Infinite Fluid Medium Research Articles

Coupling finite element and boundary element methods for the analysis of the acoustic scattering by elastic structures

Considerable interest has been expressed recently in the scattering of a plane acoustic wave by an elastic structure. In this paper, a method that couples the finite element method and a boundary element method is proposed. A finite element code (ATILA) is used to model the elastic structure and a boundary element code (EQI) to describe the propagating waves in the infinite fluid medium surrounding the body. The analysis is performed in three steps for each frequency: (i) the effect of the fluid is characterized by a nodal impedance matrix using BEM, (ii) the analysis of the structure using FEM including fluid effects via the impedance matrix provides the displacement field, (iii) from these displacements, the near-field and far-field pressures are obtained using BEM. The method is successfully applied to the scattering by spherical shells of various thicknesses for ka varying from 0 to 15 (k is the acoustic wave number in the fluid and a is the external radius of the sphere). The advantages of the method (analysis of complex geometries with various materials, direct knowledge of the displacement field, exact description of Sommerfeld’s radiation condition) and its drawbacks (irregular frequencies associated with BEM, limitation to medium values of ka) are discussed. [Work supported by D.R.E.T. Paris.]

Use of translational addition theorems for spherical wave functions in nonaxisymmetric acoustic field problems

The translational addition theorems for spherical wave functions are used to develop solutions for nonaxisymmetric acoustic field problems involving nonconcentric spherical boundaries. The forms of the addition theorems are discussed along with their use for translating modal expansions of spherical wave functions centered on one origin to another. In a typical problem involving a number of arbitrarily positioned spherical sources within a spherical‐shaped region, an infinite set of equations for the modal expansion coefficients is developed. This set must be truncated and then solved numerically. Assuming lossless media, comparison of the near‐field and far‐field‐radiated acoustic power provides a test of the accuracy of the numerical results. Other quantities of interest such as the radiation impedance of each source, including the mutual effects of all the other sources, and directivity patterns can be readily computed once a sufficient number of modal coefficients have been computed. In a companion paper [S. A. Lease and W. Thompson, Jr., J. Acoust. Soc. Am. 90, xxx–xxx (1991)] numerical results are presented for the specific case of two identical monopole sources within a fluid sphere that is embedded in another infinite fluid medium.

The Thermal Capacity Effect Upon Transient Natural Convection in a Rectangular Cavity

Transient natural convection in a rectangular enclosure is analyzed using a finite difference scheme. The enclosure is adiabatic and filled with water. The buoyancy induced flow is generated by a flat vertical uniform flux surface that has a finite thermal capacity. The full two-dimensional equations representing conservation of mass, momentum, and energy are solved in their time-dependent form. The solution technique used is a modified finite difference procedure very similar to Simple Arbitrary Lagrangian Eulerian (SALE) technique. Two values of surface thermal capacity are investigated, each resulting in a different flow regime during the transient. At short times a simple one-dimensional conduction regime is found to occur. As the leading edge effects arrive at any downstream location the conduction regime is terminated and true convection effects set in. At intermediate times a different flow regime is detected, namely a steady two-dimensional regime that approaches the steady state similarity solution for a similarly heated surface immersed in an infinite fluid medium. Excellent agreement is found with previous analyses and measurements during the early and intermediate transients.

Inelastic Response of an Infinite Cylindrical Shell to Transient Acoustic Waves

The geometrically and constitutively nonlinear response of an infinite, circular, cylindrical shell submerged in an infinite fluid medium to a transverse, transient acoustic wave is analyzed. Circumferential Fourier series solutions are obtained through the numerical integration of coupled ordinary differential equations and convolution integrals. Numerical results are presented in the form of response histories, response snapshots, and iso-damage curves for incident waves of rectangular pressure profile. Response solutions obtained with the first-order doubly asymptotic approximation are compared with their “exact” counterparts. It is found that doubly asymptotic approximations are unsuitable for two-dimensional, shock-response analysis of yielding submerged structures.

Response of a fluid‐filled spherical shell submerged in an infinite fluid medium to a transient acoustic wave

Transient response problems involving fluid‐filled shell structures submerged in infinite fluid media arise in various fields, including medicine, defense, and materials engineering. A canonical problem of this class possesses a spherical geometry, for which no solutions apparently exist. This paper delineates a relatively simple formulation and method of solution for this canonical problem and presents solutions for excitation by an incident step wave. The formulation begins with the familiar equations of motion for a thin spherical shell and the wave equation for the internal and external fluid domains. The Laplace transform is invoked, and the usual separation of variables method yields modal expressions involving Legendre polynomials and modified spherical Bessel functions. The explicit expressions for the latter are then manipulated to yield, after transform inversion, delayed differential equations in time for each response mode of the shell‐fluid system, which are integrated numerically in time. Complete response solutions then follow by modal superposition, with special techniques being employed to improve modal convergence. To validate the methodology, numerical results are first presented for an incident step wave exciting a shell with mass density and acoustic velocity identical to the corresponding properties characterizing the same internal and external fluid. Results are then shown for a step‐wave‐excited steel shell containing water and submerged in water. [Work supported by DNA.]

Transmission and reflection of acoustic waves through a composite slab using the finite element-eigenmode analysis

A numerical approach is presented for solving the interaction of acoustic waves with a composite slab that has periodically distributed substructures and is immersed in an infinite fluid medium. Reflection and transmission characteristics are studied for the case of normal plane-wave incidence. The mathematical model proposed combined the finite element method and eigenmode analysis, and can reduce the boundary value problem to a typical force vibration problem. Thus the damping of the composite slab can easily be introduced in the formulation. The results have been plotted as reflection and transmission coefficients versus frequency for different substructures and have been verified according to exact layer theory and two simple vibration models. Good agreement is observed. The presented analysis may be easily extended to the cases of oblique incidence and should be useful for future applications. [Work supported by the Research Center for the Engineering of Electronic and Acoustic Materials.]

Laminar natural convection heat transfer from sharp-edged horizontal bars with flow separation

Laminar natural convection over a sharp-edged horizontal bar situated in an infinite fluid medium has been investigated numerically and experimentally. Finite-difference solutions to the two-dimensional Navier-Stokes and energy equations were obtained with a fixed Prandtl number of 0.7 for the two configurations: a flat plate of finite thickness and a square bar. The difficulty associated with the complex physical flow domain was overcome by using the body-fitted coordinates. In the numerical study we found no indication of flow separation for the flat plate, in the range ofRayleigh number 10 3 ⩽ Ra ⩽ 10 5. For the square bar, however, the boundary layer separated easily at the upper sharp edges for Ra ⩾ 5 × 10 3 and well-defined twin vortices were identified above the upper horizontal surface. A Mach-Zehnder interferometric study was concurrently carried out in air for the square bar to determine the local temperature and Nusselt number distributions in the Rayleigh number range 1.95 × 10 4 ⩽ Ra ⩽ 1.53 × 10 5. Comparison of the two results, the numerical and the experimental, offered good agreement.

Numerical simulation of acoustic radiation from vibrating plates with slits.

The present paper deals with the problems of the steady-state radiated acoustic field generated by vibrating plates with narrow slits in an infinite fluid medium, and analytically and experimentaly investigates the effects of slit geometry on the characteristics of the acoustic field. The numerical analysis based on the Boundary Element Method, which can be applied to the Helmholtz integral equation, has been used to predict the acoustic radiation. The nodal displacements of the slit plates in the vibration problems calculated by the Finite Element Method have been transformed to the vibration velocity, which is utilized as boundary conditions in the B. E. analysis. Comparison with experimental results indicate that the numerical schemes employed are effective in correctly predicting the characteristics of acoustic radiation from vibrating arbitrarily shaped surfaces with slits.

Separated laminar natural convection above a horizontal isothermal square cylinder

Laminar natural convection over a horizontal isothermal square cylinder situated in an infinite fluid medium is studied by a finite difference technique. Difficulty associated with the complex physical domain is overcome by a body-fitted coordinate transformation. We found that there is boundary layer separation past the sharp edges of the upper horizontal surface of the square cylinder, in the range of Rayleigh number based on the one side length of a square cylinder greater than 5×10 3. Well-defined symmetric twin vortices exist above the upper horizontal surface, embedded at the base of the rising plume.

Dynamic analysis of prismatic structures surrounded by an infinite fluid medium

AbstractFor coupled vibration analyses of prismatic shell structures immersed in an infinite fluid medium, a composite element consisting of a semi‐analytical infinite fluid element and a cylindrical shell strip element is proposed in this paper. The behaviour in the infinite direction can be accurately modelled with minimum effort and great saving in computational cost is achieved using this element. Unlike many other methods which tend to take advantage of axisymmetry to simplify the analysis, this method may be used to analyse the dynamic behaviour of prismatic shell structures with arbitrary cross‐sections in offshore engineering.

Radiation and scattering from large axisymmetric structures in an infinite or semiinfinite fluid medium

A complex axisymmetric structure immersed in an infinite or semiinfinite fluid medium, excited by a plane acoustic wave or by a spherical or cylindrical acoustic wave emanating from a source in the vicinity of the structure or by a mechanical force acting on the structure, scatters and/or radiates the acoustic waves in the fluid medium. At a lower frequency range of excitation, vibrations of the structure and the radiating acoustic pressure strongly couples, whereas at higher frequency of excitation the structure can be regarded as rigid. A program called FIST (Fluid Interacting with STructures) is developed to analyze these problems. Using the Helmholtz integral, one can write a linear relation between the particle velocity normal to a cavity surface and the corresponding acoustic pressure. For efficiency and economy of calculations and for the consistency of velocity distribution between the structural elements and the corresponding fluid elements in contact, the distribution of pressure and velocity on...

Nonlinear Response of an Elastic Cylindrical Shell to a Transient Acoustic Wave

Governing equations are developed for the nonlinear response of an infinite, elastic, circular cylindrical shell submerged in an infinite fluid medium and excited by a transverse, transient acoustic wave. These equations derive from circumferential Fourier-series decomposition of the field quantities appearing in appropriate energy functionals, and from application of the “residual potential formulation” for rigorous treatment of the fluid-structure interaction. Extensive numerical results are presented that provide understanding of the phenomenology involved.

Radiation from a spherical acoustic source near a scattering sphere

The radiation from a spherical source vibrating with an arbitrary, axisymmetric, time-harmonic velocity distribution while positioned at an arbitrary point wholly outside a fluid sphere while both are embedded in another infinite fluid medium, is computed. The translational addition theorem for spherical wave functions is used to express series of wave modes centered at one sphere in terms of modes centered at the other, thereby facilitating the task of satisfying the boundary conditions. By letting the fluid sphere become either infinitely dense or infinitely compliant, the special cases of a spherical source radiating in the presence of a perfectly reflecting spherical scatterer are obtained. The source output and farfield directional response pattern are computed as a function of source and fluid sphere separation for some representative values of relative fluid density and sound speed. Numerous results are presented showing the modification of the radiation load on monopole and dipole sources of various wavelength sizes due to a nearby rigid or pressure release, spherical scatterer of the same size. The effect of the perfect scatterer is only important when the source and scatterer are very close and the pressure-release scatterer has considerably more effect than the rigid one. Subject Classification: [43]20.30, [43]20.55.

The echo signal of a finite spherical pulse from a fluid filled spherical shell: PMM vol. 40, n≗ 4, 1976, pp. 648–654

A method is proposed for determining the echo signal of a centrosymmetrical pressure pulse from a fluid filled elastic spherical shell immersed in an infinite fluid medium. The medium surrounding the shell and the fluid in it are considered to conform to the theory of perfect compressible fluid. Motion of the shell is defined in accordance with Timoshenko's linear theory of thin shells. The problem is solved with the use of Fourier's integral transformation in terms of time and Watson's transformation in terms of polar distance. Mechanisms of formation of the various echo signal components are described. The determination of the reflected component of echo signal and of its components radiated by creeping and peripheral waves from empty spherical shells was considered in [1]. The effect of the echo signal of waves which pass either a fluid or an elastic cylinder was analyzed in [2, 3] by the method of Watson's transformation. Here, the echo signal from a fluid filled spherical shell is investigated, taking into account besides the indicated echo signal components also the effect of waves which propagate partly in the shell, as peripheral waves, and partly as waves that pass through the fluid filler.

Finite−element approach to acoustic scattering from elastic structures

An approximate method is presented for solving the equations of motion that describe the scattering of an acoustic wave by an elastic structure immersed in an infinite fluid medium. The mathematical model that is developed uses the finite−element method to calculate the vibrational response of the elastic body and to calculate the acoustic pressure field of a finite volume of the fluid medium which closely surrounds the elastic body. Analytical methods are used to obtain the boundary conditions for this mathematical model. Results are presented for the total scattering cross section, axial pressure, and scattering patterns for an acoustic plane wave at normal incidence on an aluminum disk. Subject Classification: 30.30, 30.40.

Finite element approach to acoustic radiation from elastic structures

An approximation method is presented for solving the equations of motion that describe the vibration of an elastic structure immersed in an infinite acoustic fluid medium. The mathematical model that is developed uses the finite element method to calculate the vibrational characteristics of the elastic body and the acoustic pressure field of that portion of the fluid which closely surrounds the vibrating body. Analytical methods are used to obtain the boundary conditions for this mathematical model. This technique can be used to predict the response of an elastic structure over a wide range of frequencies and the acoustic pressure at a large number of field points in both the nearfield and farfields. It avoids the difficulties encountered at eigenvalues of the interior problem by determining exactly the one degree of freedom needed to overdetermine the surface equations. Experimental validation of theoretical predictions is given for a piezoelectric cylinder, and a computer-generated contour plot of a predicted nearfield pressure distribution is shown.

Acoustic radiation from a spherical source embedded eccentrically within a fluid sphere

The radiation from a spherical source vibrating with an arbitrary, axisymmetric, time-harmonic velocity distribution while positioned at an arbitrary point within a fluid sphere, which is embedded in another infinite fluid medium, is computed. This configuration is an idealization of a spherical acoustic lens, with focal point inside the lens, when used as a sound projector. The translational addition theorem for spherical wave functions is used to express series of wave modes centered at one sphere in terms of modes centered at the other, thereby facilitating the task of satisfying the boundary conditions. The radiation load on the source and the farfield directivity pattern are evaluated for representative values of the many parameters that characterize the problem, such as the wavelength size of the lens and source, the relative characteristic impedances and sound velocities of the two fluid media, the position of the source, and the velocity distribution on the source sphere. In general, no great increase in the radiation resistance is noted when the source is at the focal point of the lens. The focal points for monopole and dipole sources are usually not coincident. The size of the source has little effect upon the farfield directivity pattern.

Dynamic Stability of a Cylindrical Shell in an Acoustic Medium

A plane-strain investigation of the dynamic stability of an infinitely long cylindrical shell undergoing breathing oscillations in an infinite inviscid fluid medium is presented. Kinematic nonlinearities of the shell are retained while the fluid is regarded as a linear acoustic medium. A stability criterion which depends strongly upon the acoustic reactance is developed. When the shell is immersed in a fluid, the added mass of the fluid reduces the frequency for parametric resonance. The damping provided by the acoustic resistance proves to be negligible at parametric resonance. The validity of the neglect of kinematic nonlinearities of the fluid is demonstrated by means of an example.

Scattering of a Transient Acoustic Wave by an Elastic Cylindrical Shell

A plane acoustic step wave traveling through an infinite fluid medium is scattered by an infinite, elastic, circular cylindrical shell whose axis is parallel to the wavefront of the incident wave. The resulting transient acoustic field is studied through the use of temporal convolution techniques in conjunction with wave front analysis. Computed pressure histories are presented for various points in the fluid. The field produced by the cylindrical shell is compared with those produced by a fixed rigid cylinder and a cylindrical cavity. Attention is given to the nearfield pressure reduction capability of the cavity.

Free convection heat transfer between vertical parallel plates

A new mathematical model for the solution of the problem of free convection heat transfer between vertical parallel flat isothermal plates under isothermal boundary conditions, has been presented. The set of boundary layer equations used in the model are transformed to nonlinear coupled differential equations by similarity type variables as obtained by Ostrach for vertical flat plates in an infinite fluid medium. By utilising a parameter η w ∗ to represent the outer boundary, the governing differential equations are solved numerically for parametric values of Pr = 0.733. 2 and 3, and η w ∗ = 0.1, 0.5, 1, 2, 3, 4, ... and 8.0 . The velocity and temperature profiles are presented. Results indicate that η w ∗ can effectively classify the system into (1) thin layers where conduction predominates, (2) intermediate layers and (3) thick layers whose results can be predicted by the solutions for vertical flat plates in infinite fluid medium. Heat transfer correlations are presented for the 3 categories. Several experimental and analytical results available in the literature agree with the present correlations.