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Volume 10
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Article

Adjoint and Direct Characteristic Equations for Two-Dimensional Compressible Euler Flows

by
Kevin Ancourt
1,
Jacques Peter
1,* and
Olivier Atinault
2
1
Département Aérodynamique, Aéroélasticité et Acoustique (DAAA), ONERA, Université Paris Saclay, 29 Avenue de la Division Leclerc, Châtillon, 92322 Paris, France
2
Département Aérodynamique, Aéroélasticité et Acoustique (DAAA), ONERA, Université Paris Saclay, 8 Rue des Vertugadins, 92190 Meudon, France
*
Author to whom correspondence should be addressed.
Aerospace 2023, 10(9), 797; https://doi.org/10.3390/aerospace10090797
Submission received: 20 July 2023 / Revised: 8 September 2023 / Accepted: 10 September 2023 / Published: 12 September 2023
(This article belongs to the Special Issue Adjoint Method for Aerodynamic Design and Other Applications in CFD)
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Abstract

The method of characteristics is a classical method for gaining understanding in the solution of a partial differential equation. It has recently been applied to the adjoint equations of the 2D steady-state Euler equations and the first goal of this paper is to present a linear algebra analysis that greatly simplifies the discussion of the number of independent characteristic equations satisfied along a family of characteristic curves. This method may be applied for both the direct and the adjoint problem. Our second goal is to directly derive in conservative variables the characteristic equations of 2D compressible inviscid flows. Finally, the theoretical results are assessed for a nozzle flow with a classical scheme and its dual consistent discrete adjoint.
Keywords: continuous adjoint; inviscid flow; compressible flow; method of characteristics; characteristic equation; characteristic curve continuous adjoint; inviscid flow; compressible flow; method of characteristics; characteristic equation; characteristic curve

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MDPI and ACS Style

Ancourt, K.; Peter, J.; Atinault, O. Adjoint and Direct Characteristic Equations for Two-Dimensional Compressible Euler Flows. Aerospace 2023, 10, 797. https://doi.org/10.3390/aerospace10090797

AMA Style

Ancourt K, Peter J, Atinault O. Adjoint and Direct Characteristic Equations for Two-Dimensional Compressible Euler Flows. Aerospace. 2023; 10(9):797. https://doi.org/10.3390/aerospace10090797

Chicago/Turabian Style

Ancourt, Kevin, Jacques Peter, and Olivier Atinault. 2023. "Adjoint and Direct Characteristic Equations for Two-Dimensional Compressible Euler Flows" Aerospace 10, no. 9: 797. https://doi.org/10.3390/aerospace10090797

APA Style

Ancourt, K., Peter, J., & Atinault, O. (2023). Adjoint and Direct Characteristic Equations for Two-Dimensional Compressible Euler Flows. Aerospace, 10(9), 797. https://doi.org/10.3390/aerospace10090797

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