Theoretical models for duct acoustic propagation and radiation

The development of computational methods in acoustics has led to the introduction of analysis and design procedures which model the turbofan inlet as a coupled system, simultaneously modeling propagation and radiation in the presence of realistic internal and external flows. Such models are generally large, require substantial computer speed and capacity, and can be expected to be used in the final design stages, with the simpler models being used in the early design iterations. Emphasis is given to practical modeling methods that have been applied to the acoustical design problem in turbofan engines. The mathematical model is established and the simplest case of propagation in a duct with hard walls is solved to introduce concepts and terminologies. An extensive overview is given of methods for the calculation of attenuation in uniform ducts with uniform flow and with shear flow. Subsequent sections deal with numerical techniques which provide an integrated representation of duct propagation and near- and far-field radiation for realistic geometries and flight conditions

Eigenvalue of a semi-infinite elastic strip

A semi-infinite elastic strip, subjected to traction free boundary conditions, is studied in the context of in-plane stationary vibrations. By using normal (Rayleigh–Lamb) mode expansion the problem of existence of the strip eigenmode is reformulated in terms of the linear dependence within infinite system of normal modes. The concept of Gram's determinant is used to introduce a generalized criterion of linear dependence, which is valid for infinite systems of modes and complex frequencies. Using this criterion, it is demonstrated numerically that in addition to the edge resonance for the Poisson ratio ν=0, there exists another value of ν≈0.22475 associated with an undamped resonance. This resonance is best explained physically by the orthogonality between the edge mode and the first Lamé mode. A semi-analytical proof for the existence of the edge resonance is then presented for both described cases with the help of the augmented scattering matrix formalism

Simulation of Sound Absorption by Scattering Bodies Treated with Acoustic Liners Using a Time-Domain Boundary Element Method

Reducing aircraft noise is a major objective in the field of computational aeroacoustics. When designing next generation quiet aircraft, it is important to be able to accurately and efficiently predict the acoustic scattering by an aircraft body from a given noise source. Acoustic liners are an effective tool for aircraft noise reduction, and are characterized by a complex valued frequency-dependent impedance, Z(w). Converted into the time-domain using Fourier transforms, an impedance boundary condition can be used to simulate the acoustic wave scattering of geometric bodies treated with acoustic liners. This work uses an admittance boundary condition where the admittance, Y(w), is defined to be the inverse of impedance, i.e., Y(w) = 1/Z(w). An admittance boundary condition will be derived and coupled with a time domain boundary integral equation. The solution will be obtained iteratively using spatial and temporal basis functions and will allow for acoustic scattering problems to be modeled with geometries consisting of both unlined and soft surfaces. Stability will be demonstrated through eigenvalue analysis

Numerical Analysis of Three-dimensional Acoustic Cloaks and Carpets

We start by a review of the chronology of mathematical results on the Dirichlet-to-Neumann map which paved the way towards the physics of transformational acoustics. We then rederive the expression for the (anisotropic) density and bulk modulus appearing in the pressure wave equation written in the transformed coordinates. A spherical acoustic cloak consisting of an alternation of homogeneous isotropic concentric layers is further proposed based on the effective medium theory. This cloak is characterised by a low reflection and good efficiency over a large bandwidth for both near and far fields, which approximates the ideal cloak with a inhomogeneous and anisotropic distribution of material parameters. The latter suffers from singular material parameters on its inner surface. This singularity depends upon the sharpness of corners, if the cloak has an irregular boundary, e.g. a polyhedron cloak becomes more and more singular when the number of vertices increases if it is star shaped. We thus analyse the acoustic response of a non-singular spherical cloak designed by blowing up a small ball instead of a point, as proposed in [Kohn, Shen, Vogelius, Weinstein, Inverse Problems 24, 015016, 2008]. The multilayered approximation of this cloak requires less extreme densities (especially for the lowest bound). Finally, we investigate another type of non-singular cloaks, known as invisibility carpets [Li and Pendry, Phys. Rev. Lett. 101, 203901, 2008], which mimic the reflection by a flat ground.Comment: Latex, 21 pages, 7 Figures, last version submitted to Wave Motion. OCIS Codes: (000.3860) Mathematical methods in physics; (260.2110) Electromagnetic theory; (160.3918) Metamaterials; (160.1190) Anisotropic optical materials; (350.7420) Waves; (230.1040) Acousto-optical devices; (160.1050) Acousto-optical materials; (290.5839) Scattering,invisibility; (230.3205) Invisibility cloak

A comparison of the structureborne and airborne paths for propfan interior noise

A comparison is made between the relative levels of aircraft interior noise related to structureborne and airborne paths for the same propeller source. A simple, but physically meaningful, model of the structure treats the fuselage interior as a rectangular cavity with five rigid walls. The sixth wall, the fuselage sidewall, is a stiffened panel. The wing is modeled as a simple beam carried into the fuselage by a large discrete stiffener representing the carry-through structure. The fuselage interior is represented by analytically-derived acoustic cavity modes and the entire structure is represented by structural modes derived from a finite element model. The noise source for structureborne noise is the unsteady lift generation on the wing due to the rotating trailing vortex system of the propeller. The airborne noise source is the acoustic field created by a propeller model consistent with the vortex representation. Comparisons are made on the basis of interior noise over a range of propeller rotational frequencies at a fixed thrust

Modelling borehole wave signatures in elastic and poroelastic media with spectral method

Borehole sonic measurements are an important tool to characterize formation and completion properties of hydrocarbon or water reservoirs. Such measurements can provide direct information about rock physical parameters such as permeability or elastic moduli. These properties are obtained from guided waves propagating along boreholes. The so called tube wave or Stoneley wave is a symmetric mode which compresses the fluid column leading to a piston like motion. If the medium around the borehole wall is permeable, the radial expansion of the fluid column will result in fluid flow across the borehole wall. This results in a sensitivity of the tube wave signature to the permeability of the surrounding formation which manifests itself in a characteristic dispersion and attenuation of the tube wave. Information about the permeability of the surrounding formation provides essential knowledge for reservoir characterization.In addition to the traditional method of using tube wave signatures for formation permeability estimations, the same approach may be used for production monitoring. In sand reservoirs a complicated borehole completion is installed during the production phase for the purpose of controlling sand production. In such a setup highly permeable layers such as a sand screen or a gravel pack are used to prevent sand production.The problem with such completions is that they are very expensive to install and susceptible to plugging or corrosion. No permanent surveillance tool exists to date which allows diagnosis of problems in sand-screened deepwater completions. However, the recently proposed Real-Time Completion Monitoring (RTCM) uses the signature of tube waves to identify permeability changes: the increase of the tube wave velocity can indicate a decrease of permeability and vice versa. Therefore, RTCM has potential to identify problems in sand-screened deepwater completions.In order to understand the acoustic response of such deepwater completions, the dispersion and attenuation of tube waves in this complicated setup needs to be studied. To this end I have developed a modelling algorithm based on a spectral method. The developed algorithm computes the dispersion and attenuation of borehole modes propagating in a cylindrically layered structure with an arbitrary number of fluid, elastic and poroelastic layers. The numerical algorithm discretizes the medium along the radial axis using Chebyshev interpolation points derived from Chebyshev polynomials. The differential operators are discretized using spectral differentiation matrices. Thus, for any number of layers, the corresponding equations can be expressed as a generalized algebraic eigenvalue problem. For a given frequency, the eigenvalues correspond to the wavenumbers of different modes. The eigenvectors, computed along with the eigenvalues, correspond to the displacement potentials. They can be used to obtain the variation of displacement and stress components along the radius of the structure.In this thesis the spectral method was first developed for structures with an arbitrary number of fluid and elastic layers. Subsequently, the algorithm was extended for poroelasticity. The results produced by the modelling program are benchmarked against analytical solutions. Such analytical solutions are known for elastic and poroelastic cylinders as well as fluid filled tubes. The tube wave dispersion in a fluid-filled borehole surrounded by an elastic or poroelastic formation obtained with the spectral method was compared to the analytical low-frequency solution.I obtained the dispersion of the two tube waves propagating in a four layer completion model: fluid – permeable sand-screen – fluid – elastic casing. Varying the permeability of the sand-screen layer allowed me to account for the effect of fluid flow across this layer. Being able to obtain the acoustic response can help to identify broken fluid communication which increases the tube wave velocity. A corroded sand-screen has an extremely attenuated tube wave signature.Furthermore, I have implemented the more complex model of a borehole surrounded by an altered zone in the algorithm. Due to drilling damage the altered zone is an area of reduced permeability. In order to account for the effect of the altered zone on the tube wave signature, up to ten layers were used with stepwise increase of permeability from the borehole towards the formation. Overall, the spectral method proved to be a valuable algorithm to model wave propagation in cylindrical structures.Using borehole modes to evaluate the physical properties of the formation or completions is an important application. However, in borehole seismic modelling, such as crosshole or VSP, it is also important to account for the effect of boreholes and the associated modes. Since the borehole radius is a thousand times smaller then the investigated volume it would require a prohibitively small grid size to explicitly model the borehole. However, it is possible to effectively represent a borehole as a superposition of point sources. This mimics the presence of borehole modes. In order to implement this technique for poroelasticity, it is necessary to model source signatures in poroelastic media. To this end I have analyzed the radiation characteristics and moment tensor solutions for various source types. Together with the spectral method these point source representations can be used to model the effect of boreholes. This will pave the way for more efficient poroelastic seismic modelling in various fluid-filled boreholes and completions