Context. The single-degenerate (SD) model is one of the leading models for the progenitors of Type Ia supernovae (SNe Ia). Recently, a new version of the SD model, the common-envelope wind (CEW) model, has been proposed, which, in principle, has the potential to resolve most of the difficulties encountered by previous SD models. This model is still being developed and a number of open issues remain, such as the details of the mass-loss mechanism from the surface of the common envelope (CE), the main observational properties, and the spiral-in timescale of the binary inside the envelope. Aims. In this article, we aim to address these issues by considering hydrodynamical effects on the CE. Methods. Using the stellar evolution code MESA, we carried out a series of 1D hydrodynamical simulations of an asymptotic giant branch (AGB) star undergoing a common-envelope phase with different envelope masses (0.0007 M'0.06 M). The effect of the immersed binary was mimicked by changing the gravitational constant throughout the envelope and injecting an extra heating source at the location of the binary orbit. Results. We found that the envelopes are always dynamically unstable, leading to regular mass ejection events if the envelope is more massive than the critical value of © Y. Cui et al. 2022.