Unstable jet–edge interaction. Part 1. Instantaneous pressure fields at a single frequency

Abstract

Despite its central importance, the pressure field at a leading edge has remained uncharacterized for the classical jet-edge interaction at a single predominant frequency. This investigation shows that the force, due to the integrated instantaneous pressure field on the edge, is located a distance downstream of the tip of the edge as much as one-quarter of a wavelength (λ) of the incident instability; this distance also corresponds to about one-quarter of the geometric length ( L ) between the nozzle and tip of the edge. Consequently, the traditional assumption that the phase-locking criterion for self-sustained oscillations can be expressed as a ratio of L /λ is inappropriate for low-speed jet flows, which have been of primary interest over the past two decades. The edge pressure field is made up of two regions bounded by the maximum amplitude at the onset of separation from the surface of the edge: a near-tip region (0 ≤ x /λ [lsim ] 0.1) where the amplitude drops to a minimum as the tip is approached; and a downstream region ( x /λ [gsim ] 0.1) where the amplitude varies as x −a . Since the drop in pressure in the near-tip region does not occur over a streamwise length commensurate with the length of the edge, imposition of a Kutta condition is inappropriate in simulations of the edge region. Moreover, in the near-tip region (0 [lsim ] x /λ [lsim ] 0.2), the pressure field is non-propagating; a wave-type representation is appropriate only downstream of this region. At the tip of the edge, occurrence of the pressure minimum is due to the minimum in fluctuating angle of attack a of the approaching shear layer, deducible from the velocity eigenfunctions of linear theory; correspondingly, flow separation occurs downstream of, not at, the tip of the edge. When the tip is displaced off centreline, there is a rise in a, giving a rise in tip pressure amplitude; nevertheless, the overall x − a amplitude distribution persists. This overall x − a (a ∼ ½) variation of the pressure amplitude commences downstream of the tip of the edge near the onset of flow separation, which leads to secondary-vortex formation; in turn, it is driven by development of the primary vortex in the unstable jet shear layer, having initially distributed vorticity. The role of this flow separation and subsequent secondary-vortex formation is, therefore, not to relieve a singularity at the tip of the edge; it is simply a consequence of growth of the primary vortex along the edge.