Heat addition with variable area: Methodology for preliminary design of the scramjet combustion chamber

Well-known analytical approaches are commonly adopted for the preliminary design of scramjet engines. In this context, the combustion process in the combustion chamber can be modeled by considering heat addition to the airflow at supersonic speed. The one-dimensional Rayleigh flow theory can be applied to estimate the behavior of thermodynamic properties and velocities when the combustion chamber has a constant cross-sectional area and no mass is added within the duct. However, the temperature and pressure predicted by using constant area combustion chambers are too high, implying the necessity of modifications in the cross-sectional area of the chamber to avoid thermal choking and excessive pressure gradients. In this case, the unidimensional Rayleigh theory does not fit anymore. This work proposes an analytical methodology to estimate the airflow thermodynamic properties and velocities for scramjet combustion chambers with cross sections of variable areas by using an iterative algorithm that employs the Rayleigh flow area ratio theory. The analytical results were compared with the two-dimensional computational fluid dynamics analysis using the Reynolds-averaged Navier-Stokes method for both inviscid and viscous flow and considering turbulence effects. The proposed analytical model to estimate the flow behavior in the scramjet combustion chamber predicted results in agreement with the physics of the problem and with the results obtained via numerical simulation. The analytical model cannot predict oscillations in the flow properties caused by the expansion waves and their reflections. Still, the behavior and intensity of the properties are well captured along the entire length of three combustion chambers with variable area. The proposed algorithm is also applied to determine the angle of the combustion chamber that allows guaranteeing a constant, or a quasi-constant, static pressure along the length of the combustion chamber, approaching better the ideal thermodynamic Brayton cycle. The proposed model is suitable for preliminary scramjet designs and can be used to solve other problems involving variable area ducts.