INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS Int. J. Numer. Meth. Fluids 2013; 71:1297–1321 Published online 15 August 2012 in Wiley Online Library (wileyonlinelibrary.com/journal/nmf). DOI: 10.1002/fld.3712 Characteristics-based boundary conditions for the Euler adjoint problem Marcelo Hayashi 1, * ,† , Marco Ceze 2 and Ernani Volpe 1 1 University of São Paulo, São Paulo, 05508-970, Brazil 2 University of Michigan, Ann Arbor, MI, 48105, USA SUMMARY Over the last decade, the adjoint method has been consolidated as one of the most versatile and successful tools for aerodynamic design. It has become a research area on its own, spawning a large variety of applications and a prolific literature. Yet, some relevant aspects of the method remain relatively less explored in the literature. Such is the case with the adjoint boundary problem. In particular for Euler flows, both fluid dynamic and adjoint equations entail complementary Riemann problems, and these yield boundary conditions that are fully consistent with well-posedness. In the literature, this approach has been pursued solely in terms of Riemann variables. This work formulates the adjoint boundary problem so as to correspond precisely to that imposed on the flow, as it is given in terms of primitive variables. Test results have shown to be in agreement with the traditional approach for external flow problems. Copyright © 2012 John Wiley & Sons, Ltd. Received 6 March 2012; Revised 15 June 2012; Accepted 23 June 2012 KEY WORDS: adjoint method; aerodynamic design; boundary conditions; Euler flow 1. INTRODUCTION The problem of inverse aerodynamic design was first considered by Lighthill, in 1945, and the investigation was limited to airfoils in incompressible potential flows [1]. The interest in the topic has grown ever since, following closely on the progresses in computational resources and numerical methods for flow simulation. Over the years, a variety of methods have been proposed to tackle the problems of aerodynamic optimization and inverse design. The adjoint method has played a prominent role in that context, for a number of reasons. Among them, one could cite the great flexibility it offers with regard to the flow-physics model and to the definition of objective functionals. Originally proposed by Pirronneau [2–4] for elliptic problems, it was later extended to transonic flows by Jameson [1]. Since then, it has become the subject of extensive research activity [5–14] and spawned a wide variety of applications, ranging from nuclear reactor thermo-hydraulics to atmospheric sciences [15, 16]. In aerodynamics, the developments of the adjoint method encompass design applications regarding internal and external flows [17–22] and, more recently, unsteady flows [23–26]. To put matters into perspective, objective functionals of general interest in aerodynamics depend on flow variables and on the shape and location of the boundaries [27, 28]. These, in turn, are controlled by a set of design parameters. For all practical purposes, the set is assumed to be finite. Under these circumstances, a natural means of estimating the sensitivity of that functional to changes in flow geometry would be to perturb each design parameter individually and then to compute the sensitivity gradient by finite differences. *Correspondence to: Marcelo Hayashi, Av. Prof. Mello Moraes, 2231, São Paulo, 05508-970, Brazil. † E-mail: mhayashi@usp.br Copyright © 2012 John Wiley & Sons, Ltd.