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Trapping and Near-Trapping by Arrays of Porous Cylinders in Water Waves: A Unified Formulation Approach

This study explores trapping effects of porous cylinder arrays in water waves using the addition theorem and superposition technique. It presents a null-field approach, numerical examples, and advantages of the proposed methodology.

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Trapping and Near-Trapping by Arrays of Porous Cylinders in Water Waves: A Unified Formulation Approach

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  1. Dr. Jeng-Tzong Chen Date: September, 2009 Place: City London University National Taiwan Ocean University MSVLAB Department of Harbor and River Engineering Trapping and near-trapping by arrays of porous cylinders in water waves using the addition theorem and superposition technique National Taiwan Ocean University

  2. Outline • Motivation and literature review • Unified formulation of null-field approach • Numerical examples • Concluding remarks

  3. Outline • Motivation and literature review • Engineeringproblems • Motivation • Present approach • Unified formulation of null-field approach • Numerical examples • Concluding remarks

  4. Engineeringproblems • Platform (Offshorestructure)

  5. Constant, linear, quadratic elements C.P.V., H.P.V., M.P.V. Motivation Numerical methods for engineering problems FDM / FEM / BEM / BIEM / Mesh-less method BEM / BIEM Mesh generation Convergence rate Ill-posed model Boundary-layer effect Treatment of singularity and hypersingularity Free

  6. Motivation BEM / BIEM Improper integral Singularity & hypersingularity Regularity Fictitious BEM Bump contour Limit process Fictitious boundary Achenbach et al. (1988) Null-field approach Guiggiani (1995) Gray and Manne (1993) Collocation point CPV and HPV Ill-posed Waterman (1965)

  7. Present approach Degenerate kernel Fundamental solution No principal value CPV and HPV • Advantages of present approach • mesh-free generation • well-posed model • principal value free • elimination of boundary-layer effect • exponential convergence Kress, 1989

  8. Outline • Motivation and literature review • Unified formulation of null-field approach • Boundary integral equation and null-field integral equation • Convergence rate between present method and conventional BEM • Degenerate kernel and Fourier series • Adaptive observer system • Linear algebraic system • Flowchart of present method • Numerical examples • Concluding remarks

  9. Boundary integral equation and null-field integral equation Exterior case Interior case Degenerate (separate) form

  10. Convergence rate between present method and conventional BEM Conventional BEM Present method Degenerate kernel Two-point function Fundamental solution Constant, linear, quadratic elements Fourier series expansion Boundary density Convergence rate Exponential convergence Linear convergence

  11. x U(s,x) T(s,x) x L(s,x) M(s,x) Degenerate kernel Expand fundamental solution by using degenerate kernel Ｄegenerate kernel（Helmholtz） s O

  12. cosnθ, sinnθ boundary distributions kth circular boundary Degenerate kernel and Fourier series Expand boundary densities by using Fourier series

  13. Adaptive observer system Source point Collocation point

  14. Linear algebraic system y x

  15. Decompose two parts Free field Radiation field Expansion Degenerate kernel for fundamental solution Fourier series of boundary densities ò ò = - Î c 0 T ( s , x ) u ( s ) dB ( s ) U ( s , x ) t ( s ) dB ( s ), x D B B Flowchart of present method Collocation on the real boundary Original problem Linear algebraic system Calculation of the unknown Fourier BIE for the domain point Superposing the solution of two parts Total field

  16. Outline • Motivation and literature review • Unified formulation of null-field approach • Numerical examples • Water wave interaction with surface-piercing porous cylinders • Concluding remarks • Further studies

  17. Water wave interaction with surface-piercing porous cylinders Governing equation: Separation variable : Seabed boundary conditions : Free-surface conditions : kinematic boundary condition at free surface (KFSBC) dynamic boundary condition at free surface (DFSBC) where

  18. Problem statement Boundary condition: Dispersion relationship: , . Dynamic pressure: Force: Original problem

  19. 11 y 16 5 10 10 4 15 9 3 14 8 13 2 7 7 12 1 6 2b x Numerical examples Case 2. Five sets (ka=4.08482, a/b=0.8) Case 1. Four-cylinders array for one sets (ka=4.08482, a/b=0.8) y 3 7 1 4 2b 2 x

  20. Effect of impermeable case for contour plots (b) Null-field BIEM (M=20 ) (a) BEM ( , Chen) Contour plots of free-surface elevation of the four impermeable cylinders (G=0.0, )

  21. Effect of impermeable case for free-surface elevation (a) Williams and Li (c) Null-field BIEM (M=20) (b) BEM (Chen) Free-surface elevation of the arrays of four impermeable cylinders. (G=0.0, )

  22. Effect of disorder case for contour plots (b) Null-field BIEM (M=20 ) (a) BEM ( , Chen) Contour plots of free-surface elevation of the four porous cylinders(G=1.0, )

  23. Effect of disorder case for free-surface elevation (a) Williams and Li (c) Null-field BIEM (M=20) (b) BEM(Chen) Free-surface elevation of the arrays of four impermeable cylinders. (G=0.0, )

  24. Near-trapped mode for the four cylinders at ka=4.08482 (a/b=0.8, G=0.0, ) (no disorder and no porosity) (a) Contour by the present method (M=20)

  25. Near-trapped mode for the four cylinders at ka=4.08482 (a/b=0.8, G=0.0, ) 3 1 4 2 54 (c) Horizontal force on the four cylinders against wavenumber (b) Free-surface elevations by the present method (M=20)

  26. Near-trapped modes versus incident angle 54 2 1 3 4

  27. a 2b a random variable in the range [0,1]. maximum permissible displacement (p=b-a). global disorder parameter. Perturbation of ordered cylinder arrangements

  28. Effect of disorder and porosity Disorder cylinder (b)Contour by the present method (disorder , , impermeable) (a)Contour by the present method (no disorder, impermeable) Porous cylinder Disorder and porous cylinder (c)Contour by the present method (no disorder, porous, G =1) (d)Contour by the present method (disorder , , porous, G =1)

  29. Outline • Motivation and literature review • Unified formulation of null-field approach • Numerical examples • Concluding remarks

  30. Concluding remarks -1/2 • A general-purpose program for solving the water wave problems with arbitrary number, different size and various locations of circular cylinders was developed. • We have proposed a BIEM formulation by using degenerate kernels, null-field integral equation and Fourier series in companion with adaptive observer system.

  31. Concluding remarks -2/2 • Near trapped mode is observed in this study. • It is found that the disorder is more sensitive to suppress the occurrence of near-trapped modes than the porosity.

  32. The end Thanks for your kind attention. Your comments will be highly appreciated. Welcome to the web site of MSVLAB: http://ind.ntou.edu.tw/~msvlab

  33. Water waves containing circular and elliptical cylinders Analytical solution Semi-analytical solution Numerical solution Null-field BIEM Linton & Evan approach BEM MSVLab 陳正宗、李家瑋、李應德、林羿州 岳景雲、陳一豪、賴瑋婷 ok ok Bessel to Mathieu ? Error ?

  34. What I have done (林羿州) What I have done .doc羿州 製

  35. Trapped and near-trapped modes Trapped modes Near-trapped modes Dirichlet or Neumann modes Trapped and near-trapped modes.ppt

  36. Irregular fictitious and spurious frequency and trap modes Mathematical Nonuniqueness problem Exterior acoustics (fictitious) Interior acoustics (spurious) Floating body (water wave) mathematics Irregular frequencies Acoustics Water wave Physical resonance trap modes physics Trap2009.ppt

  37. Further studies -1/4 • The extension to Helmholtz problem with a hill can be studied by using the present approach in conjunction with the multi-domain technique by decomposing the original problem into one interior problem of circular domain and a half-plane problem with a semi-circular canyon. • In the further research, the Helmholtz problems with circular boundaries may be extended to other shapes instead of incident plane wave, shore-crested incident wave can be also considered. SH wave

  38. Further studies -2/4 • For water-wave scattering with elliptical cylinders, it deserves further study by using our approach. • We will deal with Laplace and Helmholtz problems containing circular and elliptical cylinders at the same time. Failure of Linton and Evans method

  39. Further studies -3/4 • The degenerate kernels are expanded in the polar coordinates and only problems with circular boundaries are solved. For boundary value problems with crack, further investigation should be considered. • In the further research, we may extend to mixed-type BCs by using the null-field integral equation approach. Crack Dirichlet Neumann

  40. Further studies -4/4 • Following the success of applications in two-dimensional problems, it is straightforward to extend this formulation to 3-D problems with spherical boundaries by using the corresponding 3-D degenerate kernel functions for fundamental solutions and spherical harmonic expansions for boundary densities. • Trapped modes versus incident angle F k versus incident angle 90° θ

  41. Decompose two parts ＝ ＋ Original Problem Free field Radiation field(typical BVP)

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