Acoustic theory of the many-bladed contra-rotating propeller: analysis of the effects of blade sweep on wake interaction noise

Abstract

An analytical model is presented for the wake interaction tones produced by a contra-rotating propeller. We re-cast the usual far-field radiation formulae as a double integral over a nominal propeller source annulus. Assuming that the number of blades on both propellers is large, we evaluate the integral asymptotically in terms of its leading-order contributions from interior stationary or boundary critical points which represent the specific locations on the propeller annulus that dominate the sound radiation. The asymptotic approach is powerful producing results in the form of one-line algebraic formulae that contain no integrals or special functions yet remain accurate. The asymptotics show that sweep is not necessarily beneficial and can cause the blade design to become critical for particular tones and directions in terms of a continuum of interior points distributed along a line on the propeller source annulus producing a higher-order result and thus an enhanced radiated sound field. The paper also shows how the interior points are completely consistent with the sub- or super-critical gust response of a swept blade. Tones with low and zero azimuthal mode order are treated as special cases and the asymptotics show that, as the mode order reduces, the radiated sound becomes concentrated around the flight axis where even higher-order solutions are possible, including rings and annuli of stationary points around the propeller annulus. Full numerical calculations confirm the accuracy of the asymptotic approach.

Time-domain analysis of contra-rotating propeller noise: wake interaction with a downstream propeller blade

“…Contours of constant P(r; x, t) for this case plotted against non-dimensional radius (vertical axis) and non-dimensional time (horizontal axis) with red curves indicating t n (r) = τ n + R L,τ n /c 0 superimposed (b). Note that panel (a) showing pressure versus time was also reproduced using the frequency-domain method described in Kingan & Parry (2019a). a reduction in tone levels.…”

“…It is important to check the validity of the numerical implementation of the time-domain method, even though the radiation formula of Najafi-Yazdi et al (2011) -on which our analysis is based -is well known. For this validation process we have compared the pressure time histories in the far field using both the method described in § 2.3 above and the frequency-domain method described by Hanson (1985) and Parry (1988) and utilised by Kingan & Parry (2019a) in their high blade number asymptotic approach. The frequency-domain method has been not only well documented but also well validated and well used in comparisons against model and even full-scale flight data (see Bradley 1986;Parry 1988Parry , 1997Parry & Crighton 1989;Hoff 1990;Ekoule et al 2017).…”

Acoustic theory of the many-bladed contra-rotating propeller: the event line and the precession of the interaction source

“…From the numerical point of view, Envia (2015) extended previous work on single-rotating propellers to the contra-rotating case, with the radiation expressions, as before, requiring a high-fidelity calculation of the local source region. For analytical and/or semi-numerical approaches, Kingan & Parry (2019) used the framework provided by the frequency-domain description of Hanson (1985), Parry (1988, 1997) and Parry & Crighton (1989 b ), regarding convected wake interactions, and extended the two-dimensional surface asymptotic approach of Parry (1995) to contra-rotating propellers, including the effects of blade sweep, quasi-three-dimensional blade unsteady response and sub- or supercritical wake–blade interactions. The analyses once again produced simplified, largely algebraic, results and showed that the noise is dominated in most cases by localised critical points on the surface of revolution described by the rear blade leading edges.…”

“…Kingan & Parry (2019) validated their asymptotic approach extensively by comparing their results with full numerical calculations – of a real engineering test case – over a range of combination frequencies and azimuthal modes, including the zero-mode case, and for both straight and swept propeller blades. For all cases, there was good agreement between the asymptotic results and the numerical calculations (that had, themselves, been validated via many previous comparisons with a wide range of experimental data), showing that the asymptotics could be applied to prediction and design (see Kingan & Parry, 2020 a ).…”

“…Most of these effects could be included through extensions to the approach. Despite these limitations, there are detailed discussions of the experimental validation cases in Kingan & Parry (2019) as well as of validation of the asymptotic approach via comparisons with full numerical calculations. To demonstrate the accuracy of the asymptotic (or many-bladed) approach, one of the comparisons from that paper is shown here, in figure 1, for the first interaction tone produced by a contra-rotating propeller with front and rear blade numbers of 12 and 9, respectively.…”