A global approximation to the Green function for diffraction radiation of water waves

Wu, Huiyu; Zhang, Chenliang; Zhu, Yi; Li, Wei; Wan, Decheng; Noblesse, Francis

doi:10.1016/j.euromechflu.2017.02.008

Cited by 39 publications

(32 citation statements)

References 18 publications

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“…With the rapid development of computing capacity, numerical computation of a linear hydrodynamic problem is no longer time consuming. However, the Green function evaluation due to the presence of an singular wave integral is still known to be troublesome and sophisticated mathematical treatments are supposed to be employed to attack the singularity [21,22,23,24,25]. Therefore, the purpose of the present investigation is not for reducing the numerical simulation time in solving a linear hydrodynamics problem, but to simplify the accessibility to coding a body wave motion flow.…”

Section: Discussionmentioning

confidence: 99%

“…The singular wave integral (6) has been approximated by a variety of elementary function expansions (see, for example, [21,22,23,24,25]).…”

Section: Introductionmentioning

confidence: 99%

“…One may also refer to [31] on the translating Green function for the use of (13) for determining the uniqueness of the corresponding singular integral. Recent development of [22] for polynomial function approximation to the two flow components of the singular wave integral (6) was given in [25] and applied to a body wave motion problem [5]. The straightforward integration technique is initiated from [32], on the integration of the regular wave integral of the two-dimensional vortex free surface Green function, which is developed from the understanding of the instability of viscous flow in magnetohydrodynamics [33,34] dominated by the Harmann layer friction controlled by the Hartmann number µ.…”

Section: Introductionmentioning

confidence: 99%

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Straightforward integration for free surface Green function and body wave motions

Chen

2019

European Journal of Mechanics - B/Fluids

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An alternative manner is provided for solving the classical linearised problem of the radiation and diffraction of regular water waves caused by oscillation of a floating body in deep water. It is shown that the singular wave integrals of the three-dimensional free surface Green function G and its gradient ∇G can be regarded as regular wave integrals and are integrated directly. The method is validated by comparing with benchmark data for a floating or submerged body undergoing oscillatory wave motions. The comparison shows that the evaluation is sufficiently accurate for practical purposes. As the significance of the method, the numerical approximation stability for the gradient ∇G is shown to be the same with that for G.

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Section: Discussionmentioning

confidence: 99%

“…The singular wave integral (6) has been approximated by a variety of elementary function expansions (see, for example, [21,22,23,24,25]).…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Straightforward integration for free surface Green function and body wave motions

Chen

2019

European Journal of Mechanics - B/Fluids

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“…The classical expression of G is

4 π G = - \frac{1}{r} - \frac{1}{r^{'}} - 2 \int_{0}^{\infty} \frac{\exp false(false(z^{'} - i h \cos θ false) k false)}{k - 1} normald k,

where

r = \sqrt{h^{2} + {false(z - ζ false)}^{2}}

h = \sqrt{x^{' 2} + y^{' 2}}

, x ′ = x − ξ , y ′ = y − η , and z ′ = z + ζ . Equation can be simplified as

4 π G = - \frac{1}{r} - \frac{1}{d} + W + L,

where

d = \sqrt{h^{2} + {false(z + ζ false)}^{2}} = \sqrt{h^{2} + v^{2}}

, and the terms L and W are served as non‐oscillatory local flow component and wave component separately. L is defined by

L false(h, v false) = - \frac{4}{π} \int_{0}^{π false/ 2} Re false(e^{M} E_{1} false(M false) false) normald θ,

where

M = v + i h \cos θ

and E 1 is the complex exponential–integral function.…”

Section: Methodsmentioning

confidence: 99%

An optimization on water wave diffraction approximation based on wave packets

Yang

Liu

2019

Computer Animation & Virtual

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This paper proposes a novel approach for diffraction approximation in two‐dimensional water wave simulation. Our method is based on wave packets, which ensures unconditionally stability. We introduce the Green function in order to generate diffraction patterns that are used to judge whether our method can produce realistic diffraction effects. This method also implements propagation direction optimization for diffractive wave packets to handle the efficiency problem existed in the state‐of‐the‐art approach. Various results indicate that our new method can produce more appealing diffractive effects than current particle‐based approaches. Besides, from the efficiency perspective, compared to current approaches, our implementation greatly enhances the computational efficiency for scenes with high curvature obstacle boundaries.

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“…On the basis of that, linear table interpolation fast method was proposed by Ponizy et al [4], which gave a precision of 10 -5 . In 2017, Wu et al [5] further proposed simple approximations to the local flow components of GF and its first order derivatives without discussion about the calculation error. As to a representation which was in terms of a semi-infinite integral involving a Bessel function and a Cauchy singularity, Newman [6] developed the classical fast combined method with analytical series expansions and multi-dimensional polynomial approximations, which was applied in the notable hydrodynamic analysis code-WAMIT finally.…”

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 global approximation to the Green function for diffraction radiation of water waves

Cited by 39 publications

References 18 publications

Straightforward integration for free surface Green function and body wave motions

Straightforward integration for free surface Green function and body wave motions

An optimization on water wave diffraction approximation based on wave packets

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

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