We develop a simple model for flow and suspended sediment transport in submarine channels able to describe vertical profiles of both flow field and suspended sediment concentration. We consider a conservative turbidity current flowing in a straight channel under steady and uniform flow conditions. The turbulence closure needed to account for density stratification is adapted from the model of \cite{MellorYamada82}. Solutions are found for prescribed values of the governing parameters and show the typical velocity profiles observed in turbidity currents characterized by a peak close to the bed. Results for longitudinal velocity are then compared with field observations providing excellent agreement and allowing to predict the concentration profiles and suspended sediment size. The potential use and future developments of the present approach are finally discussed.