Rigid Elliptical Cylinder Research Articles

Comment on “Radiation forces and torque on a rigid elliptical cylinder in acoustical plane progressive and (quasi)standing waves with arbitrary incidence” [Phys. Fluids 28, 077104 (2016)

Comment on “Radiation forces and torque on a rigid elliptical cylinder in acoustical plane progressive and (quasi)standing waves with arbitrary incidence” [Phys. Fluids 28, 077104 (2016)

Radiation forces and torque on a rigid elliptical cylinder in acoustical plane progressive and (quasi)standing waves with arbitrary incidence

This paper presents two key contributions; the first concerns the development of analytical expressions for the axial and transverse acoustic radiation forces exerted on a 2D rigid elliptical cylinder placed in the field of plane progressive, quasi-standing, or standing waves with arbitrary incidence. The second emphasis is on the acoustic radiation torque per length. The rigid elliptical cylinder case is important to be considered as a first-order approximation of the behavior of a cylindrical fluid column trapped in air because of the significant acoustic impedance mismatch at the particle boundary. Based on the rigorous partial-wave series expansion method in cylindrical coordinates, non-dimensional acoustic radiation force and torque functions are derived and defined in terms of the scattering coefficients of the elliptic cylinder. A coupled system of linear equations is obtained after applying the Neumann boundary condition for an immovable surface in a non-viscous fluid and solved numerically by matrix inversion after performing a single numerical integration procedure. Computational results for the non-dimensional force components and torque, showing the transition from the progressive to the (equi-amplitude) standing wave behavior, are performed with particular emphasis on the aspect ratio a/b, where a and b are the semi-axes of the ellipse, the dimensionless size parameter, as well as the angle of incidence ranging from end-on to broadside incidence. The results show that the elliptical geometry has a direct influence on the radiation force and torque, so that the standard theory for circular cylinders (at normal incidence) leads to significant miscalculations when the cylinder cross section becomes non-circular. Moreover, the elliptical cylinder experiences, in addition to the acoustic radiation force, a radiation torque that vanishes for the circular cylinder case. The application of the formalism presented here may be extended to other 2D surfaces of arbitrary shape, such as Chebyshev cylindrical particles with a small deformation, stadiums (with oval shape), or other non-circular geometries.

Acoustic backscattering and radiation force on a rigid elliptical cylinder in plane progressive waves

This work proposes a formal analytical theory using the partial-wave series expansion (PWSE) method in cylindrical coordinates, to calculate the acoustic backscattering form function as well as the radiation force-per-length on an infinitely long elliptical (non-circular) cylinder in plane progressive waves. The major (or minor) semi-axis of the ellipse coincides with the direction of the incident waves. The scattering coefficients for the rigid elliptical cylinder are determined by imposing the Neumann boundary condition for an immovable surface and solving a resulting system of linear equations by matrix inversion. The present method, which utilizes standard cylindrical (Bessel and Hankel) wave functions, presents an advantage over the solution for the scattering that is ordinarily expressed in a basis of elliptical Mathieu functions (which are generally non-orthogonal). Furthermore, an integral equation showing the direct connection of the radiation force function with the square of the scattering form function in the far-field from the scatterer (applicable for plane waves only), is noted and discussed. An important application of this integral equation is the adequate evaluation of the radiation force function from a bistatic measurement (i.e., in the polar plane) of the far-field scattering from any 2D object of arbitrary shape. Numerical predictions are evaluated for the acoustic backscattering form function and the radiation force function, which is the radiation force per unit length, per characteristic energy density, and per unit cross-sectional surface of the ellipse, with particular emphasis on the aspect ratio a/b, where a and b are the semi-axes, as well as the dimensionless size parameter kb, without the restriction to a particular range of frequencies. The results are particularly relevant in acoustic levitation, acousto-fluidics and particle dynamics applications.

Acoustic radiation force on a rigid elliptical cylinder in plane (quasi)standing waves

The acoustic radiation force on a 2D elliptical (non-circular) cylinder centered on the axis of wave propagation of plane quasi-standing and standing waves is derived, based on the partial-wave series expansion (PWSE) method in cylindrical coordinates. A non-dimensional acoustic radiation force function, which is the radiation force per unit length, per characteristic energy density and per unit cross-sectional surface of the ellipse, is defined in terms of the scattering coefficients that are determined by applying the Neumann boundary condition for an immovable surface. A system of linear equations involving a single numerical integration procedure is solved by matrix inversion. Numerical simulations showing the transition from the quasi-standing to the (equi-amplitude) standing wave behaviour are performed with particular emphasis on the aspect ratio a/b, where a and b are the ellipse semi-axes, as well as the dimensionless size parameter kb (where k is the wavenumber), without the restriction to a particular range of frequencies. It is found that at high kb values > 1, the radiation force per length with broadside incidence is larger, whereas the opposite situation occurs in the long-wavelength limit (i.e., kb < 1). The results are particularly relevant in acoustic levitation of elliptical cylinders, the acoustic stabilization of liquid columns in a host medium, acousto-fluidics devices, and other particle dynamics applications to name a few. Moreover, the formalism presented here may be effectively applied to compute the acoustic radiation force on other 2D surfaces of arbitrary shape such as super-ellipses, Chebyshev cylindrical particles, or other non-circular geometries.

Acoustic scattering of a cylindrical quasi-Gaussian beam with arbitrary incidence focused on a rigid elliptical cylinder

Using the partial-wave series expansion method in cylindrical coordinates, a formal analytical solution for the acoustical scattering of a 2D cylindrical quasi-Gaussian beam with an arbitrary angle of incidence θi, focused on a rigid elliptical cylinder in a non-viscous fluid, is developed. The cylindrical focused beam expression is an exact solution of the Helmholtz equation. The scattering coefficients for the elliptical cylinder are determined by forcing the expression of the total (incident + scattered) field to satisfy the Neumann boundary condition for a rigid immovable surface, and performing the product of matrices involving an inversion procedure. Computations for the matrices elements require a single numerical integration procedure for each partial-wave mode. Numerical results are performed with particular emphasis on the focusing properties of the incident beam and its angle of incidence with respect to the major axis a of the ellipse as well as the aspect ratio a/b where b is the minor axis (assuming a > b). The method is validated and verified against previous results obtained via the T-matrix for plane waves. The present analysis is the first to consider an acoustical beam on an elliptic cylinder of variable cross-section as opposed to plane waves of infinite extent. Other 2D non-spherical and Chebyshev surfaces are mentioned that may be examined throughout this analytical formalism assuming a small deformation parameter ε.

Numerical simulation of two dimensional vortex-induced vibrations of an elliptic cylinder at low Reynolds numbers

Flow-induced vibrations (FIV) of an elastically supported rigid elliptical cylinder, which is free to move in cross-flow and in-line directions, with equal natural frequencies in both directions, is numerically simulated. Laminar flow regime (60<Re<400) for selected cylinder cross sectional aspect ratios and inclination angles (a/b=0.5, 1, 2; α=0°, 45°, 90°), with a low non-dimensional mass (m∗=10), are considered. Also, to enhance high amplitude oscillations, the structural damping coefficient is supposed to be zero (ξ=0). The natural structural frequency of the oscillator is assumed to match the associated vortex shedding frequency of a stationary cylinder with zero inclination angle at Re=100. The unsteady fluid/cylinder interactions are accurately captured by implementing the moving mesh scheme via integration of an in-house developed User Define Function (UDF) into the main code of the commercial finite volume solver Fluent. The temporal discretization is based on the computationally efficient non-iterative Fractional Step time marching procedure. Numerical results reveal the decisive effects of cross sectional aspect ratio and inclination angle on the cylinder VIV characteristics (i.e., transverse/in-line response amplitudes, aerodynamic coefficients, Strouhal number, phase portrait, time response, orbital trajectory, and wake mode pattern). In particular, it is observed that the fluid/structure interaction effects notably amplify with the decrease (increase) in the cylinder cross sectional aspect ratio (inclination angle), linked to the decrease in the added mass effect. The validity of the simulations is established by comparison with the available experimental/numerical data.

The Cosserat eigenfunctions for the elliptic exterior problem with applications to thermoelasticity and Stokes flow

The Cosserat eigenvalue problem for the elliptic exterior is considered. It is shown that the only Cosserat eigenvalue different from the infinite multiple eigenvalues \(\tilde \omega = - 1\) and \(\tilde \omega = \infty \) is \(\tilde \omega = - 2\) which also has an infinite multiplicity. The orthogonal basis of the eigenspace corresponding to \(\tilde \omega = - 2\) is constructed. Application to thermoelasticity and Stokes flow – extensional and shear – past a rigid elliptical cylinder are presented and agreement is obtained with classical solutions for a circle. The fact that the solutions consist of only two eigenfunctions reveals the efficiency of the method.

Impact of a solid on a liquid surface: Asymmetric plane case

The vertical impact of a rigid elliptical cylinder on a compressible liquid surface is considered in a plane asymmetric formulation. Solution of the boundary problem reduces to solving an infinite system of linear Volterra integral equations of the second kind. The results are analyzed in terms of the initial asymmetry angle.

The two-dimensional acoustic standing wave and its application in coagulation of aerosols

The problem of motion of small solid or liquid particles suspended in gaseous medium (aerosols and smokes) has been studied intensively in connection with the technical application of acoustic coagulation for precipitation of gases. Along with its technical applications, the problem of motion of small particles in acoustic field (called the drift) is interesting by itself as one of the fundamental physical effects of sound. For theoretical and practical reasons, the two-dimensional standing-wave configuration seems to be of special interest. In the case when nodal and antinodal surfaces of the standing wave are convex, the particles move along some curves and gather in some particular points and not only in some planes. This means that one can obtain much greater concentration of particles in some definite regions than in the case of the plane standing wave. As a configuration of special theoretical interest it proposes the standing wave in a rigid elliptical cylinder with a strip-shaped source of uniform velocity distribution placed in the symmetry plane of the system. Some numerical examples will be presented during the conference [H. Czyż, Acustica 70, 23–29 (1990)].

Nonstationary diffraction of elastic waves on a rigid elliptical cylinder

Nonstationary diffraction of elastic waves on a rigid elliptical cylinder