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Unstable jet–edge interaction Part 2: Multiple frequency pressure fields
- Ruhi Kaykayoglu, Donald Rockwell
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- Journal:
- Journal of Fluid Mechanics / Volume 169 / August 1986
- Published online by Cambridge University Press:
- 21 April 2006, pp. 151-172
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In Part 1 of this investigation we addressed the instantaneous pressure field at a leading-edge due to single frequency jet-edge interactions; here we consider edge pressure fields and associated vortex interaction patterns at the edge having a number of (at least six) well-defined spectral components. Each of the spectral components is present along the entire extent of the approach shear layer upstream of the edge; the relative amplitudes of these components are preserved in the conversion process from the free (approach) shear layer to the surface pressure field, the key link being complex, but ordered patterns of vortex interaction at the edge. Moreover, the predominant spectral components can be reasoned on the basis of these visualized vortex interactions by considering the vortex array in the incident shear layer.
The spectral character of the surface pressure field changes dramatically with edge location in the incident shear layer, or vortex array. If the edge is symmetrically located within the vortex array of the incident jet, a low-frequency component prevails due to large-scale vortex formation; however, with an asymmetrically disposed edge, the most unstable frequency of the jet dominates due to direct vortex impingement upon the edge, and the mean-square pressure amplitude (encompassing all spectral components) is double that of the symmetrical interaction.
Irrespective of the type of interaction and the manner in which energy is partitioned between spectral components, the mean-square pressure due to the sum of all spectral components decays approximately as x−c immediately downstream of the tip of the edge. However, each spectral component tends either to a maximum or minimum amplitude as the tip of the edge is approached, depending upon the class of vortex array-edge interaction.
Unstable jet–edge interaction. Part 1. Instantaneous pressure fields at a single frequency
- Ruhi Kaykayoglu, Donald Rockwell
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- Journal:
- Journal of Fluid Mechanics / Volume 169 / August 1986
- Published online by Cambridge University Press:
- 21 April 2006, pp. 125-149
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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.
Vortices incident upon a leading edge: instantaneous pressure fields
- Ruhi Kaykayoglu, Donald Rockwell
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- Journal:
- Journal of Fluid Mechanics / Volume 156 / July 1985
- Published online by Cambridge University Press:
- 20 April 2006, pp. 439-461
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A mixing-layer flow formed by merging of high- and low-speed streams leads to successive generation of vortices of the same sense that impinge upon a leading edge. Distortion of each incident vortex, secondary-vortex shedding, and 'sweeping’ of flow about the tip of the edge are related to the instantaneous pressure fields via simultaneous flow visualization and pressure measurement. The instantaneous pressure fields are interpreted as downstream travelling waves along the upper and lower surfaces of the edge; in turn, these wavelike pressure variations are linked to the visualized vortex patterns adjacent to each surface.
Near the tip of the edge, where rapid flow distortion occurs, the pressure fields are non-wavelike; on the lower surface of the tip, negligible streamwise phase variations of fluctuating pressure are associated with secondary shedding there, while on the upper surface there is a phase jump. This jump can be as large as π when the incident vortex impinges directly upon, or passes just below, the tip of the edge. Downstream of this near-tip region, the wavelike pressure fields show short and long wavelengths on the lower and upper surfaces respectively. These wavelengths, in turn, differ substantially from the wavelength of the incident-vortex instability.
Irrespective of the transverse location of the incident vortex with respect to the leading edge, maximum pressure amplitude always occurs at the tip of the edge; it takes on its largest value when the scale of secondary shedding from the tip of the edge is most pronounced. Moreover, the fact that the net force on the edge scales with tip-pressure amplitude underscores the crucial role of the local flow distortions in the tip region.
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