This paper outlines the method of a block-structured adaptive mesh refinement (AMR) and its application to the computation of noise radiation from a realistic engine exhaust geometry with flow. The computational model described allows acoustic waves, propagating inside the bypass duct of a generic aircraft engine, to be admitted into a computational domain that includes the aft duct section, the exit plane of the duct, and the jet flow immediately downstream. Other than working on a uniformly fine mesh, the computational mesh is adaptively refined as the propagation and radiation of acoustic waves with the aim of reducing the number of computational cells and improving the computational efficiency. The method is mainly consisted of three parts: an AMR framework to manage adaptively refined meshes; a range of $4^{th}$-order schemes to compute the near-field acoustic propagation and radiation; and an integral surface for predicting far-field directivity. Propagation inside the duct, diffraction at the lip of the duct and propagation into the near field is firstly modeled by the linearised Euler equations. Hydrodynamic instabilities are discovered in the exhaust mean flow and are leading to overwhelm the acoustic solutions. In order to suppress this kind of non-physical instabilities, the original linearised Euler equations are replaced with the acoustic perturbation equations, which have been extended in the cylindrical coordinates for this case. The effect and the accuracy of the acoustic perturbation equations are validated by comparing to solutions of the linearised Euler equations through a case study of the radiation of single spinning mode from a generic engine bypass duct. The quality of the proposed AMR method is also illustrated. For this case the computational efficiency of AMR is compared with that computed on a uniformly fine mesh. The results demonstrate the efficiency of the current adaptive mesh refinement algorithm.