In this section, the effects of including the hydrodynamic surface wave modes in the MM matching procedure are assessed. The example used is Case 4, but with a rigid bottom wall. The eigenvalues and axial wavenumbers of the surface wave modes are given in table (3.3), where the unstable hydrodynamic surface wave mode is included as a decaying mode in the left-running set of modes.
TABLE3.3: Surface wave mode solutions for Case 4 with rigid hub wall Surface wave type Transverse wavenumberµn Axial wavenumberαn
right-running acoustic 3.8848+i4.1272 10.5579-i1.1921 left-running acoustic 14.9543+i6.8932 -15.0240+i10.1510 unstable hydrodynamic 171.7040-i248.3906 254.8387+i178.5613
stable hydrodynamic 180.6213-i52.1542 -58.4615-i187.5200
A finite element solution was obtained using an over-specified mesh (116 by 568 elements). Mass-momentum mode matching solutions were obtained with and without the hydrodynamic surface wave modes included in the matching procedure. The casing pressures around the matching planes obtained from the three solutions are compared in figure (3.31). Away from the matching planes the mode-matching solution including the hydrodynamic surface wave modes provide an improved comparison with the FE solution. The radial pressure profiles for
the top half of the duct are compared in figure (3.32). When the hydrodynamic surface wave modes are included the pressure profiles compare very well with the FE solutions away from the lined wall, with a strong reduction in the high order mode oscillations in the lined segment pressures. The scattered modal intensities are compared in figure (3.33) and demonstrate that the inclusion of the hydrodynamic surface wave modes provides the best comparison with the FE solution. The reflected intensity is improved by around 7 percent, but the transmitted intensity is hardly affected.
These results demonstrate that the hydrodynamic surface wave modes must be included to accurately reconstruct the field. The highly wall localised behaviour of these modes means that they are most suitable in reconstructing the singular behaviour of the field at the wall around the liner interface planes. Since the mode matching can be interpreted simply as an inverse ‘source location’ problem [67], if the source or observer are near the wall, they are in the regime of the surface wave modes. So, as Rienstra and Tester [74] noted, if they are overlooked the computed field may not be converged. This can prove problematic for numerical solutions of the field by the FE method since a mesh that is highly refined at the walls and around the liner interface planes may be required to resolve the wall localised surface wave modes and their effect upon scattering, which has been noted by Hii [38]. This is not a problem for the analytic solutions presented here.
0 0.1 0.2 0.3 0.4 0.5 8 10 12 14 16 18 20 22 24 26 28 30 10.35 10.4 10.5 10.6 10.7 10.8 6 7 8 9 10 11 12 13 14 | p | | p | x x
FIGURE 3.31: Comparison of mass-momentum mode matching with and without hydrody- namic surface waves, and finite element casing pressure solutions at liner interface matching
planes. —, finite element solution; —, mode-matching without hydrodynamic surface wave
5 10 15 20 25 30 35 0 0.2 0.4 0.6 0.8 1 5 10 15 20 25 30 35 0 0.2 0.4 0.6 0.8 1 0 10 20 30 0 0.2 0.4 0.6 0.8 1 0 10 20 30 0 0.2 0.4 0.6 0.8 1 y y y y |p| |p| |p| |p|
(a)x=xI, lefthand segment modes (b)x=xI, righthand segment modes
(c)x=xII, lefthand segment modes (d)x=xII, righthand segment modes
FIGURE 3.32: Comparison of mass-momentum mode matching with and without hydrody- namic surface waves, and finite element radial pressure solutions at liner interface matching
planes. —, finite element solution; —, mode-matching without hydrodynamic surface wave
modes;—, mode-matching with hydrodynamic surface wave modes.
1 2 3 4 5 0 0.1 0.2 0.3 0.4 0.5 1 2 3 4 5 0 0.5 1 1.5x 10 −3 Itra n s [ W m − 2] Iref [ W m − 2] radial moden radial moden ACTRAN µhydro noµhydro
FIGURE 3.33: Comparison of scattered modal intensities from mass-momentum mode match- ing with and without hydrodynamic surface waves, and finite element method due to an inci- dent plane wave of unit intensity. Top plot, Transmitted modal intensity, Bottom plot, Reflected
3.8
Summary
• A mode-matching method has been developed to model the propagation of sound in rectangular ducts with uniform mean flow, and a finite length asymmetric wall lining. • By matching mass and axial momentum equations an extra matching term, related to the
singularity in the axial gradient of wall displacement, is found relative to the standard pressure-velocity matching
• The method can use a single-mode or multi-mode description of the sound source • The method was extended to include multiple liner segments of different lengths and
impedances.
• A Wiener-Hopf solution by Koch [81], for a uniform symmetric liner in the absence of mean flow was outlined.
• A comparison of mode-matching and Wiener-Hopf solutions in the absence of mean flow showed excellent agreement.
• A comparison of mode-matching results with solutions from a Finite Element method found very good agreement. It was found that a highly refined finite element mesh is required to adequately resolve the field around the matching planes.
Optimisation of bypass duct acoustic
liners
Rearward propagating fan noise has become an increasingly important turbofan engine noise source with the current trend of high bypass-ratio designs. The increased size of the fan and outlet guide vanes (OGV) leads to higher tonal and broadband source levels, which become dominant sources at takeoff and approach [82]. A key method for mitigating the rearward propagating rotor-stator interaction tones [6] and fan-OGV broadband noise [83] is the use of passive attenuating acoustic liners in the bypass duct walls. The placement and design of the liners must be such that the attenuation performance is maximised over a wide frequency range, for different operating conditions and source content. A number of analytic (e.g. ray acoustics, modal methods) and computational techniques (e.g. finite element, time or frequency domain LEE, finite difference, discontinuous Galerkin methods) for acoustic propagation in lined flow ducts are available for evaluating the liner attenuation performance. However, computational cost is a limiting factor in the use of computational methods in the optimisation process. In fact, finite element methods for the convected wave equation are currently the most mature of the available methods and have only very recently begun to be used in the liner optimisation process [84, 85]. Analytic methods have seen greater use, since they are computationally cheap, but are limited to highly idealised models [86, 32, 33, 25]. However, they can give good approximations of attenuation performance over wider frequency ranges, for multi-modal source descriptions and multiple liner segments within reasonable time scales.
TABLE4.1: ISVR one-sixth scale no flow bypass duct rig geometry. Total liner length llined/d 10
Hub radius h[m] 0.1191 Casing radius r[m] 0.1985 Annulus half height d[m] 0.0397 Hub-to-tip ratio ~ 0.6
Speed of sound c0[ms−1] 343
Density of air ρ0[kgm−3] 1.21
In this chapter, the optimisation of uniform and axially-segmented acoustic liners for the attenuation of fan noise in turbofan bypass ducts is demonstrated. Parts of this work have contributed to a Department for Trade and Industry (DTI) sponsored research project, Aircraft Noise Disturbance Alleviation by Novel TEchnology (ANDANTE), focusing on fan noise propagation and control in bypass ducts [87]. The liner attenuation performance is calculated using the pressure-velocity mode-matching method described in chapter (3). The geometry chosen for the following studies is identical to that of the one-sixth scale bypass duct rig used at the ISVR [88], the details of which are in table (4.1). The mode-matching model is a rectangular approximation to the actual annular geometry, and the duct widthbis determined from the average radiusRato be
b/d=πRa=
π(1 +~)
2 = 2.5133. (4.1)
The non-dimensional specific liner impedance is that for a single-degree-of-freedom liner, which is determined using a commonly used semi-empirical model [89], given by
Z =R+i[k0Mr−cot(k0D)] , (4.2)
whereRis the facing sheet resistance,Mris the non-dimensional facing sheet mass inertance,
andDis the non-dimensional cavity depth.