During the last decade, most of acoustic cloak research has been done within a theoretical framework in which the medium is at rest. However, such an acoustic cloak cannot preserve its unique properties or functions to make an object acoustically invisible in the presence of flow. In this study, we propose a theoretical framework to accurately investigate the effect of compressibility and non-uniformity in flow on the scattering pattern of acoustic cloak. In the formulation, the wave operator is coupled with the non-uniform velocity vector, and the equivalent source terms due to mean flow are divided into the compressibility effect and the non-uniformity effect with their own physical meanings. Numerical simulation shows the difference in far-field directivity between previous and present formulations. The polarity of the equivalent sources in the present formulation shows hexapole and skewed quadrupole patterns for non-uniformity and compressibility effects, respectively, and their magnitudes increase with power laws of Mach number as the Mach number increases. As an application, we make use of the present formulation for predicting the acoustic scattering from newly designed convective cloaks. The simulation results show better performance compared to the existing convective cloak.