© 2025 by the authors. Licensee MDPI, Basel, Switzerland.Lozano, CarlosPonsin, J.Ponsin Roca, Jorge2025-07-102025-07-102025-05-30Aerospace 12(6) 494(2025)https://www.mdpi.com/2226-4310/12/6/494http://hdl.handle.net/20.500.12666/1052(This article belongs to the Special Issue Adjoint-Based Techniques in Computational Fluid Dynamics: Theory and Applications)The characteristic structure of the two-dimensional adjoint Euler equations is examined. The behavior is similar to that of the original Euler equations, but with the information traveling in the opposite direction. The compatibility conditions obeyed by the adjoint variables along characteristic lines are derived. It is also shown that adjoint variables can have discontinuities across characteristics, and the corresponding jump conditions are obtained. It is shown how this information can be used to obtain exact predictions for the adjoint variables, particularly for supersonic flows. The approach is illustrated by the analysis of supersonic flow past a double-wedge airfoil, for which an analytic adjoint solution is obtained in the near-wall region. The solution is zero downstream of the airfoil and piecewise constant around it except across the expansion fan, where the adjoint variables change smoothly while remaining constant along each Mach wave within the fan.engAttribution-NonCommercial-NoDerivatives 4.0 Internationalhttps://creativecommons.org/licenses/by-nc-nd/4.0/Adjoint variablesCompressible flowCharacteristic linesCompatibility conditionsSupersonic flowMach linesRiemann invariantsJump conditionsShock expansion theoryAerodynamic designinfo:eu-repo/semantics/article10.3390/aerospace120604942226-4310info:eu-repo/semantics/openAccess