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Paper P1: MATHEMATICAL and COMPUTATIONAL METHODS Coastal Engineering Professor A G L Borthwick Hilary Term 2010 Paper C2A: STRUCTURES AND FLUIDS 4CE16: COASTAL ENGINEERING Lecture 2 Coastal Hydrodynamics II 2.1 Wave Breaking Wave breaking dissipates wave energy at beaches, produces turbulence, throws sediment into suspension and can cause large impact forces on coastal structures. Excess momentum (i.e. radiation stress) from non-uniform wave breaking is responsible for longshore currents, wave set-up and set-down. Waves become unstable when their steepness (height divided by wavelength) reaches a sufficient value. A useful criterion for estimating the limiting steepness of a wave passing over a flat bed was given by Miche as 2 d H max 0.142 tanh where H is the wave height, is the wave length and d is the water depth. Waves break when the water particles begin to move faster than the wave celerity. At beaches, wave breaking depends on the bed slope, local water depth, and offshore wave conditions. The Iribarren (surf similarity) number is tan NI Ho o 13 Coastal Engineering where is the bed slope angle to the horizontal, H o and o are the height and wavelength, respectively of the offshore incident wave. Wave breaking may be classified according to the Iribarren number as follows: spilling for NI 0.4; plunging for 0.4 NI 2.4 ; collapsing for 2.4 NI 3.4; surging for NI 3.4 . Most damage is caused by plunging and collapsing waves. For shoaling and non-refracting waves, the wave height is H H o cgo cg where c is the individual wave celerity, cg is the group celerity and the subscript o refers to deep water conditions. In deep water, cgo co 2, whereas in shallow water, cg c gd . As d 0, cg 0 and H which is clearly unsustainable. In fact, waves become unstable and break. McCowan (1894) proposed that the breaking wave height is Hb db where 0.78 and db is the water depth. Later, Weggel fitted to laboratory data on breaking waves and found that depends on the beach slope, m tan . Weggel determined the following empirical formula (recommended by the US CERC Shore Protection Manual): 14 Coastal Engineering H H b b a b 1 1 2 db gT where a1 43.81 exp(19m) and b1 1.56 1 exp(19.5m). Note that Weggel’s formula matches McCowan’s criterion as m 0 (i.e. flat beds). For shoaling and refracting breaking waves, Hb Ho cgo cgb coso cosb where is the wave direction. In shallow water, b 0 and cosb 1. Also, cgb gdb . Hence, after substitution, 1/ 2 H c H o o cos . b 1/ 4 2 o gd b Equating this to the McCowan formula ( Hb db ), and rearranging, we obtain 2/5 H 2c d o o cos . b 2 1/ 2 o 2 g This equation gives an estimate of the depth of water at which breaking occurs, but does not take account of bed slope. If instead, Weggel’s formula is used, then, after rearrangement, 2 5/2 a1 2 2 a1 db db db 0 g 2T 4 gT 2 2 2 1/ 2 where Ho co coso 2b1 g . This can be solved iteratively. Goda (1970) produced a graphical approach by which the breaking wave 15 Coastal Engineering height, H b , and depth, db , may be estimated from curves based on the refracted onshore wave steepness, Ho o , where H o is the refracted wave height ( Ho KR Ho and KR is the refraction coefficient). 2.2 Wave Run-up Run-up is the height above still water level reached by the wave uprush at a beach. For small amplitude non-breaking waves, the run-up on a smooth slope is Rs Hs (2s ) where s is the angle of the beach (swash) face slope to the horizontal and H s is the wave height at the beach toe. For finite amplitude waves, Rs Hs (2s ) kHs coth(kd)/ 2. For broken waves, use Hunt’s formula, Rs Hs tan s Hs o . Note that for a vertical wall, s 2 and Rs Hs . Total run-up is R Rr Rs where the roughness factor Rr [0.9,1.0] for smooth concrete slopes and Rr [0.5, 0.7] for rip-rap. 2.3 Interaction of Waves and Currents Assume that seawater is irrotational, inviscid and incompressible and the waves are small amplitude. Consider wave propagation in the presence of a steady uniform horizontal current U iU jV of magnitude 2 2 Uc U V in which U and V are the horizontal current components acting in the onshore x and longshore y directions. Let the current be at an angle to the waves. 16 Coastal Engineering The wave speed relative to the fixed bed is ca T where T is the intrinsic wave period which would be measured by a stationary observer. The wave speed relative to the moving water mass is cr T r where Tr is the wave period measured by an observer moving with the water mass. The current component collinear with the wave is Uc cos U k/ k . The wave number vector is k ik cos jk sin where is the wave angle measured anticlockwise from the onshore x- direction. Hence, ca cr Uc cos . By definition c k , and multiplying by k 2 , the Doppler equation is a r Uck cos in which r 2 Tr is the wave angular frequency relative to the water mass and a 2 T is the (intrinsic) wave frequency relative to the sea bed. The linear dispersion equation is 2 2 r a Uck cos gk tanh(kd) 17 Coastal Engineering where the mean water depth d h where h is the still water depth and is the set-up. The wavelength is 2 k . Hence, waves are elongated when in the same direction as the current. Likewise, an opposing current shortens the wavelength. The individual wave celerities relative to the water mass and sea bed are, respectively, cr r k Tr ; ca a k T . The corresponding group celerities are d d c r ; c a . gr dk ga dk Using the linear dispersion and Doppler equations, cgr ncr ; cga cgr Uc cos where n ½1 2kd/sinh(2kd). The Doppler equation, r a Uck cos , may be written in non-dimensional form after dividing through by g / d1/ 2 as r a U c 1 (kd) cos . g / d g / d a d Substituting the linear dispersion equation into the left hand side, we have 18 Coastal Engineering a Uc kdtanh(kd) 1 (kd) cos . g / d a d This equation may be solved graphically by plotting fL (kd) kdtanh(kd) , fL (kd) kdtanh(kd) and fR (kd) a [1 (kd)(Uc (a d))cos] g / d . There are four possible solutions: A: Uc cos 0 B: cgr Uc cos 0 C: cr Uc cos cgr D: Uc cos cr 19 Coastal Engineering When A = B, Uc cos 0 and there is no current acting in the wave direction. When B = C, Uc cos cgr and cga 0 . We have a stopping current that prevents wave energy being transferred upstream For the case when 0 , Uc cgr ncr . Letting Vo g a gT (2) and making use of the Doppler and linear dispersion equations, we can derive 2 U 1 2kd c tanh(kd)1 . V 4 sinh(2kd) o In deep water, tanh(kd) 1 and sinh(2kd) , and so Uc Vo 4. In shallow water, ca and a 0 for a stopping current. Also tanh(kd) kd and so Uc gd as would be expected. When C = D, Uc cr and the waves are stationary. For waves interacting with irrotational horizontal currents, the particle velocity components are k Ag coshk(z d) u sin S cos U , r cosh(kd) k Ag coshk(z d) v cosS sin V , r cosh(kd) k Ag sinhk(z d) w cosS , r cosh(kd) 20 Coastal Engineering where A is the wave amplitude, z is measured vertically upwards from still water level, and the phase function is S at k x rt k xr . Note that x i x j y is measured from the origin of a stationary coordinate system, whereas xr i xr j yr moves with velocity U such that x xr Ut . 2.4 Long Waves Tides, storm waves, tsunamis, seiches and edge waves can be approximated as long waves (of period > 60 s) where the motions are predominantly horizontal. Consider a long wave in the Eulerian frame of reference. At a fixed location, the total depth is d h where h is the still water depth and is the free surface elevation above still water level. Assume shallow water, such that kh 10 or h 20. Assume small-amplitude waves where A ,h. Hence, vertical accelerations may be neglected, the pressure is hydrostatic (with p g( z) where z is upwards from still water level), and linearised theory applies. Ignore turbulence, Coriolis and bed friction effects. The Continuity and Euler momentum equations are then depth-averaged, linearised and approximated for small-amplitude to give the 2-D long wave equations q q x y , t x y 21 Coastal Engineering q q x gh and y gh , t x t y noting that qx Uh , qy Vh, where the depth-averaged velocity components are 1 1 U u dz and V v dz . d d h h For a horizontal bed, the long wave equations are cross-differentiated to give 2 2 1 h c2 t 2 which is the classical wave equation, where c gh .
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