Finite-wavelength scattering of incident vorticity and acoustic waves at a shrouded-jet exit

Abstract

As the vortical disturbances of a shrouded jet pass the sharp edge of the shroud exit some of the energy is scattered into acoustic waves. Scattering into upstream-propagating acoustic modes is a potential mechanism for closing the resonance loop in the ‘howling’ resonances that have been observed in various shrouded jet configurations over the years. A model is developed for this interaction at the shroud exit. The jet is represented as a uniform flow separated by a cylindrical vortex sheet from a concentric co-flow within the cylindrical shroud. A second vortex sheet separates the co-flow from an ambient flow outside the shroud, downstream of its exit. The Wiener–Hopf technique is used to compute reflectivities at the shroud exit. For some conditions it appears that the reflection of finite-wavelength hydrodynamic vorticity modes on the vortex sheet defining the jet could be sufficient to reinforce the shroud acoustic modes to facilitate resonance. The analysis also gives the reflectivities for the shroud acoustic modes, which would also be important in establishing resonance conditions. Interestingly, it is also predicted that the shroud exit can be ‘transparent’ for ranges of Mach numbers, with no reflection into any upstream-propagating acoustic mode. This is phenomenologically consistent with observations that indicate a peculiar sensitivity of resonances of this kind to, say, jet Mach number.

A model supersonic buried-nozzle jet: instability and acoustic wave scattering and the far-field sound

“…Without the inner nozzle directly represented in our model, except as a source for the incident instability modes, we do not include its role in scattering, which is expected to be of secondary importance. The upstream-propagating reflected acoustic waves that could couple with the buried-nozzle lip are finite in number and, depending upon the core jet and bypass coflow conditions, most of them decay in the upstream direction (Samanta & Freund 2008). This will be particularly true for high-speed jets with low-speed coflows, as considered here.…”

On the axisymmetric stability of heated supersonic round jets

“…The location of the principal branch cuts are determined via a causality criterion [29,30], discussed in detail elsewhere [3,31]. Basically, for harmonic time dependence of exp(−iωt), where ω in general can be complex, causality requires its imaginary part …”

“…The unstable vortex-sheet and acoustically coupled hydrodynamic modes appear as zeros in the R − plane (figure 2), and all of these are downstream propagating as we shall see §3. It is to be noted that semi-infinite ducted jets have modes identical to free jets, except for a continuous tail of poles and zeros parallel to the imaginary axis which models the acoustic scattering owing to the finite duct termination [3,31,32].…”

Instability of a vortex sheet leaving a right-angled wedge

“…This is essentially the supersonic version of Jones & Morgan's subsonic study. More recently, Samanta & Freund (2008) noted that accepted practice is to use a full Kutta condition, for which Bechert & Pfizenmaier (1975) provided experimental support. Rabchuk (2000) identified trailing edge receptivity as disturbances converted into flow instabilities at the trailing edge, where the pressure is singular, in a mixing layer or wake.…”