Computational simulations are now commonplace in the design and analysis of flight vehicles. However, in an industrial design environment we still cannot afford simulations that resolve all relevant physical scales. Instead, we must make modeling choices in which we trade resolution for uncertainty. These choices are often made ad hoc, based on limited data and experience. Furthermore, the uncertainty that follows from these choices are often taken for granted, as most current simulation tools do not provide estimates of the uncertainty associated with simulation results. In this scenario, engineers are forced to make judgement calls about how much to trust a simulation based on experience and physical reasoning.

In this work we take a step to modify this scenario, with focus on laminar airfoil design. We developed a probabilistic model of the integral boundary layer (IBL) equations in the laminar regime. In the past, the IBL equations have been coupled to inviscid flow solvers to produce computationally affordable tools for aerodynamic design of airfoils with great success. One notable example is Xfoil, the renowned airfoil design software created by Prof. Drela.

In the IBL formulation the velocity profile is replaced by a handful of integral state variables (resolved variables). To close this formulation, one must provide models for the nonlinear interactions between different physical scales in the boundary layer, which are not resolved in the integral formulation. In this work, we represent these closure terms as Gaussian Processes. The novelty in our approach is that we quantify the effects of both resolved and unresolved variables. We trained our probabilistic model using a large data set obtained by solving the thin shear layer approximation of the boundary layer on over 1,500 airfoils in a variety of angles of attack.

In a not so distant future, we will be able to run very sophisticated high-fidelity numerical experiments that will generate a tremendous amount of data, way beyond the data that we can gather with traditional physical experiments. We must be ready to leverage this capacity to create more accurate, robust, and reliable simulation tools adapted to the realities of the 21st century. We believe that the rigorous development of data-driven probabilistic models, such as the one employed here, is a viable approach to reach this goal.