Characteristics‐based boundary conditions for the Euler adjoint problem

Abstract

SUMMARYOver 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.

An Extension of Adjoint Method Capabilities

“…In order compute these inflow sensitivities, it is necessary to solve the contour problem in terms of characteristics equations. In particular for Euler flows, both fluid dynamics and adjoint equations entail complementary Riemann problems, and these yield boundary conditions that are fully consistent with well-posedness [28].…”

“…23 to zero. On comparing those terms with the usual adjoint formulation for design optimization [28], one realizes that only the inflow term (c) is different. For this reason, it will be the only case discussed here.…”

“…where P [Φ, δQ] s is the bilinear concomitant, which gives rise to the adjoint boundary problem [28]. Furthermore, L * is the adjoint operator to L. Finally, one can obtain the first variation of the augmented funcional, δG, as follows:…”

“…The applications, however, also include error analysis [21,22] and grid adaptation [23,24,22,25]. More recently, Volpe and Santos [26] developed a procedure to obtain well-posed adjoint boundary conditions for the quasi 1-D Euler equations, which was later extended to the multidimensional case by Hayashi et al [28].…”

“…On using this approach to aerodynamic applications, one could compute important non-geometric sensitivity derivatives, as those related to inflow boundary conditions. It is worth noting that, this information can be computed only if appropriate boundary conditions, as those presented by Hayashi et al [28], were implemented to obtain the adjoint solution. The usual homogeneous adjoint boundary conditions at farfield boundaries would not allow this kind of application.…”