A machine learning method for transition prediction in hypersonic flows over a cone with angles of attack Open Access

Laminar-turbulent transition in hypersonic boundary layers (HBLs) has significant effects on heat transfer and skin friction; thus, an accurate prediction of the transition location (TL) is vital for the design of a hypersonic vehicle.¹ Under realistic flight conditions, boundary layers are usually three-dimensional (3D), and the transition process is accordingly much more complicated than in extensively studied 2D boundary layers. A typical model to study 3D transition is an inclined cone due to its non-uniform distribution of the TL against its circumferential angle θ. TLs are very sensitive to the angle of attack (AOA), which promotes TLs on its lee side (causing a “tongue” shaped hotspot, see Fig. 1) and delays TLs on its windward side. Within its transition process, cross-flow effects couple with the evolution of fundamental hypersonic Mack modes^2,3 so that simulations and theoretical analysis are intolerably time-consuming.^4–6 Therefore, a data-driven prediction method is suitable for this complex problem.⁷ 

In the past few years, many machine learning methods were used to deal with fluid mechanics problems,⁸ such as turbulent models,^9–16 flow field prediction,^17–19 and aerodynamic shape optimization of airfoils.^20,21 However, there is not much literature on the application of machine learning methods in hypersonic transition. A neural network (NN) is one of the machine learning methods that can get the prediction by the input database. It is shown to be a fundamental nonlinear function approximator that can approximate any function with a sufficiently large and deep structure.²² Based on the advantages of the NN in dealing with nonlinear problems, it is used in this work to predict the TL by building models from the experimental database from the Peking University Mach 6.5 quiet wind tunnel by some algorithms, which have the ability to judge and predict. Based on the measured TL data, artificial NNs are trained to construct the mapping function between the TL and θ. With a well-trained NN, TLs can be predicted at both interpolated and extrapolated AOAs and unit Reynolds numbers Re₀.

Experiments were performed in the ϕ 300 mm Mach 6.5 quiet wind tunnel at Peking University with the suction valve open.^23,24 The wind tunnel is equipped with four replaceable germanium glass windows for thermal imaging measurements. Infrared thermography was carried out to measure the temperature increase on the surface of the model using an infrared camera (FLIR 620sc) from a side germanium glass window. During the typical test time of 30 s, the stagnation pressure remains nearly constant, with a variation of less than 3%. To avoid liquefying the air, the flow is preheated to a nominal stagnation temperature of 430 K. A test model of a cone with 5° half-angle along the axis was used to conduct the transition experiments. The model is 450 mm in length with a base diameter of 39.370 mm. The cone was constructed from phenolic plastic, which has high emissivity and low thermal conductivity, and is therefore suitable for infrared thermography. The tangent-slope-intercept method was used to determine the onset position of transition according to the temperature increase. Figures 1(a) and 1(b) show the contour of the surface temperature increase when the unit Reynolds number is 7.5 × 10⁶ L/m and AOA is 2°. Figure 1(c) shows the temperature increase along ξ at η = 22.5 mm. ξ denotes the axial location from the apex of the model, and η denotes the longitudinal location from the bottom of the model. A TL is defined as the point where a line drawn tangent to the slope of the heat transfer distribution curve (which is a fit to the entire surface dataset) through the transition region intercepts the nominal, laminar distribution level.

Figure 1(b) shows self-prediction of TLs at unit Reynolds number Re₀ = 7.5 × 10⁶ L/m and AOA = 2°. The solid line denotes the measured results, which are set as the training data of the NN. As shown, NNs can predict TLs against θ well, which are very close to the raw measured data.

In addition, the interpolated predictions of TLs agree well with the experimental data. Figure 4(a) shows interpolated prediction of TLs at an interpolated AOA = 5°, with training data from 2°, 4°, and 6°. Figure 4(b) shows interpolated prediction of TLs at an interpolated Re₀ = 6.9 × 10⁶ L/m, with training data from 1.1 × 10⁷, 9.7 × 10⁶, 8.9 × 10⁶, 6.4 × 10⁶, and 5.8 × 10⁶ L/m. As shown, the results predicted by the NN are very close to the measured data in the hypersonic wind tunnel.

Extrapolation predictions are further conducted. Figure 5 shows predictions of TLs at extrapolated angles of attack. The predicted conditions in Fig. 5(a) are AOA = 5° with training data from AOA = 2° and 4°, and those in Fig. 5(b) are AOA = 6° with training data from AOA = 4° and 5°. As shown, the NN can predict TLs that are close to the measured data for lower extrapolated AOA (5°) well but show a larger error from the measured data for a higher extrapolated AOA (6°) although the trend of the TL’s distribution is correct. The measured TLs are significantly earlier than that predicted for θ ∈ [140°, 160°] and its counterpart. The possible reason is that an increasing AOA might promote cross-flow instability or even separation.

Comparatively, extrapolation predictions of higher Re₀ are relatively more reliable (Fig. 6). The predicted conditions in Figs. 6(a) and 6(b) are Re₀ = 1.1 × 10⁷ L/m and 9.7 × 10⁶, respectively, with the same training data from Re₀ = 9.7 × 10⁶, 8.9 × 10⁶, 6.9 × 10⁶, 6.4 × 10⁶, and 5.8 × 10⁶ L/m. Both cases have little error with the measured data.

In general, based on the measured data of transition locations (TLs) over a straight cone in the hypersonic wind tunnel, this paper constructed a mapping function by the NN between TLs and θ. By comparing the measured data, the accuracy and generalization to different flow cases are validated. Furthermore, the interpolated and extrapolated prediction cases for both the angle of attack (AOA) and unit Reynolds number Re₀ are performed. It is shown that the present NN predicts for both interpolation of both the AOA and Re₀ and extrapolation of only Re₀ well whereas errors increase for the extrapolation of a higher AOA. Therefore, more AOA samples are needed to construct the NN.

This work was supported by the National Natural Science Foundation of China (Grant Nos. 91752202 and 12072002).

The authors have no conflicts to disclose.

The data that support the findings of this study are available from the corresponding author upon reasonable request.