Liquid rocket, Diesel or aircraft engines may operate in the transcritical regime. In such thermodynamic conditions, the classical phase change that occurs at subcritical pressure disappears and the mixing layer between the dense and cold jet and the outer gaseous stream is characterized by large variations of density and thermodynamic properties. Fluids show strong departure from a perfect gas behavior and a real-gas formulation is needed to model the fluid state. The extension of the unstructured AVBP solver, jointly developed by CERFACS and IFPEN, to handle high-pressure thermodynamics is presented in details. It is then validated on the experimental coaxial injectors studied with the Mascotte test rig from ONERA that operate in the transcritical range, namely the LOx/GH2 cases A60 and C60 and the LOx/GCH4 configuration G2. The flame pattern observed in experiments is properly recovered, hence validating the numerical strategy. Numerical results are then discussed focusing on the role of the momentum flux ratio on the development of transcritical flames.