MODULE VII LARGE BODY WAVE DIFFRACTION

Transcription

1 MODULE VII LARGE BODY WAVE DIFFRACTION 1.0 INTRODUCTION In the wave-structure interaction problems, it is classical to divide into two major classification: slender body interaction and large body interaction. The demarcation between the above definition of slender and large body is due to the drag dominant earlier case and inertia dominant latter..0 LINEAR DIFFRACTION PROBLEM Let us consider a large structure (fixed or floating) interacting with a monochromatic oblique linear wave of height, H, and wave angular frequency,. Waves approach the structure at an angle with the normal to the body axis. For the diffraction boundary value problem, it is generally convenient to separate the total velocity potential into incident potential, I and scattered potential, s. This relation is known as Haskind decomposition and is mathematically represented as ( x, y, z, t) (1) I s The incident velocity potential for the present wave structure interaction problem is represented by, ihg cosh k( z d) I ( xyzt,,, ) Re exp ikx ( coskysin t) () cosh kd The wave number (k) satisfies the dispersion relation, given by gk tanh kd (3) 1

2 The linear wave diffraction problem is described by a diffracted velocity potential ( D ). ( x, y, z, t) ( x, y, z, t) ( x, y, z, t) (4) D I s The boundary value problem for the diffracted potential can be defined by the governing Laplace equation ((x,y,z,t) = s (x,y,z).e -it ) and the boundary conditions as defined below. Governing equation: 0 in the fluid domain (5) s Linearised free surface condition: s z s g 0 on the free surface, F, z 0 (6) Impervious bottom condition: s z 0 on the sea bed, B, z d (7) Radiation condition: s iks 0 on the radiation boundary,, x x (8) The infinite boundary, is fixed at a finite distance, {x, y} = {x r, y r } Kinematic body boundary condition: Since the rigid body is restrained from all its degrees of freedom in diffraction problem, D r n 0 on the body surface, o (9) From Eqn.(9), Eqn.(4) becomes s I r r n n on r o ; r = 1,..., s (10)

3 where n is the unit inward normal to the body at the interface. The assumptions made in the above boundary value problem are: 1. Fluid is ideal and flow is irrotational.. Application of linear wave theory is valid. 3. There is no flow of energy through either the bottom or the free surface. 4. Energy is gained or lost by the system only through waves arriving or departing at infinity or the external forces acting on the body. 5. The body is rigid. Hydrodynamic forces The hydrodynamic pressure at any point in the fluid can be expressed as, pxyzt (,,, ) i (11) t The hydrodynamic forces can be determined by integrating the pressure over the wetted body surface o r. F r j pn r j d o (1) The hydrodynamic forces thus, evaluated, are called wave exciting forces ( e F j r ) governed by the diffraction problem. The wave exciting force, e F j r due to the diffracted potential can be expressed as e Fj r ( xyzt,,, ) i I s n r ( ) j d o (13) 3

4 Fig. 1. Three-dimensional model domain to be solved. Solution: The linear diffraction problem defined above can be solved analytically if the structure configuration is uniform (straight or circular) and the water depth is constant. However, in most cases, the bottom topography variation has to be taken into account and hence, the analytical and series solution fails to produce the solution. One needs to seek numerical solution. The most useful numerical tool is Finite Element Method due to its flexibility in fitting to any boundary. However, the numerical solutions are computationally expensive due to the three-dimensional domain. Most of the problems, it is not preferred to seek the solution by solving the entire system of equations. Either, the problem domain is simplified into twodimensional vertical plane (a two-dimensional structure such as a long caisson breakwater, a ship or a long submerged shoal) or two-dimensional horizontal plane with the assumption of vertically integrated equations. In the former, it is straightforward using the above equations in (x,z) plane. In the latter case, it forms the combined diffraction-refraction problem and will be discussed in the subsequent section. 3.0 COMBINED WAVE REFRACTION AND DIFFRACTION In the shallow water environment, it is inevitable to take into account the variation in sea bottom topography for the wave propagation to accommodate the wave refraction. In these situations, to simplify the domain, vertically integrated equations derived from Eq.(1) 4

5 are widely adopted. Berkhoff (197) derived such kind of equations, called Mild slope equation based on the assumption of mild slope. cg ccg 0 (14) c cosh k z h i Here, x, yz, xy, and x, y x, y cosh kh Based on mild slope assumption, Case (i) 0 1 that h kh (15) g 0 is neglected If water depth (or deep water) is constant, Eq (1) reduces to Helmholtz equation. 0, also c k 0, x y z 0 (16) k z (17) xy, k 0 (18) Case (ii) For shallow water case, Eq. (14) reduces to long wave equation. Fig.. Study domain for large body diffraction under long wave. 5

6 Now, it has to satisfy, two B.Cs 1. Full reflection of rigid wall. Radiation condition of infinity Consider the domain into two distinct area with the demarcation indicative the inner zone is of include when the rotten contour varies and the outer zone, or deep water Now, solution in zone inside I I, II is either of constant depth II = Incident wave + outgoing wave due to diffraction and refraction Outgoing wave = {wave coming from source, qs at B.C at infinity. }, that satisfies the radiation At, wave height& phase should be continuous. This condition leads to additional equations to determine the intensity of source distribution B.Cs In 0 at 1 n I f an orbitary function of n II, the solution must satisfy the Helmholtz equation of the diffraction problem and can be written in the form (solution at P in II ), (19) 1 1 P P qs H0 krd (0) i P = is due to incident wave q S = source intensity function along Hankel function of the first kind and zeroth order satisfying Helmholtz equation and the Sommerfeld condition r = distance between P and the point of intersection along The source intensity function, q along must satisfy the integral equation 1 1 qp qs H kr d 0 n P n ni P (1) While [ r =o], i.e P is on Continuity condition at, 6

7 and, the problem is well defined and the unknown functions q and can be found. Once q (intensity of source distribution) is known, solution II can be obtained at all P. Finite element formulation can be suitably adopted in the inner domain to fit into arbitrary body boundary and the varying bottom topography. To reduce the domain size, the infinite boundary (outer domain, II can be modelled using any one of the following condition. a. Boundary dampers (radiation boundary condition) b. Exterior boundary integral formulation using source distribution or Green s identify (Berkhoff,197) c. Exterior analytical or series solution (boundary solution) d. Infinite elements. The boundary dampers may be various types and are used depending on the order of requirement. a. Plane damper b. Cylindrical damper c. Second order damper d. Third & higher order The exterior analytical or series solution are expressed by the Hankel and trigonometrically functions.the infinite elements are similar to finite elements except that their domain extends to infinity, i.e. exponential shape functions are used. It is based on 'exponential decay' away from the boundary. These elements are 'parametric' form. 7

8 4.0 BOUSSINESQ APPROXIMATION In water wave problem, the Boussinesq approximation is valid for long waves. The approximation satisfies only weakly nonlinearity. Essentially it is obtained by approximating the vertical structure of horizontal and vertical flow velocity. The resulting non-linear partial differential equations are called as the Boussinesq equations. These equations incorporate frequency dispersion unlike in shallow water equations in which, the speed of the wave depends on the bottom topography irrespective of frequency of the wave. Due to this, Boussinesq equations can better simulate nearshore waves and also, wave penetration into harbours can be modelled by considering the effects of diffraction, bottom refraction and shoaling. The approximation of vertical structure of flow under water waves has been achieved since the waves propagate in the horizontal direction with the harmonic variation in both horizontal directions. However, the variation across the depth has different fitting. During the above process, the modelling of vertical coordinate is avoided. Thus, the three-dimensional fluid domain is simplified into two-dimensional horizontal domain. This reduces one order of the number of equations to be solved. The following procedure explains the Boussinesq approximation mathematically. 1. At an elevation, a Taylor series expansion is made on the velocity potential and/or horizontal and vertical water particle velocities.. Similar to the solution procedure of any infinite series, only a finite number of terms are selected without omitting any terms from the first term. That is, n th term was not selected without selecting (n-1) th term. 3.In the Taylor expansion, the partial derivatives corresponding to vertical coordinates are replaced with partial derivates with respect to horizontal derivatives. While doing so, the conservation of mass following incompressible fluid assumption is ensured and the irrotationality of flow is ensured by enforcing zero curl condition. 4. Final partial differential equations are represented only in terms of the horizontal coordinates and time for dynamic problems. Consider the potential flow problem in D (x-horizontal & z-vertical with the origin at left and at still water level). The water depth (h) is constant. Write the Taylor series expansion for at z = - h (let it be b ). 8

9 1 1 1 b z z z z z z z z z h 6 4 zh zh zh () For the incompressible flow, 0 h at the impermeable bed. Hence, b z z z z b 4 b x x zh 4 1 b 1 4 b b z z..., 4 x 4 x (3) The above series can be truncated according to the required accuracy. Now, apply the kinematic and dynamic free surface conditions in fully nonlinear form. u w0 t x (4) 1 u w g 0 t Limited upto quadratic terms with respect to and the velocity potential expanded at the given elevation, b, the following equations can be derived. It is here assumed that cubic and higher order terms are negligible and hence, posing weak nonlinearity ub h ub h, 3 t x 6 x (5) 3 ub ub 1 ub ub g h, t x x t x In the above equations, if the right hand side terms are set to be zero, then, the resulting equations are called as 'shallow water equations'. Combining the above two equations with a linear approximation of u b (horizontal flow velocity at z=-h) results the following: 1 0 (6) T The linear frequency dispersion characteristics of Boussinesq equations [Eqs.(5-6)] without linear approximation of bottom particle velocity is, C k h 1 gh 6 k h 1 where, C is the phase speed and k is the wave number. (7) 9

10 For an approximated Boussinesq equations [Eqs.(5-6)], the linear frequency dispersion equation can be derived as, C k gh 1 3 h (8) Considering a relative error of 4% in the phase speed compared with linear wave theory estimate, Eqs.(5-6) is valid for kh < / and Eqs.(5-6) is valid for kh < /7 for engineering applications. That is the former equation is valid for wavelengths larger than four times the water depth and the approximated form is valid for wavelengths larger than seven times the water depth. That is, the latter is valid for very long waves. Depending on the applications, the simplied form of Boussinesq equation can be adopted. However, the shallow water equations provide an estimate of wave length with less than 4% accuracy is for the condition of wavelengths larger than 13 times the water depth. The above clearly states the regime of applications of various forms of Boussinesq approximation. Any form of modification to Boussinesq equations, either in the form of domain, i.e., varying bathymetry or higher order accuracy terms in the series expansion or the incorporation of additional physics such as wave breaking, surface tension, nonlinear interaction are generally represented as Boussinesq-type equations. Some of the simplified form of Boussinesq equations based on the assumption of waves travel along one direction has specific applications as listed below. a. Equation of Wave propagation in one dimension is called Korteweg-de Vries equation: both non-periodic solitary waves and periodic cnoidal waves can be derived from kdv equation, i.e., approximated solutions of the Boussinesq equations. b. Equation of wave propagation in two dimensions is called Kadomtsev-Petviashvili equation c. The nonlinear Schrodinger equation (NLS equation) is for the complex valued amplitude of narrowband waves. 10