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#### Wave equation in cylindrical and spherical coordinates

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Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2005 SEG Annual Meeting, November 6–11, 2005

Paper Number: SEG-2005-0316

... involved plane

**wave**particle motion reflection coefficients given by Zoeppritz s**equations**and the Weyl/Sommerfeld integral for computing isotropic**spherical**-**wave**potentials. Plane-**wave**particle motion reflection coefficients for VTI media have been presented by Graebner (1992) and**in**refined form by...
Abstract

Summary The AVO response of two-layer VTI models for AVO Class I is investigated. Graebner/Rueger reflection coefficients and the “Weyl-integral for anisotropic media” are utilized for the computation. Spherical wave results are compared with the plane-wave reflectivity. Depth dependence of spherical wave AVO is found to be strongest near critical angles, as was observed in the isotropic situation. Particle motion perpendicular to the ray angle is strongest just beyond critical angles and increasing with anisotropy. Anelasticity also modifies VTI AVO responses. When reflection amplitude losses due to attenuation are compensated for by unit reflectivity scaling, AVO-characteristics similar to the elastic situation are found. Q-factor dependence of spherical wave AVO is found to be strongest near critical angles. This Qdependence, to some degree, mimics depth dependence of elastic comparisons. Particle motion perpendicular to the ray angle is also Q-factor dependent. Introduction Previous spherical wave AVO investigations by the authors are restricted to isotropic media (Haase, 2004; Haase and Ursenbach, 2004a). However it is well known that in many situations anisotropy is present either in the form of apparent anisotropy caused by layering or intrinsic anisotropy caused by, for example, shale layers. This type of anisotropy is usually called VTI (transversely isotropic with a vertical axis of symmetry). Rock fractures can cause HTI (horizontal symmetry axis TI) or also orthotropic anisotropy and these are not considered in this study. Early work on spherical wave AVO by Hron et al. (1986) investigates anisotropy using asymptotic ray theory. They note that “anisotropic media produce noticeable differences in both amplitude- and time-distance curves as a function of the degree of anisotropy”; they also show amplitudedistance plots. Previous work by the authors involved plane wave particle motion reflection coefficients given by Zoeppritz’s equations and the Weyl/Sommerfeld integral for computing isotropic spherical-wave potentials. Plane-wave particle motion reflection coefficients for VTI media have been presented by Graebner (1992) and in refined form by Rueger (1996). The Weyl-integral for anisotropic media is given by Tsvankin (2001). Their “exact” equations are utilized in this study. Approximations are introduced by performing numerical integrations Q-factor dependence has also been observed in previous investigations of isotropic spherical-wave AVO (Haase, 2004; Haase and Ursenbach, 2004b). This modeling study seeks to quantify the sensitivity of spherical-wave AVO responses with respect to finite Q-factors of VTI media. Theory The displacement from a point force located at the origin is given by the following summation over plane waves..

Proceedings Papers

Paper presented at the The Second International Offshore and Polar Engineering Conference, June 14–19, 1992

Paper Number: ISOPE-I-92-260

... introduced

**in**estimation of**wave**forces. The number of researches conducted on the diffraction force acting on the**spherical**structure is less than that on the**cylindrical**structures. Havelock (1954) discussed the diffraction force acting 316 on a submerged sphere under deep water**wave**condition. His theory...
Abstract

ABSTRACT This paper aims at experimental and analytical investigation on diffraction forces acting on a spherical structure and wave diffraction. The Source Distribution Method is applied to their theoretical estimation. The effective inertia coefficient is affected by the relative diameter and submergence of the sphere as well as the diffraction parameter. The non-dimensional wave height, however, has little effect on it. The Source Distribution Method based on the linear wave theory is revealed to evaluate well the diffraction force acting on a spherical structure due to the finite amplitude waves including the spilling breakers above the submerged sphere. The Source Distribution Method is useful also in evaluating the wave height distribution around the sphere under non-breaking condition. INTRODUCTION With increasing utilization of coastal and ocean areas, many coastal and ocean structures will be constructed in these areas. Structures with spherical shapes, like oil storage tanks, are treated here, which are considered to be one of the most suitable shapes for coastal and offshore structures against wave pressure. In designing such structures, it is necessary to accurately evaluate wave forces acting on them. The Morison equation is useful in estimating the acting wave force on a structure whose size is much smaller than the incident wavelength. If the structure is large, the Morison equation becomes rather useless and the diffraction theory should be introduced in estimation of wave forces. The number of researches conducted on the diffraction force acting on the spherical structure is less than that on the cylindrical structures. Havelock (1954) discussed the diffraction force acting on a submerged sphere under deep water wave condition. His theory, however, was derived without considering the free surface boundary condition. As a consequence, the effects of the bottom and free surface boundaries on the acting wave force cannot be evaluated.

Journal Articles

*International Journal of Offshore and Polar Engineering*3 (01).

Paper Number: ISOPE-93-03-1-007

Published: 01 March 1993

..., the Morisontype

**equation**becomes less accurate and the diffraction theory should be introduced**in**the estimation of**wave**forces. The amount of research conducted on the diffraction force acting on the**spherical**structure is less than that conducted on the**cylindrical**structures. Havelock (1954...
Abstract

ABSTRACT: This paper aims at an experimental and analytical investigation of the diffraction forces acting on a spherical structure and wave diffraction. The Source Distribution Method (SDM) is applied to their theoretical estimation. The dimensionless maximum wave force is affected by the relative diameter and submergence of the sphere as well as the diffraction parameter. The non-dimensional wave height, however, has little effect on it. The SDM, based on the linear wave theory, is revealed to evaluate well the diffraction force acting on a spherical structure due to finite amplitude waves including the spilling breakers above the submerged sphere. The SDM is useful also in evaluating wave height distribution around the sphere under non-breaking conditions. INTRODUCTION With increasing utilization of coastal and ocean areas, many coastal and ocean structures will be constructed in these areas. Treated here are structures with spherical shapes, like oil storage tanks, considered to be one of the most suitable shapes for coastal and offshore structures against wave pressure. As the size of structure increases and closes to the order of the wavelength, the Morisontype equation becomes less accurate and the diffraction theory should be introduced in the estimation of wave forces. The amount of research conducted on the diffraction force acting on the spherical structure is less than that conducted on the cylindrical structures. Havelock (1954) discussed the diffraction force acting on a submerged sphere under deepwater wave condition. His theory, however, was derived without considering the free surface boundary condition. As a consequence, the effects of the bottom and free surface boundaries on the acting wave force cannot be evaluated. Garrison and Rao (1971) and Chakrabarti and Neftzger (1974) discussed the effect of the free surface on the acting wave force in the case of a bottom-seated hemisphere.

Proceedings Papers

Paper presented at the The 30th International Ocean and Polar Engineering Conference, October 11–16, 2020

Paper Number: ISOPE-I-20-1133

... ABSTRACT

**In**this paper, we instigate numerically the performance of an oblate spheroidal heaving**Wave**Energy Converter (WEC). The diffraction/radiation problems are solved**in**the frequency domain by utilizing the conventional boundary integral**equation**method. This method is, initially...
Abstract

In this paper, we instigate numerically the performance of an oblate spheroidal heaving Wave Energy Converter (WEC). The diffraction/radiation problems are solved in the frequency domain by utilizing the conventional boundary integral equation method. This method is, initially, compared against an analytical method that utilizes the image singularity system to manipulate the underlying Green's function. Next, extended results are presented focusing on the comparison of the oblate spheroidal WEC with other WEC geometries as well as on the effect of various design parameters (i.e. equatorial and polar radius, distance from a bottom-mounted vertical wall) on the performance of the oblate spheroidal WEC. INTRODUCTION Modern technological advances seek the efficient exploitation of the abundant wave energy. Nowadays, the wave energy sector is, rapidly developing aiming at the realization of commercially competitive solutions by overcoming existing technological and economic, mainly, barriers (Magagna and Uihlein, 2015; Uihlein and Magagna, 2016). As a result, a variety of different Wave Energy Converters (WECs) in terms of power absorption mode, have been proposed and designed (de O Falcão, 2010; Aderinto and Li, 2018). Currently, these types of WECs have reached different technological development stages, with the point absorbers (heaving WECs) and the oscillating water columns representing the most advanced device types (Magagna et al., 2016). Heaving WECs correspond to single-body oscillating devices that harness wave power through the heave oscillations of their floater. This mode of operation advocates potential enhancement of the WEC's energy absorption ability by exploiting the hydrodynamic characteristics of the WEC's floater itself. Therefore, floaters of various geometries can be utilized. In this context, the performance (hydrodynamic behavior and power absorption ability) of different WEC geometries has been investigated numerically by many researchers both in frequency and time domain. Considering the implementation of a frequency-based numerical analysis, Sinha (2015) assessed and compared the performance of three different WEC geometries (hemisphere, hemispherical cylinder and cone-cylinder). The corresponding results illustrated that the cone-cylinder performs slightly better than the other examined geometries in terms of power absorption ability. Analogous conclusions have been derived by Pastor and Liu (2014), who compared cylindrical WECs having conicalshaped and hemispherical-shaped bottoms. A variety of WEC geometries (e.g. cylindrical, nailhead-shaped, disk-shaped, elliptical, egg-shaped, conical, floater-shaped and hemisphere-shaped WECs) has been compared and assessed by Belibassakis et al. (2018), who developed a novel method for estimating the performance of WECs in regions of variable bathymetry. The better performance of the conecylinder compared to a hemispherical one in the presence of a wall boundary of infinite length in the leeward side of the WEC has been illustrated in the frequency domain by Schay et al. (2013) under the action of both regular and irregular waves of various wave directions. Similar conclusions have been derived by Jin and Patton (2017), who compared in time-domain the two latter geometries along with a cylindrical WEC, while Beirão and dos Santos Pereira Malça (2014) showed that a hemispherical WEC has a very good structural behavior. In Berenjkoob et al. (2019), the hydrodynamic behavior and the power absorption ability of a conical-shaped WEC was compared in time domain with the corresponding one of a WEC having a spherical-cap, a circular-arc and a cosine-curve shape. The latter, unconventional geometry, has been proven more efficient in terms of heave oscillation velocity and absorbed power. As for oblate spheroidal heaving WECs, the relevant literature existing so far is quite limited, although this geometry depending on its slenderness, may combine the advantages of a convex structure allowing, at the same time, the excursion of relatively uniform hydrodynamic pressure on the wetted surface (upper or lower). Specifically, the enhanced power absorption ability of this geometry compared only to an elliptical-shaped WEC for two different locations in the Black sea has been reported in Erselcan et al. (2016). Furthermore, quite recently, Loukogeorgaki et al. (2020) demonstrated for the case of an array of oblate spheroidal WECs in front of a bottommounted wall of finite length, the advantage of this geometry in terms of power absorption compared to arrays consisting of either cylindrical or hemispherical WECs. The positive effect of the wall on the power absorption ability of the array and the direct impact of the array's distance from the wall on this ability were also emphasized.

Proceedings Papers

Paper presented at the The 30th International Ocean and Polar Engineering Conference, October 11–16, 2020

Paper Number: ISOPE-I-20-3245

... located

**in**the geometric center of the tanks, the first**wave**probe 1 is placed on the plane = 0 of the semi-spheroid located**in**the < 0 half of the tank, it departs and ends on the smoothing edge between the semi- spheroid and the circular**cylindrical**center body. The probe 2 cuts the plane...
Abstract

The present study investigates the resonant fluid flow within horizontal axisymmetric tanks under forced harmonic translation in the transversal direction. Experiments and numerical simulations with the MPS method were carried out for a reduced-scale model with a 2:1 aspect ratio and semi-spheres as end shells. The numerical and experimental results presented good agreement. In addition, longitudinal resonant modes were excited by the transversal motion of the tank and four different flow patterns were identified, including a double swirling flow. Furthermore, numerical simulations were performed for two additional models, with prolate and oblate semi-spheroidal end shells. The tank with prolate end shell did not show the formation of the double swirling. INTRODUCTION Liquid Natural Gas (LNG) is the most environmental-friendly fossil fuel since it produces the lowest greenhouse gas emissions per unit of energy. In such sense, the demand for LNG as energy source increased remarkably in recent years (International Gas Union, 2019). The LNG is gradually replacing traditional high sulphur bunker fuel as fuel in a number of new ship engines, known as small-scale LNG. Horizontal axisymmetric tanks are commonly used as LNG fuel containers but also could be adopted as LNG cargo tanks, such as the IMO type C tanks of pressure vessels. The violent fluid motion in partially filled containers, known as sloshing, is particularly dangerous for the case of LNG fuel tanks because it may result in a rapid drop in the system pressure and impair the operation of the gas-fuel system (Grotle, 2018). Several works in the literature investigated the sloshing in tanks with different shapes, including cylindrical and spherical containers. Budiansky (1960) proposed analytical estimations for the resonance frequencies of sloshing on horizontal cylinders and spherical containers from the solution of the boundary value problem with conformal mapping. McIver (1989) estimated analytically the lowest four symmetric and anti-symmetric modes of sloshing for cylindrical and spherical containers by using a toroidal coordinate transformation. McCarty and Stephens (1960) performed experiments to estimate the resonant frequencies of cylindrical and spherical containers.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the SEG International Exposition and Annual Meeting, September 15–20, 2019

Paper Number: SEG-2019-3214725

...-250 . Teixeira , F. L. , and W. C. , Chew , 1997 , PML-FDTD

**in****cylindrical**and**spherical**grids : IEEE Microwave and Guided**Wave**Letters , 7 , 285 – 287 , doi: 10.1109/75.622542 . Elastic reverse time migration**in**irregular tunnel environment based on polar**coordinate**...
Abstract

ABSTRACT When seismic exploration is carried out in special geological environment such as tunnel space, the traditional imaging method in Cartesian coordinate system can not discretize the air column in tunnel space accurately, so it is difficult to obtain high quality imaging results. Therefore, an elastic wave reverse time migration method based on polar coordinate system is proposed. The air column in the tunnel space is usually not an irregular circle. Therefore, the irregular tunnel space geological body in polar coordinate system is meshed into curvilinear grids and transformed into the regular one in auxiliary polar coordinate system by the mapping method. Finally, elastic reverse time migration technology is applied into auxiliary polar coordinate system. In the numerical examples, two models in complete- and incomplete tunnel space are used to test the proposed method, verifying that the proposed method can obtain accurate images from the data sets in tunnel space. Presentation Date: Wednesday, September 18, 2019 Session Start Time: 8:30 AM Presentation Start Time: 10:35 AM Location: 214C Presentation Type: Oral

Proceedings Papers

Paper presented at the ISRM International Symposium - 8th Asian Rock Mechanics Symposium, October 14–16, 2014

Paper Number: ISRM-ARMS8-2014-091

... observable. Thus, blasting design tends to be based on empirical knowledge or law. It is also well known that fracturing process envisioned here depends on the applied pressure

**wave**forms characterized by such as detonation property and amount of applied explosive. These**in**turn make the design optimization...
Abstract

Abstract In blasting, cylindrical charge is generally applied in terms of energy efficiency. The dynamic fracturing in rock due to detonation of high-explosive involves quite fast process and extremely complex fracturing pattern. In addition, detailed fracturing process is generally not observable. Thus, blasting design tends to be based on empirical knowledge or law. It is also well known that fracturing process envisioned here depends on the applied pressure wave forms characterized by such as detonation property and amount of applied explosive. These in turn make the design optimization of blasting quite difficult and, even for the simplest blasting problem with a single free face, the fracturing mechanism has not been clarified yet. Therefore, it is of paramount importance to investigate the fracturing process due to blasting in detail for various types of applied pressure wave forms. For this purpose, application of numerical simulation is one of the most promising approaches. This paper proposed a method for the simulation of dynamic fracturing process through axisymmetric finite element formulation in which the initiation, propagation, branching and coalescence of fractures in heterogeneous rock can be analyzed. In particular, blasting a cylindrical charge through bottom priming in a cylindrical rock specimen was analyzed considering the difference of load configuration characterized by length of applied explosive. It was clarified that the resultant fracturing patterns were strongly dependent on the length of applied explosive. In addition, cross-shaped fractures occurring on the free face were successfully simulated by the proposed method, which were observed in the field-scale blasting with a cylindrical charge. Therefore, the applicability of the proposed method was validated and it can give a deep insight for understanding the dynamic fracturing process in rock due to blasting with a cylindrical charge.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the SEG International Exposition and Annual Meeting, October 11–16, 2020

Paper Number: SEG-2020-3406423

...

**equation**2, the general where M is a 4×4 matrix and b and b are 4×1 vectors; For solution for the potentials of**in****equation**3**in**the -k plane**wave**incidence, their expressions are given**in**domain can be written as (Tang and Cheng, 2004; Tang et Schoenberg (1986) and Peng (1994). For the**spherical**al...
Abstract

Borehole acoustic reflection imaging is a useful application for detecting geological structures around the borehole. The recent development of shear-wave imaging using a dipole source-receiver system allows for determining the azimuth information of the structures. Existing analyses, however, focus on the data received at the borehole axis and use the elastic reciprocity theorem to model the borehole radiation and reception. This study extends the existing analyses to model the radiation, reflection, and the receiving response of the borehole for azimuthally spaced receivers off the borehole axis. By treating the mirror image of the borehole source with respect to the reflector plane as a virtual source, the borehole reception problem is equivalent to the response of the borehole to the spherical wave incidence from the virtual source, which can be solved using the cylindrical-wave expansion method. With this analysis, one can model the multi-component shear-wave reflection data from the cross dipole acoustic tool and study the azimuthal variation characteristics of the data. The results show that, while the data characteristics are dominated by those of a dipole, as predicted by the existing analyses, non-dipole responses due to the off-axis reception can be observed, the magnitude of the responses depending on the off-axis distance and frequency. The findings may shed new lights to the shear-wave reflection imaging analysis and development. Presentation Date: Tuesday, October 13, 2020 Session Start Time: 1:50 PM Presentation Time: 2:15 PM Location: 351D Presentation Type: Oral

Journal Articles

Journal:
Journal of Ship Research

*Journal of Ship Research*16 (01): 41–46.

Paper Number: SNAME-JSR-1972-16-1-41

Published: 01 March 1972

... spheroidal singularity point singularity singularity spheroid

**coordinate**system**equation**Havelock Water**Wave**Singularities**in**a Prolate Spheroidal**Coordinate**System' By J. N . Newman 2 Singular velocity potentials are derived which satisfy the linear free-surface boundary condition and appropriate...
Abstract

Singular velocity potentials are derived which satisfy the linear free-surface boundary condition and appropriate initial condition or radiation condition, but which are singular in the same form as a spheroidal harmonic. Thus these singularities can be utilized to satisfy boundary conditions on submerged spheroids without integrating conventional point sources or dipoles over the body surface or axis, in an analogous manner to the use of spheroidal harmonics in an unbounded fluid. In addition to the general unsteady case, equations are derived for the special cases of steady forward motion and of sinusoidal time-dependence, either at zero or constant forward velocity of the spheroid. Wave-free singularities are derived for the case of zero forward speed, and applications to problems in wave resistance and in seakeeping are discussed briefly.

Proceedings Papers

Paper presented at the The Second International Offshore and Polar Engineering Conference, June 14–19, 1992

Paper Number: ISOPE-I-92-393

... kg Trimonlte charge

**In**an off-centre position, generating a peak pressure of 24 MPa. To minimise the excessive CPU demand a single symmetry is assumed and only half the system modelled. As the charge Is detonated off centre and because the resulting pressure**wave**travels with a finite velocity, the...
Abstract

The Finite Element Method is used to investigate the dynamic large plastic deformation response of a spherical pressure vessel with nozzle attachments. Material and geometrical non-linearltles are incorporated In the development of the solution procedure. The system Is loaded by detonating a 3.2 kg Trimonlte charge In an off-centre position, generating a peak pressure of 24 MPa. To minimise the excessive CPU demand a single symmetry is assumed and only half the system modelled. As the charge Is detonated off centre and because the resulting pressure wave travels with a finite velocity, the pressure front arrives at different places on the Inner surface of the vessel at different times. This arrival phase lag Is Incorporated In the solution algorithm to obtain a more realistic response and the pressure peak value Is modified to take Into account the off centre location of the charge. INTRODUCTION Scaled-down laboratory experiments are often used to help develop an understanding of explosive forming processes such as welding, surface hardening, shaping, powder compaction and shock focusing for cutting purposes. Of course, for the sake of personnel safety and to prevent unwanted damage, it is extremely important that a secure and reliable confining chamber be available. One such chamber is in use at the Applied Mechanics Division of the Department of Mechanical Engineering at the University of Manchester Institute of Science and Technology. The chamber is spherical in shape with three nozzle access ports of different sizes. The geometrical details of the vessel and its attachment, together with its material characteristics are given elsewhere (Gill et ai, 1970 and Lazari et ai, 1991). Experiments with this vessel are conducted either under static loading by internal pressurisation (Gill et ai, 1970) or under dynamic loading by the detonation of an explosive charge of up to 141g (Lazari et al, 1991).

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Unconventional Resources Conference, April 1–3, 2014

Paper Number: SPE-168997-MS

... reflection of the ‘pressure

**wave**’ at the high permeability contrast boundaries was neglected**in**the previous studies because the amplitude of the reflected**waves**decays exponentially. A more rigorous treatment including the reflected amplitude into the formulation will be more complicated. However, previous...
Abstract

Current industry practice for characterization and assessment of unconventional reservoirs mostly utilizes empirical decline curve analysis or analytic rate and pressure transient analysis. High resolution numerical simulation with local PEBI grids and global corner point grids has also been used to examine complex non-planar fracture geometry, interaction between hydraulic and natural fractures and implications on the well performance. While the analytic tools require many simplified assumptions, numerical simulation techniques are computationally expensive and do not provide the more geometric understanding derived from the depth of investigation and drainage volume calculations. We propose a novel approach for rapid field-scale performance assessment of shale gas reservoirs. Our proposed approach is based on a high frequency asymptotic solution of the diffusivity equation in heterogeneous reservoirs and serves as a bridge between simplified analytical tools and complex numerical simulation. The high frequency solution leads to the Eikonal equation which is solved for a ‘diffusive time of flight’ that governs the propagation of the ‘pressure front’ in the reservoir. The Eikonal equation can be solved using the Fast Marching Method to determine the ‘diffusive time of flight’, which generalizes the concept of depth of investigation to heterogeneous and fractured reservoirs. It provides an efficient means to calculate drainage volume, pressure depletion and well performance and can be significantly faster than conventional numerical simulation. More importantly, in a manner analogous to streamline simulation, the ‘diffusive time of flight’ can also be used as a spatial coordinate to reduce the 3-D diffusivity equation into a 1-D equation, leading to a comprehensive simulator for rapid performance prediction of shale gas reservoirs (Patent Pending, 2013). The speed and versatility of our proposed method makes it ideally suited for high resolution reservoir characterization through integration of static and dynamic data. The major advantages of our proposed approach are its simplicity, intuitive appeal and computational efficiency. We demonstrate the power and utility of our method using a field example that involves history matching, uncertainty analysis and performance assessment of a shale gas reservoir located in East Texas. A sensitivity study is first carried out to systematically identify the ‘heavy hitters’ impacting the well performance. This is followed by a history matching and uncertainty analysis to identify the fracture parameters and the stimulated reservoir volume. A comparison of model predictions with the actual well performance shows that our approach is able to reliably predict the pressure depletion and rate decline.

Proceedings Papers

Paper presented at the The 30th International Ocean and Polar Engineering Conference, October 11–16, 2020

Paper Number: ISOPE-I-20-3243

... stability among different SPH schemes is made, and the improved SPH method was used to study the nonlinear sloshing. Studies have indicated that models based on Boussinesq-type

**equations**are an effective tool**in**predicting the behavior of nonlinear**waves****in**coastal and offshore environments and may be used...
Abstract

Highly accurate Boussinesq-type equations in terms of velocity potential are used for the simulation of shallow-water sloshing in a rectangular tank. The finite difference method is used for the spatial discretization of derivatives. The total velocity potential is separated into two parts: a particular solution satisfying the Laplace equation and the no-flow condition on the walls while the other part is solved by the Boussinesq-type model. Distinct features of the free surface can be observed under the selected external exciting frequencies. Comparisons are made between different cases, the results are analyzed and discussed. INTRODUCTION Sloshing is defined as the movement of fluid in a partially filled container, and has been studied by numerous scholars with different methods. A series of analyses about sloshing in spherical and cylindrical tanks including linear and nonlinear situations were given in the pioneering work of Abramson (1966). A study of the sloshing model and frequency in a spherical tank in two dimensions, as well as a multidimensional modal method of establishing the mode and calculating the frequency of natural sloshing was developed by Faltinsen(2000). A complete discussion of various aspects of liquid sloshing was given by Ibrahim (2005). Most analytical methods are based on potential flow theory and focus on the mechanism of fluid motion. Because of the limitations of the theoretical method, computational fluid dynamic methods are more widely used for researching sloshing phenomena. In Gedikli and Erguven (2003), a Variational Boundary Element Method (VBEM) based on the Hamilton's principle was presented to investigate the effect of a rigid baffle on the natural frequencies of the sloshing in a cylindrical container. Firouz et al. (2010) developed a modal approach for the nonlinear analysis of sloshing in a tank of arbitrary-shape under both horizontal and vertical excitations. An empirical formula for calculating the natural frequency of sloshing in a rectangular tank with baffle is presented in Hu et al. (2018) by using the Boundary Element Method (BEM) with linear free surface condition considered. The boundary element method has the advantages of less meshes and reducing dimension.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2015 SEG Annual Meeting, October 18–23, 2015

Paper Number: SEG-2015-5853357

... plates, etc.) accurately; 5) no linear system needs to be solved for ghost zone wavefield estimation, thus avoiding the singularity issue

**in**a matrix inversion. Conventional Finite-Difference vs. IBFD Method The acoustic**wave**propagation**equations**are as follows: qp j jp v fv (1) where p is the...
Abstract

Summary To accurately simulate seismic wave propagation for the purpose of land data acquisition, processing, and interpretation, especially for land data full waveform inversion (FWI), we developed an efficient high-order finite-difference modeling algorithm with the capability of handling arbitrarily shaped surface topography. Unlike most of the existing modeling algorithms dealing with irregular surface topography, this finite-difference algorithm, based on an immersed boundary (IB) method, uses regular Cartesian staggered or collocated grid system without suffering from the well-known staircasing error. In this immersed boundary finite-difference (IBFD) algorithm, arbitrary surface topography is accounted for by imposing the free surface boundary conditions at the exact boundary locations instead of using body-conforming grids, thus greatly reducing the complexity of preprocessing procedures. Furthermore, local continuity and curvatures (including sub-cell curvatures) are represented precisely through the employment of a local cylindrical or a spherical coordinate system. Wavefield values in a ghost zone required for boundary condition enforcement are obtained using a special recursive interpolation technique, which simplifies the boundary treatment and further improves the accuracy, as validated by the numerical simulation. Another unique feature of this algorithm is that the stencil length for ghost zone wavefield interpolation is adaptively determined by the local curvature to maintain the accuracy and stability. This method is a general algorithm applicable to acoustic, elastic, 2D, 3D, and anisotropic cases. A numerical example is presented to show its excellent performances compared with the conventional finitedifference method. Introduction Full waveform inversion (FWI) method developed rapidly in recent years because it is regarded as a powerful tool for subsurface structure and property reconstruction. Because it utilizes the full waveform information contained in recorded seismic data, FWI is able to provide high resolution velocity model and other geophysical property models such as anisotropic parameters, attenuation, density, and etc. Unlike conventional processing tools such as velocity tomography using traveltime information only, FWI is mainly based on amplitude fitting between simulated seismic data and recorded seismic data. Consequently, a highly accurate forward modeling engine is necessary for successful FWI inversions. Topography has a significant effect on recorded seismic data.

Proceedings Papers

Paper presented at the The 29th International Ocean and Polar Engineering Conference, June 16–21, 2019

Paper Number: ISOPE-I-19-005

... from the empty tank data). Two different geometries are considered: a hemispherical-bottomed buoy; and a

**cylindrical**structure with a moon-pool (Figure 1). For both structures, the effect of**wave**steepness on the motion of the structure, along with the load**in**the mooring, is examined. tank data...
Abstract

ABSTRACT This work concerns focused wave interactions with floating structures and represents an individual contribution to the CCP-WSI Blind Test Series 3, in which the submitted results are compared against both physical and alternative numerical solutions for varying wave steepness. The present numerical results are obtained using open-source CFD with waves generated via linear superposition of first order wave components, derived from empty tank data. Two geometries are considered and the effect of wave steepness on surge, heave and pitch motion, along with the load in the mooring, is examined. A ‘blind’ estimation of error is presented, thought to be 10% or less in peak motion and mooring loads. INTRODUCTION Two key issues that are limiting the routine use of computational fluid dynamics (CFD), are the uncertainty in the accuracy and the required time to obtain numerical results. The time taken to run a simulation is notoriously large, but this can be decreased through use of a larger computational resource. However, an often-overlooked factor is the man-hours required to set up a case through processes such as mesh design, and this has the potential to be considerably more time-consuming than the simulation time. For industry to benefit from the strengths of CFD models, the efficiency of the set up process needs to be increased, and this could be achieved through increased confidence in prediction by parametrically understanding numerical accuracy and providing standardised, ‘best practice’ implementations. Therefore, the scope of this work is to provide a ‘blind’ estimation of numerical accuracy, based purely on the reproduction of empty tank data. This approach was previously utilised for a case with a fixed structure (Brown et al., 2019), but here the focus is on the interaction of focused wave events with floating structures that represent a simplified wave energy converter (WEC). The presented work represents an individual contribution to the CCP-WSI Blind Test Series 3 (CCP-WSI, 2018), in which the submitted results are compared against both physical and alternative numerical solutions for varying wave steepness. The numerical results reported here are obtained using the open-source CFD software OpenFOAM (version 5.0). Wave generation is achieved via an expression-based boundary condition based on a linear superposition of first order wave components (derived from the empty tank data). Two different geometries are considered: a hemispherical-bottomed buoy; and a cylindrical structure with a moon-pool (Figure 1). For both structures, the effect of wave steepness on the motion of the structure, along with the load in the mooring, is examined.

Journal Articles

*International Journal of Offshore and Polar Engineering*30 (01): 70–77.

Paper Number: ISOPE-20-30-1-070

Published: 01 March 2020

... moment of inertia, and ID is the case ID number. Table 1 Mass properties for the WECs: Case 1 is for the hemi-

**spherical**-bottomed cylinder**in**focused**wave**groups, and Case 2 is the hollow cylinder with a moonpool**in**focused**wave**groups Case A [m] Tp [s] Hs [m] h [m] kpA 1 0.20 2.5 0.274 3.00 0.1287 2...
Abstract

This paper presents the numerical modelling of two point absorber wave energy converters (WECs) with and without a moonpool under focused wave conditions. The numerical model applies the overset mesh technique in order for the mesh to conform with the large-amplitude WEC motion induced by the focused wave groups. The incident wave group is first examined by a mesh convergence test and by comparing with the experimental data. The simulations are then carried out with the presence of the WEC. In total, three wave conditions are considered, each with the same wave period but with different wave heights. Nonlinear effects on the WEC motion are clearly exhibited when the wave steepness increases. The accuracy of the numerical results is carefully assessed against experimental data. Furthermore, the effects of the moonpool on the dynamics of the WEC are also discussed, where the WEC motion is compared for the case with and without a moonpool under the same wave conditions. INTRODUCTION In recent years, the possibility of harnessing energy from ocean wave resources has gained great interest, where different design concepts of wave energy converters (WECs) have been proposed, such as oscillating water columns, bottom-hinged pitching devices, floating pitching devices, overtopping devices, and point absorbers. Point absorbers are one of the simplest WECs. Their characteristic length is generally smaller than the typical wavelength at the peak wave frequency. Meanwhile, they are typically subjected to large-amplitude motions close to resonance. In such a condition, a highly nonlinear wave-structure interaction is expected, where local wave breaking and overtopping may occur. Moreover, the damping coefficient of the WEC can be composed of not only the radiation wave damping but also the power takeoff (PTO) damping, as well as the viscous damping. The viscous damping force itself can be important in many cases, which are due to vortex shedding and shear stress force; see, for example, Gu et al. (2018), Palm et al. (2018), Wei et al. (2015), and Giorgi and Ringwood (2017). Such characteristics make the wave-structure interaction process highly complex and distinct from the traditional large-volume offshore structures.

Proceedings Papers

Paper presented at the The Fourteenth International Offshore and Polar Engineering Conference, May 23–28, 2004

Paper Number: ISOPE-I-04-005

... introduced,

**in**accordance with**Equations**(20) and (21), to express the incident**wave****in**the**coordinate**system of body q. We note, that the present method has been also used**in**the past (Mavrakos et al., 1997a, 1997b) to assess the hydrodynamic characteristics of arrays of offshore**wave**energy converters.**In**...
Abstract

ABSTRACT The present paper is dealing with the numerical prediction of the performance characteristics of an array of five wave energy heaving converters placed in front of a reflecting vertical breakwater. At a first stage, the appropriate mechanical system modeling is presented. Detailed insight into the system's kinematical and power production characteristics is given. In the second part, the analytical method used for the evaluation of the hydrodynamic characteristics of the cylindrical floats moving in front of the breakwater is outlined, supplemented with representative numerical results concerning the hydrodynamic parameters in frequency and in time domain. Hydrodynamic interactions among the floats and the adjacent breakwater are exactly taken into account using the method of images. The third part of the paper is devoted to the numerical integration algorithm for solving the coupled non-liner equations of motion taking into account both linear and non-linear couplings with the power take-off mechanism. INTRODUCTION The work presented here has been carried out within the framework of the already finished first phase of the LABBUOY research project, supported by the EU, which has dealt with the mathematical and physical modeling of an economically efficient floating device for wave energy conversion into electricity. The operational principle of the LABBUOY concept comprises a row of half-immersed spherical or truncated cylindrical floaters attached on a lever with appropriate power transmission system. The system is mounted on a pier or breakwater above water level, as it is shown in Fig. 1, where some indicative geometrical dimensions of the system are also given. In the context of the present contribution, the numerical system modeling will be presented giving at a first stage insight into the hydrodynamics of the floaters placed in front of the breakwater together with the system's coupled non-linear governing equations of motion.

Journal Articles

Journal:
Journal of Ship Research

*Journal of Ship Research*33 (04): 318–325.

Paper Number: SNAME-JSR-1989-33-4-318

Published: 01 December 1989

... design of a submersible hull [6].

**In**the second phase, the structures are verified and opti- mized using direct procedures generally based on numerical methods. Submarine strength ulls are generally constructed using**cylindrical**shells for central sections and**spherical**shells for head domes. Because of...
Abstract

Submersible structures consist merely of simple and double curvature thin-walled shells. For this kind of structure, collapse occurs due to the combined nonlinear action of buckling and plasticity of material. Load-carrying capacity may then be assessed mainly by two approaches: experimental investigations and step-by-step numerical procedures. In nonlinear analyses, the results obtained are influenced by the magnitude of the load increment adopted. Solution procedures are then required in order to choose adequate parameters for material failure description as well as elastic nonlinearity. The aim of this paper is to carry out a suitable numerical procedure whose reliability does not depend on the finite-element code adopted.

Journal Articles

*International Journal of Offshore and Polar Engineering*18 (01).

Paper Number: ISOPE-08-18-1-043

Published: 01 March 2008

... circumferential

**wave**number n for half- filled and fully filled**cylindrical**containers, respectively. From these figures, it can be seen that the present RD element results are**in**agreement with results obtained by Xi et al. (1997), using a semi-analytical procedure.**In**particular, for a given circumfer- ential...
Abstract

ABSTRACT This paper is concerned with the free vibration analyses of axisymmetric multi-layer storage containers filled with liquid. A new finite element method (FEM) based on the relative displacement concept is presented for the analysis. The multilayer cylindrical tank walls are modeled by shell elements which have been developed based on the concept of relative displacements instead of the conventional degrees of freedom involving rotations. The liquid in the containers is discretised by specially designed quadrilateral fluid elements that are kinematically compatible with the shell elements. Some examples are presented to demonstrate the accuracy and effectiveness of the proposed method for the free vibration analysis of multilayer storage containers containing liquid. INTRODUCTION The dynamic interaction between the shell container and the liquid inside it has been a subject of intense study, more so in recent years due to the transportation of liquefied natural gas (LNG) in specially designed LNG vessels as well as the storage of LNG in floating terminals. Many studies on coupled liquid container interaction have been carried out. For example, Ang (1980) solved this problem by developing an axisymmetric thin shell element, and the effect of liquid sloshing was considered by using a coupling matrix for the shell and fluid elements. Chiba et al. (1985) analyzed a clamped-free cylindrical shell partially filled with an incompressible, inviscid liquid, and the effects of the initial hoop stresses and surface condition on the natural frequencies were investigated. Subhash and Bhattacharyya (1996) employed the FEM that made use of 2-node thin elastic shell elements and 8-node fluid elements for the coupled vibration analysis. Amabili (2000) used the Rayleigh-Ritz method to analyse the vibration problem of cylindrical shells filled with liquid. Further, Cho et al. (2002) addressed analytical and numerical studies on the free vibration of fluid-structure interaction problems considering the fluid compressibility.

Proceedings Papers

Publisher: Offshore Technology Conference

Paper presented at the Offshore Technology Conference, April 21–23, 1970

Paper Number: OTC-1209-MS

... upstream end of the diaphragm, a results of this type of analysis agrees distance of 12.0

**in**. from the coupling point. accurately with the other experiments conducted h additional 0.635 millisec is required for with different geometries. Thus, one can the**spherical**diverging**wave**to reach the correctly...
Abstract

ABSTRACT An investigation of the transient character of the near-field acoustic radiation from a waterhammer-excited, cylindrical diaphragm was conducted. The investigation was principally experimental focused on relating the radiated waveforms to pressure transients within the pipe. The basic experimental system consisted of a length of rigid pipe extending from a flow source into a large sonar tank. The rigid pipe was interrupted by a cylindrical section of elastomeric tubing which served as a diaphragm and, thus, as an acoustic coupler between the water inside the pipe and water surrounding the pipe. A quick closing valve downstream of the diaphragm produced water hammer transients which were coupled to the surrounding water through the diaphragm and were monitored in the near-field by hydrophones. The physical system geometry was varied to include data for a range of pipe sizes from 0.5 in. to 4.0 in. ID and from 10 ft to 34 ft in length. The experiment revealed that two distinct major wave could always be identified in each radiated wave pattern. One was found to originate from the primary water hammer compressive wave downstream of the diaphragm. The other major wave was attributed to a precursor wave which originated upstream of the diaphragm. The propagation paths of the two waves are identified, and it is shown that their relation to one another in time space is a References and illustrations at end of paper predictable function of geometry. It is estimated, by extrapolating from the experimental data, that an acoustic pressure in excess of 88 psi at 1 yd is possible from a system of realistic mechanical design. INTRODUCTION Acoustic energy is an important and necessary energy form in the marine sciences. It serves as the fundamental vehicle for ascertaining the vast majority of all geophysical knowledge of the ocean floor and underlying shallow Substrata. The investigation described in this paper was principally motivated by a need to generate acoustic energy more efficiently from a low-cost, readily available source of energy. The most basic limitation imposed by any type of current acoustic energy source lies in the tremendous amount of energy which must be made available in order to produce the minimum necessary amount of acoustic energy. Powdered explosives, for example, have far less than 1 percent of their chemical potential energy converted to acoustic energy. Most of the energy is consumed in heating the combustion products and in surface blowout. The air gun can be made to produce acoustic energy in the desired low frequency bandwidth, but the poor acoustic impedance match between air and water reduces its efficiency to the order of 1 percent. The kinetic energy of flow had here to fore never been extensively investigated as a possible source of acoustic energy-applicable to the marine sciences. It is the purpose of this writing to report the results of such an investigation. Both experimental and theoretical modeling were employed. The modeling techniques, data analysis, results and conclusions are presented.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 1988 SEG Annual Meeting, October 30–November 3, 1988

Paper Number: SEG-1988-0672

... McGowan (1986) have developed a formulation of the

**cylindrical**slant stack as a weighted sum of Cartesian slant stacks. P z,P) = 2Jdp (p2-pr2) -112, J 1: dr l e 0 0 [U(ztp r,r,O)tU(T-p r,r,O)] (2) where U is the**wave**field, r is the offset**in****cylindrical****coordinates**, 7 is the intercept ime p is the ray...Advertisement