R.J. Astley

Infinite elements for wave problems: a review of current formulations and an assessment of accuracy

Infinite element schemes for unbounded wave problems are reviewed and a procedure is presented forassessing their performance. A general computational scheme is implemented in which orthogonal functions are used for the transverse interpolation within the infinite element region. This is used as a basis for numerical studies of the effectiveness of various combinations of the radial test and trial functions which give rise to different conjugated and unconjugated formulations. Results are presented for the test case of a spherical radiator to which infinite elements are directly attached. Accuracy of the various schemes is assessed for pure multipole solutions of arbitrary order. Previous studies which have indicated that the conjugated and unconjugated schemes are more effective in the far and near fields, respectively, are confirmed by the current results. All of the schemes tested converge to the exact solution as radial order increases. All are however susceptible to ill conditioning. This places practical restrictions on their effectiveness at high radial orders. A close relationship is demonstrated between the discrete equations which arise from first-order infiniteelement schemes and those derived from the application of more traditional, local non-reflecting boundary conditions.

Theoretical model for sound radiation from annular jet pipes: Far- and near-field solutions

An analytical model is presented for sound radiation from a semi-infinite unflanged annular duct. The duct carries a jet which issues into a uniform mean flow while an inner cylindrical centre body extends downstream from the duct exit. This geometrical arrangement forms an idealized representation of a turbofan exhaust where noise propagates along the annular bypass duct, refracts through the external bypass stream and radiates to the far field. The instability wave of the vortex sheet and its interaction with the acoustic field are accounted for in an exact way in the current solution. Efficient numerical procedures are presented for evaluating near-field and far-field solutions, and these are used as the basis for a parametric study to illustrate the effect of varying the hub–tip ratio, and the ratio of jet velocity to external flow velocity. Since the ‘Kutta’ condition can be turned on and off in the current solution, this capability is used to assess the effect of vortex shedding on noise radiation. Far-field directivity patterns are presented for single modes and also for a multi-mode ‘broadband’ source model in which all cut-on modes are assumed to be present with equal modal power. Good agreement is found between analytical solutions and experimental data. Near-field pressure maps of the acoustic and instability portions of the solution are generated for selected tones.

Finite element formulations for acoustical radiation

Finite and infinite element techniques are applied to linear acoustical problems involving infinite anechoic boundaries. Theory is presented for a simple one dimensional model based on Websters horn equation. Results are then presented both for the one dimensional model and for two axisymmetric test cases. Comparisons with exact solutions indicate that both the infinite element and wave envelope schemes are effective in correctly predicting the near field. The wave envelope scheme is also shown to be capable of resolving the far field radiation pattern.

A finite element method for transmission in non-uniform ducts without flow: Comparison with the method of weighted residuals

A finite element method is developed for the study of transmission of sound in non-uniform ducts without flow. The formulation is based on a weighted residual approach and eight noded isoparametric elements are used. Two computational schemes are described, one based on the Helmholtz equation obtained by combining the basic conservation equations and one based on the conservation equations themselves. The latter case is considered because in future extensions to problems involving mean flow a single governing equation is not readily obtainable except for irrotational flows. Both two-dimensional and circular duct geometries are considered. Comparisons are made with a Method of Weighted Residuals in the form of a Modified Galerkin Method in the two-dimensional case to assess both accuracy and computational efficiency. It is found that the finite element method produces results for transmission and reflection coefficients nearly identical to those from the Galerkin approach. Used to its best advantage the finite element method is of comparable efficiency.

Acoustic transmission in non-uniform ducts with mean flow, part II: The finite element method

This second paper in a two part series describes the implementation of the finite element method for the solution of the problem of acoustic transmission through a non-uniform duct carrying a high speed subsonic compressible flow. A finite element scheme based on both the Galerkin method and the residual least squares method and with eight noded isoparametric elements is described. Multi-modal propagation is investigated by coupling of the solution in the duct non-uniform section to modal expansions in uniform sections. The accuracy of the finite element results for both the eigenvalue and transmission problems is assessed by comparison with exact solutions and with results from the method of weighted residuals in the form of a modified Galerkin method as introduced in Part I of this pair of papers. The results of calculations show that modal interactions, particularly in transmitted modes, become increasingly important with increasing duct flow Mach number. Power transmission coefficient calculations for the geometries studied reveal no indication of a linear basis for the phenomenon of subsonic acoustic choking.

A finite element scheme for attenuation in ducts lined with porous material: Comparison with experiment

A general finite element formulation for analysis of the sound field in a uniform flow duct with a bulk-reacting porous liner is given. Numerical results on rectangular ducts lined on all four sides are shown, and some of the lower order acoustic modes are described. Comparison is made between numerically predicted and measured values of modal axial attenuation rate, phase speed and transverse pressure profile both with and without mean airflow. Generally good agreement between prediction and measurement is observed.

A three-dimensional boundary element scheme for acoustic radiation in low mach number flows

A boundary element scheme is proposed for acoustical radiation in moving flows. The scheme relies upon a transformation, valid at low Mach numbers, which permits the case with flow to be restated as an equivalent problem, in the absence of flow but with modified boundary conditions. This restatement of the original problem is then amenable to boundary integral representation and boundary element solution. Results obtained by using this approach are presented for the test case of a pulsating or juddering sphere of finite dimensions in a uniform low Mach number flow. A fully three-dimensional boundary element model is used. The validity of the results is established by comparison with an analytic perturbation solution (where applicable) and by comparison with results from an alternative, axisymmetric numerical scheme.

Conditioning of infinite element schemes for wave problems

While a number of infinite element schemes have been implemented for time-harmonic unbounded wave problems in two and three dimensions, stability and conditioning of these schemes can limit the radial order at which they can be applied. In this paper, the choice of radial basis functions is shown to influence the condition number of such schemes. This effect is illustrated for three formulations; the Bettess-Burnett formulation, the conjugated Burnett formulation and the Astley-Leis formulation. Calculated values for the condition number are presented for infinite element schemes based on an orthogonal modal decomposition at the finite element/infinite element interface, and for more conventional infinite element schemes based on a transverse finite element discretization. Ill-conditioning can be avoided in the two conjugated formulations by the selection of a suitable radial basis. In the case of the Bettess-Burnett formulation, the condition number increases rapidly irrespective of the radial basis. The effect of the condition number on the convergence of the various schemes is also discussed.

A finite element formulation of the eigenvalue problem in lined ducts with flow

A finite element method is used to formulate the eigenvalue problem for a lined duct with flow. Either two dimensional or axially symmetric ducts with sheared flows can be studied, although the examples cited are two dimensional. The primitive variables of velocity and pressure are used with quadratic shape functions in each element. Results indicate that a useful level of accuracy can be achieved with a modest number of elements. Spurious eigenvalues, easily identified by obviously inconsistent eigenvectors, occur in certain instances. If the number of elements is not too small, these spurious modes are among the higher order eigenvalues of otherwise questionable accuracy. The possibility of using higher order elements which have slope continuity is proposed for future investigations to eliminate the spurious results.

Modelling the elastic properties of softwood part 2: The cellular microstructure

Numerical Finite Element models are presented which relate the macroscopic elastic properties of softwood to local cell characteristics such as cell size, wall thickness, moisture content and microfibril angle. Preliminary results show good agreement with reported values. The model is used to assess the effects of S2 microfibril angle and spiral grain on orthotropic wood stiffness, and to predict the stiffening effect of latewood bands.ZusammenfassungDer Beitrag präsentiert numerische FE-Modelle, welche die makroskopischen elastischen Eigenschaften von Nadelholz auf lokale Zellmerkmale wie Größe, Wanddicke, Feuchte und Winkel der Mikronbrillen zurückführen. Erste Ergebnisse zeigen gute Übereinstimmung mit Literturwerten. Mit Hilfe des Modells wird der Einfluß des Winkels der Mikronbrillen in der S2 und der Faserorientierung auf die Biegesteifigkeit des Holzes abgeschätzt sowie ein Versteifungseffekt der Spätholzzonen vorhergesagt.