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Liquid motion in cylindrical containers with elastic covers under external excitation
No full textThe coupled motion of liquid with an elastic plate or membrane cover in a cylindrical container under external excitation is investigated. Unlike self-oscillation problem at a natural frequency, the problem is fully transient, and it is first converted from the time domain to the s-domain through the Laplace transform. For each given s, velocity potential for the fluid flow and cover deflection are obtained through the Bessel-Fourier series. The solution in the time domain is then obtained through the inverse Laplace transform with respect to s. When doing so analytically, it is necessary to find singularities of the integrand in the entire complex plane s. It is shown that these singularities are only on the imaginary axis, corresponding precisely to the natural frequencies of the system and the excitation frequencies. This allows that the final solution to be obtained explicitly, which gives insight how the motion behaves. Extensive results are presented for the time history of the cover deflection and the energy components under various external excitation, including tank motion and external pressure on the cover. The frequency components of the solutions are analysed both at resonance and off-resonance. The energy transfer into the system from external forcings and its redistribution during vibration within the system are analysed
Hydroelastic wave interaction with a circular crack of an ice-cover in a channel
No full textHydroelastic wave interaction with a circular crack of an ice-cover in a channel together with some related problems is considered, based on the linearized velocity potential theory and Kirchhoff plate theory. The domain decomposition method is adopted in the solution procedure. Two sub-domains are divided by the crack, one below the inner ice sheet and the other below the outer ice sheet. By using the Green function of an ice-covered channel, the velocity potential in the outer domain is established from the source distribution formula over an artificial vertical surface extended from the crack. The source distribution is expanded in both vertical and circumferential directions, which allows the velocity potential to be obtained in an explicit form with unknown coefficients. The velocity potential in the inner domain is expanded into a double series. An orthogonal inner product is used to impose continuity conditions on the artificial vertical surface and the edge conditions at the crack. The derived formulation is not just limited to the circular crack problem but can also be readily used in a variety of other problems, including wave diffraction by a surface-piercing vertical cylinder, polynya and circular disc floating on the free surface in a channel. Extensive results are provided for the forces on the inner ice sheet, the transmission and reflection coefficients. In particular, a detailed analysis is made on their behaviours near the natural frequencies of the channel, and the natural frequencies corresponding to the motion of the inner ice sheet.</p
Interaction between a uniform current and a submerged cylinder in a marginal ice zone
No full textThe interaction between a uniform current with a circular cylinder submerged in a fluid covered by a semi-infinite ice sheet is considered analytically. The ice sheet is modelled as an elastic thin plate, and the fluid flow is described by the linearised velocity potential theory. The Green function or the velocity potential due to a source is first obtained. As the water surface is divided into two semi-infinite parts with different boundary conditions, the Wiener-Hopf method (WHM) offers significant advantages over alternative approaches and is consequently adopted. To do that, the distribution of the roots of the dispersion equation for fluid fully covered by an ice sheet in the complex plane is first analysed systematically, which does not seem to have been done before. The variations of these roots with the Froude number are investigated, especially their effects or factorisation and decomposition required in the WHM. The result is verified by comparing with that obtained from the matched eigenfunction expansion method. Through differentiating the Green function with respect to the source position, the potentials due to multipoles are obtained, which are employed to construct the velocity potential for the circular cylinder. Extensive results are provided for hydrodynamic forces on the cylinder and wave profiles, and some unique features are discussed. In particular, it is found that the forces can be highly oscillatory with the Froude number when the body is below the ice sheet, whereas such an oscillation does not exist when the body is below the free surface.</p
Corrigendum. Three-dimensional interaction between uniform current and a submerged horizontal cylinder in an ice-covered channel (Journal of Fluid Mechanics (2021) 928 (A4) DOI: 10.1017/jfm.2021.792)
No full textIn the Appendix B of Yang, Wu & Ren (2021), we made the statement that the Green function G has the symmetry property with respect to the field point P(x
1, y
1, z
1) and field point P
0(x
0, y
0, z
0), or G(x
1, y
1, z
1; x
0, y
0, z
0) = G(x
0, y
0, z
0; x
1, y
1, z
1). This is not always correct. The mistake arose from the statement below (B2) that 'Although G and ξ involve only the real part, we may use the whole complex function here'. In the derivations followed, the full complex functions of G
i and ξ
i (i = 0, 1) in (B1) and (B2) were directly used without taking their real parts, which led to an incorrect conclusion. However, it should be noted that when 0 < Fn < Fn
(1)
c , G
i and ξ
i contain only the k
0 component. G
(0)
i is fully real and ξ
(0)
i is fully imaginary, and they can be taken out of the operator Re{}. Therefore, the symmetry property is satisfied within this range. In summary, the symmetry property G(x
1, y
1, z
1; x
0, y
0, z
0) = G(x
0, y
0, z
0; x
1, y
1, z
1) holds only when 0 < Fn < Fn
(1)
c, and it is incorrect when Fn > Fn
(1)
c. This mistake is confined solely to the Appendix B, and it does not affect any other formulas or results presented in the paper.</p
Wave diffraction and radiation by a vertical circular cylinder standing in a three-dimensional polynya
No full textThe wave diffraction and radiation problem of a body in a polynya surrounded by an ice sheet extending to infinity is considered through a vertical circular cylinder. The ice sheet is modelled through the elastic thin-plate theory and the fluid flow through the linearized velocity potential theory. In particular, when the polynya is of the circular shape, eigenfunction expansion method is applied to the two regions below the ice sheet and the free surface respectively, and the velocity and pressure continuity conditions are imposed on the interface of the two regions. The wave motion in the polynya, the hydrodynamic coefficients as well as the exciting forces on a body located arbitrarily in the polynya are calculated. The nature of highly oscillatory behaviour of the results is investigated and their physical implications are discussed
Hydroelastic waves propagating in an ice-covered channel
No full textThe hydroelastic waves in a channel covered by an ice sheet, without or with crack and subject to various edge constraints at channel banks, are investigated based on the linearized velocity potential theory for the fluid domain and the thin-plate elastic theory for the ice sheet. An effective analytical solution procedure is developed through expanding the velocity potential and the fourth derivative of the ice deflection to a series of cosine functions with unknown coefficients. The latter are integrated to obtain the expression for the deflection, which involves four constants. The procedure is then extended to the case with a longitudinal crack in the ice sheet by using the Dirac delta function and its derivatives at the crack in the dynamic equation, with unknown jumps of deflection and slope at the crack. Conditions at the edges and crack are then imposed, from which a system of linear equations for the unknowns is established. From this, the dispersion relation between the wave frequency and wavenumber is found, as well as the natural frequency of the channel. Extensive results are then provided for wave celerity, wave profiles and strain in the ice sheet. In-depth discussions are made on the effects of the edge condition, and the crack
Diffraction of hydroelastic waves by multiple vertical circular cylinders
No full textThe diffraction problem of hydroelastic waves beneath an ice sheet by multiple bottom-mounted circular cylinders is considered. The elastic thin-plate theory is adopted to model the ice sheet, while the linearized velocity potential theory adopted for the fluid flow. The velocity potential corresponding to each cylinder is expanded into a series of eigenfunctions, and the total potential is expressed as a summation of these expansions over the entire NC number of cylinders. For each cylinder, the Green’s second identity is used outside its domain to obtain a set of linear equations. For each different cylinder, the domain used is different. NC cylinders give NC sets of coupled linear equations. Investigations are made for different arrangements of cylinders, piercing through ice sheets. Results for the wave forces on the cylinders with clamped and free conditions of the ice edge are obtained. Physical phenomena corresponding to cylinders arranged in square, in an array, in a double-array and in a staggered double array are discussed
Surface wave interaction with floating elastic plates in channels
No full textThe interaction between surface waves and a finite rectangular floating plate in a channel is considered analytically, while the location of the plate is not restricted. The mathematical model is based on the linear velocity potential flow theory for the fluid and the Kirchhoff-Love plate theory for the plate. The problem is converted into an integral equation through using the Green function. The second-order singularity associated with a body with no thickness is treated with the Dirac delta function. The developed scheme is used for case studies of various edge constraints. Extensive results are provided for the hydrodynamic forces acting on the plate and the wave reflection and transmission coefficients. The effects of wave frequency, channel width, plate length, and edge conditions are analyzed, and their physical implications are highlighted. Significant findings comprise the highly oscillatory nature of force curves, influenced by the natural frequencies of the channels and the length of the plate, and substantial effects of edge conditions and the plate position on the results.</p
Coupled free vibrations of liquid in a three-dimensional rectangular container with an elastic cover
No full textThe coupled free vibration of liquid and its elastic cover, such as a plate or a membrane, in a three-dimensional rectangular tank is investigated through an analytical scheme based on the velocity potential theory for the flow and the linear elastic theory for the cover. For the fluid domain, the velocity potential is expanded into double cosine series along the longitudinal and transverse directions, respectively, with the corresponding eigenvalues determined from the impermeable conditions on the side walls. The vertical modes of the potential are obtained from the Laplace equation. The deflection of the rectangular cover is expanded into the same double cosine series to match the potential, together with additional terms for satisfying the edge conditions. The polynomials are used for these additional terms, which are then expanded into cosine series. For the expansions of the higher-order derivatives of the deflection, the derivatives of these polynomial terms are expanded into cosine series directly, rather than being obtained through differentiating the cosine series of the deflection, to avoid the non-convergent series. Through imposing the boundary conditions on the fluid–plate interface and edge conditions, an infinite matrix equation for the unknown coefficients can be established. The natural frequencies can be obtained when the determinant of the matrix is zero. In practical computation, the infinite matrix equation is truncated into finite size. Results are first provided for natural frequencies. This is followed by the corresponding natural mode shapes and principal strains distribution on the cover. The underlying physics of these results is then provided
Natural Modes of Liquid Sloshing in a Cylindrical Container with an Elastic Cover
Full text linkLiquid sloshing and its interaction with an elastic cover in a cylindrical tank is considered. The velocity potential for the fluid flow is expanded into the Bessel-Fourier series as commonly used. An efficient scheme is then developed, which allows the plate deflection to use the same type of expansion as the potential. When these two series are matched on the interface of the fluid and the plate, the unknown coefficients in the two expansions can be easily obtained. This is much more convenient than the common procedure where a different expansion is used for the plate and upon matching each term in the series of the plate is further expanded into the series used for the potential. Through the developed method, an explicit equation is derived for the natural frequencies and extensive results are provided. The corresponding natural mode shapes and principal strains distribution of the elastic cover are also investigated. Results are provided and the underlining physics is discussed. To verify the obtained results, the problem is also solved through a different method in which the potential is first expanded into vertical modes. Another explicit equation for the natural frequencies is derived. While the equation may be in a very different form, through the residual theorem, it is found that the second equation is identical to the first one
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