia*, https://encyclopedia.pub/entry/31129 (accessed December 10, 2023).

Le, T., Kang, H., & Kim, H.(2022, October 25). Computational Fluid Dynamics Simulations. In *Encyclopedia*. https://encyclopedia.pub/entry/31129

Le, Thi-Thu-Huong, et al. "Computational Fluid Dynamics Simulations." *Encyclopedia*. Web. 25 October, 2022.

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Computational fluid dynamics (CFD) simulations are employed in several fields, such as mechanical engineering, medicine, and civil engineering. CFD solvers are numerical tools for simulating fluid flow characteristics to design, analyze, or optimize fluid flow behavior.

CFD (computational fluid dynamics)
laminar flow
Naiver–Stokes equation

Computational fluid dynamics (CFD) simulations are employed in several fields, such as mechanical engineering, medicine, and civil engineering. CFD solvers are numerical tools for simulating fluid flow characteristics to design, analyze, or optimize fluid flow behavior. However, high temporal and spatial resolution is required to achieve high accuracy in state-of-the-art CFD simulations. These solvers require expensive computational resources, especially with iterative problems. To this end, data-driven machine learning models not only estimate accurate approximation fluid flow fields, but also require fewer computational resources ^{[1]}. To reduce computational costs, a trained neural network (NN) might take the role of a portion of the numerical resolution process. As an illustration, several NN applications have been produced to solve and predict flow in terms of the large eddy simulation (LES) ^{[2]} and Reynolds-averaged Navier–Stokes (RANS) computations ^{[3]}^{[4]}.

CFD simulations’ complexity might be reduced by using reduced order model (ROM) techniques, such as simplified physics methods, reduced basis (RB), or proper orthogonal decomposition (POD). In particular, deep learning (DL)-enabled ROM can be used to set up a nonlinear relationship between various inputs and outputs of a target system. The DL-enabled ROM, along with the training model, can generate a low-dimensional subspace that records the average behavior of flows. Through this training process, complex features that cannot be expressed explicitly in a functional form can be represented ^{[5]}. In practice, the DL-enabled ROM accurately captures fluid flow’s temporal and spatial nonlinear features. For example, Wang et al. ^{[6]} presented a model recognition of reduced-order fluid dynamic systems by DL. The researchers proved that their framework could capture complex fluid dynamics features with less computational cost. Furthermore, Fukami et al. ^{[7]} performed a super-resolution analysis of evidently under-resolved turbulent flow data based on the DL model and then reconstructed the high-resolution flow field. This successful model built a nonlinear mapping between low and high resolutions of the turbulent flow fields.

The various proposed DL-ROM algorithms ^{[8]}^{[9]} were evaluated on both linear and nonlinear time-dependent parameters to demonstrate the flexibility of this methodology and its incredible computing savings. After executing a prior dimensional reduction by POD, Fresca et al. ^{[10]} suggested that DL-based ROMs rely on DNNs, significantly reducing their training times. The blood flow in a cerebral aneurysm, the fluid-structure interaction between an elastic beam attached to a fixed, rigid block, and the flow around a cylindrical benchmark are all accurately predicted by the resulting POD-DL-ROMs in almost real-time. A novel DL framework called DL-ROM was also developed by Pant et al. ^{[11]} to build a neural network that can make non-linear projections to lower-order states. They then employ a 3D autoencoder and 3D U-Net-based architectures to effectively forecast future time steps of the simulation using the learned reduced state. By traversing time in the learned reduced state, their model DL-ROM can efficiently anticipate future time steps by building highly accurate reconstructions from the learned ROM. Recently, Kang et al. ^{[12]} presented POD-ROM, which quickly and precisely describes the flow status of the fluid field in rod bundles.

DL-based CFD models have recently attracted the attention of fluid flow and thermal engineering research as a reduced-order modeling method. To learn the solutions of parametric partial differential equations (PDEs) over irregular domains, including Navier–Stokes and heat transfer equations, Gao et al. ^{[13]} presented a physics-constrained convolutional neural network (CNN) architecture. Their findings showed how well the DL technique predicted temperature and velocity fields. The DL-based CFD model was employed by San et al. ^{[14]} to precisely resolve the spatial–temporal nonlinear characteristic in a fluid dynamic system. In addition, a data-driven DL model was also used by Sekar et al. ^{[15]} to measure laminar flow on an airfoil dataset. The experimental findings demonstrated that the model accurately predicted laminar flow fields using the airfoil geometry, Reynolds number, and attack angles as learning parameters. Jin et al. ^{[16]} subsequently suggested a CFD approach based on DL that directly maps the relationship between the pressure and velocity distribution on the surface of a cylinder to determine the fluctuating velocity field around it. The aforementioned studies proved and validated the capacity of DL-based CFD to offer substitute numerical solutions to physical issues.

Guo et al. ^{[17]} and Ribeiro et al. ^{[18]} are representative pioneers of DL-based CFD model approaches. The researchers used CNN and U-Net models to evaluate the proposed methods to predict steady flow around obstacles and different loss types.

Recent advances in ML have impacted CFD research owing to its significant advantages. ML models estimate approximate thermal or fluid flow fields with low cost and accuracy compared with conventional CFD simulations. Sarghini et al. ^{[19]} developed ML model to estimate steady-state velocity flows. Next, Lee et al. ^{[20]} built an NN to predict unsteady flow around a cylinder. The researchers minimized the physical loss function comprising conservation laws and regression errors. Kashefi et al. ^{[21]} created an artificial neural network with modest geometry alterations to achieve various velocity and pressure fields. Furthermore, additional recent publications have demonstrated a variety of effective uses for CFD-based DL models, including physics-informed NN ^{[3]}, airfoil design optimization ^{[22]}, and acceleration of sparse linear system solutions ^{[23]}^{[24]}.

The modern DL technique has recently played a vital role in CFD simulations. Deep neural network models have been used as data-driven surrogate models that efficiently approximate the velocity and pressure fields. Regarding the DL-based CFD approach, the direct estimation of fluid flow fields comprises two representative models: CNN-based CFD and variant autoencoder (AE)-based CFD.

In terms of a CNN-based CFD prediction approach, Guo et al. ^{[17]} proposed a CNN model for predicting stationary flow fields around solid objects. Moreover, previous studies ^{[25]}^{[26]} have used CNN models to learn arbitrary geometry representations. Georgiou et al. ^{[27]} developed a CNN application for reconstructing fluid force and flow prediction. Jin et al. ^{[16]} proposed a fusion CNN model to predict velocity snapshots around a cylinder. Furthermore, Zhang et al. ^{[28]} predicted the lift and drag coefficients of 2D airfoils using a CNN model. In addition, the CNN model was applied to measure flow in arbitrary shapes by Viquerat et al. ^{[29]}.

In terms of AE and its variant U-Net model-based CFD prediction approach, AE models were used for supervised learning to predict various full-field flows in ^{[16]}^{[17]}^{[20]}. Especially among the various AE architectures, the U-Net model was recently applied successfully to estimate CFD flows. According to Ronneberger et al. ^{[30]}, U-Net models might achieve the best segmentation accuracy by fusing high-level latent-space representation with low-level characteristics. Thuerey et al. ^{[31]} applied the U-Net model to estimate turbulent flow around airfoils, including the velocity and pressure flows, which were computed using RANS. Fukami at el. ^{[7]} applied the U-Net model to reconstruct turbulence with remarkable accuracy from rough flow field images. A recurrent U-Net architecture was investigated and developed by Kamrava et al. ^{[32]} to predict stationary velocity and pressure fields in porous membranes. Wang at el. ^{[33]} proposed a gated U-Net-based pixel CNN++ architecture to simulate fluids in porous media. Ribeiro et al. ^{[18]} applied the high-performance accuracy of the U-Net model to steady-state laminar flow approximation around simple obstacles. Chen et al. ^{[34]} proposed a twin-decoder based on the U-Net model to reconstruct incompressible laminar flow on 2D obstacle data.

In addition, the flow field feature information is crucial for enhancing the performance accuracy of flow estimation models. For example, Ribeiro et al. ^{[18]} and Alvaro et al. ^{[32]} reported remarkable prediction results through deep CFD models to predict the velocities and pressure by generating features based on the signed distance function (SDF) and flow region channel. Peng et al. ^{[35]} generated network input learning using the SDF and temperature field from numerical simulation data as output learning to feed the CNN model. SDF and binary features were developed for CFD input learning of CNN and U-Net models in ^{[36]}. Li et al. ^{[37]} proposed a wall distance field and space coordinate field for the U-Net model’s input features.

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