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This paper demonstrates a methodology for time-domain and time-accurate nonlinear, direct and adjoint simulations of unsteady flows and aeroacoustics for multi-component systems in relative motion. Here, the principal effort is directed towards mitigating the problem of distortion and contamination of the adjoint field at the moving interface, through a computationally lightweight, high-order sliding plane approach for which the adjoint equivalent is simple to obtain. This effort requires an attentive treatment of the interface conditions that surpasses the requirements of the more common forward (primary) problem. Sensitivity of a given quantity of interest from a time-varying flow with respect to a large number of parameters is then obtained through the adjoint operator, which is evaluated using nonlinear-adjoint looping. This technique is implemented using checkpointing and the PETSc TSAdjoint library and, after validation, applications including a rotor–stator interaction problem are presented. [ABSTRACT FROM AUTHOR]