Development of Computational Aeroacoustics Equations for Subsonic Flows Using a Mach Number Expansion Approach

Computational aeroacoustics equations are developed using a Janzen–Rayleigh expansion of the compressible flow equations. Separate expansions are applied to an inner region characterized to lowest order by an incompressible flow field and an outer region characterized by propagating acoustic waves. Several perturbation equation sets are developed in the inner and outer regions by truncating the expanded equations using different orders in the perturbation variable, ε, where ε2 is proportional to the square of the Mach number characterizing the flow. Composite equation sets are constructed by matching the equations governing the inner and outer regions. The highest-order perturbation continuity and momentum equations include an infinite series in ε2 and are shown to be identical to the equations used in the expansion about incompressible flow approach. As such, the perturbation analysis is used to interpret the physical meaning of the perturbation variables and to highlight the assumptions inherent in this approach. Differences between numerical solutions obtained with the composite equation sets are evaluated for two unsteady flow problems. The lowest-order perturbation equation set is shown to yield adequate acoustic predictions for low Mach number flows. This equation set is considerably simpler to implement into a numerical solver and reduces the required CPU time relative to the highest-order equation set.