  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 novel 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.
