Nonlinear Wave Theory

Transcription

1 Nonlinear Wave Theory Weakly Nonlinear Wave Theory (WNWT): Stokes Expansion, aka Mode Coupling Method (MCM) 1) Only applied in deep or intermediate depth water ) When truncated at a relatively high order, it is burdened by tedious and extremely lengthy algebraic work. 3) It was discovered by Schwartz (1974) that the small wave steepness expansion is not convergent for deep or intermediatedepth water waves before reaching their breaking limits. 4) It cannot be applied to shallow water waves (Ursell number).

2 Nonlinear Wave Theory Finite Amplitude Wave Theory Finite Amplitude Wave Theory (FAWT) was developed to overcome the above shortcomings. Schwartz (1974), Cokelet (1977) and Hogan (1980) Stream Function Wave Approach, Chappelear (1961) Dean (1965)

3 Key Differences b/w WNWT and FAWT 1) The two free-surface boundary conditions are satisfied exactly at the free surface in FAWT, while they are satisfied at the still water level in WNWT. ) FAWT can be applied to shallow water waves while Stokes expansion is limited to deep or intermediate-depth water waves. 3) A recursive relation between low-order coefficients and high-order Fourier coefficients has been derived in FAWT, which eliminates similar computational burden in WNWT. 4) FAWT is very powerful tool for computing waves, but it limited to -D periodic wave trains. On the other hand, WNWT can be applied to 3-D and Irregular Waves

4 Weakly Nonlinear Wave Theory for Periodic Waves (Stokes Expansion) Introduction The solution for Stokes waves is valid in deep or intermediate water depth. It is assumed that the wave steepness is much smaller than one. ak 1 kh O(1) where k is the wavenumber and h is the water depth which is assumed constant.

5 Governing Equation & Boundary Conditions Governing Equation: 0 h z x y z Bottom B. C. 0 at z h z Free-Surface K.B.C. h h at z t z 1 Free-Surface K.B.C. g C at z t where and stand for gradient and horizontal gradient, respectively. h

6 Nondimensional Variables X xk, Z zk, Y yk, tt, / a, C gk, C, D, ag ag where x, z, t, h, and C are dimensional variables and X, Z, t,, and C are corresponding nondimensional variables.

7 Nondimensional Governing Equation & Boundary Conditions 0 kh Z (3.1.1) X Y Z 0 at Z kh (3.1.) Z Dh h D at Z (3.1.3) t Z 1 D C at Z (3.1.4) t where and h stand for gradient and horizontal gradient, respectively.

8 Perturbation (Stokes Expansion) Assuming the wave train is weakly nonlinear ( ak 1), its potential and elevation can be perturbed in the order of : 1 3 j1 j 1 3 j1 j 1 3 j1 j C C C C C

9 Hierachy Equations Using the Taylor expansion, the free-surface boundary conditions (Equations (3.1.3) and (3.1.4) are expanded at the still water level (Z = 0). Then we substitute perturbation forms of potential and elevation into the Laplace Equation, bottom and free-surface boundary conditions. The equations are sorted and grouped according to the order in (j) wave steepness. The governing equations for j - th order solutions is given by:

10 ( j) 0 hk Z 0 (3.1.5) j j j1 j j1 P, at Z0 (3.1.6) t j j j1 j j1 D Q, at Z0 (3.1.7) t Z j 0 at Z kh Z (3.1.8)

11 ( j) ( j) where the P and Q can be derived in terms of the solutions for the potential and elevation of order ( j -1) or lower. Therefore, the above hierarchy equations must be solved sequentially from lower to higher order until the required accuracy is reached. To derive the third-order solution for a Stokes wave train, it is adequate to truncate the equations at j 3. ( j ) ( j) Up to j 3, P and Q are given below.

12 1 (1) 1 P C and Q D 1 () P C Z t Q D D h h Z D 1 P D Z (3) C Z t Z t D 1 1 Q D Z 3 Z 1 1 D h h h h D h h Z Z t

13 Solving the non-dimensional Equations from lower order (j=1) to higher order (j=3) for the non-dimensional solutions (wave advances in the x-direction). 1 cosh( kh Z ) sin( X t ), cosh kh 1 1 cos( X t), C 0, D 1 / tanh kh 3 cosh(kh Z ) sin( X t ) 8 3 sinh kh cosh kh (3 1) cos( X t) 4 1 C sinhkh

14 3 1 ( 1)( 3)(9 13) 64 cosh(3kh 3 Z ) sin(3 X 3 t ) cosh kh 3 3 ( 4 3 3)cos( X t ) 8 C 3 3 (8 6 ( 1) ) cos(3 X 3 t ) 64 0 D where coth kh 1 1 9

15 The non-dimensional solutions are then transferred back to the dimensional form. (1) First-order: cosh[ k( z cosh( kh) (1) where A h)] sin a cos, gk tanh kh kx t and a A. g

16 Second-order: 3akA ( 1) cosh[ kz ( h)]sin 8 () 1 (3 1) ak cos 4 () Bernoulli Constant: C o sinhkh 4 1 akg akg( 1)

17 Third-order: (3) 1 ( 1)( 3)(9 13) 64 cosh[3 k( z h)] a k A sin 3 cosh 3kh (3) 3 ( 4 3 3) 3 a k cos 8 3 (8 6 ( 1) ) 3 cos 3 a k 64 Nonlinear Dispersion Relation: 9 gk tanh( kh) 1 k a 1 8

18 Convergence For the fast convergence of the perturbed coefficient,, must be much smaller than unity, which is consistent with weakly nonlinear assumption. However, when the ratio of depth to wave length is small, the Stokes perturbation may not be valid. Convergence rate: R R () (1) mag mag, is the ratio of the potential magnitude of second-order to that of first order solution at z 0.

19 3 ( 1) R. 8 For fast convergence, R should be << 1. This is true when kh ~ O(1). When kh 1, we have : 3 ~ ( kh), hence R ~ O ( kh) 8 R may be much greater than unity Ursell number 1 3 U 8 For R 1, then Ur. 3 r a 1 = h ( kh) ( kh) 3

20 A few striking features of a nonlinear wave train can be described for the above equation: The crests are steeper and troughs are flatter; (see applet (Nonlinear Wave Surface)). Phase velocity increases with the increase in wave steepness. Non-closed trajectories of particles movement. (see applet (N-Trajectory)). Nonlinear wave characteristics (up to nd order).

21 Wave advancing in the x-direction V iukw Particle velocity akg cosh[ k( zh)] 3 a k g cosh[ k( zh)] u cos cos 3 cosh kh 4 sinh kh cosh kh akg sinh[ k( z h)] 3 a k g sinh[ k( z h)] w sin sin 3 cosh kh 4 sinh kh cosh kh Acceleration a iax kaz u (1) (1) cosh[ k( zh)] ax V u akg sin t cosh kh 3 cosh[ k( z h)] 1 akg sin 3 sinh kh cosh kh sinh kh w (1) (1) sinh[ k( zh)] az V w akg cos t cosh kh 3sinh[ k( z h)] sinh[ k( z h)] akg cos 3 sinh kh coshkh sinh kh

22 Particle Trajectory Denoting the mean position of a particle by ( xz, ), and its instantaneous displacement from the mean position by (, ), the Lagrangian velocities of the particle are hence ux (, z) and wx (, z), they are related to the Eulurian velocities through a Taylor Expansion: (1) (1) (1) () (1) (1) (1) (1) (1) (1) () u (1) u (1) (1) ux (, z) u ( xz, ) u ( xz, ) O( ) u x z w w wx (, z) w ( xz, ) w ( xz, ) O( ) w x z (1) () (1) () where u ( x, z), u ( x, z), w ( x, z) and w ( x, z) are first- and secondorder horizontal and vertical velocities.

23 (, ) are calculated by integrating the related Lagrangian velocities. () t u( x, z, ) d ; () t w( x, z, ) d (1) () (, t We intend to compute (, ) up to second order in wave steepness t x () () (1) zt,) (,.) xzt (,) xz O( ) () t (,,) x z t (,,) x z t O( ) (1) () (1) where superscripts stand for orders and overbar denotes a secular term. At leading-order, the solution is the same as that in LWT, t (1) (1) (1) cosh[ kz ( h)] u ( x, z, ) d 0 0 a sin sinh kh t (1) (1) sinh[ kz ( h)] w ( x, z, ) d 0 0 a cos sinh kh t

24 (1) (1) u (1) u (1) a k gcosh [ k( zh)] sinh [ k( zh)] sin cos x z sinh khcoshkh sinh khcoshkh akg 1 cosh kz ( h) cos sinh kh sinh kh (1) (1) w (1) w (1) 0 x z (1) (1) () u (1) u (1) a k g3 cosh[ k( zh)] 1 u (,) xz cos 3 x z 4sinh khcosh kh sinhkh w akg cosh kz ( h) +, sinh kh w w 3 akgsinh[ kz ( h)] (,) x z 3 sin x z 4 sinh khcoshkh (1) (1) () (1) (1)

25 The leading-order trajectory of a particle is an ellipse of the center at ( xz, ) cosh[ kz ( h)] sinh[ kz ( h)] and a major-axis a and minor-axis a. sinh kh sinh kh (1) (1) ( ) ( ) 1. cosh[ kz ( h)] sinh[ k( zh)] a a sinh kh sinh kh The secon-order solutions for the displacement are calculated by integrating the related second-order lagrangian velocities. () () 3 cosh[ kz ( h)] 1 ak 8 sinh kh 4sinh cosh[ kz ( h)] ak t sinh kh 3 sinh[ kz ( h)] ak 4 cos 8 sinh kh () 4 sin kh

26 The secular term ( () ) in the horizontal displacement indicates the particles will continuously move in the wave direction. Hence, the trajectory of a particle is no longer an ellipse. Becasue the horiztonal mean position of a particle is not fixed at x but change with time, we re-define the horizontal mean position by cosh[ kz ( h)] x' xa k t and ' kx' t. sinh kh Correspondingly, the displacement with respect to the instantaneous mean position ( x', z) is given by, cosh[ kz ( h)] sinh[ kz ( h)] a sin ', a cos ', sinh kh sinh kh 3 cosh[ kz ( h)] 1 ak sin ', 4 8 sinh kh 4sinh kh 3 sinh[ kz ( h)] ak cos '. 8 sinh kh (1) (1) () () 4

27 The trajectory of a particle based on the solution is plotted in Applet (N-trajctory). The time average Lagragian velocity of a particle is equal to the derivative of the secular term ( ) with respe ct to time. cosh[ kz ( h)] ul a k sinh kh The integral of the average Lagragian velocity with respect to water depth renders the average mass flux induced by a periodic wave train over a unit width Mass flux udz l = a akg/ E/ Cp, h which is consistent with the result derived using Eulurian approach. ()

28 (1) p cosh[ k( z h Dynamic Pressure Using the Bernoulli equation, dynmaic pressure head induced by a periodic wave train can be calculated up to second-order, (1) () p 1 1 (1) C g g t t g g () a cosh( kh) )] co s, () p 3a k a k ( 1) cosh[ kz ( h)]cos ( 1)cos, g 4 4 p g ak( 1) 1cosh[ kz ( h)]. 4 0,

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