Coordinate independent forms of the compressible potential flow equations

The compressible motion of gases has long been connected with high energy yield devices such as combustors, jet engines, turbines, and rocket motors. Most available formulations for this high speed gaseous environment lead to coupled nonlinear PDEs. While their integration requires the use of numerical methods, their stability and convergence remain conditional on the stiffness of the system. In seeking reduced-order models, it has become customary to introduce specific assumptions that simplify the problem's complexity without sacrificing its fundamental physical attributes. Examples include the variety of low-Mach number models for subsonic combustion and atmospheric flows, or the high-Mach number schemes used in the treatment of hypersonic motions. The ensuing approximations constitute limiting process expressions of the original equations and are often accompanied by their own stability and convergence criteria. Another reduction in complexity may be accomplished by relaxing certain physical constraints or implementing simplifying assumptions that correspond, if the situation permits, to inviscid, adiabatic, and non-reactive models. A classic example is the study of acoustic instability in solid rocket motors where some analyses consider the flow to be irrotational. For the variety of reasons mentioned, we consider in this work the limiting potential case of the compressible gaseous motion not only in propulsion devices, such as rockets and other con?ned thrust chambers, but also in external aerodynamic flows as well. In this spirit, we derive the equations of motion for a steady, irrotational, compressible fluid in coordinate-independent vector form. We accomplish this using two approaches, one based on the velocity potential, and the other using the streamfunction formulation. In both instances, the spatial dependence of the speed of sound is eliminated in favor of its stagnation value. This key aspect of the present framework enables us to systematically apply the Rayleigh-Janzen asymptotic technique to explicitly solve the compressible potential equations in the low-Mach number limit. While the procedure is rather straightforward with the velocity potential, auxiliary relations are required when using the streamfunction approach. By way of example, the two formulations developed here are implemented to the extent of producing analytical approximations for two geometric settings associated with the irrotational compressible flow fields in planar and cylindrical rocket motors.