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1 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 Key Laboratory of Coastal and Offshore Engineering Dalian University of Technology, Dalian A time-domain numerical model is established by a higher-order boundary element method to study the problem of wave-current action on 3D bodies. Under the assumption of small flow velocity, the velocity potential is expanded by a perturbation method. The boundary value problem is decomposed into a steady double-body flow problem at the zero-order of wave steepness and an unsteady wave problem at the first-order of wave steepness. A 4th-order Runge-Kutta method is applied for the time marching. The velocity potential on the body surface and the derivative of the velocity potential on the free surface are then given as the solution of a higher order boundary element integral equation, which is solved by a numerical code. An artificial damping layer is adopted to dissipate the scattering waves. Validation of the numerical method is carried out on wave forces, wave run-up and mean drift forces of wave-current acting on a bottom-mounted vertical cylinder. The results are all in close agreement with the results of a frequency-domain method and a published time-domain method. I. INTRODUCTION The problem of wave-current interaction with 3D bodies, which is equivalent to wave action on forward moving bodies in mathematics, has triggered a boom of all theoretical, numerical and experimental researches. Boundary conditions of wave-body interaction problem are changed with considering the effect of uniform steady currents. Consequently, diffraction potential, forces and run-up of wavecurrent action on 3D bodies are also changed. Many researches have been carried out to solve the problem mostly in the frequency domain (e.g. Grue and Palm, 985; Wu and Eatock Taylor, 987; Matsui et al, 99; Zhao and Faltinsen, 988; Nossen et al, 99; Teng and Eatock Taylor, 993; Teng and Eatock Taylor, 995; Teng et al, ). Although the frequency-domain analysis is ed for a long time, it is in general mathematically complicated and cannot be easily extended to fully nonlinear problems. A time domain analysis based on boundary element method is then developed. Time domain analysis was applied to solve wave-current-body interaction problem using constant panel method by Isaacson and Cheung (99). Using regular perturbation with steepness parameter ε to the mean position of the non-stationary boundaries, a solution of time domain higher-order boundary element method (THOBEM) for 3D wave radiation and diffraction in a current was given by Buchmann et al (998). They ed the results of the diffractive run-up of wave-current interaction with a flume. Using the solution of THOBEM and applying a perturbation with two small parameters ε and δ associated with wave slope and current velocity, Kim and Kim (997) investigated exciting forces, mean drift forces and run-up of wave-current action on a vertical cylinder. The time domain solution can be better than frequency domain solutions while considering irregular seas, transient effects or highly non-linear effects. In this paper, a time domain model of wavecurrent action on 3D bodies is set up by applying a THOBEM. Under the assumption of small flow velocity, the velocity potential and wave run-up are expanded by a perturbation method. Using perturbation with two small parameters ε and δ associated with wave slope and current velocity, the boundary value problem is decomposed into a steady double-body flow problem at the zero-order of wave steepness and an unsteady wave problem at the first-order of wave steepness. An artificial damping layer is adopted to dissipate the scattering waves, as Ferrant (993) applied. The boundary surface is discretized into a set of higher-order boundary elements, which are composed of quadrangular elements with eight nodes and triangular elements with six nodes. The elements can be transformed to equivalent parameter elements of parameter coordinates ( ξ, ς ). Based on the theory of equivalent parameter elements, the velocity potential and geometry coordinates can be expressed by the values of nodes and shape functions. Validation of the numerical method is carried out for a bottom-mounted vertical cylinder. The effects of uniform steady current (or small forward velocity) on exciting forces, wave run-up and mean drift forces are investigated by the solution of the THOBEM. The model is then applied to compute the wave-current force and wave run-up on a complex bridge pier. 46

2 Chinese-German Joint ymposium on Hydraulic and Ocean Engineering, August 4-3, 8, Darmstadt II. NUMERICAL IMULATION L A. Boundary Conditions We consider wave interaction with a body moving forward with a small constant forward speed U, or a body under the action of waves in a weak current directing in the negative x direction with velocity U. A right-handed co-ordinate system oxyz is founded, as shown in Fig.. The z = plane locates on the undisturbed free surface and the z-axis is positive upwards. Under the assumption of ideal fluid and no surface tension, the fluid velocity vector V( x, t) can be defined by φ, with the velocity potential φ ( x, yzt,, ) satisfying the following Laplace equation and boundary conditions: φ =, in Ω () U + φ φ + gη =, on z = η () t x η U η + φ η =, on z = η (3) t x z φ = V n, on (4) where g res the gravitational acceleration, η is the instantaneous wave profile, is the instantaneous body surface, V n res the normal velocity of the body. Equations () and (3) are fully non-linear dynamic and kinematic free surface conditions, respectively. Incident Wave U Z b Figure. Definition of sketch To make the solution of the above boundary value problem unique, we have to impose an extra condition, which is called the radiation condition at far field. In numerical methods an artificial damping layer is adopted to dissipate the scattering waves, as shown in Fig.. The hatched part is the artificial damping layer, where the damping terms are imposed to the boundary conditions. f h Figure. ketch of the damping layer olving this fully nonlinear problem directly by time-domain method, we should discretize the instantaneous free surface and the body surface, set up and invert the influence matrix at each time step, which requires tremendous computational time. To save computation time, a perturbation procedure is used under the assumption of small wave amplitudes and body motions. The influence matrix can be set up only once and used repeatedly at all time steps. Consequently, it makes the time-domain method more efficient and attractive. By introducing a small parameter ε = H / λ (wave height/wave length) associated with wave steepness and a small parameter δ = U / gl (or Froude number F n ) associated with the current velocity, where l is the characteristic length of the body, the velocity potential and the wave profile can be expressed by the following perturbation series: () () φ( xt, ) = φ( x) + φ ( xt, ) + φ ( xt, ) +..., O( δ ) O( εεδ, ) O( ε,...) () () η( xt, ) = η( x) + η ( xt, ) + η ( xt, ) +..., O( δ ) O( εεδ, ) O( ε,...) r (5) (6) where φ is the zeroth-order steady disturbance () potential, φ and φ () are the time-dependent firstorder and second-order potentials, respectively. The terms higher than O( δ ) and O( εδ ) can be neglected when δ is assumed small. o the water surface boundary can be considered as the wall boundary on z = while the steady ship wave system η at O( δ ) is neglected, which is called double-body flow problem. By substituting (5), (6) into (), (3) and collecting terms of orders O( εδ ) and O( ε ), we can obtain the boundary value problem at the first-order of wave steepness. φ ( η Uη ) = F, (7) () () () () z t x φ φ + η = (8) () () () () t U x g F, where 46

3 Chinese-German Joint ymposium on Hydraulic and Ocean Engineering, August 4-3, 8, Darmstadt η η φ φ ( ), x x y y x y () () () () = + η + F () F = x x y y () () (9). () The total velocity potential can be divided into a known incident potential and an unknown scattering potential, while the wave elevation can be divided into a known incident wave elevation and an unknown scattering wave elevation φ = φ + φ () () () () w s, η = η + η. () () () () w s The fully nonlinear dynamic and kinematic free surface conditions are applied to solve the unknown scattering potential problem, which can be written in the form: () () () () η s η w φ = + + U η F t t z x t t x (), () () () s w () () = gη + U + F (3). (4) The damping terms are imposed to the boundary conditions: where k (k = π / λ ) is the wave number, β is the wave heading with respect to the x axis, ω is the original circular frequency of the incident wave, which satisfies the dispersion relation ω = gktanh kh, h is the water depth. The incident wave potential, which including the effect of the current, can be written in the form: () ga cosh k( z + h) φw = sin( kxcosβ + ky sin β ωt ), (9) ω cosh kh where A is the wave amplitude, and h is the water depth. For the diffraction problem, the boundary condition (4) can be written in the form: + =. () () () w s B. Numerical Method Rankine sources and their images about the seabed are selected as Green function: Gx (, ξ ) = / r+ / r () where x res the field point, and ξ res the source point. The integral equation for the scattering potential can be obtained by Green s second identity, s ( x) Gx (, ξ ) αφs( ξ) = Gx (, ξ) φs( x) d π s () () () () () ηs ηw η = + + U F r t t z x t t x () () ν () ηs, () () () s w () () () = gη + U + F ν r φs (), (5) (6) where α is the solid angle coefficient. The boundary of the computing domain includes the average body surface and the average free surface. According to source point s position, the integral equation can be written into two categories. When the source points are distributed on the body surface, () can be written by where r r αω = + ν () r = βλ ( r < r ) ( ) ( r r r r βλ) (7) α is the damping coefficient, β is the beach breadth coefficient, λ is the characteristic wave length. In the computation we take α =. and β =.. The encounter frequency ω is given by ω = ω Uk cos β, (8) G αφs + φ d G d = b f G φ b G d d f (3) When the source points are distributed on the free surface, () can be written by b b φ = G d G d f G φ G d d f αφ s (4) 463

4 Chinese-German Joint ymposium on Hydraulic and Ocean Engineering, August 4-3, 8, Darmstadt After discretizing the integral equations, a group of linear equations can be given by Z φ b A A B φ A A =. (5) B f -.5 Z ince the integral boundary and Green function are time-independent, the coefficient matrix [ A ] is invariable at each time step. o the coefficient matrix [ A ] needs to be set up only once and it can be used repeatedly at each time step. The normal derivative of the potential at the free surface is obtained by the integral equation method at the moment t. The dynamic and the kinematic free surface conditions are used to solve the problem of the unknown potential and wave profile at the next moment t+ t. A 4th-order Runge-Kutta method is applied for the time marching. - 5 Figure 3. Meshes of the circular cylinder - - C. Forces and Moments Having obtained the unknown velocity potentials, we are able to calculate the forces and moments acting on the body. The pressure is given by the Bernoulli equation -5 p = ρ( U + φ φ + gz), t x (6) where ρ is the fluid density. The forces on a body can be calculated by integration of the pressure over the body, which can be written by () () () () F = ρ ( U + φ φ) nd. (7) t x s b The mean drift forces are obtained from the timeaverage of the second-order forces () () F = ρg ( ) NdW η W ρ φ φ b () () ( ) nd (8) where N = n/ n3, and W is the water-line, which is the intersection line of the body surface and the free surface. III. REULT AND DICUION A model of vertical cylinder is proposed to validate the reliability of this method. The water depth h is m, and the radius of the cylinder is a = m. The forces and the wave run-up in following current or adverse current are investigated. The meshes on the body surface and the free surface are shown in Figs. 3 and 4, respectively. - - Figure 4. Meshes of the free surface Due to the symmetry of the model, only a quarter of the body surface and the free surface need to be discretized. A quarter of the body surface is equally divided into eight parts in the circular direction while six parts in depth. The free surface is discretized by non-structural meshes. The time step for the time marching is t = T/8 (T is wave period). Fig. 5 shows the free-surface-profile time series at a free surface point (-.98,.) when ka=.5, Fn=-.. Fig. 6 shows the time history of the wave exciting force, where ka=.6, Fn= -.. It can be observed from the periodic results that the model is stable and there is no reflection in the computation domain. η/α - Fn= Time Figure 5. Time history of wave elevations (ka=.5) 464

5 Chinese-German Joint ymposium on Hydraulic and Ocean Engineering, August 4-3, 8, Darmstadt Fx/(ρga A) Fn= Time Figure 6. Time history of wave exciting forces (ka=.6) Fig. 7a, 7b show the comparison of the run-up of wave-current interaction with a vertical cylinder at the wave number ka =.5, where F n =. and F n =-., respectively. The angle θ = res the lee side of the vertical cylinder when wave acts on the body, and θ = π res the front side. It shows obviously that the run-up at the front side is higher than that at the lee side. The results are all in close agreement with the results of Teng et al (). η/α η/α θ/π (a) F n = θ/π (b) F n =. Figure 7. Run-up around the circular cylinder at ka =.5 Fig. 8a, 8b show the exciting forces as functions of wave number ka at different F n. The results are compared with the results of Matsui et al. (99) and that of Teng et al. (). The results are all in close agreement with the published results. fx () /(ρga A) fx () /(ρga A) ka (a) Fn= -. Matsui's ka Matsui's (b) Fn=. Figure 8. Exciting forces at different wave numbers Fig. 9 shows the mean drift forces as functions of wave number ka. It shows clearly that the values of mean drift forces with waves directing in following currents are larger than that in adverse currents. The results are also in close agreement with the results of Matsui et al (99). fx () /(ρga A) Fn=.5 Matsui's Fn=.5 Fn=. Matsui's Fn=. Fn= -.5 Matsui's Fn= ka Figure 9. Mean drift force at different F n IV. CONCLUION A time-domain higher order boundary element method has been developed to study wave-currentbody interaction problem. Under the assumption of potential theory, the velocity potential and wave runup can be expressed by perturbation series. The Rankine sources are distributed over the entire boundary, and integral equation methods are used to solve the problem of the unknown potential at the moment t. Thereafter, the dynamic and kinematic 465

6 Chinese-German Joint ymposium on Hydraulic and Ocean Engineering, August 4-3, 8, Darmstadt free surface conditions and a 4th-order Runge-Kutta method are applied to calculate the wave run-up and velocity potential at the next moment. An artificial damping layer is adopted to dissipate the scattering waves. The results are all in close agreement with the published results. Wave forces, run-up, and mean drift forces are investigated in this paper. When waves direct in currents of different F n, the effects of current make a big change to the maximal wave run-up of both the back side and the front side of body. The values of exciting force of no current are not always between the values of waves directing in adverse currents and the values in following currents. The values of mean drift forces with waves directing in following currents are larger than that in adverse currents regardless of incident wave frequencies. The research of this paper can apply some reference to engineering practice. REFERENCE [] B. Buchmann, J. kourup, K.F. Cheung. (998). Run-up on A trucrure Due to econd-order Waves and Current in A Numerical Wave Tank, Applied Ocean Research, Vol., [] P. Ferrant. (993). Three-dimensional Unsteady Wave-body Interactions by A Rankine Boundary Element Method, hip Technology Research, Vol.4, [3] J. Grue, E. Palm. (985). Wave Radiation and Wave Diffraction from A ubmerged Body in A Uniform Current, Journal of Fluid Mechanics, Vol.5, [4] M. Isaacson, K.F. Cheung. (99). Wave-current Interaction with A Large tructure, Proc. of International Conference on Civil Engineering in Oceans, ACE, College tation, Texas, Nov. [5] D.J. Kim, M.H. Kim. (997). Wave-Current-Body Interaction by A Time-domain High-order Boundary Element Method, International offshore and polar engineering conference, Vol.5, 5-3. [6] T. Matsui,.. Lee, K. ano. (99). Hydrodynamic Forces on A Vertical Cylinder in Current and Waves, Jour. of the ociety of Naval Architects of Japan, Vol.7, [7] J. Nossen, J. Grue, E. Plam. (99). Wave Forces on Three Dimensional Floating Bodies with mall Forward peed, Journal of fluid Mechanics, Vol.7, [8] B. Teng, R. Eatock Taylor. (993). The Effect of Corners on Diffraction/Radiation Forces and Wave Drift Damping, Proc. of Offshore Technology Conference, OTC787. [9] B. Teng, R. Eatock Taylor. (995). Application of A Higher Order BEM in the Calculation of Wave Run-up on Bodies in A Weak Current, Int. Jour. of Offshore and Polar Engineering, Vol. 5, No. 3, 9-4. [] B. Teng, M. Zhao, W. Bai. (). Wave Diffraction in A Current over A Local hoal, Coastal Engineering, Vol.4, [] G.. Wu, R. Eatock Taylor. (987). Hydrodynamic Forces on ubmerged Oscillating Cylinders at Forward peed, Proceeding of the Royal ociety of London, eries A, Vol.44, [] R. Zhao, O.M. Faltinsen. (988). Interaction between Waves and Current on A Two-dimensional Body in The Free urface, Applied Ocean Research, Vol.(),

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