We consider turbulent flows in a differentially heated Taylor-Couette system with an axial Poiseuille flow. The numerical approach is based on the Reynolds Stress Modeling (RSM) of Elena and Schiestel widely validated in various rotor-stator cavities with throughflow. To show the capability of the present code, our numerical predictions are compared very favorably to the velocity measurements of Escudier and Gouldson in the isothermal case, for both the mean and turbulent fields. The RSM model improves, in particular, the predictions of the k-eps model of Naser. Then, the second order model is applied for a large range of rotational Reynolds and Prandtl numbers, flow rate coefficient in a very narrow cavity of radius ratio s=0.961 and aspect ratio L=0.013. Temperature gradients are imposed between the incoming fluid and the inner and outer cylinders. The mean hydrodynamic and thermal fields reveal three distinct regions across the radial gap with a central region of almost constant axial and tangential mean velocities and constant mean temperature. Turbulence, which is weakly anisotropic, is mainly concentrated in that region and vanishes towards the cylinders. The mean velocity distributions are not clearly affected by the rotational Reynolds number and the flow rate coefficient. The effects of the flow parameters on the thermal field are more noticeable and considered in details. Correlations for the averaged Nusselt numbers along both cylinders are finally provided according to the flow control parameters.