Deep-learning prediction and uncertainty quantification for scramjet intake flowfields

Abstract

Scramjet is a promising propulsion technology that provides efficient and flexible access-to-space and high-speed point-to-point transportation. Since the design process of scramjet (supersonic combustion ramjet) engines requires numerous flowfield evaluations, fast and accurate flow predictions play a key role in promoting the development of knowledge and technologies. Deep learning techniques are increasingly used for flow prediction, and the present study applies them to viscous supersonic flowfields inside scramjet intakes. The capability and limitations of deep learning prediction for such flowfields have been investigated from the viewpoints of both physics and machine learning by means of uncertainty quantification and principal component analysis. The results indicate that the flowfields with complex aerodynamic phenomena such as boundary layer separation and Mach disks are difficult to predict. It has been attributed to lack of similar flowfields as well as the sensitivity of the fluid phenomena to the geometries. Uncertainty quantification effectively allows for the detection of such difficult cases without model verification prior to utilization. It has been further employed to address the issues in the prediction of flowfields with boundary-layer separations by increasing the number of training data effectively.

Introduction

Scramjet engines, a class of hypersonic airbreathing propulsion systems, are expected to enable more efficient, safe, and flexible access-to-space in lieu of conventional rocket systems, removing the need for oxidizers and moving parts. High-speed point-to-point transportation is also a suitable application of scramjet engines. The research and development of scramjets started in the 1940s [1] and have advanced to a level that enabled successful flight tests [2], [3]. A typical scramjet engine is constituted by an air intake, fuel injectors, a combustor, and a nozzle, as shown schematically in Fig. 1. Among scramjet components, air intakes, which are located the most upstream in the scramjet flow path, play an important role to determine the overall performance, being responsible for the entire flowfield. Since scramjets operate in severe hypersonic flow conditions, the internal flowfields of intakes are characterized by various complex aerothermal phenomena such as shock/shock interactions, shock/boundary-layer interactions, and high-temperature effects. To provide suitable flowfields for downstream components (i.e., fuel injectors and combustors) in consideration of such flow phenomena, the performance is commonly assessed by using various interrelated parameters [4], [5]. Therefore, the design of scramjet intakes requires sophisticated approaches to assure desirable efficiency and robustness as well as to elucidate underlying physics that determines the performance and flowfields.

Data-driven knowledge discovery is receiving increasing recognition as an effective and promising approach for various purposes including scientific research and design exploration. In the field of aerospace fluid engineering, which requires sophisticated design and technologies, data-driven design exploration is drawing attention due to its capability of discovering innovative design. The applications have a wide variety, including airfoils [6], fly-back boosters [7], [8], and scramjet engines [9], [10]. Despite demonstrated utility and effectiveness of data-driven knowledge discovery and design exploration in fluid engineering, data-driven approaches tend to be subject to huge computational costs that are incurred for a large number of numerical (computational fluid dynamics, CFD) simulations. While efforts have been dedicated to mitigating the computational costs by employing surrogate models that predict target scaler variables (i.e., performance parameters) from features (i.e., design variables) [11], [12], such models do not offer detailed insights that might be useful for design as well as thorough comprehension of underlying physical grounds. More informative surrogate models are thus required for effective data-driven knowledge discovery in fluid engineering.

To enable data-driven knowledge discovery and design exploration in fluid engineering, fast flow prediction is becoming a key research topic lately [13]. Owing to the developments of computer technologies and machine learning techniques, the research has been accelerated, employing various machine learning techniques including proper orthogonal decomposition (POD) [14], dynamic mode decomposition (DMD) [15], and deep learning [16]. POD and DMD are techniques for reduced-order modeling (ROM), and enable flow prediction by coupling regression techniques. While such prediction models feature high computational efficiency owing to the reduced-order nature, the information in lower modes is inherently discarded. On the other hand, deep learning-based approaches perform both feature extraction and predictions by themselves, and the computational cost for the modeling tends to be relatively expensive, but the structure of the system is simpler. Moreover, the loss of information is smaller, as compared to ROM-based approaches, since the approach does not involve order reduction of dataset [17]. Further, deep learning is capable of dealing with nonlinear problems and discontinuity, making itself particularly suitable for fluid prediction [18].

Deep learning-based flow prediction has often been applied to flowfields around cylinders and airfoils in subsonic and transonic regimes. Kashefi et al. proposed a point-cloud deep learning framework for flow prediction on irregular geometries, and demonstrated its applicability to a range of prediction problems of incompressible viscous flow [19]. Sekar et al. employed a convolutional neural network (CNN) for the parameterization of airfoils and multilayer perceptron (MLP) for flow prediction for various geometries, Reynolds numbers, and angles of attack [16]. The model could accurately predict the flowfields even on the airfoil surface, while increases in prediction errors were observed in the separation region behind airfoils. Thuerey et al. used U-net, which is a kind of CNN, for the prediction of the Reynolds-averaged Navier-Stokes solution of airfoil flow as well [20]. The study discussed the influence of the number of training data and the model size. While errors were typically observed in the wakes behind walls, the results indicated the need to elucidate the underlying grounds of prediction errors to further improve prediction accuracy. Turbulence research and modeling are also the fields where deep learning is actively applied. Deep learning was first applied to turbulence modeling by Ling et al. to model the Reynolds stress anisotropy tensor from high-fidelity simulation data [21]. Fukami et al. conducted a super-resolution analysis using CNN and a hybrid downsampled skip-connection/multi-scale (DSC/MS) model to reveal the subgrid-scale physics of turbulent flows [22]. Other applications of deep learning techniques for fluid dynamics include studies that concern nonlinear mode decomposition [23] and physics-informed modeling [24], [25].

As discussed in the preceding studies [20], comprehension of prediction errors is becoming increasingly important for flow prediction via deep learning not only to improve the prediction accuracy but also to avoid unrealistic prediction. Uncertainty quantification (UQ) is a useful technique to estimate and manage the risk and safety margin of model prediction [26]. In the field of deep learning, various UQ methods have been proposed and used for various applications such as image analysis, natural language processing, and bioinformatics [27]. The UQ methods for deep learning prediction can be classified into 3 groups, namely ensemble-based methods, distance-based methods, and mean-variance estimation [28]. Ensemble-based methods employ multiple prediction models rather than a single model and the variance of target properties predicted by multiple models is considered to estimate the prediction uncertainty. Distance-based methods are one of the most intuitive strategies and assess the dissimilarity between the inputs for prediction and those of the training dataset. Mean-variance estimation replaces the model output layer to predict the mean and variance of the target properties and employs negative log likelihood loss so as to behave as a Gaussian likelihood model. In the field of fluid dynamics, the quantification of uncertainties in deep-learning predictions has been employed for data-driven turbulence modeling in several preceding studies [29], [30], [31]. Despite the importance of risk management in deep-learning flow prediction for practical use, few studies have discussed the results of uncertainty quantification for prediction of flowfields.

While deep learning techniques are becoming common in the field of fluid science and engineering, they have scarcely been applied to supersonic and hypersonic flowfields despite their significance for aerospace engineering. Supersonic/hypersonic flows are observed in various situations such as exhaust jets, high-speed transportations, and scramjet engines, characterized by complex flow phenomena such as shock waves, boundary layers, and their interactions. A few studies have been reported on flow reconstruction in scramjet isolaters [32], [33] and pressure field prediction in an inviscid regime [34]. Kong et al. studied the prediction of future distributions of static pressure in a scramjet isolator [35] and flame in a scramjet combustor [36]. However, the prediction of entire flowfields by employing geometries as the inputs in supersonic/hypersonic viscous regimes is yet to be reported, despite its crucial importance from the viewpoint of design exploration and knowledge discovery for high-speed aerospace applications. The capabilities and limitations of deep learning prediction of supersonic/hypersonic flow are yet to be understood, and there have been no studies on uncertainty quantification for such flow prediction problems to date.

The present study is thus undertaken to investigate the applicability of deep learning-based flow prediction for viscous supersonic internal flowfields, aiming at application to scramjet intake design. The flow prediction model is constructed by employing multilayer perceptron neural networks. Uncertainty quantification is conducted to identify the flowfields and flow phenomena that are difficult to predict accurately, and principal component analysis is then performed to scrutinize the sources of prediction errors from the viewpoints of both fluid dynamics and machine learning. Finally, an approach to address the difficulties has been proposed and its effectiveness has been demonstrated to allow for reliable use of deep learning flow prediction for scramjet intake design as well as to verify the causes of the difficulties.