1. Specialization in Ocean Energy MODELLING OF WAVE ENERGY CONVERSION António F.O. Falcão Instituto Superior Técnico, Universidade de Lisboa 2014

2. PART 2 LINEAR THEORY OF OCEAN SURFACE WAVES

3.  The free-surface is unknown, which makes the problem non-linear. FLUID MOTION IN WAVES Surface tension neglected (no ripples) Perfect fluid (no viscosity)  Incompressible flow  Irrotational flow Boundary conditions At the free-surface: (atmospheric pressure)  At the bottom: (normal velocity is zero) In general the boundary condition is applied at the undisturbed free-surface (flat surface): LINEAR THEORY. Laplace equation

4. Excess pressure due to waves x y z Euler’s equation (perfect fluid)

5. Linear theory: the products of small quantities are neglected Equation of the disturbed water free-surface

6. Kinematic boundary condition on the free surface

8. Sinusoidal waves Complex variable 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.

9. Sinusoidal waves in deep water T = period (s), f = 1/T = frequency (Hz or c/s), = radian frequency (rad/s), λ = wavelength (m), = wave number (m -1 ) crest through

10. Sinusoidal waves in deep water Wave velocity Dispersion relationship

11. Sinusoidal waves in deep water Particle orbits: circles of radius = Wave amplitude: Radius decreases exponentially with distance to free surface Disturbance practically vanishes at depth

12. In deep water, the water particles have circular orbits. The orbit radius decreases exponentially with the distance to the surface.

13. Sinusoidal waves in water of arbitrary, but uniform, depth h boundary condition at bottom

15. Sinusoidal waves in water of arbitrary, but uniform, depth h Wave speed Shallow water or long-wave limit: kh small

16. Sinusoidal waves in water of arbitrary, but uniform, depth h

19. Orbits of water particles: elipses with semi-axes: (major)(minor)

20. wave crests 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 shoreline

21. rays crests shoreline Dispersion of energy at a bay. Concentration of energy at a headland.

22. Standing waves. Reflection on a vertical wall

23. Horizontal velocity component: Satisfies condition for reflection at vertical wall antinodes nodes Surface elevation

24. 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)  Potential energy (sea surface is not plane) v Potential energy per unit horizontal surface area (time averaged) Kinetic energy per unit horizontal surface area (time averaged) Total energy per unit horizontal surface area (time averaged, any water depth)

25. 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). 1 Time average: The group velocity may be regarded as the velocity at which the wave energy is propagated. group velocity

26. Wave energy and wave energy flux group velocity In general, the energy travels at a velocity smaller than the wave crests. Deep water

27. 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. As above, with the back wall extending to the sea bottom.

28. Irregular waves

29. Real ocean waves are not regular: they are irregular and random Sea surface elevation at one location as a function of time

30. 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.

31. We consider a wave record with duration D (typically 15 to 30 min). We can exactly reproduce that record as the sum of a large (theoretically infinite) number of harmonic wave components (a Fourier series) as With a Fourier analysis, we can determine the amplitudes and phases for each frequency. For wave records, the phases have any value between 0 and without any preference for any one value. So we will ignore the phase spectrum.

32. 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 However, it is more meaningful to use the variance 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).

33. The variance spectrum is discrete, i.e., only the frequencies are present, whereas in fact all frequencies are present at sea. This is resolved by letting the frequency interval. The variance density spectrum is defined as Units for are or

34. Typical variance density spectrum

35. 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

36. We recall that the time-averaged total (potential plus kinetic energy) of a regular wave per unit horizontal surface is If we multiply by we obtain the energy density spectrum

37. The variance density was defined in terms of frequency (where T is the period of the harmonic wave). It can also be formulated in terms of radian frequency We may write 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.

39. When the random sea-surface elevation is treated as a stationary, Gaussian process, then all statistical characteristics are determined by the variance density spectrum. These characteristics will be expressed in terms of the moments of the spectrum (moment of order m) For example, the mean-square or variance of surface elevation:

40. Significant wave height and mean wave period The significant wave height is the mean value of the highest one-third of wave heights in the wave record. It is given approximately by Several different definitions of “mean” period for irregular waves are used. One is the peak period Another is the energy period

41. 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 Exercise Establish a relationship between the peak period and the energy period for the Pierson-Mokowitz spectrum.

42. In regular waves: energy per unit horizontal surface area energy flux per unit wave crest length In deep water Energy flux of irregular waves In irregular waves: In By integration

43. irregular waves

44. 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

45. Example of annual joint relative frequency of occurrence of and

46. END OF PART 2 LINEAR THEORY OF OCEAN SURFACE WAVES