Airfoil leading-edge suction and energy conservation for compressible flow

Abstract

When a flat-plate airfoil at zero angle of attack encounters a vertical gust in an otherwise uniform flow, it experiences a force along the chord. This leading-edge suction force is examined for compressible flow with a time-dependent gust. A simple derivation of the thrust force is based on the fact that the leading edge is a singular point so that the flow here is dominated by the leading-edge dipole strength. From the viewpoint of a fluid-fixed observer the fluid does work on the airfoil, and this energy must come from the incident gust. Demonstrating energy conservation is not surprising, but it gives a better understanding of the relationship between the individual energy terms. The derivation shows that the acoustic energy can be calculated using compact assumptions at low frequency, but that it must be calculated non-compactly at high frequency. For a general gust the work done on the airfoil is shown to equal the energy taken from the fluid, the energy transfer occurring at the leading edge. For a sinusoidal gust the energy contained in the incident gust is shown to equal the sum of the energy remaining in the wake, the work done on the airfoil and the acoustic energy radiated away. The relative proportions of the incident energy going to these three energy types depends on the gust frequency, the acoustic radiation becoming more efficient as the frequency increases. For a fixed gust frequency, the thrust force goes to zero at a Mach number of one, and for an incident gust consisting of vorticity on the airfoil axis, the entire energy of the gust is radiated as acoustic energy at this Mach number.