The propagation of acoustic waves in a slowly varying duct with radially sheared axial mean flow

We consider the propagation of acoustic waves along a cylindrical duct carrying radially sheared axial mean flow, in which the duct radius is allowed to vary slowly along the axis. In previous work [A.J. Cooper & N. Peake, Journal of Fluid Mechanics 445 (2001) 207-234.] radially sheared axial mean flow with nonzero swirl in a slowly varying duct was considered, but in this paper we set the swirl to zero, thereby allowing simplification of the calculations of both the mean and unsteady flows. In this approach the acoustic wavenumber and corresponding eigenfunction are determined locally, while the wave amplitude is found by solving an evolution equation along the duct. Sample results are presented, including a case in which, perhaps surprisingly, the number of cut-on modes increases as the duct radius decreases. © 2013 Elsevier Ltd. All rights reserved.

Towards robust design of axial compressors with uncertainty quantification

Operational uncertainties such as throttle excursions, varying inlet conditions and geometry changes lead to variability in compressor performance. In this work, the main operational uncertainties inherent in a transonic axial compressor are quantified to deter- mine their effect on performance. These uncertainties include the effects of inlet distortion, metal expansion, ow leakages and blade roughness. A 3D, validated RANS model of the compressor is utilized to simulate these uncertainties and quantify their effect on polytropic efficiency and pressure ratio. To propagate them, stochastic collocation and sparse pseudospectral approximations are used. We demonstrate that lower-order approximations are sufficient as these uncertainties are inherently linear. Results for epistemic uncertainties in the form of meshing methodologies are also presented. Finally, the uncertainties considered are ranked in order of their effect on efficiency loss. © 2012 AIAA.

Sound transmission in strongly-curved slowly-varying cylindrical and annular lined ducts with flow

In this paper we consider the propagation of acoustic waves along a curved hollow or annular duct with lined walls. The curvature of the duct centreline and the wall radii vary slowly along the duct, allowing application of an asymptotic multiple scales analysis. This generalises Rienstra's analysis of a straight duct of varying cross-sectional radius. The result of the analysis is that the modal wavenumbers and mode shapes are determined locally as modes of a torus with the same local curvature, while the amplitude of the modes evolves as the mode propagates along the duct. The duct modes are found numerically at each axial location using a pseudo-spectral method. Unlike the case of a straight duct, there is a fundamental asymmetry between upstream and downstream propagating modes, with some mode shapes tending to be concentrated on either the inside or outside of the bend depending on the direction of propagation. The interaction between the presence of wall lining and curvature is investigated in particular; for instance, in a representative case it is found that the curvature causes the first few acoustic modes to be more heavily damped by the duct boundary than would be expected for a straight duct. Analytical progress can be made in the limit of very high mode order, in which case well-known 'whispering gallery' modes, localised close to the wall, can be identified.

Modal scattering at an impedance transition in a lined flow duct

An explicit Wiener-Hopf solution is derived to describe the scattering of duct modes at a hard-soft wall impedance transition in a circular duct with uniform mean flow. Specifically, we have a circular duct r = 1, - ∞ < x < ∞ with mean flow Mach number M > 0 and a hard wall along x < 0 and a wall of impedance Z along x > 0. A minimum edge condition at x = 0 requires a continuous wall streamline r = 1 + h(x, t), no more singular than h = Ο(x1/2) for x ↓ 0. A mode, incident from x < 0, scatters at x = 0 into a series of reflected modes and a series of transmitted modes. Of particular interest is the role of a possible instability along the lined wall in combination with the edge singularity. If one of the "upstream" running modes is to be interpreted as a downstream-running instability, we have an extra degree of freedom in the Wiener-Hopf analysis that can be resolved by application of some form of Kutta condition at x = 0, for example a more stringent edge condition where h = Ο(x3/2) at the downstream side. The question of the instability requires an investigation of the modes in the complex frequency plane and therefore depends on the chosen impedance model, since Z = Z (ω) is essentially frequency dependent. The usual causality condition by Briggs and Bers appears to be not applicable here because it requires a temporal growth rate bounded for all real axial wave numbers. The alternative Crighton-Leppington criterion, however, is applicable and confirms that the suspected mode is usually unstable. In general, the effect of this Kutta condition is significant, but it is particularly large for the plane wave at low frequencies and should therefore be easily measurable. For ω → 0, the modulus fends to |R001| → (1 + M)/(1 -M) without and to 1 with Kutta condition, while the end correction tends to ∞ without and to a finite value with Kutta condition. This is exactly the same behaviour as found for reflection at a pipe exit with flow, irrespective if this is uniform or jet flow.

Multimode radiation from an unflanged, semi-infinite circular duct with uniform flow.

Multimode sound radiation from an unflanged, semi-infinite, rigid-walled circular duct with uniform subsonic mean flow everywhere is investigated theoretically. The multimode directivity depends on the amplitude and directivity function of each individual cut-on mode. The amplitude of each mode is expressed as a function of cut-on ratio for a uniform distribution of incoherent monopoles, a uniform distribution of incoherent axial dipoles, and for equal power per mode. The directivity function of each mode is obtained by applying a Lorentz transformation to the zero-flow directivity function, which is given by a Wiener-Hopf solution. This exact numerical result is compared to an analytic solution, valid in the high-frequency limit, for multimode directivity with uniform flow. The high-frequency asymptotic solution is derived assuming total transmission of power at the open end of the duct, and gives the multimode directivity function with flow in the forward arc for a general family of mode amplitude distribution functions. At high frequencies the agreement between the exact and asymptotic solutions is shown to be excellent.

INVISCID BLADE-TO-BLADE PREDICTION OF A WAKE-GENERATED UNSTEADY FLOW.

A description is presented of a time-marking calculation of the unsteady flow generated by the interaction of upstream wakes with a moving blade row. The inviscid equations of motion are solved using a finite volume technique. Wake dissipation is modeled using an artificial viscosity. Predictions are presented for the rotor mid-span section of an axial turbine. Reasonable agreement is found between the predicted and measured unsteady blade surface static pressures and velocities. These and other results confirm that simple theories can be used to explain the phenomena of rotor-stator wake interactions.

METHOD FOR ASSESSING EFFECTS OF CIRCUMFERENTIAL FLOW DISTORTION ON COMPRESSOR STABILITY.

This paper describes the development of a new analysis to predict the onset of flow instability for an axial compressor operating in a circumferentially distorted inlet flow. A relatively simple model is used to examine the influence of various distortions in setting this instability point. It is found that the model reproduces known experimental trends for the loss of stability margin with increasing distortion amplitude and with changes in reduced frequency.

NUMERICAL STUDY OF THE 3D FLOWFIELD IN A TRANSONIC COMPRESSOR ROTOR WITH A MODELLING OF THE TIP CLEARANCE FLOW.

This paper describes a computer code aimed at solving the equations of three dimensional viscous compressible flow in turbomachine geometries. The code is applied to the study of the flowfield in a transonic axial compressor rotor at design speed at both maximum flow and towards stall. The predicted flowfield is compared with the laser measurements and the performance of the code discussed. In addition the discussion highlights the change in the predicted endwall and tip clearance flows as the rotor operating point is moved towards stall.

Experimental study of tip clearance flow in a radial inflow turbine

This paper describes an experimental investigation of tip clearance flow in a radial inflow turbine. Flow visualization and static pressure measurements were performed. These were combined with hot-wire traverses into the tip gap. The experimental data indicates that the tip clearance flow in a radial turbine can be divided into three regions. The first region is located at the rotor inlet, where the influence of relative casing motion dominates the flow over the tip. The second region is located towards midchord, where the effect of relative casing motion is weakened. Finally a third region exists in the exducer, where the effect of relative casing motion becomes small and the leakage flow resembles the tip flow behaviour in an axial turbine. Integration of the velocity profiles showed that there is little tip leakage in the first part of the rotor because of the effect of scraping. It was found that the bulk of tip leakage flow in a radial turbine passes through the exducer. The mass flow rate, measured at four chordwise positions, was compared with a standard axial turbine tip leakage model. The result revealed the need for a model suited to radial turbines. The hot-wire measurements also indicated a higher tip gap loss in the exducer of the radial turbine. This explains why the stage efficiency of a radial inflow turbine is more affected by increasing the radial clearance than by increasing the axial clearance.