Effect of Turbulence-Closure Consistency on Airfoil Identification

TL;DR

This paper tackles the inverse problem of inferring airfoil geometry from wake measurements by formulating a PDE-constrained optimization using an adjoint-based RANS framework and a Free-Form Deformation parameterization. It demonstrates that single-condition inversions are ill-posed, while combining wake signatures across multiple angles of attack and Reynolds numbers substantially reduces non-uniqueness and improves forward predictions. A central finding is that turbulence-closure inconsistency can yield order-of-magnitude differences in recovered shapes and gradients, underscoring the need for sensitivity-consistent closures and uncertainty quantification. The work proposes a sensitivity-consistency metric and advocates training and evaluating closures not only on predictive accuracy but also on adjoint behavior, guiding the development of data-driven and hybrid turbulence models for reliable inverse-design applications.

Abstract

We consider an inverse flow problem in which the airfoil shape is inferred from its wake signature, namely the velocity field in the wake of a target airfoil. This is an ill-posed problem and highly sensitive to the accuracy and consistency of the employed turbulence closure. We first demonstrate that shape identification based on a single flow condition is ill-posed, whereas incorporating multiple wake signatures obtained at different angles of attack substantially mitigates this ill-posedness. We further show that aggregating wake profiles across multiple Reynolds numbers provides an additional and practically relevant source of information that can further constrain the inverse problem and improve reconstruction robustness. We then compare the inferred geometries obtained using different turbulence closures and find that inconsistencies among the models lead to markedly divergent shapes. These findings underscore that turbulence-closure consistency is essential for reliable shape identification and further suggest that effective turbulence models must ensure not only accurate predictions but also physically consistent sensitivities - a principle that should guide the development of both classical and data-driven closure models.

Figures (9)

• Figure 1: Concept of the present paper. The airfoil shape is identified from the wake signature using the adjoint method. Two issues of the inverse shape determination are discussed: ill-posedness, and the effect of the inconsistent turbulence closures. Ill-posedness is alleviated by incorporating wake signatures across multiple operating conditions, including multiple angles of attack and multiple Reynolds numbers.
• Figure 2: Problem sketch. The origin point is located at the airfoil leading edge. The airfoil chord length is 1. The domain $\Omega (x\in[1.1,4.5], y\in[-0.5,0.5])$ is where the wake signature is taken from and the $L_2$ error in \ref{['eq:obj']} is calculated.
• Figure 3: FFD control points and geometry constraints on the airfoil. The FFD control points are designed to move in the y-direction only.
• Figure 4: Velocity magnitude fields for airfoils at different angles of attack. “Multiple” refers to the airfoil obtained through inverse shape optimization using three angles of attack, whereas “Single” denotes the airfoil optimized using only one angle of attack ($0^\circ$). All simulations are performed with the Spalart–Allmaras (S–A) turbulence model, which is also employed in the adjoint-based shape optimization.
• Figure 5: Comparison among the target, initial, and two inversely obtained airfoil profiles. The target profile is NACA16021, and the initial guess of optimization is NACA0012. The S-A model is used to calculate the wake signatures and determine the shapes. Two inversely obtained airfoils are based on one and three angles of attack, respectively.