Convolutional neural network for transition modeling based on linear stability theory

Transition prediction is an important aspect of aerodynamic design because of its impact on skin friction and potential coupling with flow separation characteristics. Traditionally, the modeling of transition has relied on correlation-based empirical formulas based on integral quantities such as the shape factor of the boundary layer. However, in many applications of computational fluid dynamics, the shape factor is not straightforwardly available or not well-defined. We propose using the complete velocity profile along with other quantities (e.g., frequency, Reynolds number) to predict the perturbation amplification factor. While this can be achieved with regression models based on a classical fully connected neural network, such a model can be computationally more demanding. We propose a convolutional neural network inspired by the underlying physics as described by the stability equations. Specifically, convolutional layers are first used to extract integral quantities from the velocity profiles, and then fully connected layers are used to map the extracted integral quantities, along with frequency and Reynolds number, to the output (amplification ratio). Numerical tests on classical boundary layers clearly demonstrate the merits of the proposed method. More importantly, we demonstrate that, for Tollmien-Schlichting instabilities in two-dimensional, low-speed boundary layers, the proposed network encodes information in the boundary-layer profiles into an integral quantity that is strongly correlated to a well-known, physically defined parameter—the shape factor.

Figure 1

Fully connected neural networks, (a) with input features including a scalar boundary-layer parameter, shape factor $H$, and (b) with boundary-layer profiles as direct input features. Other scalar parameters in both architectures represent disturbance characteristics of instability waves, i.e., frequency ($\omega$) and Reynolds number (${\text{Re}}_{\theta }$).

Figure 2

Proposed hybrid convolutional neural network architecture including (a) a regular CNN, which encodes the boundary-layer profiles to a set of latent physical parameters $\mathrm{\Psi }$, and (b) a fully connected neural network, which maps the CNN-extracted features $\mathrm{\Psi }$ along with other physical parameters (frequency of the instability wave $\omega$ and Reynolds number ${\text{Re}}_{\theta }$) to the output (instability amplification rate $\sigma$).

Figure 3

$N$-factor curves for non-self-similar boundary-layer profiles over the upper surface of a symmetric HSK airfoil section and an asymmetrical NLF-0416 airfoil at different angles of attack ($\alpha$). Transition location corresponds to the critical value of $N=9$ (marked by a dashed line). Corresponding transition onset locations are mentioned on the upper left corner and marked on the horizontal axis as predicted by the proposed neural network ${\text{C}}_1$ model (blue arrow) and computed by linear stability theory (LST) (red arrow). The neural network model was trained using the Falkner-Skan database (with self-similar boundary layers). Red and blue lines correspond to $N$-factor curves. (a) HSK airfoil, α = 0°, (b) NLF-0416, α = 0°, (c) NLF-0416, α = 2°, (d) NLF-0416, α = 5°.

Figure 4

Correlation between normalized CNN learned parameter ($\tilde{\mathrm{\Psi }}$) from non-self-similar boundary-layer profile $U$ and normalized shape factor ($\tilde{H}$), at 111 locations along the upper surface of an asymmetric NLF-0416 airfoil section at 0 degrees angle of attack. The learned parameter $\mathrm{\Psi }$ and the shape parameter $H$ are normalized to within the range $[0,1]$ to facilitate comparison. The Falkner-Skan database (with self-similar boundary layers) has been used for training.

Figure 5

Comparison of correlation between normalized CNN learned parameter ($\tilde{\mathrm{\Psi }}$) for varying sets of non-self-similar boundary layer profiles (corresponding to networks ${\text{C}}_1^{*}$, ${\text{C}}_2^{*}$, and ${\text{C}}_3^{*}$) and normalized shape factor ($\tilde{H}$), at 111 locations along the upper surface of an asymmetric NLF-0416 airfoil section at 0 degrees angle of attack. Falkner-Skan database (with self-similar boundary layers) has been used for training.

Figure 6

Comparison of Blasius profile and a confluent boundary layer involving the wake deficit due to an upstream element. The shape factor value of both profiles is 2.59, but the instability characteristics are expected to be very different from each other.

Figure 7

Comparison between fully connected network B and proposed convolutional neural network ${\text{C}}_1$. Validation plots of local instability amplification rates $\sigma$ and $N$-factor curves for an asymmetric NLF-0416 airfoil section at 0 degrees angle of attack are given for both networks. Transition location corresponds to the critical value of $N=9$ (marked by a dashed line). Corresponding transition onset locations are mentioned below the legends in (b) and (d) and marked on the horizontal axis as predicted by the neural network model (NN) (blue arrow) and computed by linear stability theory (LST) (red arrow). (a) Validation plot, growth rate predictions for CNN ${\mathrm{C}}_1$, (b) N-factor curves, CNN ${\mathrm{C}}_1$, (c) Validation plot, growth rate predictions for fully connected network B and (d) N-factor curves, fully connected network B.

Figure 8

Comparison of testing error for the fully connected neural network (FC NN) [Fig. 1, Network B from Table I] and the convolutional neural network (Fig. 2, Network ${\text{C}}_1$ from Table 1) as a function of the total number of learnable model parameters. Input features for both models include boundary-layer profiles. The Falkner-Skan database (with self-similar boundary layers) was used for training while the testing error was evaluated for an asymmetric NLF-0416 airfoil section with non-self-similar boundary layers at 0 degrees angle of attack.