A semi-analytical computation of the Kelvin kernel for potential flows with a free surface

In the computation of a three–dimensional steady creeping flow around a rigid body, the total body force and torque are well predicted using a boundary integral equation (BIE) with a single concentrated pair Stokeslet- Rotlet located at an interior point of the body. However, the distribution of surface tractions are seldom considered. Then, a completed indirect velocity BIE of Fredholm type and second-kind is employed for the computation of the pointwise tractions, and it is numerically solved by using either collocation or Galerkin weighting procedures over flat triangles. In the Galerkin case, a full numerical quadrature is proposed in order to handle the weak singularity of the tensor kernels, which is an extension for fluid engineering of a general framework (Taylor, 2003, “Accurate and Efficient Numerical Integration of Weakly Singulars Integrals in Galerkin EFIE Solutions,” IEEE Trans. on Antennas and Propag., 51(7), pp. 1630–1637). Several numerical simulations of steady creeping flow around closed bodies are presented, where results compare well with semianalytical and finite-element solutions, showing the ability of the method for obtaining the viscous drag and capturing the singular behavior of the surface tractions close to edges and corners. Also, deliberately intricate geometries are considered.

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“…( 4), and φ j (y y y) is the single-layer surface density given in Eq. (8). It should be noted that Eq.…”

Section: The Hebeker Alternativementioning

confidence: 99%

“…Closed forms derived from a side local frame strategy are also commonly employed [6], where the surface integral over an element is replaced by its closed contour integration, and a side local frame is used for each side contribution, with applications, e.g. in free surface flows [7], seakeeping [8] and exterior flows [9].…”

Section: Introductionmentioning

confidence: 99%

Galerkin Boundary Elements for a Computation of the Surface Tractions in Exterior Stokes Flows

D’Elı́a

Battaglia

Cardona

et al. 2014

Journal of Fluids Engineering

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“…Peter and Meylan [8] proposed an eigenfunction expansion representation for the free surface Green function which is easy to evaluate numerically. In 2011, a semi-analytical method was used with Haskind-Havelock kernel calculated by a singularity substractive technique [9]. The multipole expansion of the free surface Green's function and its derivative has been extended in a 3D fast multipole algorithm [10].…”

Section: Introductionmentioning

confidence: 99%

Use of Clement’s ODEs for the Speedup of Computation of the Green Function and its Derivatives for Floating or Submerged Bodies in Deep Water

Xie

Babarit

Rongère

et al. 2018

Volume 7A: Ocean Engineering

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A new acceleration technique for the computation of first order hydrodynamic coefficients for floating bodies in frequency domain and in deep water is proposed. It is based on the classical boundary element method (BEM) which requires solving a boundary integral equation for distributions of sources and/or dipoles and evaluating integrals of Kelvin’s Green function and its derivatives over panels. The Kelvin’s Green function includes two Rankine sources and a wave term. In present study, for the two Rankine sources, analytical integrations of strongly singular kernels are adopted for the linear density distributions. It is shown that these analytical integrations are more accurate and faster than numerical integrations. The wave term is obtained by solving Clément’s ordinary differential equations (ODEs) [1] and an adaptive numerical quadrature is performed for integrations over the panels. It is shown here that the computational time of the wave term by solving the ODEs is greatly reduced compared to the classical integration method [7].

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“…As to the other representations of GF, in 2004, Peter et al [10] proposed the eigenfunction expansion method in which the truncation terms number should reach not less than 60 in order to obtain 10 -6 precision. In 2011, Elia et al [11] advocated a semi-analytical method that divided the integral into two terms, an adaptive quadrature was used for the regular term, and the singular term was completed by an approximation function. Clement [12] introduced the pioneering method using classical fourthorder Runge-Kutta method to solve a second-order ordinary differential equation of frequencydomain GF.…”

Section: Introductionmentioning

confidence: 99%

Highly Precise Approximation of Free Surface Green Function and Its High Order Derivatives Based on Refined Subdomains

Shan¹,

Wu²

2017

brod

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SummaryThe infinite depth free surface Green function (GF) and its high order derivatives for diffraction and radiation of water waves are considered. Especially second order derivatives are essential requirements in high-order panel method. In this paper, concerning the classical representation, composed of a semi-infinite integral involving a Bessel function and a Cauchy singularity, not only the GF and its first order derivatives but also second order derivatives are derived from four kinds of analytical series expansion and refined division of whole calculation domain. The approximations of special functions, particularly the hypergeometric function and the algorithmic applicability with different subdomains are implemented. As a result, the computation accuracy can reach 10 -9 in whole domain compared with conventional methods based on direct numerical integration. Furthermore, numerical efficiency is almost equivalent to that with the classical method.

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A semi-analytical computation of the Kelvin kernel for potential flows with a free surface

Cited by 14 publications

References 14 publications

Galerkin Boundary Elements for a Computation of the Surface Tractions in Exterior Stokes Flows

Galerkin Boundary Elements for a Computation of the Surface Tractions in Exterior Stokes Flows

Use of Clement’s ODEs for the Speedup of Computation of the Green Function and its Derivatives for Floating or Submerged Bodies in Deep Water

Highly Precise Approximation of Free Surface Green Function and Its High Order Derivatives Based on Refined Subdomains

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