Complex Axial Wavenumbers Research Articles

A numerical method for modeling ultrasonic guided waves in thin-walled waveguides coupled to fluids

The paper presents a hybrid numerical method for the computation of the dispersion characteristics of ultrasonic guided waves propagating in isotropic and viscoelastic thin-walled waveguides in contact with fluids along their inner or outer wall. To this end, the solid waveguide is modeled by means of a Semi-Analytical Finite Element method implementing the first order shear deformation theory, while the effects of the fluid are taken into account by means of a regularized 2.5D Boundary Element Method with Helmholtz-type kernels. By coupling the two methods along the mid-surface of the solid waveguide, the dispersion relations of the elastic-acoustic system are obtained as the solution of a nonlinear eigenvalue problem in the complex axial wavenumbers for any fixed circular frequency. Two different case studies are presented: an oil-filled pipe and a water-loaded steel rectangular waveguide.

Dispersion analysis of leaky guided waves in fluid-loaded waveguides of generic shape

A fully coupled 2.5D formulation is proposed to compute the dispersive parameters of waveguides with arbitrary cross-section immersed in infinite inviscid fluids.The discretization of the waveguide is performed by means of a Semi-Analytical Finite Element (SAFE) approach, whereas a 2.5D BEM formulation is used to model the impedance of the surrounding infinite fluid. The kernels of the boundary integrals contain the fundamental solutions of the space Fourier-transformed Helmholtz equation, which governs the wave propagation process in the fluid domain. Numerical difficulties related to the evaluation of singular integrals are avoided by using a regularization procedure. To improve the numerical stability of the discretized boundary integral equations for the external Helmholtz problem, the so called CHIEF method is used.The discrete wave equation results in a nonlinear eigenvalue problem in the complex axial wavenumbers that is solved at the frequencies of interest by means of a contour integral algorithm. In order to separate physical from non-physical solutions and to fulfill the requirement of holomorphicity of the dynamic stiffness matrix inside the complex wavenumber contour, the phase of the radial bulk wavenumber is uniquely defined by enforcing the Snell–Descartes law at the fluid–waveguide interface.Three numerical applications are presented. The computed dispersion curves for a circular bar immersed in oil are in agreement with those extracted using the Global Matrix Method. Novel results are presented for viscoelastic steel bars of square and L-shaped cross-section immersed in water.

Finite element computation of elliptical vocal tract impedances using the two-microphone transfer function method

A two-microphone transfer function (TMTF) method is adapted to a numerical framework to compute the radiation and input impedances of three-dimensional vocal tracts of elliptical cross-section. In its simplest version, the TMTF method only requires measuring the acoustic pressure at two points in an impedance duct and the postprocessing of the corresponding transfer function. However, some considerations are to be taken into account when using the TMTF method in the numerical context, which constitute the main objective of this paper. In particular, the importance of including absorption at the impedance duct walls to avoid lengthy numerical simulations is discussed and analytical complex axial wave numbers for elliptical ducts are derived for this purpose. It is also shown how the direct impedance of plane wave propagation can be computed beyond the TMTF maximum threshold frequency by appropriate location of the virtual microphones. Virtual microphone spacing is also discussed on the basis of the so-called singularity factor. Numerical examples include the computation of the radiation impedance of vowels /a/, /i/, and /u/ and the input impedance of vowel /a/, for simplified vocal tracts of circular and elliptical cross-sections.

A 2.5D boundary element formulation for modeling damped waves in arbitrary cross-section waveguides and cavities

A regularized 2.5D boundary element method (BEM) is proposed to predict the dispersion properties of damped stress guided waves in waveguides and cavities of arbitrary cross-section. The wave attenuation, induced by material damping, is introduced using linear viscoelastic constitutive relations and described in a spatial manner by the imaginary component of the axial wavenumber. The discretized dispersive wave equation results in a nonlinear eigenvalue problem, which is solved obtaining complex axial wavenumbers for a fixed frequency using a contour integral algorithm. Due to the singular characteristics and the multivalued feature of the wave equation, the requirement of holomorphicity inside the contour region over the complex wavenumber plane is fulfilled by the introduction of the Sommerfeld branch cuts and by the choice of the permissible Riemann sheets. A post processing analysis is developed for the extraction of the energy velocity of propagative guided waves. The reliability of the method is demonstrated by comparing the results obtained for a rail and a bar with square cross-section with those obtained from a 2.5D Finite Element formulation also known in literature as Semi Analytical Finite Element (SAFE) method. Next, to show the potential of the proposed numerical framework, dispersion properties are predicted for surface waves propagating along cylindrical cavities of arbitrary cross-section. It is demonstrated that the attenuation of surface waves approaches asymptotically the attenuation of Rayleigh waves.

A coupled SAFE-2.5D BEM approach for the dispersion analysis of damped leaky guided waves in embedded waveguides of arbitrary cross-section

The paper presents a Semi-Analytical Finite Element (SAFE) formulation coupled with a 2.5D Boundary Element Method (BEM) for the computation of the dispersion properties of viscoelastic waveguides with arbitrary cross-section and embedded in unbounded isotropic viscoelastic media. Attenuation of guided modes is described through the imaginary component of the axial wavenumber, which accounts for material damping, introduced via linear viscoelastic constitutive relations, as well as energy loss due to radiation of bulk waves in the surrounding media. Energy radiation is accounted in the SAFE model by introducing an equivalent dynamic stiffness matrix for the surrounding medium, which is derived from a regularized 2.5D boundary element formulation.The resulting dispersive wave equation is configured as a nonlinear eigenvalue problem in the complex axial wavenumber. The eigenvalue problem is reduced to a linear one inside a chosen contour in the complex plane of the axial wavenumber by using a contour integral method. Poles of leaky and evanescent modes are obtained by choosing appropriately the phase of the wavenumbers normal to the interface in compliance with the nature of the waves in the surrounding medium. Finally, the obtained eigensolutions are post-processed to compute the energy velocity and the radiated wavefield in the surrounding domain.The reliability of the method is first validated on existing results for waveguides of circular cross sections embedded in elastic and viscoelastic media. Next, the potential of the proposed numerical framework is shown by computing the dispersion properties for a square steel bar embedded in grout and for an H-shaped steel pile embedded in soil.

Twisted absolute instability in lifted flames

The theory of resonant modes is extended to finite length systems containing pinch points of complex axial wavenumber k0 and frequency ω0 with arbitrary ωkk0=∂2ω/∂k2. The quantity ωkk0 is shown to be an important indicator of how streamwise boundary conditions modify the local absolute mode at (k0,ω0). In particular, when Im(ωkk0)>0, the pinch point is twisted, and resonant modes owing to streamwise boundary conditions may then have growth rates greater than that of the unbounded absolute mode. In this case, global instability may occur while the flow is only convectively unstable. The premixing zone between the nozzle and a lifted flame on a variable-density jet is an example of a streamwise-confined system containing a twisted pinch point. For this system, linear stability analysis is employed to locate resonant modes along a solution curve in the complex k and ω planes. The orientation of the solution curve predicts destabilization owing to streamwise confinement as well as increasing global frequency with decreasing lift-off height as observed in previous direct numerical simulations. The theory also suggests that low-frequency fluctuations observed in the simulations may be explained by beating between two resonant modes of slightly differing frequencies.

Analysis and design of pod silencers

Parallel baffle mufflers or split silencers are used extensively in heating, ventilation and air conditioning systems for increased attenuation of noise within a short or given length. Acoustic analysis of rectangular parallel baffle mufflers runs on the same lines as that of a rectangular duct lined on two sides. This simplification would not hold for circular configurations. Often, a cylindrical pod is inserted into a circular lined duct to increase its attenuation (or transmission loss), thereby making the flow passage annular and providing an additional absorptive layer on the inner side of this annular passage. This configuration, called a pod silencer, is analyzed here for the four-pole parameters as well as transmission loss, making use of the bulk reaction model. The effect of thin protective film or a highly perforated metallic plate is duly incorporated by means of a grazing-flow impedance. Use of appropriate boundary conditions leads to a set of linear homogeneous equations which in turn lead to a transcendental frequency equation in the unknown complex axial wave number. This is solved by means of the Newton–Raphson method, and the axial wave number is then used in the expressions for transmission loss as well as the transfer matrix parameters. Finally, results of a parametric study are reported to help the designer in optimization of a pod silencer configuration within a given overall size for minimal cost.

Spiral and Taylor vortex fronts and pulses in axial through flow.

The influence of an axial through flow on the spatiotemporal growth behavior of different vortex structures in the Taylor-Couette system with radius ratio eta=0.5 is determined. The Navier-Stokes equations (NSE) linearized around the basic Couette-Poiseuille flow are solved numerically with a shooting method in a wide range of through flow strengths Re and different rates of co-rotating and counter-rotating cylinders for toroidally closed vortices with azimuthal wave number m=0 and for spiral vortex flow with m=+/-1. For each of these three different vortex varieties we have investigated (i) axially extended vortex structures, (ii) axially localized vortex pulses, and (iii) vortex fronts. The complex dispersion relations of the linearized NSE for vortex modes with the three different m are evaluated for real axial wave numbers for (i) and over the plane of complex axial wave numbers for (ii) and (iii). We have also determined the Ginzburg-Landau amplitude equation (GLE) approximation in order to analyze its predictions for the vortex structures (ii) and (iii). Critical bifurcation thresholds for extended vortex structures are evaluated. The boundaries between absolute and convective instability of the basic state for vortex pulses are determined with a saddle-point analysis of the dispersion relations. Fit parameters for power-law expansions of the boundaries up to Re4 are listed in two tables. Finally, the linearly selected front behavior of growing vortex structures is investigated using saddle-point analyses of the dispersion relations of NSE and GLE. For the two front intensity profiles (increasing in positive or negative axial direction) we have determined front velocities, axial growth rates, and the wave numbers and frequencies of the unfolding vortex patterns with azimuthal wave numbers m=0,+/-1, respectively.

Inviscid instability of the Batchelor vortex: Absolute-convective transition and spatial branches

The main objective of the study is to examine the spatio-temporal instability properties of the Batchelor q-vortex, as a function of swirl ratio q and external axial flow parameter a. The inviscid dispersion relation between complex axial wave number and frequency is determined by numerical integration of the Howard–Gupta ordinary differential equation. The absolute-convective nature of the instability is then ascertained by application of the Briggs–Bers zero-group-velocity criterion. A moderate amount of swirl is found to promote the onset of absolute instability. In the case of wakes, transition from convective to absolute instability always takes place via the helical mode of azimuthal wave number m=−1. For sufficiently large swirl, co-flowing wakes become absolutely unstable. In the case of jets, transition from absolute to convective instability occurs through various helical modes, the transitional azimuthal wave number m being negative but sensitive to increasing swirl. For sufficiently large swirl, weakly co-flowing jets become absolutely unstable. These results are in good qualitative and quantitative agreement with those obtained by Delbende et al. through a direct numerical simulation of the linear response. Finally, the spatial (complex axial wave number, real frequency) instability characteristics are illustrated for the case of zero-external flow swirling jets.

PROPAGATING WAVES AND END MODES IN PRETWISTED BEAMS

A method is presented for studying propagating waves and end modes in a uniformly pretwisted beam. The variationally derived equations of motion are based on three-dimensional elasticity and finite element modelling of the cross-section. This discretization procedure accommodates arbitrarily shaped cross-sections of inhomogeneous anisotropic material properties that follow the pretwist rotation rate. An harmonic solution form in both time tand axial co-ordinate ζ is introduced, i.e., exp {i( kζ−ο t)}, resulting in a two parameter algebraic eigensystem. By specifying the axial wavenumber k, the eigenproblem permits real frequencies of propagating modes to be determined. By specifying real frequency ο, both real and complex axial wavenumbers can be extracted, where real values pertain to propagating modes and the complex ones to edge vibrations or end modes. Due to pretwist, the effect of extension, torsion and flexural are coupled. Examples of homogeneous, isotropic beams with rectangular and square cross-sections are given to illustrate the method of analysis and the physical behavior.

Wave propagation on coated cylindrical shells

This presentation describes methods of using an approximate elasticity formulation to compute natural wave numbers of waves in coated cylindrical shells. The efficient computation of these wave numbers is critical to coating design. A previously developed displacement-based variational formulation for coated shells [J. G. McDaniel and J. H. Ginsberg, J. Appl. Mech. 60, 463–469 (1993)] retains the accuracy of analytical formulations, but avoids the computational burdens associated with special functions of complex argument. This formulation, which was previously applied to two-dimensional problems, is extended to address wave propagation in the axial coordinate for each circumferential harmonic. Because the formulation is energy based, one has ready access to the strain energy distributions of each wave. For a specified real axial wave number, one obtains an easily solvable generalized eigenvalue problem for complex natural frequency. For a specified real frequency, the search for complex axial wave number is iterative. A fast algorithm for finding the complex axial wave numbers, which facilitates the design process, will also be discussed.

Electron density and collision frequency of microwave-resonant-cavity-produced discharges

A review of perturbation diagnostics applied to microwave resonant cavity discharges is presented. The classical microwave perturbation technique examines the shift in the resonant frequency and cavity quality factor of the resonant cavity caused by low-electron density discharges. However, the modifications presented allow the analysis to be applied to discharges with electron densities beyond the limit predicted by perturbation theory. An ‘‘exact’’ perturbation analysis is presented which models the discharge as a separate dielectric, thereby removing the restrictions on electron density imposed by the classical technique. The ‘‘exact’’ method also uses measurements of the shifts in the resonant conditions of the cavity. Third, an electromagnetic analysis is presented which uses a characteristic equation, based upon Maxwell’s laws, and predicts the discharge conductivity based upon measurements of a complex axial wave number. By allowing the axial wave number of the electromagnetic fields to be complex, the fields are experimentally and theoretically shown to be spatially attenuated. The diagnostics are applied to continuous-wave microwave (2.45 GHz) discharges produced in an Asmussen resonant cavity. Double Langmuir probes, placed directly in the discharge at the point where the radial electric field is zero, act as a comparison with the analytic diagnostics. Microwave powers ranging from 30 to 100 W produce helium and nitrogen discharges with pressures ranging from 0.5 to 6 Torr. Analysis of the data predicts electron temperatures from 5 to 20 eV, electron densities from 1011 to 3×1012 cm−3, and collision frequencies from 109 to 1011 s−1.

Extended reacting impedance characteristics of polyurethane foam in a duct with flow

Extensive use at the Boeing Company of Scottfelt, Grade 900, polyurethane foam as a relatively inexpensive test material for jet-engine nacelle acoustic studies prompted the development of a polyurethane-foam impedance math mode. A math model for an extended reacting foam liner in a duct with flow has been formulated, based on results of previous normal incidence impedance tube tests and math model studies reported at the 86th Meeting of the Acoustical Society of America. Using this model, predictions of the complex axial wavenumber are compared to measurements of firmness 4 Scottfelt in the Boeing/Wichita Grazing Flow Impedance Tube. Results for thicknesses of 1/2 and 1 in. at duct Mach numbers of 0, 0.35, and 0.5 are reported. The extended reacting impedance model has also been incorporated into a duct attenuation prediction program. Predicted attenuations are compared with polyurethane-foam flow duct test results as a further check on the math model.

Theoretical evaluation of sound fields in circular or annular ducts with rigid or nonrigid radial splitters

In aircraft engines, extra attenuation can be obtained by installing lined radial or circumferential splitters or both. A study of the governing wave equation and its solution in the lined radial splitter environment is presented. It is shown that the general solution can be written in the terms of modified Bessel and Neumann functions of complex fractional orders and complex arguments. This is due to the fact that the circumferential wavenumber is in general complex, noninteger, and a function of the radial coordinate. In the analysis, the circumferential wave vector is evaluated first as a function of the radial coordinate and in terms of the angular separation between the splitters and the frequency parameter; this function is then used in the radial integration, yielding the mode shapes and the corresponding complex axial wavenumbers. The model attenuations are higher for two reasons: the increased surface area of absorptive materials and the much higher cutoff frequencies of the modes. This analysis can be extended to the evaluation of sound field between blades.