Second-order wave diffraction by a circular cylinder using scaled boundary finite element method
|
|
- Edmund Singleton
- 6 years ago
- Views:
Transcription
1 IOP Conference Series: Materials Science and Engineering Second-order wave diffraction by a circular cylinder using scaled boundary finite element method To cite this article: H Song and L Tao 010 IOP Conf. Ser.: Mater. Sci. Eng Related content - The scaled boundary finite element method applied to electromagnetic field problems Jun Liu, Gao Lin, Fuming Wang et al. - Schur decomposition in the scaled boundary finite element method in elastostatics M Li, H Song, H Guan et al. - Isogeometric analysis based on scaled boundary finite element method Y Zhang, G Lin and Z Q Hu View the article online for updates and enhancements. This content was downloaded from IP address on 9/1/017 at 1:33
2 Second-order wave diffraction by a circular cylinder using scaled boundary finite element method H Song 1 and L Tao 1 Griffith School of Engineering, Gold Coast campus, Griffith University, QLD4, Australia h.song@griffith.edu.au School of Marine Science and Technology, Newcastle University, NE1 7RU, England, UK l.tao@newcastle.ac.uk Abstract. The scaled boundary finite element method (SBFEM) has achieved remarkable success in structural mechanics and fluid mechanics, combing the advantage of both FEM and BEM. Most of the previous works focus on linear problems, in which superposition principle is applicable. However, many physical problems in the real world are nonlinear and are described by nonlinear equations, challenging the application of the existing SBFEM model. A popular idea to solve a nonlinear problem is decomposing the nonlinear equation to a number of linear equations, and then solves them individually. In this paper, second-order wave diffraction by a circular cylinder is solved by SBFEM. By splitting the forcing term into two parts, the physical problem is described as two second-order boundary-value problems with different asymptotic behaviour at infinity. Expressing the velocity potentials as a series of depth-eigenfunctions, both of the 3D boundary-value problems are decomposed to a number of D boundary-value sub-problems, which are solved semi-analytically by SBFEM. Only the cylinder boundary is discretised with 1D curved finite-elements on the circumference of the cylinder, while the radial differential equation is solved completely analytically. The method can be extended to solve more complex wave-structure interaction problems resulting in direct engineering applications. 1. Introduction Recently, a new semi-analytical method, namely scaled boundary finite element method has been successfully applied to many engineering problems, combining the advantages of both finite element method (FEM) and boundary element method (BEM) [10]. The method only discretises the body boundary surface with finite elements, then transforms the governing partial differential equations to ordinary matrix differential equations in radial direction which are solved analytically. The method solves the problems of singularities more accurately and the problems of unbounded domains more efficiently compared to FEM. Unlike BEM, it does not require a fundamental solution and is free from the difficulties caused by irregular frequencies and sharp corners. Fewer elements are needed to obtain very accurate solution. SBFEM was first proposed to solve the problems in soil-structure interaction by Song and Wolf [6], and later was applied to a variety of engineering fields. For example, Ekevid and Wiberg applied SBFEM and FEM to analyse wave propagation related to moving loads in railway engineering, demonstrating the performance of the method by presenting the dynamical response of a railroad section []. Teng et al. used SBFEM to simulate the water sloshing in a rectangular water container, finding that the SBFEM method gives much better results than the FEM method in the case of same c 010 Published under licence by Ltd 1
3 mesh size [9]. By coupling SBFEM and FEM, Li et al. obtained semi-analytical solution for the characteristics of a two-dimensional dam-reservoir system with absorptive reservoir bottom in the frequency domain [3]. For short-crested waves diffracted by a vertical circular cylinder, Tao et al. obtained a high accurate and efficient result in unbounded domain by SBFEM with a small number of surface finite elements [8]. Song et al. extended their solution to water wave interaction with multiple cylinders of arbitrary shape recently [7]. However, most of the previous works focus on linear problems, in which superposition principle is applicable. In fact, many physical problems in the real world are nonlinear and are described by nonlinear equations, challenging the application of the existing SBFEM model. In this paper, second-order wave diffraction by a circular cylinder is solved by SBFEM. As an important attempt of the application of SBFEM in nonlinear dynamics, the method in this paper has a promising extension to solve more complex wave-structure interaction problems resulting in direct engineering applications.. Theoretical consideration.1. General governing equations The governing equation of the wave diffracted by a circular cylinder is Laplace equation. By separating the variables in direction of the water depth, the governing equations to be solved are homogeneous Helmholtz equation and non-homogeneous Helmholtz equation. The equation can be expressed in a general form φ(x, y) + k φ(x, y) + f = 0, in Ω, (1) where is the horizontal Laplacian, φ(x, y) is the velocity potential to be solved, k is the wave number, f is a known item, and Ω is the solution domain. Defining the Dirichlet boundary and Neumann boundary as Γ φ and Γ v respectively, we have φ = φ, on Γ φ, () φ,n = φ n, on Γ v, (3) where φ n is the prescribed normal derivative of the velocity potential, the overbar denotes a prescribed value, and comma in the subscript designates the partial derivative with respect to the variable following the comma... Scaled boundary finite-element transformation The finite-element method requires the weighted residuals of the governing equation to be zero. Hence Eqs. (1), () and (3) are multiplied by a weighting function w and integrated over the solution domain and the boundary. Performing integration by parts, the resulting equation becomes T w φdω wfdω w φ n dγ = 0. (4) Ω Ω Γ v SBFEM defines the solution domain Ω by scaling a single piecewise-smooth curve S relative to a scaling centre (x 0, y 0 ) (see Fig. 1). The circumferential coordinate s is anticlockwise along the curve S and the normalised radial coordinate ξ is a scaling factor, defined as 1 at curve S and 0 at the scaling centre. The whole solution domain Ω is in the range of ξ 0 ξ ξ 1 and s 0 s s 1. The two straight sections s = s 0 and s = s 1 are called side-faces. They coincide, if the curve S is closed. For bounded domain, ξ 0 = 0 and ξ 1 = 1; whereas, for unbounded domain, ξ 0 = 1 and ξ 1 =. Therefore the Cartesian coordinates are transformed to the scaled boundary coordinates ξ and s with the scaling equations x = x 0 + ξx s (s), y = y 0 + ξy s (s). (5)
4 Typical element S ξ S(ξ=1) ξ=ξ 0 O (x 0, y 0 ) s=s 0 ξ=ξ 1 s=s 1 Figure 1. The coordinate definition of SBFEM. By employing SBFEM, an approximate solution of φ is sought as φ A (ξ, s) = N(s)a(ξ), (6) where N(s) is the shape function, the vector a(ξ) is analogous to the nodal values same as in FEM. By performing scaled boundary transformation, the final scaled boundary finite-element equations are q(ξ 1 ) = N(s) T ( v n (ξ 1, s))ξ 1 ds, (7) S q(ξ 0 ) = N(s) T ( v n (ξ 0, s))ξ 0 ds, (8) where S E 0 ξ a(ξ),ξξ + (E 0 + E T 1 E 1 )ξa(ξ),ξ E a(ξ) + k ξ M 0 a(ξ) + ξ F b = ξf s (ξ), (9) q(ξ) = E 0 ξa(ξ),ξ + E T 1 a(ξ), (10) and the meaning of all the coefficients can be found in [7, 10]. In the present study, the side-faces coincide so that the flow across the side-faces is equal and opposite, thus the term F s (ξ) vanishes. Therefore, the final governing equation (9) is a homogeneous second-order ordinary matrix differential equation of rank m. The solution of equation (9) for F b = 0 is found as [8] a S 1 (ζ) = THC, (11) where matrices T and H are related to the geometry coefficients and C is related to the boundary conditions. 3. Wave diffraction 3.1. The first-order problem Consider a monochromatic wave train propagating in the direction of the positive x axis. A fixed vertical cylindrical structure extends from the sea bottom to the free surface of the ocean 3
5 y Γ Ω Γ c O x Γ Incident waves z η Γ s O z= 0 x Γ Ω h Ω Γ Γ b a Figure. Definition sketch of wave diffraction by a cylindrical structure along z axis. The origin (x 0, y 0 ) is placed at the centre of the cylinder on the mean surface level (see Fig. ). The following notation have been used in the paper: Φ = total velocity potential, Φ I = velocity potential of incident wave, Φ S = velocity potential of scattered wave, k = total wave number, ω = wave frequency, h = water depth, A = amplitude of incident wave, a = cylinder radius, t = time, ρ = mass density of water, and g = gravitational acceleration. Assuming the fluid to be inviscid, incompressible and the flow to be irrotational, the fluid motion can be described by potential flow theory. The velocity potential of the first order incident wave is [4] Φ I 1 = iga cosh k(z + h) e ikx e iωt in Ω. (1) ω cosh kh The first order solution of the diffraction wave was given by [8] a S 1 (ζ) = THC = 1 k TH ht 1 v S 1, (13) where the vector a S 1 (ξ) is analogous to the nodal values of the velocity potential of the scattered wave, T is a matrix related to the coefficient matrices of the discretisation, H h is a matrix related to the Hankel functions of the first kind, v S 1 is the vector of nodal normal velocity of the first-order scattered wave on the body boundary. 3.. The second-order problem At the second order, O(ɛ ), the velocity potential Φ satisfies [5] Φ + Φ = 0, h < z < 0, (14) subject to the bottom condition Φ = 0, z = h, (15) 4
6 body boundary condition Φ = 0, r r = a, (16) and combined free surface boundary condition Φ + 1 Φ g t = 1 Φ 1 g t [ g ( ) ] Φ1 + Φ 1 t 1 g [ ( Φ 1 ) + t ( ) ] Φ1, z = 0. (17) Eq. (17) can be simplified as where the forcing function W is Φ + 1 Φ g t = W e iωt, (18) with W = β 1 η 1 η 1 + β η 1 η 1, (19) β 1 = igk ω where η 1 is the first order elevation without the time component The second-order forcing The forcing term W can be split into two parts, where 3iω3 g, β = ig ω. (0) W = I + S, (1) I = β 1 η I 1η I 1 + β η I 1 η I 1, () represents the self-interaction of the progressive incident wave, and with S = S IS + S SS, (3) S IS = [β 1 η I 1η S 1 + β η I 1 η S 1 ], (4) being the cross-interaction between incident and scattered waves, and the self-interaction of the scattered wave. For plane incident waves, I can be explicitly expressed as S SS = [β 1 η S 1 η S 1 + β η S 1 η S 1 ], (5) I = 3iωkA 4 sinh kh cosh kh eikr cos(θ), (6) η I 1 and ηs 1 can be expressed for short as η I 1 = T m e imθ, η S 1 = m= m= B m e imθ, (7) 5
7 where T m (r) = Ai m J m (kr), B m (r) = Ai m α m H m (kr), (8) with α m = J m(ka)/h m(ka). J m and H m are the Bessel function of the first kind and the Hankel function of the first kind respectively. The quadratic products of η1 I and ηs 1 can be written in an abbreviation form with η I 1η S 1 = m= T m e imθ + n= B n e inθ = Therefore, S can be expressed as a Fourier series S m = n= S = m= S m e imθ = m=0 m= n= T m n B n e imθ, (9) ɛ m S m cos mθ, (30) {[ ] ( n(m n) B n β 1 r β B n (T m n + B m n ) + β T m n + B )} m n, r r r where ɛ m = for m 1, and ɛ 0 = Second-order Boundary-value problem The second-order wave potential can be decomposed as (31) Φ = ( φ II + φ SS ) e iωt, (3) where Φ II and ΦSS represent the responses to I and S respectively. Both of Φ II and ΦSS satisfy governing equation (14), bottom condition (15) and body condition (16). They also satisfy the following specific free surface boundary conditions respectively φ II φ SS and suitable radiation conditions at infinity Response to I Φ II can be further decomposed as 4ω g φii = I, z = 0, (33) 4ω g φss = S, z = 0. (34) φ II = φ I + φ IS, (35) where φ I is the second-order component of the incoming wave and φis is the diffracted wave corresponding to φ I. Both of them satisfy governing equation (14), bottom condition (15). Additionally, φ I only satisfies the inhomogeneous free surface condition (33). The solution is the second-order component of the incoming wave, written as φ I = 3iωA cosh[k(z + h)] 16 sinh 4 e ikr cos(θ), (36) kh 6
8 Expression cosh[k(z + h)] can be expanded as cosh[k(z + h)] = n=0 cos[κ n (z + h)] a n, (37) cos(κ n h) where κ n is the nth positive real root of 4ω = gκ n tan(κ n h)(n = 1,, 3...), κ 0 = iκ, 4ω = gκ tanh(κh), and 16ω κ n sinh (kh) a n = g(4k + κ n)[sin(κ n h) + κ n h] (38) Thus (36) can be written in polar coordinate as φ I = 3iωA cos[κ m (z + h)] 16 sinh 4 a m ɛ n i n J n (kr) cos(nθ). kh cos(κ m h) (39) m=0 The part φ IS satisfies the homogeneous free-surface condition but the inhomogeneous condition on the cylinder φ IS n=0 r So it can be regarded as a free wave radiating energy to infinity. The solution of φ IS can be sought as From Eq. (14), we have φ IS = m=0 = φi, r = a, (40) r σ m (r, θ) cos κ m(z + h), (41) cos κ m h σ m κ mσ m = 0, m = 0, 1,,..., r > a. (4) Defining κ m = κ m, The solution can be easily found similar to equation (11) where C 1 is determined by boundary conditions. a σ = TH 1 C 1, (43) 3.6. Response to S Φ SS can also be expressed as a series of depth-eigenfunctions for frequency ω [1], h Φ SS = ig ω m=0 Define f n = cos κ n (z + h) and employ Green s formula, we have 0 (f Φ SS n Φ SS ) f n Φ dz = [f SS n Using the orthogonality of f n, we obtain where ζ m (r, θ) cos κ m(z + h), (44) cos κ m h Φ SS ] 0 f n, (45) h ζ n κ nζ n = C s S, n = 0, 1,..., (46) C s = 8iωκ n cos (κ n h) g[κ n h + sin(κ n h)], (47) The solution of equation (46) can be formulated as a ζ = TH 1C (ξ), (48) where C (ξ) is a function of the radial coordinate ξ and is determined by boundary conditions. 7
9 3.7. Other physical quantities The pressure p(r, θ, z, t) at any point (r, θ, z) is determined by the Bernoulli s equation: p ρ + gz + Φ t + 1 [ Φ r + 1 ] r Φ θ + Φ z = 0, (49) Based on the perturbation theory, the first- and second-order terms of the hydrodynamic pressure are p 1 = ρφ 1t, (50) [ p = ρ Φ t + 1 ( Φ 1r + 1 )] r Φ 1θ + Φ 1z, (51) The total horizontal force per unit length in the direction of wave propagation can be determined by integrating the pressure along the cross-sectional contour at a given depth z. Therefore, we have ( ) π dfx = a p j r=a cos(θ)dθ, j = 1,. (5) dz j 0 The total horizontal force F x on the cylinder can be calculated by vertically integrating the df x dz from the seabed to the free surface. To the second-order, the upper limit of the above z integral can be taken at z = 0 rather than z = η 1 + η as the latter would only create terms of order higher than ɛ. The total horizontal force can be integrated as F xj = 0 h The second-order elevation can be obtained as ζ (t) = ( ) dfx dz, j = 1,. (53) dz j [ 1 g ( Φ 1) + 1 Φ 1 Φ 1 g t t 1 g ] Φ. (54) t z=0 4. Conclusions In this paper, second-order wave diffraction by a circular cylinder is solved by SBFEM. By splitting the forcing term into two parts, the diffraction problem is described as two secondorder boundary-value problems with different asymptotic behaviour at infinity. Both of the 3D boundary-value problems are decomposed to a number of D boundary-value sub-problems by separating variables in the direction of the water depth, and then are solved semi-analytically by SBFEM. Only 1D curved finite-elements are needed on the circumference of the cylinder, while the matrix differential equation in radial direction is solved completely analytically. The method can be extended to solve more complex wave-structure interaction problems resulting in direct engineering applications. 5. Acknowledgments The first author is grateful for the postdoctoral fellowship from Griffith University to support this research. References [1] Chau F P and Eatock Taylor R 199 J Fluid Mech [] Ekevid T and Wiberg N-E 00 Comput. Methods Appl. Mech. Engrg [3] Li S, Liang H and Li A 008 Journal of Hydrodynamics, Ser. B 0 77 [4] Mei C C 1989 The applied dynamics of ocean surface waves (Singapore: World Scientific) 8
10 [5] Mei C C, Stiassnie M and Yue D K-P 005 Theory and applications of ocean surface waves (Singapore: World Scientific) [6] Song C and Wolf J P 1997 Comput. Methods Appl. Mech. Engrg [7] Song H, Tao L and Chakrabarti S 010 J. Comput. Phys [8] Tao L, Song H and Chakrabarti S 007 Comput. Method Appl. M [9] Teng B, Zhao M and He G H 006 Int. J. Numer. Meth. Fluids [10] Wolf J P 003 The scaled boundary finite element method (Chichester: Wiley) 9
Semi-analytical solution of Poisson's equation in bounded domain
Semi-analytical solution of Poisson's equation in bounded domain Author Song, Hao, Tao, L. Published 21 Journal Title ANZIAM Journal Copyright Statement 21 Australian Mathematical Society. The attached
More informationSchur decomposition in the scaled boundary finite element method in elastostatics
IOP Conference Series: Materials Science and Engineering Schur decomposition in the scaled boundary finite element method in elastostatics o cite this article: M Li et al 010 IOP Conf. Ser.: Mater. Sci.
More informationScaled boundary FEM solution of short-crested wave diffraction by a vertical cylinder
Scaled boundary FEM solution of short-crested wave diffraction by a vertical cylinder Author Tao, Longbin, Song, Hao, Chakrabarti, Subrata Published 27 Journal Title Computer Methods in Applied Mechanics
More informationReview of Fundamental Equations Supplementary notes on Section 1.2 and 1.3
Review of Fundamental Equations Supplementary notes on Section. and.3 Introduction of the velocity potential: irrotational motion: ω = u = identity in the vector analysis: ϕ u = ϕ Basic conservation principles:
More information1 POTENTIAL FLOW THEORY Formulation of the seakeeping problem
1 POTENTIAL FLOW THEORY Formulation of the seakeeping problem Objective of the Chapter: Formulation of the potential flow around the hull of a ship advancing and oscillationg in waves Results of the Chapter:
More informationLocalisation of Rayleigh-Bloch waves and stability of resonant loads on arrays of bottom-mounted cylinders with respect to positional disorder
Localisation of Rayleigh-Bloch waves and stability of resonant loads on arrays of bottom-mounted cylinders with respect to positional disorder Luke Bennetts University of Adelaide, Australia Collaborators
More informationSecond-order diffraction by two concentric truncated cylinders
Second-order diffraction by two concentric truncated cylinders by Spyros A. Mavrakos and Ioannis K. Chatjigeorgiou National Technical University of Athens Laboratory for Floating Bodies and Mooring Systems,
More informationThe purpose of this lecture is to present a few applications of conformal mappings in problems which arise in physics and engineering.
Lecture 16 Applications of Conformal Mapping MATH-GA 451.001 Complex Variables The purpose of this lecture is to present a few applications of conformal mappings in problems which arise in physics and
More informationNumerical comparison of two boundary meshless methods for water wave problems
Boundary Elements and Other Mesh Reduction Methods XXXVI 115 umerical comparison of two boundary meshless methods for water wave problems Zatianina Razafizana 1,2, Wen Chen 2 & Zhuo-Jia Fu 2 1 College
More informationL e c t u r e. D r. S a s s a n M o h a s s e b
The Scaled smteam@gmx.ch Boundary Finite www.erdbebenschutz.ch Element Method Lecture A L e c t u r e A1 D r. S a s s a n M o h a s s e b V i s i t i n g P r o f e s s o r M. I. T. C a m b r i d g e December
More informationPeriodic wave solution of a second order nonlinear ordinary differential equation by Homotopy analysis method
Periodic wave solution of a second order nonlinear ordinary differential equation by Homotopy analysis method Author Song, Hao, Tao, L. Published 2010 Journal Title ANZIAM Journal Copyright Statement 2010
More informationA MODIFIED DECOUPLED SCALED BOUNDARY-FINITE ELEMENT METHOD FOR MODELING 2D IN-PLANE-MOTION TRANSIENT ELASTODYNAMIC PROBLEMS IN SEMI-INFINITE MEDIA
8 th GRACM International Congress on Computational Mechanics Volos, 2 July 5 July 205 A MODIFIED DECOUPLED SCALED BOUNDARY-FINITE ELEMENT METHOD FOR MODELING 2D IN-PLANE-MOTION TRANSIENT ELASTODYNAMIC
More informationOn spherical-wave scattering by a spherical scatterer and related near-field inverse problems
IMA Journal of Applied Mathematics (2001) 66, 539 549 On spherical-wave scattering by a spherical scatterer and related near-field inverse problems C. ATHANASIADIS Department of Mathematics, University
More informationTheory of Ship Waves (Wave-Body Interaction Theory) Quiz No. 2, April 25, 2018
Quiz No. 2, April 25, 2018 (1) viscous effects (2) shear stress (3) normal pressure (4) pursue (5) bear in mind (6) be denoted by (7) variation (8) adjacent surfaces (9) be composed of (10) integrand (11)
More informationYanlin Shao 1 Odd M. Faltinsen 2
Yanlin Shao 1 Odd M. Faltinsen 1 Ship Hydrodynamics & Stability, Det Norsk Veritas, Norway Centre for Ships and Ocean Structures (CeSOS), NTNU, Norway 1 The state-of-the-art potential flow analysis: Boundary
More informationHydrodynamic Forces on Floating Bodies
Hydrodynamic Forces on Floating Bodies 13.42 Lecture Notes; c A.H. Techet 1. Forces on Large Structures For discussion in this section we will be considering bodies that are quite large compared to the
More informationAn adaptive fast multipole boundary element method for the Helmholtz equation
An adaptive fast multipole boundary element method for the Helmholtz equation Vincenzo Mallardo 1, Claudio Alessandri 1, Ferri M.H. Aliabadi 2 1 Department of Architecture, University of Ferrara, Italy
More informationSimplified formulas of heave added mass coefficients at high frequency for various two-dimensional bodies in a finite water depth
csnak, 2015 Int. J. Nav. Archit. Ocean Eng. (2015) 7:115~127 http://dx.doi.org/10.1515/ijnaoe-2015-0009 pissn: 2092-6782, eissn: 2092-6790 Simplified formulas of heave added mass coefficients at high frequency
More informationHigher-order spectral modelling of the diffraction force around a vertical circular cylinder
Downloaded from orbit.dtu.dk on: Apr 10, 2018 Higher-order spectral modelling of the diffraction force around a vertical circular cylinder Bredmose, Henrik; Andersen, Søren Juhl Publication date: 2017
More informationDRBEM ANALYSIS OF COMBINED WAVE REFRACTION AND DIFFRACTION IN THE PRESENCE OF CURRENT
54 Journal of Marine Science and Technology, Vol. 10, No. 1, pp. 54-60 (2002) DRBEM ANALYSIS OF COMBINED WAVE REFRACTION AND DIFFRACTION IN THE PRESENCE OF CURRENT Sung-Shan Hsiao*, Ming-Chung Lin**, and
More informationTIME-DOMAIN SIMULATION OF THE WAVE-
Chinese-German Joint ymposium on Hydraulic and Ocean Engineering, August 4-3, 8, Darmstadt TIME-DOMAIN IMULATION OF THE WAVE- CURRENT DIFFRACTION FROM 3D BOD Abstract: Zhen Liu, ing Gou and Bin Teng tate
More informationImproved near-wall accuracy for solutions of the Helmholtz equation using the boundary element method
Center for Turbulence Research Annual Research Briefs 2006 313 Improved near-wall accuracy for solutions of the Helmholtz equation using the boundary element method By Y. Khalighi AND D. J. Bodony 1. Motivation
More informationHigher Orders Instability of a Hollow Jet Endowed with Surface Tension
Mechanics and Mechanical Engineering Vol. 2, No. (2008) 69 78 c Technical University of Lodz Higher Orders Instability of a Hollow Jet Endowed with Surface Tension Ahmed E. Radwan Mathematics Department,
More informationComputation of the scattering amplitude in the spheroidal coordinates
Computation of the scattering amplitude in the spheroidal coordinates Takuya MINE Kyoto Institute of Technology 12 October 2015 Lab Seminar at Kochi University of Technology Takuya MINE (KIT) Spheroidal
More informationFree-surface potential flow of an ideal fluid due to a singular sink
Journal of Physics: Conference Series PAPER OPEN ACCESS Free-surface potential flow of an ideal fluid due to a singular sink To cite this article: A A Mestnikova and V N Starovoitov 216 J. Phys.: Conf.
More informationDiffraction of ocean waves around a hollow cylindrical shell structure
University of Wollongong Research Online Faculty of Informatics - Papers (Archive) Faculty of Engineering and Information Sciences 29 Diffraction of ocean waves around a hollow cylindrical shell structure
More informationCHAPTER 3 CYLINDRICAL WAVE PROPAGATION
77 CHAPTER 3 CYLINDRICAL WAVE PROPAGATION 3.1 INTRODUCTION The phase and amplitude of light propagating from cylindrical surface varies in space (with time) in an entirely different fashion compared to
More informationThe Evolution of Large-Amplitude Internal Gravity Wavepackets
The Evolution of Large-Amplitude Internal Gravity Wavepackets Sutherland, Bruce R. and Brown, Geoffrey L. University of Alberta Environmental and Industrial Fluid Dynamics Laboratory Edmonton, Alberta,
More informationSecond-order water wave diffraction by an array of vertical cylinders
J. Fluid Mech. (1999), vol. 39, pp. 39 373. Printed in the United Kingdom c 1999 Cambridge University Press 39 Second-order water wave diffraction by an array of vertical cylinders By Š. MALENICA 1, R.
More informationThe effect of a background shear current on large amplitude internal solitary waves
The effect of a background shear current on large amplitude internal solitary waves Wooyoung Choi Dept. of Mathematical Sciences New Jersey Institute of Technology CAMS Report 0506-4, Fall 005/Spring 006
More informationOn Vertical Variations of Wave-Induced Radiation Stress Tensor
Archives of Hydro-Engineering and Environmental Mechanics Vol. 55 (2008), No. 3 4, pp. 83 93 IBW PAN, ISSN 1231 3726 On Vertical Variations of Wave-Induced Radiation Stress Tensor Włodzimierz Chybicki
More informationarxiv: v1 [physics.flu-dyn] 28 Mar 2017
J. Indones. Math. Soc. (MIHMI) Vol., No. (006), pp. 4 57. arxiv:703.09445v [physics.flu-dyn 8 Mar 07 LINEAR THEORY FOR SINGLE AND DOUBLE FLAP WAVEMAKERS W.M. Kusumawinahyu, N. Karjanto and G. Klopman Abstract.
More information1. Froude Krylov Excitation Force
.016 Hydrodynamics eading #8.016 Hydrodynamics Prof. A.H. Techet 1. Froude Krylov Ecitation Force Ultimately, if we assume the body to be sufficiently small as not to affect the pressure field due to an
More informationRF cavities (Lecture 25)
RF cavities (Lecture 25 February 2, 2016 319/441 Lecture outline A good conductor has a property to guide and trap electromagnetic field in a confined region. In this lecture we will consider an example
More informationCOMPUTATION OF ADDED MASS AND DAMPING COEFFICIENTS DUE TO A HEAVING CYLINDER
J. Appl. Math. & Computing Vol. 3(007), No. 1 -, pp. 17-140 Website: http://jamc.net COMPUTATION OF ADDED MASS AND DAMPING COEFFICIENTS DUE TO A HEAVING CYLINDER DAMBARU D BHATTA Abstract. We present the
More informationWave-Body Interaction Theory (Theory of Ship Waves) Lecture Notes for Graduate Course
Wave-Body Interaction Theory (Theory of Ship Waves) Lecture Notes for Graduate Course ( April 018 ) Lab of Seakeeping & Floating-Body Dynamics in Waves Dept of Naval Architecture & Ocean Engineering Osaka
More informationGraduate School of Engineering, Kyoto University, Kyoto daigaku-katsura, Nishikyo-ku, Kyoto, Japan.
On relationship between contact surface rigidity and harmonic generation behavior in composite materials with mechanical nonlinearity at fiber-matrix interface (Singapore November 2017) N. Matsuda, K.
More informationCONTINUED-FRACTION ABSORBING BOUNDARY CONDITIONS FOR THE WAVE EQUATION
Journal of Computational Acoustics, Vol. 8, No. 1 (2) 139 156 c IMACS CONTINUED-FRACTION ABSORBING BOUNDARY CONDITIONS FOR THE WAVE EQUATION MURTHY N. GUDDATI Department of Civil Engineering, North Carolina
More informationKelvin Helmholtz Instability
Kelvin Helmholtz Instability A. Salih Department of Aerospace Engineering Indian Institute of Space Science and Technology, Thiruvananthapuram November 00 One of the most well known instabilities in fluid
More informationLiquid Sloshing in a Rotating, Laterally Oscillating Cylindrical Container
Universal Journal of Mechanical Engineering 5(3): 97-101, 2017 DOI: 10.13189/ujme.2017.050304 http://www.hrpub.org Liquid Sloshing in a Rotating, Laterally Oscillating Cylindrical Container Yusuke Saito,
More informationSeakeeping Models in the Frequency Domain
Seakeeping Models in the Frequency Domain (Module 6) Dr Tristan Perez Centre for Complex Dynamic Systems and Control (CDSC) Prof. Thor I Fossen Department of Engineering Cybernetics 18/09/2007 One-day
More informationPHY321 Homework Set 10
PHY321 Homework Set 10 1. [5 pts] A small block of mass m slides without friction down a wedge-shaped block of mass M and of opening angle α. Thetriangular block itself slides along a horizontal floor,
More informationWave Theory II (7) Physical Optics Approximation
Wave Theory II (7) Physical Optics Approximation Jun-ichi Takada (takada@ide.titech.ac.jp) In this lecture, the physical optics approximation (), which is classified as a semi-analytical technique, is
More information13.42 LECTURE 13: FLUID FORCES ON BODIES. Using a two dimensional cylinder within a two-dimensional flow we can demonstrate some of the principles
13.42 LECTURE 13: FLUID FORCES ON BODIES SPRING 2003 c A. H. TECHET & M.S. TRIANTAFYLLOU 1. Morrison s Equation Using a two dimensional cylinder within a two-dimensional flow we can demonstrate some of
More informationStrongly nonlinear long gravity waves in uniform shear flows
Strongly nonlinear long gravity waves in uniform shear flows Wooyoung Choi Department of Naval Architecture and Marine Engineering, University of Michigan, Ann Arbor, Michigan 48109, USA Received 14 January
More informationOn the evaluation quadratic forces on stationary bodies
On the evaluation quadratic forces on stationary bodies Chang-Ho Lee AMIT Inc., Chestnut Hill MA, USA June 9, 006 Abstract. Conservation of momentum is applied to finite fluid volume surrounding a body
More information. (70.1) r r. / r. Substituting, we have the following equation for f:
7 Spherical waves Let us consider a sound wave in which the distribution of densit velocit etc, depends only on the distance from some point, ie, is spherically symmetrical Such a wave is called a spherical
More informationCommun Nonlinear Sci Numer Simulat
Commun Nonlinear Sci Numer Simulat 6 (2) 63 75 Contents lists available at ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns The scaled boundary FEM for nonlinear
More informationSome negative results on the use of Helmholtz integral equations for rough-surface scattering
In: Mathematical Methods in Scattering Theory and Biomedical Technology (ed. G. Dassios, D. I. Fotiadis, K. Kiriaki and C. V. Massalas), Pitman Research Notes in Mathematics 390, Addison Wesley Longman,
More informationCombined diffraction and radiation of ocean waves around an OWC device
J Appl Math Comput (211) 36: 41 416 DOI 1.17/s1219-1-41-y JAMC Combined diffraction and radiation of ocean waves around an OWC device Song-Ping Zhu Lewis Mitchell Received: 19 February 21 / Published online:
More information7 EQUATIONS OF MOTION FOR AN INVISCID FLUID
7 EQUATIONS OF MOTION FOR AN INISCID FLUID iscosity is a measure of the thickness of a fluid, and its resistance to shearing motions. Honey is difficult to stir because of its high viscosity, whereas water
More informationFourier Coefficients of Far Fields and the Convex Scattering Support John Sylvester University of Washington
Fourier Coefficients of Far Fields and the Convex Scattering Support John Sylvester University of Washington Collaborators: Steve Kusiak Roland Potthast Jim Berryman Gratefully acknowledge research support
More informationFourier spectra of radially periodic images with a non-symmetric radial period
J. Opt. A: Pure Appl. Opt. 1 (1999) 621 625. Printed in the UK PII: S1464-4258(99)00768-0 Fourier spectra of radially periodic images with a non-symmetric radial period Isaac Amidror Laboratoire de Systèmes
More informationPostprint.
http://www.diva-portal.org Postprint This is the accepted version of a paper presented at 3th International Workshop on Water Waves and Floating Bodies (IWWWFB), 12-1th April, 1, Bristol, UK. Citation
More informationSurface Tension Effect on a Two Dimensional. Channel Flow against an Inclined Wall
Applied Mathematical Sciences, Vol. 1, 007, no. 7, 313-36 Surface Tension Effect on a Two Dimensional Channel Flow against an Inclined Wall A. Merzougui *, H. Mekias ** and F. Guechi ** * Département de
More informationHydrodynamical Analysis of Bottom-hinged Oscillating Wave Surge Converters
Hydrodynamical Analysis of Bottom-hinged Oscillating Wave Surge Converters Inês Furtado Marques Mendes da Cruz Alves inesfmmendes@tecnico.ulisboa.pt Centre for Marine Technology and Ocean Engineering,
More informationANALYSIS AND NUMERICAL METHODS FOR SOME CRACK PROBLEMS
INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING, SERIES B Volume 2, Number 2-3, Pages 155 166 c 2011 Institute for Scientific Computing and Information ANALYSIS AND NUMERICAL METHODS FOR SOME
More informationFigure 1. adiabatically. The change in pressure experienced by the parcel is. dp = -ρ o gξ
6. Internal waves Consider a continuously stratified fluid with ρ o (z) the vertical density profile. z p' ξ p ρ ρ ο (z) Figure 1. Figure by MIT OpenCourseWare. At a point P raise a parcel of water by
More informationCHAPTER 5. RUDIMENTS OF HYDRODYNAMIC INSTABILITY
1 Lecture Notes on Fluid Dynamics (1.63J/.1J) by Chiang C. Mei, 00 CHAPTER 5. RUDIMENTS OF HYDRODYNAMIC INSTABILITY References: Drazin: Introduction to Hydrodynamic Stability Chandrasekar: Hydrodynamic
More informationModel of non-ideal detonation of condensed high explosives
Journal of Physics: Conference Series PAPER OPEN ACCESS Model of non-ideal detonation of condensed high explosives To cite this article: E B Smirnov et al 2016 J. Phys.: Conf. Ser. 774 012076 View the
More information13.42 LECTURE 2: REVIEW OF LINEAR WAVES
13.42 LECTURE 2: REVIEW OF LINEAR WAVES SPRING 2003 c A.H. TECHET & M.S. TRIANTAFYLLOU 1. Basic Water Waves Laplace Equation 2 φ = 0 Free surface elevation: z = η(x, t) No vertical velocity at the bottom
More informationA new approach to uniqueness for linear problems of wave body interaction
A new approach to uniqueness for linear problems of wave body interaction Oleg Motygin Laboratory for Mathematical Modelling of Wave Phenomena, Institute of Problems in Mechanical Engineering, Russian
More informationMATH 311 Topics in Applied Mathematics Lecture 25: Bessel functions (continued).
MATH 311 Topics in Applied Mathematics Lecture 25: Bessel functions (continued). Bessel s differential equation of order m 0: z 2 d2 f dz 2 + z df dz + (z2 m 2 )f = 0 The equation is considered on the
More informationCut-on, cut-off transition of sound in slowly varying flow ducts
Cut-on, cut-off transition of sound in slowly varying flow ducts Sjoerd W. Rienstra 19-3-1 Department of Mathematics and Computing Science, Eindhoven University of Technology, The Netherlands, s.w.rienstra@tue.nl
More information/01/04: Morrison s Equation SPRING 2004 A. H. TECHET
3.4 04/0/04: orrison s Equation SPRING 004 A.. TECET. General form of orrison s Equation Flow past a circular cylinder is a canonical problem in ocean engineering. For a purely inviscid, steady flow we
More informationAPPLIED PARTIM DIFFERENTIAL EQUATIONS with Fourier Series and Boundary Value Problems
APPLIED PARTIM DIFFERENTIAL EQUATIONS with Fourier Series and Boundary Value Problems Fourth Edition Richard Haberman Department of Mathematics Southern Methodist University PEARSON Prentice Hall PEARSON
More informationMAE 101A. Homework 7 - Solutions 3/12/2018
MAE 101A Homework 7 - Solutions 3/12/2018 Munson 6.31: The stream function for a two-dimensional, nonviscous, incompressible flow field is given by the expression ψ = 2(x y) where the stream function has
More informationHelmholtz Equation Applied to the Vertical Fixed Cylinder in Wave Using Boundary Element Method
Helmholtz Equation Applied to the Vertical Fixed Cylinder in Wave Using Boundary Element Method Hassan Ghassemi, a,* Ataollah Gharechahi, a, Kaveh Soleimani, a, Mohammad Javad Ketabdari, a, a) Department
More informationPrediction of the radiated sound power from a fluid-loaded finite cylinder using the surface contribution method
Prediction of the radiated sound power from a fluid-loaded finite cylinder using the surface contribution method Daipei LIU 1 ; Herwig PETERS 1 ; Nicole KESSISSOGLOU 1 ; Steffen MARBURG 2 ; 1 School of
More informationHigh-velocity collision of particles around a rapidly rotating black hole
Journal of Physics: Conference Series OPEN ACCESS High-velocity collision of particles around a rapidly rotating black hole To cite this article: T Harada 2014 J. Phys.: Conf. Ser. 484 012016 Related content
More information1. Reflection and Refraction of Spherical Waves
1. Reflection and Refraction of Spherical Waves Our previous book [1.1] was completely focused on the problem of plane and quasi-plane waves in layered media. In the theory of acoustic wave propagation,
More informationExact solutions to model surface and volume charge distributions
Journal of Physics: Conference Series PAPER OPEN ACCESS Exact solutions to model surface and volume charge distributions To cite this article: S Mukhopadhyay et al 2016 J. Phys.: Conf. Ser. 759 012073
More informationA note on interior vs. boundary-layer damping of surface waves in a circular cylinder
J. Fluid Mech. (1998), vol. 364, pp. 319 323. Printed in the United Kingdom c 1998 Cambridge University Press 319 A note on interior vs. boundary-layer damping of surface waves in a circular cylinder By
More informationComputational Modelling of Acoustic Scattering of a Sound Source in the Vicinity of the Ground
Computational Modelling of Acoustic Scattering of a Sound Source in the Vicinity of the Ground Dr. Panagiota Pantazopoulou and Prof. Dimitris Drikakis Members, IAENG Abstract The paper presents a computational
More informationarxiv:gr-qc/ v1 11 May 2000
EPHOU 00-004 May 000 A Conserved Energy Integral for Perturbation Equations arxiv:gr-qc/0005037v1 11 May 000 in the Kerr-de Sitter Geometry Hiroshi Umetsu Department of Physics, Hokkaido University Sapporo,
More informationA Second-order Finite Difference Scheme For The Wave Equation on a Reduced Polar Grid DRAFT
A Second-order Finite Difference Scheme For The Wave Equation on a Reduced Polar Grid Abstract. This paper presents a second-order numerical scheme, based on finite differences, for solving the wave equation
More information9 Fluid Instabilities
9. Stability of a shear flow In many situations, gaseous flows can be subject to fluid instabilities in which small perturbations can rapidly flow, thereby tapping a source of free energy. An important
More informationDynamics of Strongly Nonlinear Internal Solitary Waves in Shallow Water
Dynamics of Strongly Nonlinear Internal Solitary Waves in Shallow Water By Tae-Chang Jo and Wooyoung Choi We study the dynamics of large amplitude internal solitary waves in shallow water by using a strongly
More information2 Wave Propagation Theory
2 Wave Propagation Theory 2.1 The Wave Equation The wave equation in an ideal fluid can be derived from hydrodynamics and the adiabatic relation between pressure and density. The equation for conservation
More informationHydrodynamics for Ocean Engineers Prof. A.H. Techet Fall 2004
13.01 ydrodynamics for Ocean Engineers Prof. A.. Techet Fall 004 Morrison s Equation 1. General form of Morrison s Equation Flow past a circular cylinder is a canonical problem in ocean engineering. For
More informationHydrodynamics of the oscillating wave surge converter in the open ocean
Loughborough University Institutional Repository Hydrodynamics of the oscillating wave surge converter in the open ocean This item was submitted to Loughborough University's Institutional Repository by
More informationGrid Generation and Applications
Grid Generation and Applications Acoustic Scattering from Arbitrary Shape Obstacles.5 Fictitous infinite boundary.5 C.5.5 Ω δ.5.5.5.5 Boundary Value Problem: Acoustic Scattering (p sc ) tt = c p sc, x
More informationWater wave radiation by a submerged rough disc
radiation by a Leandro Farina Institute of Mathematics & Post-Graduation Program in Geosciences UFRGS Porto Alegre, Brazil BCAM, Bilbao, 14 June 2012. Outline 1 2 3 TLP Mars platform. Projected for suporting
More informationLecture notes 5: Diffraction
Lecture notes 5: Diffraction Let us now consider how light reacts to being confined to a given aperture. The resolution of an aperture is restricted due to the wave nature of light: as light passes through
More informationA Hybrid Formulation Using The Transition Matrix Method and Finite Elements in a 2D Eddy Current Interaction Problem
4th International Symposium on NDT in Aerospace 2012 - Poster 5 A Hybrid Formulation Using The Transition Matrix Method and Finite Elements in a 2D Eddy Current Interaction Problem Lars LARSSON, Anders
More informationSurvival Guide to Bessel Functions
Survival Guide to Bessel Functions December, 13 1 The Problem (Original by Mike Herman; edits and additions by Paul A.) For cylindrical boundary conditions, Laplace s equation is: [ 1 s Φ ] + 1 Φ s s s
More informationChapter 1. Condition numbers and local errors in the boundary element method
Chapter 1 Condition numbers and local errors in the boundary element method W. Dijkstra, G. Kakuba and R. M. M. Mattheij Eindhoven University of Technology, Department of Mathematics and Computing Science,
More information1. Comparison of stability analysis to previous work
. Comparison of stability analysis to previous work The stability problem (6.4) can be understood in the context of previous work. Benjamin (957) and Yih (963) have studied the stability of fluid flowing
More informationLecture 4: Geometrical Theory of Diffraction (continued) and the Shallow Water Theory
Lecture 4: Geometrical Theory of Diffraction continued) and the Shallow Water Theory Joseph B. Keller 1 Introduction In this lecture we will finish discussing the reflection from a boundary Section 2).
More informationWave Driven Free Surface Motion in the Gap between a Tanker and an FLNG Barge
Wave Driven Free Surface Motion in the Gap between a Tanker and an FLNG Barge L. Sun a, R. Eatock Taylor a,b and P. H. Taylor b a Centre for Offshore Research & Engineering and Department of Civil & Environmental
More informationTwo special equations: Bessel s and Legendre s equations. p Fourier-Bessel and Fourier-Legendre series. p
LECTURE 1 Table of Contents Two special equations: Bessel s and Legendre s equations. p. 259-268. Fourier-Bessel and Fourier-Legendre series. p. 453-460. Boundary value problems in other coordinate system.
More informationCoupled Heave-Pitch Motions and Froude Krylov Excitation Forces
Coupled Heave-Pitch Motions and Froude Krylov Excitation Forces 13.42 Lecture Notes; Spring 2004; c A.H. Techet 1. Coupled Equation of Motion in Heave and Pitch Once we have set up the simple equation
More information21 Laplace s Equation and Harmonic Functions
2 Laplace s Equation and Harmonic Functions 2. Introductory Remarks on the Laplacian operator Given a domain Ω R d, then 2 u = div(grad u) = in Ω () is Laplace s equation defined in Ω. If d = 2, in cartesian
More informationGreen s functions for planarly layered media
Green s functions for planarly layered media Massachusetts Institute of Technology 6.635 lecture notes Introduction: Green s functions The Green s functions is the solution of the wave equation for a point
More informationNew Developments of Frequency Domain Acoustic Methods in LS-DYNA
11 th International LS-DYNA Users Conference Simulation (2) New Developments of Frequency Domain Acoustic Methods in LS-DYNA Yun Huang 1, Mhamed Souli 2, Rongfeng Liu 3 1 Livermore Software Technology
More informationPhysics 506 Winter 2004
Physics 506 Winter 004 G. Raithel January 6, 004 Disclaimer: The purpose of these notes is to provide you with a general list of topics that were covered in class. The notes are not a substitute for reading
More informationNatural frequency analysis of fluid-conveying pipes in the ADINA system
Journal of Physics: Conference Series OPEN ACCESS Natural frequency analysis of fluid-conveying pipes in the ADINA system To cite this article: L Wang et al 2013 J. Phys.: Conf. Ser. 448 012014 View the
More informationScalar electromagnetic integral equations
Scalar electromagnetic integral equations Uday K Khankhoje Abstract This brief note derives the two dimensional scalar electromagnetic integral equation starting from Maxwell s equations, and shows how
More informationProceedings of the 5th International Conference on Inverse Problems in Engineering: Theory and Practice, Cambridge, UK, 11-15th July 2005
Proceedings of the 5th International Conference on Inverse Problems in Engineering: Theory and Practice, Cambridge, UK, 11-15th July 2005 SOME INVERSE SCATTERING PROBLEMS FOR TWO-DIMENSIONAL SCHRÖDINGER
More informationScattering of electromagnetic waves by many nano-wires
Scattering of electromagnetic waves by many nano-wires A G Ramm Department of Mathematics Kansas State University, Manhattan, KS 66506-2602, USA ramm@math.ksu.edu Abstract Electromagnetic wave scattering
More information