Generalizable physics-constrained modeling using learning and inference assisted by feature-space engineering

This work presents a formalism to improve the predictive accuracy of physical models by learning generalizable model augmentations from sparse data. Building on recent advances in data-driven turbulence modeling, the present approach, referred to as “learning and inference assisted by feature-space engineering”, is based on the hypothesis that robustness and generalizability demand a meticulously designed feature space that is informed by the underlying physics and a carefully constructed features-to-augmentation map. The critical components of this approach are: (1) maintaining consistency across the learning and prediction environments to make the augmentation case-agnostic; (2) tightly coupled inference and learning by constraining the augmentation to be learnable throughout the inference process to avoid loss of inferred information (and hence accuracy); (3) identification of relevant physics-informed features in appropriate functional forms to enable significant overlap in feature space for a wide variety of cases to promote generalizability; (4) localized learning, i.e., maintaining explicit control over feature space to change the augmentation function behavior only in the vicinity of available data points. To demonstrate the viability of this approach, it is used in the modeling of bypass transition. The augmentation—in the form of model coefficient functions—is developed using skin friction data from two flat plate cases from the ERCOFTAC dataset. Piecewise linear interpolation on a structured grid in feature space is used as a sample functional form for the augmentation to demonstrate the capability of localized learning. The impact of using a different function class (neural network) is also assessed. The augmented model is then applied to a variety of flat plate cases which are characterized by different freestream turbulence intensities, pressure gradients, and Reynolds numbers. The predictive capability of the augmented model is also tested on single-stage high-pressure-turbine cascade cases, and the model performance is analyzed from the perspective of information contained in the feature space. The results show consistent improvements across these cases, as long as the physical phenomena in question are well-represented in the training.

Figure 1

Schematic of feature-space discretization.

Figure 2

Mesh used for RANS simulation of T3A and T3B cases.

Figure 3

Mesh used for RANS simulation of T3C cases.

Figure 4

$U_{\infty }/U_{\text{in}}$ vs $x$ for T3C1.

Figure 5

Decay of freestream turbulence intensity. (a) T3A and (b) T3C1.

Figure 6

Field Inversion: Optimization plots. (a) T3A and (b) T3C1.

Figure 7

$\rho E$ Residual histories for all optimization iterates. (a) Residual convergence for energy and (b) Residual convergence for energy adjoints.

Figure 8

Skin friction coefficients along the length of the flat plate. (a) T3A and (b) T3C1.

Figure 9

Intermittency contours (white line marks $U=0.95U_{\infty }$), $40\times$ scaling in the $y$ direction. (a) T3A (Flat plate length: 1.5 m) and (b) T3C1 (Flat plate length: 1.7 m).

Figure 10

Augmentation contours on feature-space slices after training on T3A and T3C1 datasets. (a) ${\eta }_1=0.05$, (b) ${\eta }_1=0.15$, (c) ${\eta }_1=0.25$, (d) ${\eta }_1=0.35$, (e) ${\eta }_1=0.45$, (f) ${\eta }_1=0.55$, (g) ${\eta }_1=0.65$, (h) ${\eta }_1=0.75$, and (i) ${\eta }_1=0.85$.

Figure 11

Mesh used for RANS simulation of VKI turbine cascade.

Figure 12

Decay of freestream turbulence intensity. (a) T3B, (b) T3C2, (c) T3C3, and (d) T3C5.

Figure 13

Verification of velocity profiles for MUR224 using LES data [53] ($C_{\text{ax}}$ refers to chord length in the $x$ direction). (a) Normalized coordinates and (b) Wall coordinates.

Figure 14

Predicted skin friction coefficients for flat plate cases using the model trained on T3A and T3C1. (a) T3B, (b) T3C2, (c) T3C3, and (d) T3C5.

Figure 15

Predicted heat transfer coefficients for VKI turbine cases using model trained on T3A and T3C1. (a) MUR116, (b) MUR129, (c) MUR224, and (d) MUR241.

Figure 16

Contours near the transition location on suction surface for MUR116. (a) ${\eta }_1=min\left(\frac{R{e}_{\mathrm{\Omega }}}{\overline{R{e}_{\theta ,t}}},3\right)$, (b) ${\eta }_3=\frac{\nu }{\nu +{\nu }_t}$, and (c) Intermittency ($\gamma$).

Figure 17

Contours near the transition location on suction surface for MUR241. (a) ${\eta }_1=min\left(\frac{R{e}_{\mathrm{\Omega }}}{\overline{R{e}_{\theta ,t}}},3\right)$, (b) ${\eta }_3=\frac{\nu }{\nu +{\nu }_t}$, and (c) Intermittency ($\gamma$).

Figure 18

Zoomed out views to indicate predicted transition locations on upper surface. (a) MUR116 and (b) MUR241.

Figure 19

Augmentation training using data from T3A only. (a) Optimization convergence and (b) Skin friction prediction.

Figure 20

Feature maps ($x$ axis: ${\eta }_2$, $y$ axis: ${\eta }_3$, uniform color-bar range [0,1] across all plots). (a) ${\eta }_1=0.05$, (b) ${\eta }_1=0.15$, (c) ${\eta }_1=0.25$, (d) ${\eta }_1=0.35$, (e) ${\eta }_1=0.45$, (f) ${\eta }_1=0.55$, (g) ${\eta }_1=0.65$, (h) ${\eta }_1=0.75$, (i) ${\eta }_1=0.85$, and (j) ${\eta }_1=0.95$.

Figure 21

Skin friction coefficients for T3 cases. (a) T3B, (b) T3C1, (c) T3C2, (d) T3C3, and (e) T3C5.

Figure 22

Heat transfer coefficients for VKI cases. (a) MUR116, (b) MUR129, (c) MUR224, and (d) MUR241.

Figure 23

Comparison between different preset distance intervals. (a) MUR116, (b) MUR129, (c) MUR224, and (d) MUR241.

Figure 24

Training results with different neural network architectures. (a) Objective reduction (T3A), (b) Skin friction profile (T3A), (c) Objective reduction (T3C1), and (d) Skin friction profile (T3C1).

Figure 25

Training results on a finer feature-space grid. (a) Optimization history (T3A), (b) Optimization history (T3C1), (c) Skin friction profile (T3A), and (d) Skin friction profile (T3C1).

Figure 26

Augmentation contours on feature-space slices. (a) ${\eta }_1=0.05$, (b) ${\eta }_1=0.15$, (c) ${\eta }_1=0.25$, (d) ${\eta }_1=0.35$, (e) ${\eta }_1=0.45$, (f) ${\eta }_1=0.55$, (g) ${\eta }_1=0.65$, and (h) ${\eta }_1=0.75$.

Figure 27

Skin friction coefficient profiles for the T3 test cases. (a) T3B, (b) T3C2, (c) T3C3, and (d) T3C5.

Figure 28

Heat transfer coefficient profiles for the VKI test cases. (a) MUR116, (b) MUR129, (c) MUR224, and (d) MUR241.