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Although a lot of research and development has been done to understand and master the major physics involved in cryogenic rocket engines (combustion, feeding systems, heat transfer, stability, efficiency, etc.), the injection system and wall heat transfer remain critical issues due to complex physics, leading to atomization in the subcritical regime and the interactions of hot gases with walls. In such regimes, the fuel is usually injected through a coaxial annulus and triggers the atomization of the central liquid oxidizer jet. This type of injector is often referred to as air-assisted, or coaxial shear, injector, and has been extensively studied experimentally. Including such injection in numerical simulations requires specific models as simulating the atomization process is still out of reach in practical industrial systems. The effect of the injection model on the flame stabilization process and thus on wall heat fluxes is of critical importance when it comes to the design of wall-cooling systems. Indeed, maximizing the heat flux extracted from the chamber can lead to serious gain for the cooling and feeding systems for expander-type feeding cycles where the thermal energy absorbed by the coolant is converted into kinetic energy to drive the turbo-pumps of the feeding system. The methodology proposed in this work to numerically predict the flame topology and associated heat fluxes is based on state-of-the-art methods for turbulent reactive flow field predictions for rocket engines, including liquid injection, combustion model, and wall treatment. For this purpose, high-fidelity Large Eddy Simulation Conjugate Heat Transfer, along with a reduced kinetic mechanism for the prediction of H2/O2 chemistry, liquid injection model LOx sprays, and the use of a specific wall modeling to correctly predict heat flux for large temperature ratio between the bulk flow and the chamber walls, is used. A smooth and a longitudinally ribbed combustor configuration from JAXA are simulated. The coupling strategy ensures a rapid convergence for a limited additional cost compared to a fluid-only simulation, and the wall heat fluxes display a healthy trend compared to the experimental measurements. An increase of heat transfer coherent with the literature is observed when walls are equipped with ribs, compared to smooth walls. The heat transfer enhancement of the ribbed configuration with respect to the smooth walls is coherent with results from the literature, with an increase of around +80% of wall heat flux extracted for the same chamber diameter.