Fluid Elastic Interface Research Articles

Application of elastic parabolic equation solutions to acoustic reverberation in ice-covered underwater environments

Parabolic equation solutions can be obtained for underwater acoustic environments where elastic boundaries affect the acoustic propagation. One example of this situation is in the Arctic where a (potentially rough) ice layer is on top of the water column. The parabolic equation method shown here rigorously applies the zero-traction and fluid-elastic interface boundary conditions to obtain full field Green’s function estimates near the ice-water interface. This solution is crucial for calculation of recently derived reverberation estimates that require both horizontal and vertical derivatives of the complex acoustic field. Monostatic reverberation estimates for a rough under-ice interface are calculated for an upward refracting water sound speed profile and an elastic ocean bottom with a nearly fluid mud layer. These calculations could be used, for example, to estimate areal distribution of the ice layer roughness from azimuth-time dependence of acoustic reverberation measured in Arctic environments. Performance of the Pade rational function approach in the presence of thin ice cover and low shear speed ocean bottom layers will be addressed.

A 3D common-refinement method for non-matching meshes in partitioned variational fluid–structure analysis

We present a three-dimensional (3D) common-refinement method for non-matching unstructured meshes between non-overlapping subdomains of incompressible turbulent fluid flow and nonlinear hyperelastic structure. The fluid flow is discretized using a stabilized Petrov–Galerkin method, and the large deformation structural formulation relies on a continuous Galerkin finite element method. An arbitrary Lagrangian–Eulerian formulation with a nonlinear iterative force correction (NIFC) coupling is achieved in a staggered partitioned manner by means of fully decoupled implicit procedures for the fluid and solid discretizations. To begin, we first investigate the accuracy of common-refinement method (CRM) to satisfy the traction equilibrium condition along the fluid-elastic interface with non-matching meshes. We systematically assess the accuracy of CRM against the matching grid solution by varying grid mismatch between the fluid and solid meshes over a tubular elastic body. We demonstrate the second-order accuracy of CRM through uniform refinements of fluid and solid meshes along the interface. We then extend the error analysis to transient data transfer across non-matching meshes between the fluid and solid solvers. We show that the common-refinement discretization across non-matching fluid–structure grids yields accurate transfer of the physical quantities across the fluid–solid interface. We next solve a 3D fluid–structure interaction (FSI) problem of a cantilevered hyperelastic plate behind a circular bluff body and verify the accuracy of coupled solutions with respect to the available solution in the literature. By varying the solid interface resolution, we generate various non-matching grid ratios and quantify the accuracy of CRM for the nonlinear structure interacting with a laminar flow. We illustrate that the CRM with the partitioned NIFC treatment is stable for low solid-to-fluid density ratio and non-matching meshes for the 3D FSI problem. Finally, we demonstrate the 3D parallel implementation of the common-refinement with the NIFC method for a realistic engineering problem of drilling riser undergoing complex vortex-induced vibration with strong added mass effects and turbulent wake flow.

Elastic parabolic equation solutions for oceanic T-wave generation and propagation from deep seismic sources.

Oceanic T-waves are earthquake signals that originate when elastic waves interact with the fluid-elastic interface at the ocean bottom and are converted to acoustic waves in the ocean. These waves propagate long distances in the Sound Fixing and Ranging (SOFAR) channel and tend to be the largest observed arrivals from seismic events. Thus, an understanding of their generation is important for event detection, localization, and source-type discrimination. Recently benchmarked seismic self-starting fields are used to generate elastic parabolic equation solutions that demonstrate generation and propagation of oceanic T-waves in range-dependent underwater acoustic environments. Both downward sloping and abyssal ocean range-dependent environments are considered, and results demonstrate conversion of elastic waves into water-borne oceanic T-waves. Examples demonstrating long-range broadband T-wave propagation in range-dependent environments are shown. These results confirm that elastic parabolic equation solutions are valuable for characterization of the relationships between T-wave propagation and variations in range-dependent bathymetry or elastic material parameters, as well as for modeling T-wave receptions at hydrophone arrays or coastal receiving stations.

COUPLED 3D WAVE EQUATIONS WITH IRREGULAR FLUID-ELASTIC INTERFACE: THEORETICAL DEVELOPMENT

A coupled three-dimensional fluid-elastic wave propagation mathematical model has been developed to handle environmental interactions in the ocean between a fluid medium and an elastic bottom. The existing model combines three-dimensional fluid and elastic wave propagation models with the incorporation of a set of horizontal fluid-elastic interface conditions. This paper extends the above model to consider an irregular fluid-elastic interface. The theoretical development of the irregular fluid-elastic interface equations is presented. Comparison of the fluid-elastic irregular interface to the horizontal case is one of the major objectives of this paper. Another major objective of this paper is the construction of a complete set of three-dimensional fluid-elastic wave equations, including the irregular interface, in a form suitable for stable numerical solution.

An irregular interface model for coupled fluid-elastic parabolic equations

In a realistic three-dimensional ocean environment involving both fluid and elastic media, a treatment of an irregular fluid-elastic interface is needed for the analysis of wave propagation. This paper introduces a model whcih possesses the necessary cpability. A complete mathematical decelopment of the model is presented with adaptable to the application of the parabolic equation method

A T-MATRIX PERTURBATION FORMALISM FOR SCATTERING FROM A ROUGH FLUID-ELASTIC INTERFACE

A perturbative representation of the null field T-matrix for scattering a pressure wave from a fluid elastic interface with surface roughness is developed. It is shown that thenth order T-matrix may be calculated recursively and expressed in terms of the elements of a zeroth order T-matrix. Further, it is shown that the expression for thenth order T-matrix is general in form and may be specialized to scattering from rigid and soft rough surfaces and from the rough surface of a fluid sediment. The T-matrix for thenth order spectral amplitude of the scattered pressure field in the fluid is calculated and its diagrammatic representation is constructed. For sinusoidal surface roughness, the perturbative representation of the T-matrix is used to calculate the spectral amplitudes of the Floquet modes of the pressure field scattered in the fluid. The results are compared with those obtained from a non-perturbative representation of the T-matrix and the accuracy and region of applicability of the formalism is determined. Then the relative scattering amplitudes of some of the individual scattering processes that occur in the diagrammatic representation of the T-matrix are calculated through second order. It is shown that diagrams that correspond to scalar processes in which surface pressure fields are excited and propagated tend to dominate diagrams that correspond to vector and tensor processes in which surface displacement fields are excited and propagated. A “scalar” approximation to the perturbation series is defined in which only scalar diagrams in the perturbation series are maintained. Numerical results for the perturbation series in the scalar approximation to fourth order are compared with exact results and it is shown that the scalar approximation provides reasonably accurate results for some applications.

A perturbation theory for scattering from targets near rough surface sediments: Numerical results

A null field T-matrix perturbation theory developed for plane-wave scattering from a fluid loaded elastic spherical shell near a rough fluid elastic interface [J. Acoust. Soc. Am. 101, 767–788 (1997)] is generalized to include scattering from spherical, spheroidal, and endcapped cylindrical targets near rough surface rigid, soft, fluid, and elastic sediments. Numerical results are obtained that demonstrate some of the effects of surface roughness on resonant and nonresonant target-boundary scattering.

Scattering from fluid-loaded elastic targets near penetrable sediments

A T-matrix formalism developed for plane-wave scattering from a fluid-loaded elastic spherical shell near a rough fluid elastic interface [J. Acoust. Soc. Am. 101, 767–788 (1997)] is generalized to include scattering from targets for which a T-matrix is known and constructed on a spherical wave basis, and from the interface of penetrable sediments for which a T-matrix is known and constructed on a Weyl plane-wave basis. A variety of numerical results are obtained for plane-wave scattering from elastic spheres and elastic spherical shells, as well as a finite elastic cylinder with hemispherical end caps near rigid, soft, fluid, elastic, poroelastic, and layered fluid sediments with planar boundaries that demonstrate some of the effects’ target-boundary scattering on resonant and nonresonant scattering.

A coupled 3D fluid-elastic wave propagation model: mathematical formulation and analysis

A three-dimensional mathematical model is developed for handling environmental interactions in the ocean between a fluid medium and an elastic bottom. The model is a combination of three-dimensional wave propagaion models in both the fluid and the elastic media, together with the incorporation of a set of fluid-elastic interface conditons. A complete theory with regard to the mathematical formulation of the fluid and elastic models is discussed, along with an error analyiss, and a description of the derivation of the fluid-elactic interface equations is given. A ordinary differential equation (ODE) method in conjunction with a finite difference marching scheme to obtain the numerical solution of the complete representative system of coupled fluid-elastic wave equations is outlined; moreover, the theory with regard to the stability and consistency of the marching scheme is discussed. The major development of this paper is the combination of the fluid equations, the elastic equations and the set of fluid-elastic interface conditions into a single unified three-dimensional model capable of applications to realistic fluid-elastic interactions in the ocean environment.

Scattering from fluid loaded targets in proximity to a sediment interface: Numerical results

A null-field T-matrix formalism developed for plane-wave scattering from a fluid-loaded elastic spherical shell in proximity to a rough fluid elastic interface [J. Acoust. Soc. Am. 101, 767–788 (1997)] is extended to include scattering from sound-hard and -soft spherical targets and a sound-hard finite cylinder with hemispherical end caps and, for a planar interface, sound-hard, sound-soft, and fluid-saturated poroelastic sediments. A variety of numerical results are obtained that demonstrate some of the effects of target and sediment parameters on target–boundary acoustic coupling and target strength. Then, for small amplitude surface roughness, a perturbation formalism is used to indicate some of the effects of interface roughness.

Scattering from an elastic shell and a rough fluid–elastic interface: Theory

A null-field T-matrix formalism is developed for scattering a pressure wave from a stationary elastic shell immersed in a homogeneous and isotropic fluid half-space and in proximity to a rough fluid–elastic interface. Helmholtz–Kirchhoff integral representations of the various scattered pressure and displacement fields are constructed. The surface fields are required to satisfy the elastic tensor boundary conditions and the scattered fields are required to satisfy the extended boundary condition. Spherical basis functions are used to construct a free-field T-matrix for the elastic shell and rectangular vector basis functions are used to construct a representation of the free-field T- matrix for the rough fluid–elastic interface. The free-field T-matrices are introduced into the Helmholtz–Kirchhoff and the null-field equations for the shell-interface system and a general system of equations for the spectral amplitudes of the various fields is obtained. The general system of equations is specialized to scattering from periodic surface roughness and an exact solution for the scattered pressure field in the fluid is obtained. Then the general system of equations is specialized to scattering from small-amplitude arbitrary roughness profiles and a perturbative solution is obtained. It is shown that the formalism contains multiple scattering effects on the rough surface and between the rough surface and the shell.

Scattering from a fluid‐loaded elastic spherical shell in proximity to a rough fluid–elastic interface: Numerical results

A null field T‐matrix formalism is developed and used to calculate scattering of a pressure wave from a fluid‐loaded elastic spherical shell in proximity to a rough fluid–elastic interface. The Helmholtz–Kirchhoff integral representations of the various scattered pressure and displacement fields are constructed. The surface fields are required to satisfy the elastic boundary conditions and the scattered fields are required to satisfy the extended boundary condition. The free‐field T matrices for the elastic shell and the rough fluid–elastic interface are constructed and used in the Helmholtz–Kirchhoff and the null field equations for the shell‐interface system. It is shown that the T matrix for the system is simply related to the free‐field T matrices for the shell and the rough fluid–elastic interface. A perturbative solution is obtained for scattering from arbitrary roughness, and an ‘‘exact’’ solution is obtained for scattering from periodic roughness. The ‘‘exact’’ solution is used to obtain a variety of numerical results that show some of the effects of the acoustic coupling that occurs between the rough interface and the elastic spherical shell.

A model for wave propagation in an inhomogeneous elastic medium

A perturbation method is used to derive equations for wave propagation in an inhomogeneous elastic medium. The formulation is three dimensional, and is written explicitly in cylindrical coordinates so that it can include property variations in the depth, range, and azimuth directions. A factorization is used to convert the equations into an operator form which distinguishes outgoing waves from incoming waves. This factorization removes the requirement for a far-field boundary condition. For large-scale numerical problems, the factorization produces a tremendous reduction in computation time and in memory requirements. The elastic equations are written in a way that is directly analogous to previous work in ocean acoustics. With appropriate fluid–elastic interface conditions, the elastic and acoustic equations may be used to study the fluid–elastic interactions which are important in shallow-water environments, especially the propagation of shear waves in the ocean bottom. In this paper, the theoretical development of the elastic equations are presented and their merits and applications are discussed. [This work was jointly supported by the U.S. Office of Naval Research and the U.S. Naval Undersea Warfare Center.]

Do pseudo-Scholte resonances exist?

In a pioneering work Talmant etal. [J. Acoust. Soc. Am. 86, 278 (1989)] established the presence of pseudo-Stonely resonances excited when acoustical signals scatter from shells at or near coincidence frequency. This notion was supported by the argument that Stoneley waves exist at the fluid–elastic interface of a plate when water is on one side and the other side is evacuated. It is known that nondispersive waterborne waves are excited at the fluid–water interface for that case in the frequency region around coincidence frequency. This coincided with the bounded shell case in which very narrow strong resonances corresponded with the waterborne waves and a broad envelope function corresponded with the onset of the flexural resonances which become manifest at coincidence frequency when the flexural waves become supersonic and thus radiated into the water. The envelope effect corresponds to an abrupt phase change of pressure at coincidence frequency. An analogous argument predicts the existence of pseudo-Scholte resonances. The analog for that case is a plate in which fluids of like properties exist on both sides. In that case waterborne waves exist over the entire frequency range. The implication is that if one scatters from such an object there should be a proliferation of waterborne waves and thus for closed shells an abundance of resonances associated with waterborne waves circumnavigating the shell should by present. Can the abundance of sharp peaks excited on fluid filled shells be explained in terms of this mechanism? This issue is examined and the question is answered. [Work sponsored by NRL and ONR.]

Time-frequency analysis of model based pulse scattering from a rough fluid-elastic interface

A null field T-matrix formalism for plane-wave scattering from a doubly infinite fluid–solid interface with doubly periodic surface roughness [Bishop and Smith, J. Acoust. Soc. Am. 94, 1560–1583 (1993)] is extended to include scattering of broadband acoustic pulses. The scattered pressure field is expressed in terms of a Fourier integral over the spectra of the incident pulse and the T matrix. Time series are calculated for the scatter of a Gaussian amplitude-modulated linear frequency-modulated pulse from a fluid–elastic interface with sinusoidal and triangular roughness. The time series are analyzed using short time Fourier, Wigner, wavelet packet, and local cosine T-F transforms. The results are displayed graphically in a T-F plot in which the T-F plane is covered by rectangular boxes of dimensions Δt×Δω. These results are used to evaluate the ability of each of the transforms to address inverse scattering and identification problems. It is shown that the T-F analysis provided by the wavelet packet may be used to distinguish between the scatter from surfaces of different roughness and is superior to the T-F analysis provided by both the short time Fourier and Wigner transforms.

A null field T‐matrix perturbation formalism for scattering from a fluid–elastic interface with random surface roughness

A null field perturbation formalism for scattering a pressure wave from a fluid–elastic interface with random surface roughness is developed. Helmholtz–Kirchhoff integral equations and the elastic tensor boundary conditions are used to represent the unknown scattered and surface pressure and displacement fields. The null field hypothesis is used to obtain a system of coupled integral equations for the surface fields. Perturbative representations of the scattered and surface fields and of the matrix elements of the Helmholtz–Kirchhoff integral equations are constructed and used to develop equations for the nth‐order spectral amplitudes of the unknown fields. It is shown that the nth‐order spectral amplitude of a scattered field is coupled to the nth‐order spectral amplitudes of all scattered fields and to all lower‐order spectral amplitudes of all surface fields. Consequently, the nth‐order T‐matrix may be calculated recursively and expressed in terms of the zeroth‐order off‐shell T‐matrix elements. The T matrix for the nth‐order spectral amplitude of the scattered pressure field in the fluid is calculated, and it is shown that scattering process is mediated by excitation of all possible intermediate surface states created by mode conversion of all lower‐order surface field states.

Three-dimensional scattering from a rough fluid-elastic interface

Out-of-plane scattering is of interest to the Arctic noise community because of the observation of significant out-of-plane motion in ice excited by explosive sources in the water column. Because low-frequency fluid-elastic scattering apparently couples energy into subsonic flexural waves as well as horizontally polarized shear (SH) waves in the ice, radiating in all directions, some impact on Arctic transmission loss and horizontal coherence properties is anticipated. The self-consistent boundary perturbation formulation for rough surface scattering developed by Kuperman and Schmidt [J. Acoust. Soc. Am. 86, 1511–1522 (1989)] is used to estimate the mean and scattered fields for a two-dimensionally rough fluid-elastic interface excited by an incoming plane wave. The reflection and scattering coefficients determined by the full 3-D formulation are compared to earlier results, where the surface roughness was assumed to be polarized in the plane of the exciting wave only. In addition, the kernel of the wave-number integral for the scattered field is examined and it is found that significant energy is scattered into Scholte waves as well as SH waves at all azimuths relative to the incoming plane wave, consistent with the experimental observations. [Work supported by ONR.]

A fourth-order accurate finite-difference scheme for the computation of elastic waves

AbstractA finite-difference model for elastic waves is introduced. The model is based on the first-order system of equations for the velocities and stresses. The differencing is fourth-order accurate on the spatial derivatives and second-order accurate in time. The model is tested on a series of examples including the Lamb problem, scattering from a plane interfaces and scattering from a fluid-elastic interface. The scheme is shown to be effective for these problems. The accuracy and stability is insensitive to the Poisson ratio. For the class of problems considered here, we find that the fourth-order scheme requires from two-thirds to one-half the resolution of a typical second-order scheme to give comparable accuracy.

Sound generation by turbulence near an elastic wall

The Lighthill theory has been extended to describe the sound generated by turbulence near an elastic wall. The case of a thin elastic slab with identical fluid in contact with both faces is investigated in detail by solving the elastic equations in the slab together with the acoustic boundary-layer equations in the fluid, with all stresses and displacements continuous across the fluid-elastic interface. The low-wavenumber elements of the pressure spectrum under the plate are determined, and it is found that the surface pressure spectrum has considerable structure which is not predicted by simple bending plate theory. This is because admitting that the elastic medium can support general elastic deformations introduces new modes of vibration that are not present in the simple plate theory. In addition to the flexural waves displayed by bending plate theory the elastic region can support a symmetric mode which propagates with the compressive wave speed √ E ϱ M (1 − ν 2) , where E is Young's modulus, ν is Poisson's ratio and ϱ M is the density of the elastic medium. The surface pressure spectrum under the plate is found to have a local maximum for spectral elements whose phase speeds are equal to either the compressive or the flexural wave speeds. The spectrum of the fluctuating shear stress is also investigated. The low-wavenumber spectrum of the surface shear stress is found to have considerable structure with peaks for spectral elements whose phase speeds are nearly equal to speed of sound in the fluid, to the compressive wave speed in the plate and to the bending wave speed if it is subsonic.