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Straightforward discretisation of Green function and free-surface potential flow around a three-dimensional lifting body

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Abstract

A straightforward evaluation approach of free-surface Green function is developed to solve the potential flow around a three-dimensional lifting body. The free-surface waves generated by the movement of the lifting body is presented in an expansion of plane regular waves traveling in \(\theta \) directions with wave number magnitudes \(k>0\). A boundary element method is combined with the evaluation approach and Hess-Smith panel integral formulae to predict hydrodynamic performance of a three-dimensional lifting body. Numerical results produced by the proposed method are compared favourably with experimental measurements.

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Abbreviations

b :

Hydrofoil span

c :

Hydrofoil chord

C\(_d\) :

Drag coefficient

\(C_l\) :

Lift coefficient

\(C_p\) :

Pressure coefficient

\((x',y',z')\) :

Dimensional coordinates

(x, y, z):

Non-dimensional coordinates

\(c_{l,m}\) :

Regular wave expansion coefficients

\(\mathrm {Fn}\) :

Froude number \(U/\sqrt{gc}\)

h :

(NACA4412) Vertical distance between \(z=0\) and a mid chord point

h :

(Joukowski) Vertical distance between \(z=0\) and a trailing edge point

g :

Gravitational acceleration

G :

Free-surface Green function

\(G^\mu \) :

Dissipative free-surface Green function

\(i \) :

Pure imaginary number \(\sqrt{-1}\)

K :

Singular wave integral

\(K^\mu \) :

Regular wave integral

\({\mathbf {n}}_{i,j}\) :

Panel normal vector

\(N_c\) :

Number of panels in chordwise direction

\(N_s\) :

Number of panels in the spanwise direction

\(N_k\) :

Number of panels of \((0, k_{\mathrm {max}})\)

\((0, k_{\mathrm {max}})\) :

Approximate domain of the wave number integral domain \((0,\infty )\)

\(N_\theta \) :

Number of panels of the \(\theta \) direction domain \((-\frac{\pi }{2},\frac{\pi }{2})\)

\({\mathbf {p}}_{i,j}\) :

Panel grid points

\({\mathbf {q}}_{i,j}\) :

Panel control points

U :

Uniform stream speed

\({\mathbf {t}}_{i,j}^c\) :

Panel chordwsie tangential vector

\({\mathbf {t}}_{i,j}^s\) :

Panel spanwsie tangential vector

D :

Linearised fluid domain

\(\alpha \) :

Angle of attack

\(\mu \) :

Energy dissipation number

\(\nu \) :

Wave number \(1/{\mathrm {Fn}}^2\)

\(\phi \) :

Dimensionless perturbed velocity potential

\(\chi \) :

Wave elevation around a hydrofoil

\(\chi _s\) :

Wave elevation around a single source

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Acknowledgments

This research was partially supported by NSF (Grant No. 11571240) of China.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Shenzhen University, Shenzhen, 518052, China

    Zhi-Min Chen & Qiang Liu

  2. Ship Science, University of Southampton, Southampton, SO17 1BJ, UK

    Zhi-Min Chen & W. G. Price

  3. WUT-UoS High Performance Ship Technology Joint Center, Wuhan, 430063, China

    W. G. Price

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  1. Zhi-Min Chen
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  2. Qiang Liu
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Correspondence to Zhi-Min Chen.

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Chen, ZM., Liu, Q. & Price, W.G. Straightforward discretisation of Green function and free-surface potential flow around a three-dimensional lifting body. J Mar Sci Technol 22, 149–161 (2017). https://doi.org/10.1007/s00773-016-0400-3

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