  We present a new high-resolution, non-oscillatory semi-discrete
central-upwind scheme for one-dimensional two-layer shallow-water flows with
friction and entrainment along channels with arbitrary cross sections and
bottom topography. These flows are described by a conditionally hyperbolic
balance law with non-conservative products. A detailed description of the
properties of the model is provided, including entropy inequalities and
asymptotic approximations of the eigenvalues of the corresponding coefficient
matrix. The scheme extends existing central-upwind semi-discrete numerical
methods for hyperbolic conservation and balance laws and it satisfies two
properties crucial for the accurate simulation of shallow-water flows: it {\it
preserves the positivity} of the water depth for each layer, and it is {\it
well balanced}, i.e., the source terms arising from the geometry of the channel
are discretized so as to balance the non-linear hyperbolic flux gradients.
Along with the description of the scheme and proofs of these two properties, we
present several numerical experiments that demonstrate the robustness of the
numerical algorithm.
