An analysis of the acoustic energy in a flow duct with a vortex sheet

摘要

Modelling the acoustic scattering and absorption at an area expansion in a flow duct requires the incorporation of the flow-acoustic interaction. One way to quantify the interaction is to study the energy in the incident and the scattered field respectively. If the interaction is strong, energy may be transferred between the acoustic and the main flow field. In particular, shear layers, that may he the result of the flow separation, are unstable to low frequency perturbations such as acoustic waves. The vortex sheet model is an analytical linear acoustic model, developed to study scattering of acoustic waves in duct with sharp edges including the interaction with primarily the separated flows that arise at sharp edges and corners. In the model the flow field at an area expansion in a duct is described as a jet issuing into the larger part of the duct. In this paper, the flow-acoustic interaction is described in terms of energy flow. The linear convective wave equation is solved for a two-dimensional, rectangular flow duct geometry. The resulting modes are classified as "hydrodynamic" and "acoustic" when separating the acoustic energy from the part of the energy arising from the steady flow field. In the downstream duct, the set of modes for this complex flow field are not orthogonal. For small Strouhal numbers, the plane wave and the two hydrodynamic waves are all plane, although propagating with different wave speeds. As the Strouhal numbers increases, the hydrodynamic modes changes to get a shape where the amplitude is concentrated near the vortex sheet. In an intermediate Strouhal number region, the mode shape of the first higher order mode is very similar to the damped hydrodynamic mode. A physical interpretation of this is that we have a strong coupling between the flow field and the acoustic field when the modes are non-orthogonal. Energy concepts for this duct configuration and mean flow profile are introduced. The energy is formulated such that the vortex sheet turns out as a sink for the acoustic field, but a source for the unstable hydrodynamic wave. This model is physical only close to the edge, due to an exponentially growing hydrodynamic mode. In a real flow, non-linearities will limit the growth, but this is not included in the model.