Turbulence closure for high Reynolds number airfoil flows by deep neural networks

https://doi.org/10.1016/j.ast.2020.106452Get rights and content

Abstract

The combination of turbulence big data with artificial intelligence is an active research topic for turbulence study. This work constructs black-box algebraic models to substitute the traditional turbulence model by the artificial neural networks (ANN), rather than correcting the existing turbulence models in most of current studies. We mainly focused on flows past airfoils at high Reynolds (Re) numbers. Our previous work has developed a turbulence model for flows at different Mach (Ma) number and angles of attack (AOA) with fixed Re number and achieved satisfying results. Nevertheless, for turbulence with variable Re numbers, the generalization ability of the model can not be enhanced effectively by simply increasing the train data. To model the nonlinearity of various turbulent effects at high Re number, prior knowledge about scaling analysis is integrated into the model design and deep neural networks (DNN) is adopted as the framework. Considering the different scaling characteristics, the flow field is divided into different regions and two individual ANN models are built separately. Besides, the combination of regularization, limiters, and stability training is adopted to enhance the robustness of the proposed model. The results of Spallart-Allmaras (SA) model are used as the datasets and reference to the modeling evaluation. The proposed model is trained by six flows around NACA0012 airfoil and applicative to different free stream conditions and airfoils. It is found that the results calculated by the proposed model, such as eddy viscosity, velocity profile, drag coefficient and so on, agree well with reference data, which validate the generalization ability of the model. This work shows the prospect of turbulence modeling by machine learning methods.

Introduction

Currently, understanding the essence of turbulence is still an open question [1]. In aeronautics and aerodynamics, the expensive cost of flow simulation at high Reynolds number (Re) limits the use of high-fidelity methods such as direct numerical simulation (DNS) and large eddy simulation (LES) to a large extend. Thus, the traditional Reynolds averaged Navier–Stokes simulation (RANS) methods will remain the mainstream of turbulence computation in the near future. However, RANS models have their own shortcomings. The uncertainty of the model structure and parameters affects the model applicability to some extent. For some complex flows with strong nonlinearity such as the secondary flows and massively separated flows, it is difficult to describe the anisotropy by RANS models. In summary, the development of RANS models is much slower than before, although there are some new innovative achievements [2].

With the dramatic development of machine learning (ML), many investigators have also begun to apply the artificial intelligence (AI) to fluid mechanics, such as flow field reconstruction [3] and reduced-order models [4], [5]. For the aeronautical engineering, machine learning has been used for shape [6] and robust [7] optimization and predicting the pressure distribution over airfoils [8]. Besides, Habibnia and Pascoa [9] interpreted the operating conditions in ground effect by artificial neural network. Pesce et al. [10] combined neural network and Kalman filter for state estimation. Some practical tasks can also be improved or solved innovatively by artificial intelligence, like air traffic flow prediction [11], prognostic and health management of aircraft [12], etc. In recent years, data-driven methods also play more and more important role in turbulence computation. The research tendency has evolved from calculating local flow field [13], [14] or some terms of the equation [15] to improving the Reynolds stress directly and even substituting the traditional turbulence models. In general, research on machine learning in turbulence computation can be divided into two categories. The first is to detect and quantify the uncertainty of the RANS models. The uncertainty of the RANS models mainly comes from the ensemble-average operation, the functional form, the representation of the Reynolds stress and the empirical parameters in the model [16]. The uncertain region of the RANS models is affected by the model applicability of the specific flow state. Based on high-fidelity data, a classifier is constructed to distinguish the uncertain region in the flow field, and then more accurate methods in the corresponding region are adopted. Ling et al. [17] constructed three classifiers to mark the uncertain region in the flow field. Singh et al. [18] quantified the uncertainty of the RANS model by the field inversion and Bayesian method. More discussions of uncertainty analysis of RANS model or sub-grid model in LES are detailed in review [19]. The second is to construct a regression network to improve or substitute the traditional RANS models. For some separated or secondary flows which are beyond the capabilities of the RANS model, more accurate results are expected by modifying the equation of baseline RANS models [20], [21], [22] or decreasing the discrepancy between RANS models and high-fidelity simulation methods [23], [24], [25]. Generally, the improvement achieved by the proposed approach is based on the specific RANS model. The feasibility needs to be demonstrated whether the proposed approach is applicable to other RANS models. In addition, some work attempts to substitute the traditional RANS models [26] or sub-grid models in LES [27], [28], [29]. Some encouraging results have been achieved in both kinds of research and the feasibility is well demonstrated.

The investigation combining machine learning with turbulence modeling is an emerging domain in fluid mechanics. The existing research results indicate the positive prospect of machine learning in future turbulence model applications [30]. Meanwhile, there are also many problems and challenges. On the one hand, most of the present investigations are driven by high-fidelity data calculated by DNS or LES. Therefore, the related cases are mainly about flow with low/medium Re number and simple geometry, such as the flat plate and channel flow. However, the high-fidelity turbulence simulation with high Re number in engineering problems, such as flow around wings and airfoils, is costly and time-consuming. How to apply the machine learning to turbulence computation in this field requires further exploration. On the other hand, extra consideration should be given to the model robustness and stability if the proposed model couples the CFD solver. However, it is difficult to achieve high accuracy and robustness at the same time. Higher accuracy is often at the cost of less robustness [31]. In most cases, it is a tradeoff. Static training and testing completely neglect the interaction between model deviation and the mean flow field during the iteration process. In fact, for models that explicitly affects the Reynolds stress or sub-grid stress, the slight perturbations can be cumulatively amplified into significant errors in the mean flow field. The instability has been mentioned in many research works [32], [33]. Wu et al. [33] proposed a method to measure the model conditions and recommended the implicit expression of Reynolds stress to overcome the ill-conditioning of RANS equations. Recently, Cruz et al. [34] proposed to take the Reynolds force vector as the model target. In addition, more available public turbulent flow databases are also needed [35].

The target of our researches has always been constructing a data-driven model to substitute the traditional RANS models rather than modify the base turbulence models for high Re number turbulent flows. Our previous work has built such a model for the attached flows around different airfoils at different angles of attack and Ma numbers, and achieved satisfying results [26]. Nonetheless, the generalization to airfoil flows with different Re numbers is not effective. To embody the nonlinear characteristics of the high Re number turbulent flows, we improved the model from three aspects: modifying the input features; designing the modeling strategy from the perspective of scaling analysis;using DNN framework, which is good at complex mappings. The proposed model is expected to generalize to different airfoil flows at various angles of attack, Re number and Ma number.

Section snippets

Modeling strategy

The scaling analysis of wall-bounded turbulent flows is of great interest over the past decades for both theoretical and practical applications. Traditionally, the wall-bounded turbulent flows are assumed to have a four-layer structure: viscous sublayer (y+<5), buffer layer (5<y+<30), logarithmic layer (30<y+<0.15δ+) and wake layer (y+>0.15δ+), see Fig. 1. In this paper, the “+” indicates normalization by friction velocity uτ and wall friction length unit δυ. Generally, the mixing length of the

Dataset

The CFD solver and Spallart-Allamaras (SA) model [54] adopted in our work are developed independently. The benchmarks of subsonic flow around NACA0012 airfoil and transonic flow around RAE2822 airfoil are calculated to validate the CFD code with the cell-centered finite volume method. Spatial discretization uses an AUSM+Up scheme with second-order accuracy. The governing equations are nondimensionalized by the mean aerodynamic chord, speed of sound and freestream density. More information about

Training process

The robustness of neural network is simply defined as the ability to resist adversarial examples. More detailed definition can be found in [57]. If the robustness of the model is low, the slight perturbation of the input can cause the significant change of the output. The research about the adversarial examples and the robustness of the neural network can be referred to [57], [58], [59], [60]. The proposed model in this paper is embedded in a CFD solver and couples with the solver during the

Accuracy

Fig. 10 shows the comparison of the skin friction coefficient Cf. It can be found that the Cf distribution varies a lot with the airfoils and freestream conditions. The result of ML model can reflect these differences qualitatively and have a good agreement with that of SA model. The comparison of correlation coefficient of the skin friction coefficient C.C.(Cf) and drag coefficient Cd is shown in Fig. 11 and Fig. 12, respectively. Specifically, the correlation coefficient is defined as followsC

Conclusion

In this paper, based on the data from high Re number turbulent flows around NACA0012 airfoil, the fully-connected DNN is adopted to construct a surrogate turbulence model for subsonic flows. By comparing the results from the proposed model with those from the SA model, the accuracy and generalization to different freestream conditions and airfoil geometries are validated. It is found that the model performance of the wall-bounded inner region is better than that of the more chaotic wake region.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This paper was supported by the National Natural Science Foundation of China (no. 91852115), the National Numerical Wind tunnel Project (no. NNW2018-ZT1B01).

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