Eigenfunction Expansion Method Research Articles

Interaction between a uniform current and a submerged cylinder in a marginal ice zone

The 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.

Phase change in a one-dimensional functionally graded material

Functionally graded materials (FGMs) are a class of materials with engineered microstructure to achieve spatial variation in properties so as to serve a desired function. While FGMs have been well-researched for their structural properties, there is a lack of investigation of FGMs in solid-liquid phase change heat transfer processes. A well-designed functionally graded phase change material (FG-PCM) may overcome some of the well-known limitations of the phase change process. In order to investigate this possibility, this work presents a theoretical model to predict phase change propagation in a one-dimensional FG-PCM containing an arbitrary spatial variation in thermal conductivity, thermal diffusivity and latent heat. The problem is solved using the method of eigenfunction expansion to determine the temperature distribution, wherein, spatial variation in thermophysical properties listed above is accounted for by deriving a first-order matrix ordinary differential equation in the coefficients of the series solution. Results are found to correctly reduce to the well-known Stefan solution under special conditions. The impact of variation in thermophysical properties is investigated in detail. It is found that while spatial variation in thermal conductivity and latent heat has a significant effect on the rate of phase change, the impact of the thermal diffusivity function is negligible for realistic values of the Stefan number. The effect of the Stefan number governing this problem on the rate of melting is investigated. The nature of phase change under linear (1+aξ), polynomial (1+aξn) and exponential (eaξ) functional forms of thermophysical properties is investigated. It is shown that an appropriate choice of these spatial functions may help overcome the well-known self-limiting nature of phase change and produce a nearly-linear phase change propagation curve. For example, results show 33% increase in phase change propagation for a linear (1+5ξ) variation in thermal conductivity compared to baseline. The theoretical tools developed here may help design and optimize functionally graded PCMs towards improved phase change heat transfer in applications of practical interest.

TOPOGRAPHICAL EFFECTS ON WAVE SCATTERING BY AN ELASTIC PLATE FLOATING ON TWO-LAYER FLUID

This article illustrates the hydroelastic interactions between surface gravity waves and a floating elastic plate in a two-layer liquid with variable bottom topography under the assumptions of small amplitude waves and potential flow theory. In this study, semi-infinite and finite-length plates are considered. The eigenfunction expansion method is applied in the fluid region with uniform bottom topography. A system of differential equations (mild-slope equations) is solved in the fluid region with variable bottom topography. From the matching and jump conditions, the solution is expressed as a linear algebraic system from which all the unknown constants are computed. We explored the effects of density ratio, depth ratio, and bottom topography on the bending moment, shear force, and the deflection of the elastic plate. Results show that when the density ratio becomes closer to one, the occurred bending moment and shear forces to the elastic plates tend to diminish. The bending moment and shear forces to the pates are higher and lower at a smaller depth ratiofor the incident surface wave and interfacial waves, respectively. The variations in the bending moment, shear force, and plate deflection, caused by surface and interfacial waves, are observed to be in opposite trends, respectively. Bottom profiles similarly affect semi-infinite and finite-length plates when they undergo free-edge conditions. These effects, however, are substantial when the plate is simply supported at the edges. Elastic plate with free edges experiences lower deflection for concave-up and plane-sloping bottoms for incident surface and interfacial waves, respectively.

Analytical Model of an E-core Driver-pickup Coils Probe Applied to Eddy Current Testing of Multilayer Conductor

An analytical model of an E-core driver-pickup coils probe located above a multilayer conductor containing a hidden cylindrical conductor is presented. The truncated region eigenfunction expansion (TREE) method is used to deal with the axial symmetry problem, and the closed-form final expression of the induced voltage in the pickup coil is derived. The changes of the induced voltage in the pickup coil due to the hidden cylindrical conductor are examined and calculated in Mathematica. Experiments and finite element simulations are performed and the results are compared with the analytical results, and they are in good agreement.

A sectional method to estimate the transmission coefficients of a new type pile-supported permeable breakwater

This article investigates the transmission coefficient of a novel pile-supported permeable breakwater featuring inclined porous plates under regular waves. To analyze the transmission coefficient, we propose a sectional method that involves multiplying the transmission coefficients of three distinct sections: the upper retaining wall and porous plates, the horizontal platform, and the piles. The calculation employs the eigenfunction expansion method and empirical formula for pile-supported breakwaters. This method is applicable for water levels that fall between the upper and lower edges of the horizontal platform. The analytical results are compared with experimental tests and show good agreement. Further analysis reveals that the transmission coefficient is significantly influenced by the increasing length of the horizontal platform. Moreover, it is observed that the effect of porosity ε is not particularly noticeable when ε>20%.

Model of an E-cored probe over layered conductor containing corrosion for eddy current testing

PurposeIn eddy current nondestructive testing, ferrite-cored probes are usually used to detect and locate defects such as cracks and corrosion in conductive materials. However, the generic analytical model for evaluating corrosion in layered conductor using ferrite-cored probe has not yet been developed. The purpose of this paper is to propose and verify the analytical model of an E-cored probe for evaluating corrosion in layered conductive materials.Design/methodology/approachA cylindrical coordinate system is adopted and the solution domain is truncated in the radial direction. The magnetic vector potential of each region excited by a filamentary coil is derived first, and then the expansion coefficients of the solution are obtained by matching the boundary and interface conditions between the regions and the subregions. Finally the closed-form expression of the impedance of the multi-turn coil is derived by using the truncated region eigenfunction expansion (TREE) method, and the impedance calculation is carried out in Mathematica. In the frequency range of 100 Hz to 10 kHz, the impedance changes of the E-cored coil and air-cored coil due to the layered conductor containing corrosion are calculated, respectively, and the influences of corrosion on the coil impedance change are investigated.FindingsAn analytical model for the detection and evaluating of corrosion in layered conductive materials using E-cored probe is proposed. The model can quickly and accurately calculate the impedance change of E-cored coil due to corrosion in layered conductor. The correctness of the analytical model is verified by finite element method and experiments.Originality/valueAn accurate theoretical model of E-cored probe for evaluating corrosion of multilayer conductor is presented. The analytical model can be used to detect the inhomogeneity of layered conductor, design ferrite-cored probe or directly evaluate the corrosion defects of layered conductors.

Oblique Wave Scattering by a Combination of Two Asymmetric Trenches of Finite and Infinite Depths

Abstract The focal point of the current study lies in investigating oblique wave scattering within the framework of linear potential theory, with particular attention to scenarios involving asymmetric trenches of both finite and infinite depths. By employing the eigenfunction expansion method, the physical problem undergoes a transformation into an equivalent boundary value problem. This newly formulated problem is characterized by a system of four weakly singular integral equations, which pertain to the horizontal component of velocity across the gaps situated above the edges of the trenches. The solution to these integral equations is achieved through the utilization of a multi-term Galerkin approximation method. This approach involves expansions using ultraspherical Gegenbauer polynomials as basis functions, coupled with the appropriate weight functions tailored to address the one-third singularity. Graphical representations are employed to depict the numerical evaluations of reflection and transmission coefficients across various non-dimensional parameters. These visualizations offer insight into the behavior and dependencies of these coefficients under different conditions. To validate the accuracy of the current model, it is compared against previously published results available in the literature.

Analysis of Wave-Induced Forces on a Floating Rectangular Box with Analytical and Numerical Approaches

A three-dimensional mathematical hydrodynamic model associated with surface wave radiation by a floating rectangular box-type structure due to heave, sway, and roll motions in finite water depth is investigated based on small amplitude water wave theory and linear structural response. The analytical expressions for the radiation potentials, wave forces, and hydrodynamic coefficients are presented based on matched eigenfunction expansion method (MEFEM). The correctness of the analytical results of wave forces is compared with the construction of a numerical model using the open-source boundary element method code NEMOH. In addition, the present result is compared with the existing published experimental results available in the literature. The effects of the different design parameters on the floating box-type rectangular structure are studied by analyzing the vertical wave force, horizontal wave force, torque, added mass, and damping coefficients due to the heave, sway, and roll motions, and the comparison analysis between the forces is also analyzed in detail. Further, the effect of reflection and transmission coefficients by varying the structural width and drafts are analyzed.

Oblique wave trapping by periodic array of two-layer pile rock breakwaters placed on uneven seabed

ABSTRACT This research work evaluates the wave trapping efficiency of the periodic array of two-layer pile-rock breakwaters (TPRBs) placed far away from vertical rigid wall. The analytical model is developed to study the effect of TPRBs on the variation of wave reflection and reduction of fluid force experienced by wall based on the classical potential flow and porous media flow theory. The effect of the uneven seabed on wave trapping by TPRBs is added to show its significance, and the porous structure consists of a pair of layers with individual porosities. The seaside and leeside protection walls are introduced to secure the two-layer rock from vertical and horizontal failures. Thus, the continuity of velocity and pressure is applied in the vertical and horizontal directions. The eigenfunction expansion method based on orthogonal mode-coupling relation is used to solve the unknown coefficients. The effect of incident wave properties and TPRBs physical properties on wave trapping is reported as a function of breakwater width, gap region, and trapping chamber length. The correctness of the present study results is verified with the available results. The study strongly suggests that the variation of wave reflection by TPRBs is a trivariate function, which would depend upon the width, spacing, and porosity of the TPRBs. Around 30–40% reduction of wave reflection is obtained after introducing the TPRBs.

Flexural-gravity wave scattering by an array of bottom-standing partial porous barriers in the framework of Bragg resonance and blocking dynamics

Flexural-gravity wave scattering by an array of vertical porous barriers of various configurations is investigated in finite water depth from the viewpoint of blocking dynamics. A scattering matrix is introduced for the velocity potentials using the canonical eigenfunction expansion method developed for a single propagating wave mode to account for the multiple propagating wave modes. Subsequently, appropriate matching conditions are applied at the interface boundaries and edges to solve the physical problem. Apart from multiple barriers of equal length, the efficiency of four different barrier configurations of unequal lengths is investigated. This study shows that out of these four barrier configurations, the convex and increasing order of the barrier arrangements are more effective as wave-dissipating systems than the concave and decreasing order of the barriers. Bragg reflection occurs in the case of two or more barriers for a specific value of porosity and suitable barrier configuration, whose amplitude decreases with an increase in the number of barriers due to the dissipation of wave energy. The presence of three propagating wave modes in the blocking paradigm leads to mode conversion within a certain range of the frequency space. Both the scattering and dissipation coefficients are influenced by the wave energy transfer rates and the amplitudes of incident, reflected, and transmitted wave modes. This investigation exhibits the presence of discontinuities in the scattering coefficients at frequencies where blocking and mode conversion occur. The frequency domain results are used to simulate the plate displacement in the time domain by applying the Fourier transform.

Second-order wave loads on a floating bridge of T-type cross section

An analytical model for the complete second-order wave loads on a floating bridge of T-type cross section is established within the framework of potential flow theory. The first-order problem is solved using the matched eigenfunction expansion method (MEEM). Meanwhile, the second-order wave loads are divided into two distinct parts which are computed separately. The first part is dependent on the quadratic products of the first-order velocity potential and its derivative. It has been evaluated through two different ways. One is based on a direct pressure integration, i.e., a near-field method. To ensure a high accuracy of the computation, a local expansion is applied to determine the spatial derivative of the first-order velocity potential at a sharp corner. The other is based on the momentum conservation over a control volume. The effect of the control surface location on the computation has been discussed. In addition, the two proposed methods have been applied to both the time-independent wave-load component as well as the second-harmonic component, respectively. The second part of the complete second-order wave loads is due to the second-order velocity potential. It is evaluated based on an indirect method by introducing an auxiliary potential. Detailed numerical investigation is then conducted. The influence law of the sectional geometric parameters on the second-order wave loads is investigated. The numerical results reveal that the cantilever plate imposes an apparent protection effect on the inner web beam. Such effect gets obviously enhanced as the section net width or section height decreases or as the plate width or draft increases.

Assessment of viscous damping of a solid heave disk for motion reduction of a floating circular cylinder

In the present study, we developed an analytical model incorporating viscous damping of a solid heave disk for the motion reduction of a floating circular cylinder in three degrees of freedom (3DOF) (surge, heave, pitch). The matched eigenfunction expansion method (MEEM) based on a potential theory is employed for solving the diffraction and radiation problem of a floating cylinder with a solid disk. To implement viscous damping due to vortex formation at the disk edge, a novel approach using the drag force and moment in the Morison equation is presented along with the conventional method using free decay tests. The developed analytical model is validated through the comparison with the regular wave experiments and computational fluid dynamics (CFD) turbulent model. Attaching the solid heave disk to the bottom of a cylinder reduces the maximum values of the heave and pitch RAOs (Response of Amplitude Operator) to 34 % and 53 %, compared to the cylinder model with no disk. Increasing the size of the heave disk decreases heave motion accordingly and significantly, however, its radius ratio must exceed the specified value of a/b=1.5 for affecting the reduction of a pitch motion.

Asymptotic stress field for the blunt and sharp notches in bi-material media under mode III loading

Abstract This study introduces a novel approach to analyze the stress and displacement fields around blunt notches in bi-material media, focusing on mode III loading conditions. The eigenfunction expansion method is used to derive a simplified yet accurate solution, satisfying the boundary conditions for bi-material blunt V-notches. The robustness of the proposed asymptotic solution is validated through several finite element analyses, encompassing a range of notched geometries such as blunt V-notches, VO-notches, and circular holes. Notably, it is demonstrated that when the notch tip radius approaches zero, the solution coincides with the existing sharp V-notch model, illustrating the robustness of the methodology.

Effect of Sloping Bottom on Wave Interaction With Multiple Flexible Moored Breakwaters

Abstract An analytical framework is developed to analyze the interaction of oblique waves with multiple flexible porous breakwaters under the consideration of bottom undulation. The mathematical problem is tackled using the small amplitude water-wave theory, with Darcy’s law being applied to account for wave interaction with porous media. The bottom topography is considered to have a finite length, flanked by two semi-infinite sections of uniform bottom. The solution to the boundary value problem is approached by employing the eigenfunction expansion method within the uniform bottom regions. For the varying bottom topography, a modified mild-slope equation (MMSE) is utilized. To address the solution at the slope discontinuity at the bottom, a mass-conserving jump condition is applied. By matching solutions at the interfaces, a set of equations is derived. This system of equations encapsulates the behavior of reflection and transmission coefficients, as well as the force exerted on the breakwaters. These parameters are then investigated across various factors such as the length of the varying bottom, depth ratio, angle of the mooring line, angle of incidence, and flexural rigidity. Graphical representations of the reflection and transmission coefficients, along with the breakwater force, provide insights into the system’s behavior under different conditions. The water wave energy can be dissipated for the optimum values of flexural rigidity. The transmission coefficient is observed to be least for higher mooring angle.

Wave diffraction and radiation studies of an array of coaxial double-layer porous cylindrical structures placed on a porous seabed

This article theoretically investigates the diffraction and radiation of an array of coaxial double-layer porous cylindrical structures placed on a porous seabed. The array of the structures is considered a bottom-mounted cylindrical structure with vanishing thickness. Under the assumption of a small amplitude wave theory, the hydrodynamic characteristics of an array of coaxial double-layer porous cylindrical structures are investigated. The diffraction and the radiation of the structure array are investigated using the eigenfunction expansion method, Bessel function addition theory, and Darcy's law. Analytical expressions for the velocity potentials and free-surface elevations for the diffraction and radiation of the structural array are derived. Finally, the unknown coefficients of the velocity potential are determined using the matching feature function method. After completing the validation, a detailed parameter study is conducted on the structural array, and the hydrodynamic characteristics of the structure array under the influence of different parameters are discussed and explained. Taking the horizontal layout as an example, the results show that in addition to the influence of structure parameters such as porosity, the radius ratio, and the cylinder distance on the hydrodynamic presence of the array of porous structures, the number of cylinders has a greater influence on the hydrodynamic characteristics of the array of porous structures. The calculation results of the diffraction and radiation of an array of porous structures placed on a porous seabed are closer to the actual situation, which can better understand the interaction between waves and porous structure and provide effective theoretical reference for engineers to design ocean structures.

Short crested wave–current interaction with a cylinder and two concentric asymmetric arc-shaped walls

The diffraction problem of the interaction of short-crested incident waves and currents with a combined structure is studied. The combined structure consists of two concentric asymmetric exterior porous arc-shaped cylinders and an impermeable interior cylinder. A variable separation and an eigenfunction expansion method are used to derive the velocity potential for the entire fluid domain. The accuracy of the current model is verified by comparing the results of this study with published calculations. Numerical results reveal the variation of wave force and run-up on the combined structure with current speed, opening angle, and the locations of the two arc exterior cylinders. Moreover, the specific parameter conditions under which the resonance of the wave and structure may occur are analyzed, and the resonance phenomenon may be enhanced owing to the presence of a uniform current. Furthermore, the hydrodynamic performance of segmented arcs in the limiting case is examined. The results show that the segmented arc is not effective in protecting the interior structure when the wavelength is short. The findings of this study are anticipated to provide theoretical direction for the design of nearshore architecture.

Effect of vertical porous baffle on sloshing mitigation of two-layered liquid in a swaying tank

The sloshing behavior of two-layered liquid in a rectangular sloshing tank with a vertical porous baffle under sway excitation has been investigated using both analytical and numerical methods. The analytical model is developed using the matched eigenfunction expansion method in conjunction with linear potential flow theory. The existence of vertical porous baffle results in energy dissipation by applying the boundary condition that pressure drop across the baffle is proportional only to the quadratic drag term. For comparison, numerical simulations are conducted using computational fluid dynamics (CFD) with unsteady, implicit, and incompressible fluid equations. The study explores the effects of the liquid density ratio, the baffle's porosity, and its submergence depth, on various parameters including the free surface and interfacial elevations, hydrodynamic force on the tank surface, and horizontal force on the baffle. The trends observed in the analytical model align well with the CFD results and experiments. It is found that a fully submerged vertical porous baffle with a porosity close to P = 0.1 leads to stable and less dynamically active sloshing behavior in a two-layered liquid tank.

Hydrodynamic coefficients on a deeply submerged heaving solid/porous disk: in analytical and experimental approaches

ABSTRACT The hydrodynamic coefficients on a deeply submerged solid/porous disk subjected to forced heave oscillation have been investigated in analytical and experimental approaches. The analytical model is developed by means of a matched eigenfunction expansion method (MEEM) using a quadratic relationship between the pressure drop and traversing fluid velocity across the porous disk. For the improvement and validation of the analytical model, a series of experiments with the solid/porous disks is conducted for various porosities, Keulegan-Carpenter (KC) numbers, and oscillation periods. In addition to the small-scale hole vortices across the porous disk, the effect of edge vortices on the hydrodynamic coefficients is empirically implemented by curve-fitting with experimental data. The inclusion of edge vortices in the analytical model results in good agreement with experimental data of the damping coefficient for a deeply submerged disk. However, there are some discrepancies in the added mass, with noticeable differences occurring in the disk with low porosity.

Time-dependent flexural gravity wave scattering due to uneven bottom in the paradigm of blocking dynamics

The present study investigates the scattering of flexural gravity waves due to uneven bottom topography in the context of wave blocking. Emphasis is given to analyzing the effects of multiple propagating wave modes on the solution procedures. The mathematical model is developed for two scenarios: a bottom step and a submerged rectangular breakwater. For the bottom step case, the complete solution in terms of the velocity potential is obtained using the eigenfunction expansion method. Subsequently, the solution associated with the wave transformation by the bottom step is extended to the case of a submerged rectangular breakwater using symmetry characteristics of the velocity potential. The energy balance relation is derived in both cases using the conservation of energy flux in the presence of multiple propagating wave modes. Wave blocking occurs for four different frequencies in both the cases of the bottom step and the submerged breakwater due to variations in water depth. This makes the problem more complex as, depending on the frequency, multiple propagating wave modes can exist in either the reflected region, the transmitted region, or both. The transmitted wave amplitude associated with the lower wavenumber within the blocking frequencies exceeds unity, and this excess energy is balanced by the corresponding energy transfer rate. Additionally, removable discontinuities are observed at the blocking frequencies in the scattering coefficients, where group velocity ceases. In the context of floating ice sheets, the deflection is analyzed in the time domain for frequencies within and outside the blocking limits.

Wave interactions with a cylinder surrounded by an arc-shaped breakwater

Inspired by the energy island used for energy storage, conversion, and transmission, this study explores the water wave interactions with a bottom-standing cylinder surrounded by an arc-shaped breakwater of negligible thickness. The arc-shaped breakwater can be either impermeable or porous. To analyze the problem, a semi-analytical matched eigenfunction expansion method is developed within the framework of linear potential flow theory. The fluid domain is divided into subdomains, with the velocity potentials in each subdomain expressed as eigenfunction expansions. Matching conditions, ensuring continuity in pressure and normal velocity, are imposed at the juncture boundary. The accuracy of the developed semi-analytical method is verified through comparison with results obtained using the numerically-based boundary element method. Extensive discussions are made on hydrodynamic responses, including wave exciting force and free surface elevations. When the breakwater is impermeable, fluid resonance occurs in the gap between the cylinder and the arc-shaped breakwater, leading to significantly large responses in both free surface elevation and hydrodynamic forces. A simple method is developed to estimate the resonant frequencies and mode shapes associated with these resonances. On the other hand, it is observed that the presence of the porous breakwater considerably reduces the free surface responses in the gap as well as the hydrodynamic loads on the cylinder. The porous breakwater effectively mitigates the occurrence of fluid resonance and offers an efficient means to dampen wave energy, providing valuable insights for wave energy dissipation and coastal engineering applications.