Bubbly flows appear in a large number of engineering applications including bubble column reactors (chemical reactions, mixing, …) and nuclear reactors in operating conditions as well as in the case of hypothetical severe accident conditions. We present in this paper an Eulerian drift-flux model for the simulation of such flows. This model deals with the compressibility of the dispersed (gaseous) phase while assuming incompressibility for the (liquid) continuous phase. The closure relation for the drift velocity is implemented in order to obtain an inconditionally hyperbolic system of conservation equations, as far as only first order terms are regarded. Connections of the proposed model with other well known drift-flux models are discussed. We take benefit from the hyperbolic nature of the system to build a family of upwind Godunov type schemes, using exact or approximate Riemann solvers and based on an unstructured finite volumes/finite elements discretization. Then, following recently obtained results in the case of single phase flows, a preconditioning scheme is designed to keep the same accuracy in the low Mach number limit. Numerical applications are presented, to assess the capabilities of the model.