Interpretation of Adjoint Solutions for Turbomachinery Flows

Abstract

While the mathematical derivation of the adjoint equations and their numerical implementation is well established, there is a scant discussion on the understanding of the adjoint solution by itself. As this is a field solution of similar resolution of the flow-field, there is a wealth of data that can be used for design guidance. This paper addressess this specific topic. In particular, we take representative cases from turbomachinery aerodynamic problems and use the adjoint solution to identify the "physical insight" it provides. We aim to tie the adjoint solution to the flow-field which has physical properties. Towards this end, we first look at three problems 1) a fan, 2) a compressor rotor and stator, 3) a low pressure turbine. In all three of them, we focus on changes related to geometry, but one can also realize the changes using other inputs to the flow solver (eg. boundary conditions). We show how the adjoint counter-part of the density, the velocity fields and the turbulence quantities can be used to provide insights into the nature of changes the designer can induce to cause improvement in the performance metric of interest. We also discuss how to use adjoint solutions for problems with constraints to further refine the changes. Finally, we use a problem where it is not immediately apparent what geometry changes need to be used for further evaluation with optimization algorithms. In this problem, we use the adjoint and flow solution on a turbine strut, to determine the kind of end-wall treatments that reduce the loss. These changes are then implemented to show that the loss is reduced by close to 8%.