Sw Sjoerd Rienstra

A classification of duct modes based on surface waves

Abstract For the relatively high frequencies relevant in a turbofan engine duct, the modes of a lined section may be classified in two categories: genuine acoustic 3D duct modes resulting from the finiteness of the duct geometry, and 2D surface waves that exist only near the wall surface in a way essentially independent of the rest of the duct. Per frequency and circumferential order there are at most four surface waves. They occur in two kinds: two acoustic surface waves that exist with and without mean flow, and two hydrodynamic surface waves that exist only with mean flow. The number and location of the surface waves depends on the wall impedance Z and mean flow Mach number. When Z is varied, an acoustic mode may change via small transition zones into a surface waves and vice versa. Compared to the acoustic modes, the surface waves behave—for example as a function of the wall impedance—rather differently as they have their own dynamics. They are therefore more difficult to find. A method is described to trace all modes by continuation in Z from the hard-wall values, by starting in an area of the complex Z -plane without surface waves.

Impedance Models in Time Domain including the Extended Helmholtz Resonator Model

The problem of translating a frequency domain impedance boundary condition to time domain involves the Fourier transform of the impedance function. This requires extending the definition of the impedance not only to all real frequencies but to the whole complex frequency plane. Not any extension, however, is physically possible. The problemshouldremain causal, the variables real, andthe wall passive. This leads to necessary conditions for an impedance function. Various methods of extending the impedance that are available in the literature are discussed. A most promising one is the so-called z-transform by Ozyoruk & Long, which is nothing but an impedance that is functionally dependent on a suitable complex exponent e −iωκ . By choosing κ a multiple of the time step of the numerical algorithm, this approach fits very well with the underlying numerics, because the impedance becomes in time domain a delta-comb function and gives thus an exact relation on the grid points. An impedance function is proposed which is based on the Helmholtz resonator model, called Extended Helmholtz Resonator Model. This has the advantage that relatively easily the mentioned necessary conditions can be satisfied in advance. At a given frequency, the impedance is made exactly equal to a given design value. Rules of thumb are derived to produce an impedance which varies only moderately in frequency near design conditions. An explicit solution is given of a pulse reflecting in time domain at a Helmholtz resonator impedance wall that provides some insight into the reflection problem in time domain and at the same time may act as an analytical test case for numerical implementations, like is presented at this conference by the companion paper AIAA-2006-2569 by N. Chevaugeon, J.-F. Remacle and X. Gallez. The problem of the instability, inherent with the Ingard-Myers limit with mean flow, is discussed. It is argued that this instability is not consistent with the assumptions of the Ingard-Myers limit and may well be suppressed.

ACOUSTIC RADIATION FROM A SEMI-INFINITE ANNULAR DUCT IN A UNIFORM SUBSONIC MEAN FLOW

Using a Wiener-Hopf approach, ain analytical description is derived of the scattered field of a harmonic sound wave coming out of an open ended annular duct (a semi-infinite cylinder inside of which, coaxially, a doubly infinite hub), submerged in a subsonic, coaxial, uniform mean flow. The possibility of vortex shedding from the pipe exit edge is included.Explicit expressions are given of the acoustic power inside the pipe, in the acoustic far field, and, in the presence of vortex shedding, in the hydrodynamic far field and of the power absorbed by the vortex sheet. The formulae are evaluated with the aid of asymptotic expansions, and a method utilizing complex contour deformation, more convenient than those usually employed for this type of diffraction problems. The equality of power appeared to be an important check on the calculations. A numerical survey is made of the behaviour of the acoustic power loss, due to vortex shedding from the trailing edge, at frequencies near cut-off, as a function of Mach number, mode number of the incident wave, and hub radius. The power loss appears to increase with increasing Mach number, increasing hub radius and with decreasing frequency. Only in case of the plane wave (where k-K)) the ratio of radiated and transmitted power becomes zero, for the other modes (at their cutoff frequencies) this ratio tends to a finite value. Somewhat surprising is that, in comparison with the jet, the power loss in a uniform flow is much higher. As a typical example for higher frequencies, the far field radiation pattern of a k=50, m=U wave is considered as a function of Kutta condition and hub radius.

Boundary-layer thickness effects of the hydrodynamic instability along an impedance wall

The Ingard―Myers condition, modelling the effect of an impedance wall under a mean flow by assuming a vanishingly thin boundary layer, is known to lead to an ill-posed problem in time domain. By analysing the stability of a linear-then-constant mean flow over a mass-spring-damper liner in a two-dimensional incompressible limit, we show that the flow is absolutely unstable for h smaller than a critical h c and convectively unstable or stable otherwise. This critical h c is by nature independent of wavelength or frequency and is a property of liner and mean flow only. An analytical approximation of h c is given, which is complemented by a contour plot covering all parameter values. For an aeronautically relevant example, h c is shown to be extremely small, which explains why this instability has never been observed in industrial practice. A systematically regularised boundary condition, to replace the Ingard―Myers condition, is proposed that retains the effects of a finite h, such that the stability of the approximate problem correctly follows the stability of the real problem.

A numerical comparison between the multiple-scales and finite-element solution for sound propagation in lined flow ducts

An explicit, analytical, multiple-scales solution for modal sound transmission through slowly varying ducts with mean flow and acoustic lining is tested against a numerical finite-element solution solving the same potential flow equations. The test geometry taken is representative of a high-bypass turbofan aircraft engine, with typical Mach numbers of 0.5–0.7, circumferential mode numbers m of 10–40, dimensionless wavenumbers of 10–50, and both hard and acoustically treated inlet walls of impedance Z = 2 − i. Of special interest is the presence of the spinner, which incorporates a geometrical complexity which could previously only be handled by fully numerical solutions. The results for predicted power attenuation loss show in general a very good agreement. The results for iso-pressure contour plots compare quite well in the cases where scattering into many higher radial modes can occur easily (high frequency, low angular mode), and again a very good agreement in the other cases.

Sound propagation in slowly varying lined flow ducts of arbitrary cross-section

Sound transmission through ducts of constant cross-section with a uniform inviscid mean flow and a constant acoustic lining (impedance wall) is classically described by a modal expansion, where the modes are eigenfunctions of the corresponding Laplace eigenvalue problem along a duct cross-section. A natural extension for ducts with cross-section and wall impedance that are varying slowly (compared to a typical acoustic wavelength and a typical duct radius) in the axial direction is a multiple-scales solution. This has been done for the simpler problem of circular ducts with homentropic irrotational flow. In the present paper, this solution is generalized to the problem of ducts of arbitrary cross-section. It is shown that the multiple-scales problem allows an exact solution, given the cross-sectional Laplace eigensolutions. The formulation includes both hollow and annular geometries. In addition, the turning point analysis is given for a single hard-wall cut-on, cut-off transition. This appears to yield the same reflection and transmission coefficients as in the circular duct problem.

A small Strouhal number analysis for acoustic wave-jet flow-pipe interaction

Asymptotic expansions for small Strouhal number, valid for arbitrary subsonic Mach number, are derived for the solution of a simple problem of the interaction between an acoustic wave, a jet flow and a pipe, based on Munts (1977) exact formal solution. These expansions relate to the pressure and velocity fluctuations in the jet flow and in the far field, and to the reflection coefficient, end-impedance and end-correction for the reflected wave in the pipe. In particular the influence of a Kutta condition at the lip of the pipe is shown to be highly significant.

Mode-matching strategies in slowly-varying engine ducts

A matching method is proposed to connect the computational fluid dynamics (CFD) source region to the computational aeroacoustics propagation region of rotor-stator interaction sound produced in a turbofan engine. The method is based on a modal decomposition across three neighbouring axial interfaces adjacent to the matching interface. The modal amplitudes are determined by a least-squares fit. When slowly varying modes are taken, the interface may be positioned in a duct section of varying cross section. Furthermore, the spurious reflections back into the CFD domain, which result from imperfect reflection-free CFD boundary conditions, can be filtered out by including both left- and right-running modes in the matching. Although the method should be applicable to a wider range of acoustic models, it is implemented and favourably tested for the recently available relatively simple case of slowly varying modes in homentropic potential flow in lined ducts. Homentropic potential flow is a very relevant model for the inlet side and a good model for the bypass side if swirl or other types of vorticity are not dominant in the mean flow. By matching with density or pressure perturbations, any contamination of residual nonacoustical vorticity is avoided.

Spatial Instability of Boundary Layer Along Impedance Wall

A numerical analysis is made of the hydro-acoustical spatial instability, apparently occurring in a mean flow with thin boundary layer along a locally reacting lined d uct wall. This problem is of particular interest because unstable behaviour of liner and mean flow has been obs erved only very rarely. It is found that this instability quickly disappears for inc reasing boundary layer thickness. Specifically, for boundary-layer-thickness based Helmholtz numbers !�/c0 of the order of 0.1 the growth rate vanishes and the instability disappears. This corresponds to very thin boundary layers for practical values of frequencies that occur in aero-engine applications, which is in turn in good agreement with the fact that in industrial practice no instabilities are observed.

Kelvin-Helmholtz instabilities occuring at a Nacelle exhaust

The difference of flow velocity between the free stream and the jet stream at a nacelle exhaust generates a shear layer where some hydrodynamic instabilities, called Kelvin-Helmholtz instabilities, can occur. Some instabilities occur in the shear layer in the computations with Actran/DGM a time domain code which solves the LEE equations (see,1,2 and3) but it is often difficult to conclude whether they are physical or not. This document recalls the theory on the Kelvin-Helmholtz instabilities and gives guidelines to sort out the physical and numerical instabilities. This methodology is put into practice on concrete cases studied in the frame of the TURNEX project.