A methodology for the rapid development of adjoint solvers for computational fluid dynamics (CFD) models is presented. The approach relies on the use of automatic differentiation (AD) tools to almost completely automate the process of development of discrete adjoint solvers. This methodology is used to produce the adjoint code for two distinct 3D CFD solvers: a cell-centred Euler solver running in single-block, single-processor mode and a multi-block, multi-processor, vertex-centred, magneto-hydrodynamics (MHD) solver. Instead of differentiating the entire source code of the CFD solvers using AD, we have applied it selectively to produce code that computes the transpose of the flux Jacobian matrix and the other partial derivatives that are necessary to compute sensitivities using an adjoint method. The discrete adjoint equations are then solved using the Portable, Extensible Toolkit for Scientific Computation (PETSc) library. The selective application of AD is the principal idea of this new methodology, which we call the AD adjoint (ADjoint). The ADjoint approach has the advantages that it is applicable to any set of governing equations and objective functions and that it is completely consistent with the gradients that would be computed by exact numerical differentiation of the original discrete solver. Furthermore, the approach does not require hand differentiation, thus avoiding the long development times typically required to develop discrete adjoint solvers for partial differential equations, as well as the errors that result from the necessary approximations used during the differentiation of complex systems of conservation laws. These advantages come at the cost of increased memory requirements for the discrete adjoint solver. However, given the amount of memory that is typically available in parallel computers and the trends toward larger numbers of multi-core processors, this disadvantage is rather small when compared with the very significant advantages that are demonstrated. The sensitivities of drag and lift coefficients with respect to different parameters obtained using the discrete adjoint solvers show excellent agreement with the benchmark results produced by the complex-step and finite-difference methods. Furthermore, the overall performance of the method is shown to be better than most conventional adjoint approaches for both CFD solvers used.

Although the mathematical derivation of the adjoint equations and their numerical implementation is well established, there is scant discussion on the understanding of the adjoint solution by itself. As this is a field solution of similar resolution of the flowfield, there is a wealth of data that can be used for design guidance. The aim is to tie the adjoint solution to the flowfield, which has physical properties. The adjoint solution of four representative cases taken from turbomachinery aerodynamic problems are used to identify the physical insight it provides. The focus is on changes related to geometry, but the changes can also be realized using other inputs to the flow solver (e.g., boundary conditions). It is shown how the adjoint counterpart of the density and velocity field can be used to provide insights into the nature of changes the designer can induce to cause improvement in the performance metric of interest. Discussion on how to use adjoint solutions for problems with constraints to furthe...

The adjoint method is extended to assess the sensitivity of turbomachinery performance with respect to inlet and exit boundary conditions. The derivation of the adjoint and sensitivity equations are briefly derived in general form, such that they can be applied to any set of flow-governing equations. In this paper, the Reynolds-averaged Navier–Stokes equations have been used, where the $k\text{-}\omega$ turbulence model has been selected. A compressor rotor blade test case is studied for verification and demonstration of the methodology. The adjoint-based sensitivities are verified using finite differences. The sensitivities of efficiency and pressure ratio with respect to the boundary condition parameters prescribed at each computational grid node at the inlet and exit faces of the blade passage are illustrated in the form of contour plots. Circumferentially averaged values and sensitivities demonstrate that fuller, more uniform profiles lead to improved performance, in line with basic thermodynamic principles. Two examples of modifying either inlet or exit boundary profiles, according to the sensitivity data obtained, show that performance tuning can be achieved. The sensitivity assessment approach presented is shown to be accurate and extremely efficient, while providing the designer with valuable information and insights to achieve a robust design.