N. Peake

Propagation of unsteady disturbances in a slowly varying duct with mean swirling flow

The propagation of unsteady disturbances in a slowly varying cylindrical duct carrying mean swirling flow is described. A consistent multiple-scales solution for the mean flow and disturbance is derived, and the effect of finite-impedance boundaries on the propagation of disturbances in mean swirling flow is also addressed. Two degrees of mean swirl are considered: first the case when the swirl velocity is of the same order as the axial velocity, which is applicable to turbomachinery flow behind a rotor stage; secondly a small swirl approximation, where the swirl velocity is of the same order as the axial slope of the duct walls, which is relevant to the flow downstream of the stator in a turbofan engine duct. The presence of mean vorticity couples the acoustic and vorticity equations and the associated eigenvalue problem is not self-adjoint as it is for irrotational mean flow. In order to obtain a secularity condition, which determines the amplitude variation along the duct, an adjoint solution for the coupled system of equations is derived. The solution breaks down at a turning point where a mode changes from cut on to cut off. Analysis in this region shows that the amplitude here is governed by a form of Airys equation, and that the effect of swirl is to introduce a small shift in the location of the turning point. The reflection coefficient at this corrected turning point is shown to be exp (iπ/2). The evolution of axial wavenumbers and cross-sectionally averaged amplitudes along the duct are calculated and comparisons made between the cases of zero mean swirl, small mean swirl and O (1) mean swirl. In a hard-walled duct it is found that small mean swirl only affects the phase of the amplitude, but O (1) mean swirl produces a much larger amplitude variation along the duct compared with a non-swirling mean flow. In a duct with finite-impedance walls, mean swirl has a large damping effect when the modes are co-rotating with the swirl. If the modes are counter-rotating then an upstream-propagating mode can be amplified compared to the no-swirl case, but a downstream-propagating mode remains more damped.

Noise generation by the interaction between ingested turbulence and a rotating fan

An important criterion in the development of modern aeroengines is the identication of the dominant noise sources under typical aircraft take-o and approach conditions, and also in ground-based tests in which the engine is stationary. In this paper, we develop a theoretical model for unsteady distortion noise, which results from the interaction of ingested atmospheric turbulence with the rotating fan, with a view to providing a better understanding of the important physical mechanisms in this particular aspect of sound generation. The theory, developed in the frequency domain, is applicable for any arbitrary spectral form of atmospheric turbulence upstream of the fan, and as a simple model we take the von K arm an spectra for isotropic turbulence. The key fluid dynamical process in unsteady distortion is the deformation of turbulent eddies into long, narrow laments as they enter the engine, due to the strong streamtube contraction experienced by the steady, non-uniform mean flow generated by the fan. Simple models of the steady flow elds are provided for both open and ducted rotor geometries. The distorted turbulent eld at the fan face can be obtained using rapid distortion theory, and considerable simplication is made here by noting that the number of blades in typical aeroengine fans is large, allowing the application of asymptotic analysis and the derivation of closed-form expressions for those parts of the turbulence spectrum at the fan face which dominate the radiation. The unsteady forces exerted on the rotating fan blades are then calculated via a strip-theory approach. The resulting sound scattered to the far eld is then evaluated using asymptotic theory for open and ducted rotors. Results are presented in the form of frequency spectra for the turbulent eld at the fan face, the blade forces and the radiated sound for typical testing and aircraft operating conditions. High tonal noise levels are obtained under static conditions, whereas the sound is generally broadband in flight. The dependence on turbulence parameters such as the integral lengthscale is highlighted.

On sound generation by the interaction between turbulence and a cascade of airfoils with non-uniform mean flow

The sound generated by the interaction between a turbulent rotor wake and a stator is modelled by considering the gust response of a cascade of blades in non-uniform, subsonic mean flow. Previous work by Hanson & Horan (1998) that considers a cascade of flat plates at zero incidence is extended to take into account blade geometry and angle of attack. Our approach is based on the work of Peake & Kerschen (1997), who calculate the forward radiation due to the interaction between a single vortical gust and a cascade of flat plates at non-zero angle of attack. The extensions completed in this present paper are two-fold: first we include the effects of small but non-zero camber and thickness; and second we produce uniformly valid approximations which predict the upstream radiation near modal cut-off. The thin-airfoil singularity in the steady flow at each leading edge is crucial in our model of the sound generation. A new analytical expression for the coefficient of this singularity is derived via a sequence of conformal mappings, and it turns out that in our asymptotic limit this is the only quantity which needs to be calculated from the steady flow in order to predict time-averaged noise levels. Once the response to a single gust has been completed, we use Hanson & Horan (1998)s approach to determine the response to an incident turbulent spectrum, and find that as well as the noise corresponding to the auto-correlation of the gust velocity component normal to the blade, there is also a contribution from the cross-correlation of the normal and tangential velocities. Predictions are made of the effects of blade geometry on the upstream acoustic power level. The blade geometry can have a very significant effect on the noise generated by interaction with a single gust, with changes of up to 10 dB from the flat-plate noise levels. However, once these gust results have been integrated over a full incident turbulence spectrum the effects of the geometry are rather smaller, although still potentially significant, leading to changes of up to about 2 dB from the flat-plate results. The implication of all this is that the blade geometry can have a significant effect on the tonal noise components generated by rotor–stator interaction (i.e. by single harmonic gusts), but that the broadband part of the noise spectrum is relatively unaffected.

Algebraic and exponential instability of inviscid swirling flow

In this paper we consider the spectrum and stability properties of small-amplitude waves in three-dimensional inviscid compressible swirling flow with non-zero mean vorticity, contained in an infinitely long annular circular cylinder. The mean flow has swirl and sheared axial components which are general functions of radius. We describe the form of the spectrum, in particular the three distinct types of disturbance: sonic (or acoustic) modes; nearly-convected modes; and the non-modal continuous spectrum. The phenomenon of accumulation of infinitely many eigenvalues of the nearly-convected type in the complex wavenumber-plane is classified carefully: we find two different regimes of accumulating neutral modes and one regime of accumulating instability modes, and analytic conditions for the occurrence of each type of behaviour are given. We also discuss the Greens function for the unsteady field, and in particular the contribution made by the continuous spectrum. We show that this contribution can grow algebraically downstream, and is responsible for a new type of convective instability. The algebraic growth rate of this instability is a complicated function of the mean flow parameters, and can be arbitrarily large as a function of radius in cases in which the local convected wavenumber has a local extremum. The algebraic instability we describe is additional to any conventional modal instability which may be present, and indeed we exhibit cases which are convectively stable to modes, but which nevertheless grow algebraically downstream.

The stability and transition of the boundary layer on a rotating sphere

This paper is concerned with convective and absolute instabilities in the boundarylayer flow over the outer surface of a sphere rotating in an otherwise still fluid. Viscous and streamline-curvature eects are included and the analysis is conducted between latitudes of 10 and 80 from the axis of rotation. Both convective and absolute instabilities are found at each latitude within specic parameter spaces. The results of the convective instability analysis show that a crossflow instability mode is the most dangerous below =6 6. Above this latitude a streamline-curvature mode is found to be the most dangerous, which coincides with the appearance of reverse flow in the radial component of the mean flow. At low latitudes the disturbances are considered to be stationary, but at higher latitudes they are taken to rotate at 76% of the sphere surface speed, as observed in experimental studies. Our predictions of the Reynolds number and vortex angle at the onset of convective instability are consistent with existing experimental measurements. Results are also presented that suggest that the occurrence of the slowly rotating vortices is associated with the dominance of the streamline-curvature mode at =6 6. The local Reynolds number at the predicted onset of absolute instability matches experimental data well for the onset of turbulence at =3 0; beyond this latitude the discrepancy increases but remains relatively small below =7 0. It is suggested that this absolute instability may cause the onset of transition below =7 0. Close to the pole the predictions of each stability analysis are seen to approach those of existing rotating disk investigations.

Upstream-radiated rotor-stator interaction noise in mean swirling flow

A major component of the noise in modern aeroengines is rotor–stator interaction noise generated when the wake from the rotating fan impinges on a stator row downstream. An analytically based model for the prediction of upstream-radiated rotor–stator interaction noise is described, and includes the important effect of mean swirling flow on both the rotor wake evolution and the acoustic response. The analytic nature of the model allows for the inclusion of all wake harmonics and enables the response at all blade passing frequencies to be determined. An asymptotic analysis based on large rotor blade number is used to model the evolution of the rotor wake downstream in a cylindrical duct carrying mean swirling flow. The equations governing the axial evolution of the wake simplify to three coupled first-order differential equations in the interior, while close to the duct walls, a boundary-layer correction is required in order to satisfy the impermeability conditions at the boundaries. At the stator location, the wake is used as input into a local linear cascade model at each radius. The interaction of each wake harmonic gives rise to acoustic waves of multiple azimuthal order which contribute to the pressure field radiated back upstream. This enables the total acoustic response to be determined in terms of cylindrical duct modes in mean swirling flow. The effect of stator blade geometry (thickness, camber, angle of attack) and rotor–stator separation on the total upstream-radiated noise is determined. Blade geometry is shown to have a significant effect on the noise generated, and increasing the rotor–stator gap can lead to large reductions in noise levels. Asymptotic treatment of the acoustic field, based on large azimuthal order, is also considered and used to identify the dominant contributions to the total pressure field resulting from the rotor–stator interaction. The ray structure of the acoustic modes in swirl is shown to be very different in some cases from that in uniform flow.

Transient growth in vortices with axial flow

We investigate transient growth in high-Reynolds-number vortices with axial flow. Manycases of vortex instability are not fully explained by strong exponential instability modes, and transient growth could offer an alternative route to breakdown in such cases. Strong transient growth is found, in agreement with previous studies. We first discuss the problem by reference to ducted vortices which model aeroengine flow. The transient growth is inviscid in character, and in this paper we specifically interpret it as an effect of the inviscid continuous spectrum. The relevant inviscid theory explains new scalings which we find for the transient growth, which are generalizations of the quadratic scaling seen previously in two-dimensional flows and non-swirling pipe flows. We then turn to a second case, of interest for vortex breakdown, the Batchelor vortex, and present calculations of the transient growth. Large growth is possible, especially for the helical modes (with azimuthal wavenumber | m | = 1). The general trends are complicated by a number ofissues, including a long-wavelength effect and a resonance effect, both of which were recently discovered for a vortex without axial flow and are found here to be present in the Batchelor vortex also. Overall, the results suggest that strong transient effects are present in the moderate- to high-swirl regime of practical interest (swirl number q ≳ 2). Foraxisymmetric ( m = 0) and higher (| m | > 1) modes, however, transient effects are not found to be significant.

On the behaviour of a fluid-loaded cylindrical shell with mean flow

The unsteady behaviour of an infinitely long fluid-loaded elastic plate which is driven by a single-frequency point-force excitation in the presence of mean flow is known to exhibit a number of unexpected features, including absolute instability when the normalized flow speed, U, lies above some critical speed U 0 , and certain unusual propagation effects for U a 1 (U), but in practical situations it turns out that a 1 (U) is exceedingly large, and indeed seems much larger than radii of curvature achievable in engineering practice. Other negative-energy waves are seen to exist down to a smaller, but still very large, critical radius a 2 (U), while the existence of a real modal coalescence point, leading to a divergence in the driver admittance, occurs down to a slightly smaller critical radius a 3 (U). The transition through these various flow regimes as U and a vary is fully described by numerical investigation of the dispersion relation and by asymptotic analysis in the (realistic) limit of small U. The inclusion of plate dissipation is also considered, and, in common with Abrahams & Wickham (1994) for the flat plate, we show how the flow then becomes absolutely unstable at all flow speeds provided that a > a 2 (U).

Influence of mean loading on noise generated by the interaction of gusts with a flat-plate cascade: upstream radiation

The sound generated by the interaction between convected vortical and entropic disturbances and a blade row is a significant component of the total noise emitted by a modern aeroengine, and the blade geometry has an important effect on this process. As a first step in the development of a general prediction scheme, we model in this paper just the action of the blade mean loading by treating the blades as flat plates aligned at a non-zero incidence angle, δ, to the oncoming stream, and consider harmonic components of the incident field with reduced frequency k. We then use asymptotic analysis in the realistic limit k >> 1, δ << 1 with kδ = 0(1) to make a consistent asymptotic expansion of the compressible Euler equations. The flow is seen to consist of inner regions around each leading edge, in which sound is generated by the local gust-airfoil and gust-flow interactions, and an outer region in which both the incident gust is distorted according to rapid distortion theory and the out-going sound is refracted by the non-uniform mean flow. The complicated multiple interactions between the sound and the cascade are included to the appropriate asymptotic order, and analytical expressions for the forward radiation are derived. It is seen that even a relatively small value of δ can have a significant effect, thanks to both the O(δk 1/2 ) change in the amplitudes and the O(kδ) change in the phases of the various radiation components, corresponding to the additional source mechanisms associated with the flow distortion around each leading edge and the effects of propagation through the non-uniform flow, respectively. Further work will extend this analysis to include the effects of camber and thickness.

Acoustic propagation and scattering in the exhaust flow from coaxial cylinders

In this paper we present an analytical solution to the problem of sound radiation from semi-infinite coaxial cylinders, as a model for rearward noise emission by aeroengines. The cylinders carry uniform subsonic flows, whose Mach numbers may differ from each other and from that of the external flow. The incident field takes the form of a downstream-going acoustic mode in either the outer cylinder (the bypass flow) or the inner cylinder (the jet). The key geometrical ingredient of our problem is that the two open ends are staggered by a finite axial distance, so that the inner cylinder can be either buried upstream inside the outer cylinder, or can protrude downstream beyond the end of the outer cylinder (sometimes called the ‘half-cowl’ configuration). The solution is found by solving a matrix Wiener–Hopf equation, which involves the factorization of a certain matrix in the form − = + , with ± analytic, invertible and with algebraic behaviour at infinity in the upper and lower halves of the complex Fourier plane respectively. It turns out that the method of solution is different for the buried and protruding cases. In the buried case the well-known pole removal technique can be applied to a certain meromorphic function (denoted k 11 ), but in the protruding case the corresponding function k 22 is no longer meromorphic. Progress is made, however, by using a Pade representation of k 22 to yield a meromorphic problem which can then be solved using the pole removal technique as before. A range of results is presented, for both buried and protruding systems and with and without mean flow, and it becomes clear that the stagger of the two open ends can have a very significant effect on the far-field noise. We also obtain reasonable agreement between our predictions and some experimental results. One particular noise mechanism we identify in the presence of mean shear is the way in which a Kelvin–Helmholtz instability mode launched from the upstream trailing edge can be scattered into sound by its interaction with the downstream edge, provided that the separation between the edges is sufficiently large in a way which we identify.