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Specialization in Ocean Energy MODELLING OF WAVE ENERGY CONVERSION António F.O. Falcão Instituto Superior Técnico, Universidade de Lisboa 2014 PART 2 LINEAR THEORY OF OCEAN SURFACE WAVES FLUID MOTION IN WAVES • Surface tension neglected (no ripples) • Perfect fluid (no viscosity) Incompressible flow u 0 Irrotational flow u 0 or u Laplace equation 2 0 Boundary conditions • At the free-surface: p pa (atmospheric pressure) At the bottom: u n 0 (normal velocity is zero) The free-surface is unknown, which makes the problem non-linear. In general the boundary condition is applied at the undisturbed free-surface (flat surface): LINEAR THEORY. pa atmospheric pressure z y p0 undisturbed pressure(no waves) p0 pa g z x Excess pressure due to waves pe p p0 D derivative following the flow Dt Du accelerati on of fluid particle Dt Euler’s equation (perfect fluid) Du p gk Dt Du pe Dt Du pe Dt Du u (u )u Dt t Linear theory: the products of small quantities are neglected u pe t Equation of the disturbed water free-surface z ( x, y, t ) p p0 pe pa on thefreesurface pe g p0 pa g z p p0 pe pa on thefreesurface pe g u pe t u z ( x, y, t ) pe g pe t g t z g t z 0 Kinematic boundary condition on the free surface D u Dt t z z t z z 0 g t z 0 t z z 0 2 g on z 0 (boundarycondition) 2 z t 2 0 g on z 0 (boundary condition) z t 2 2 Sinusoidal waves Complex variable f (t ) Re(F ei t ) F F0 ei cost Re ei t , sin t Im ei t ei t cost i sin t f (t ) Re F0 ei( t ) F0 cos(t ) • In most cases, we will omit the notation Re( ). • It will be assumed that, whenever a complex expression is equated to a physical quantity, the real part of it is to be taiken. f (t ) Re(F ei t ) f (t ) F eit Sinusoidal waves in deep water 1. 0 crest Aw 0. 5 x 0. 5 1. 0 1. 5 through 0. 5 1. 0 ( z) expi (t x c) ( z) expi(t k x) 2 2 , k T c T = period (s), cT . f = 1/T = frequency (Hz or c/s), 2f 2 T = radian frequency (rad/s), λ = wavelength (m), k 2 / = wave number (m-1) ( z) k 2( z) 0. ( z ) 0 e kz , 2. 0 Sinusoidal waves in deep water ( z) expi (t x c) ( z) expi(t k x) ( z ) 0 e kz , Dispersion relationship 12 g Wave velocity c k k 12 g . 2 30m c 6.8m/s T 4.4s 300m c 21.6m/s T 13.9s Sinusoidal waves in deep water 1. 0 Aw 0. 5 0. 5 x 1. 0 1. 5 2. 0 0. 5 1. 0 Particle orbits: circles of radius = 1k 0ekz Radius decreases exponentially with distance to free surface Disturbance practically vanishes at depth 2 Wave amplitude: Aw 1k 0 0 k Aw In deep water, the water particles have circular orbits. The orbit radius decreases exponentially with the distance to the surface. Sinusoidal waves in water of arbitrary, but uniform, depth h 1. 0 Aw 0. 5 0. 5 x 1. 0 1. 5 2. 0 0. 5 1. 0 ( z) expi (t x c) ( z) expi(t k x) ( z) k 2( z) 0 ( z) 1ekz 2ekz 1ekh 2ekh (h) 0 boundary condition at bottom ( z) 0 coshk ( z h) cosh x 1 2 e e x x d cosh x sinh x dx 1 2 e e cosh x tanh x x x 4 3 2 cosh x sinh x 1 tanh x 0 0.0 0.5 1.0 x 1.5 2.0 Sinusoidal waves in water of arbitrary, but uniform, depth h 2 g on z 0 2 z t c k 2 gk tanhkh ( gk 1 tanh kh )1 2 12 Wave speed 2h g c tanh 2 c g tanh h c Shallow water or long-wave limit: kh small c ( gh)1 2 for kh 1 tanh kh kh Sinusoidal waves in water of arbitrary, but uniform, depth h Sinusoidal waves in water of arbitrary, but uniform, depth h Numerical example: T 8 s, g 9,8 m/s2 Deep water gT 9,8 8 c 12,5 m/s 2 2 cT 12,5 8 100m Shallow water h 1 m cc gH gh 9,8 1 3,1 m/s cT 3,1 8 25,0 m/s Intermediate water depth h 15 m 2 2 0,785 rad/s T 8 h 0,785 0,785 15 c tanh c tanh g c 9,8 c cT 10,2 8 81,8 m c 10,2 m/s Sinusoidal waves in water of arbitrary, but uniform, depth h T 8s h (m) 1 3 5 10 15 20 25 30 40 50 c (m/s) 3,10 5,25 6,63 8,86 10,22 11,09 11,65 12,00 12,33 12,44 12,48 (m) 24,8 42,0 53,0 70,9 81,8 88,7 93,2 96,0 98,6 99,5 99,8 Sinusoidal waves in water of arbitrary, but uniform, depth h Orbits of water particles: elipses with semi-axes: 1k0 coshk ( z h) 1k0 sinhk ( z h) (major) (minor) Aw 1k0 sinh kh h 0.16 Refraction effects due to bottom bathymetry The propagation velocity c decreases with decreasing depth h. As the waves propagate in decreasing depth, their crests tend to become parallel to the shoreline wave crests shoreline crests rays shoreline Dispersion of energy at a bay. shoreline Concentration of energy at a headland. Standing waves. Reflection on a vertical wall Standing waves. Reflection on a vertical wall 1 ( z) expi(t kx), 2 ( z) expi(t kx) 1 ( z) eit eikx eikx 2 ( z) eit coskx Surface elevation ( x, t ) 2 Aw coskx sin t Horizontal velocity component: x 2k ( z) eit sin kx 0 at kx n , n 0, 1, 2, ... Satisfies condition for reflection at vertical wall antinodes nodes Wave energy and wave energy flux Unlike wind, waves permit the transport of energy without the need for any net transport of material. Kinetic energy (circular or elliptic orbits) v Potential energy (sea surface is not plane) Potential energy per unit horizontal surface area (time averaged) Kinetic energy per unit horizontal surface area (time averaged) Epotential Ekinetic 1 gAw2 4 1 gAw2 4 Total energy per unit horizontal surface area (time averaged, any water depth) 1 E Epotential Ekinetic gAw2 2 Wave energy and wave energy flux Wave energy flux (or transmitted power) We are more interested in the energy flux across a vertical plane parallel to the wave crests (from bottom to surface). Ppotential ( gz ) dz h x Pkinetic ( 12 h ) dz x 2 Time average: Pwave Ppotential Pkinetic E cg 1 2kh 1 2kh cg 1 c 1 2 k sinh 2kh 2 sinh 2kh 1 E 1 gAw2 2 group velocity The group velocity may be regarded as the velocity at which the wave energy is propagated. Wave energy and wave energy flux 1 2kh 1 2kh cg 1 c 1 2 k sinh 2kh 2 sinh 2kh group velocity Deep water limit kh sinh 2kh cg c 2 Shallow water limit kh 0 sinh 2kh 2kh cg c In general, the energy travels at a velocity smaller than the wave crests. Deep water c gT 2 4 1 E gAw2 2 cg 1 Pwave E cg g 2 Aw2 T 8 Exercise Consider a two-dimensional OWC subject to regular waves. The submergence of the OWC walls is small so that the wave diffraction they produce may be neglected. The air pressure p(t) inside the chamber is a sinusoidal function of time, and its amplitude and phase may be controlled. From the interference between the incident wave and the radiated waves, determine the maximum fraction of the incident wave power that can be absorbed by the OWC. p(t ) As above, with the back wall extending to the sea bottom. p(t ) Irregular waves Real ocean waves are not regular: they are irregular and random 1.5 1 m 0.5 0 0.5 1 100 150 200 t s 250 300 Sea surface elevation at one location as a function of time We want to describe the sea surface as a stochastic process, i.e. to characterize all possible observations (time records) that could have been made under the conditions of the actual observation. An observation is thus formally treated as one realization of a stochastic process. 1.5 We consider a wave record with duration D (typically 15 to 30 min). 1 m 0.5 We can exactly reproduce that record as the sum of a large (theoretically infinite) number of harmonic wave components (a Fourier series) as 0 0.5 1 100 N (t ) ai cos(2 fi t i ), i 1 150 200 t s 250 fi i D With a Fourier analysis, we can determine the amplitudes ai and phases i for each frequency. For wave records, the phases have any value between 0 and 2 without any preference for any one value. So we will ignore the phase spectrum. 300 Only the amplitude spectrum remains to characterize the wave record. To remove the sample character of the spectrum, we should repeat the experiment many times (M) and take the average over all these experiments, to find the average amplitude spectrum 1 ai M M ai,m m 1 for all frequencies f i However, it is more meaningful to use the variance 1 a2 1 2 i M M m 1 1 a2 2 i,m of each wave component . An important reason is that the wave energy is proportional to the square of the wave amplitude (not to the amplitude). The variance spectrum is discrete, i.e., only the frequencies fi i D are present, whereas in fact all frequencies are present at sea. This is resolved by letting the frequency interval f 1 D 0. The variance density spectrum is defined as 1 S f ( f ) lim f 0 f Units for S f ( f ) are m2s or m2 /Hz 1 a2 2 i 4 3 2 1 0.05 0.10 0.15 0.20 Frequency f (Hz) Typical variance density spectrum 0.25 The variance density spectrum gives a complete description of the surface elevation of ocean waves in statistical sense, provided that the surface elevation can be seen as a stationary Gaussian process. To use this approach, a wave record needs to be divided into segments that are each assumed to be approximately stationary (a duration of about 30 min is commonly used). The sea surface elevation is a random function of time. Its total variance is S f ( f ) df 2 0 We recall that the time-averaged total (potential plus kinetic energy) of a regular wave per unit horizontal surface is E 12 gAw2 If we multiply S f ( f ) by g we obtain the energy density spectrum 1 2 E f ( f ) gS f ( f ) g lim ai . f 0 f 1 2 The variance density S f ( f ) was defined in terms of frequency f 1 T (where T is the period of the harmonic wave). It can also be formulated in terms of radian frequency 2 f 2 T We may write d 2 df 0 0 S f ( f ) df S () d 2 1 S ( ) S f ( f ). 2 The overall appearance of the waves can be inferred from the shape of the spectrum: the narrower the spectrum, the more regular the waves are. When the random sea-surface elevation is treated as a stationary, Gaussian process, then all statistical characteristics are determined by the variance density spectrum S f ( f ) . These characteristics will be expressed in terms of the moments of the spectrum (moment of order m) mn f n S f ( f )df 0 (m ...,3, 2, 1, 0,1, 2, 3,...). For example, the mean-square or variance of surface elevation: S f ( f ) df m0. 2 0 Significant wave height and mean wave period The significant wave height H s is the mean value of the highest one-third of wave heights in the wave record. It is given approximately by H s H m0 4 m0 Several different definitions of “mean” period for irregular waves are used. One is the peak period Tp 1 f p Another is the energy period m1 Te m0 fp The characteristics of the frequency spectra of sea waves have been fairly well established through analyses of a large number of wave records taken in various oceans and seas. The spectra of fully developed waves in deep water can be approximated by the Pierson-Moskowitz equation exp 1052(T ) S f ( f ) 0.1688H s2 Te 4 f 5 exp 0.675(Te f ) 4 S ( ) 262.6 H s2 Te 4 5 4 e Exercise Establish a relationship between the peak period and the energy period for the Pierson-Mokowitz spectrum. Energy flux of irregular waves In regular waves: • energy per unit horizontal surface area E 12 gAw2 • energy flux per unit wave crest length Pwave E cg • In deep water cg 12 c g ( 4f ) . In irregular waves: • In ( f , f df ) g2 1 dPwave cg E f ( f )df gcg S f ( f ) df S f ( f ) df 4 f • By integration g2 1 g2 Pwave S f ( f ) df m1 4 0 f 4 g2 Pwave m1 4 Te m1 m0 g2 2 Pwave H s Te . 64 H s 4 m0 irregular waves Pwave in kW/m 2 g 9.8 ms 3 1025kgm Pwave 0.490H s2Te H s in m Te in s Wave climate So far, the statistical characteristics of the waves were considered for short term, stationary conditions, usually for the duration of a wave record (15 to 30 min). • For long-term statistics, over durations of several years (possibly tens of years) the conditions are not stationary. • For these long time scales, each stationary condition (with a duration of 15 to 30 min) is replaced with its values of the significant wave height and period. This gives a long-term sequence of these values with a time interval of typically 3 h, which can be analysed to estimate the long-term statistical characteristics of the waves. • The number of observations in then presented (instead of the probability density) in bins of size (H s , Te ) Example of annual joint relative frequency of occurrence of H s and Te END OF PART 2 LINEAR THEORY OF OCEAN SURFACE WAVES