Wind-driven outflows are observed around a broad range of accreting objects throughout the universe, ranging from forming low-mass stars to supermassive black holes. We study the interaction between a central isotropic wind and an infalling, rotating envelope, which determines the steady-state cavity shape formed at their interface under the assumption of weak mixing. The shape of the resulting wind-blown cavity is elongated and self-similar, with a physical size determined by the ratio between wind ram pressure and envelope thermal pressure. We compute the growth of a warm turbulent mixing layer between the shocked wind and the deflected envelope, and calculate the resultant broad-line profile, under the assumption of a linear (Couette-type) velocity profile across the layer. We then test our model against the warm broad velocity component observed in CO J=16-15 by Herschel/HIFI in the protostar Serpens-Main SMM1. Given independent observational constraints on the temperature and density of the dust envelope around SMM1, we find an excellent match to all its observed properties (line profile, momentum, temperature) and to the SMM1 outflow cavity width for a physically reasonable set of parameters: a ratio of wind to infall mass flux of;4%, a wind speed of v w ; 30 km s −1 , an interstellar abundance of CO and H 2 , and a turbulent entrainment efficiency consistent with laboratory experiments. The inferred ratio of ejection to disk accretion rate, ;6%-20%, is in agreement with current disk wind theories. Thus, the model provides a new framework to reconcile the modest outflow cavity widths in protostars with large observed flow velocities. Being self-similar, it is applicable over a broader range of astrophysical contexts as well.