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Specialization in Ocean Energy

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.

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Specialization in Ocean Energy

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  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. In general the boundary condition is applied at the undisturbed free-surface (flat surface): LINEAR THEORY. FLUID MOTION IN WAVES • Surfacetensionneglected (no ripples) • Perfectfluid (no viscosity) • Incompressibleflow • Irrotational flow Laplace equation • Boundaryconditions • Atthe free-surface: (atmosphericpressure) • Atthebottom: (normal velocityis zero) • The free-surface is unknown, which makes the problem non-linear.

  4. z y x Excesspressuredue to waves Euler’sequation (perfectfluid)

  5. Linear theory: theproductsofsmallquantities are neglected Equation of the disturbed water free-surface

  6. Kinematicboundaryconditiononthe free surface

  7. Sinusoidal waves Complexvariable • In most cases, wewillomitthenotationRe( ). • Itwillbeassumedthat, whenever a complexexpressionisequated to a physicalquantity, the real partofitis to betaiken.

  8. Sinusoidal waves in deepwater crest through T = period (s), f = 1/T = frequency (Hz or c/s), = radianfrequency (rad/s), λ = wavelength (m), = wavenumber (m-1)

  9. Sinusoidal waves in deepwater Wavevelocity Dispersionrelationship

  10. Sinusoidal waves in deepwater Particleorbits: circlesofradius = Radiusdecreasesexponentiallywithdistance to free surface Disturbancepracticallyvanishesatdepth Wave amplitude:

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

  12. Sinusoidal waves in waterofarbitrary, butuniform, depthh boundaryconditionatbottom

  13. Sinusoidal waves in waterofarbitrary, butuniform, depthh Wave speed Shallowwaterorlong-wavelimit: khsmall

  14. Sinusoidal waves in waterofarbitrary, butuniform, depthh

  15. Sinusoidal waves in waterofarbitrary, butuniform, depthh

  16. Sinusoidal waves in waterofarbitrary, butuniform, depthh

  17. Sinusoidal waves in waterofarbitrary, butuniform, depthh Orbitsofwaterparticles: elipses withsemi-axes: (major) (minor)

  18. 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

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

  20. Standing waves. Reflection on a vertical wall

  21. Standing waves. Reflection on a vertical wall Surfaceelevation Horizontal velocitycomponent: Satisfiescondition for reflectionat vertical wall nodes antinodes

  22. Wave energy and wave energy flux Unlike wind, waves permit the transport of energy without the need for any net transport of material. • Kineticenergy (circular orellipticorbits) • Potentialenergy (seasurfaceisnot 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)

  23. 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: groupvelocity The group velocity may be regarded as the velocity at which the wave energy is propagated.

  24. Wave energy and wave energy flux In general, theenergytravelsat a velocitysmallerthanthewavecrests. groupvelocity Deepwater

  25. Exercise Consider a two-dimensional OWC subject to regular waves. Thesubmergenceofthe OWC wallsissmallsothatthewavediffractiontheyproducemaybeneglected. Theairpressurep(t)insidethechamberis a sinusoidal functionof time, andits amplitude andphasemaybecontrolled. Fromtheinterferencebetweentheincidentwaveandtheradiatedwaves, determine themaximumfractionoftheincidentwavepowerthat can beabsorbedbythe OWC. As above, withthebackwallextending to theseabottom.

  26. Irregular waves

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

  28. 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 realizationof a stochastic process.

  29. 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.

  30. 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).

  31. 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

  32. Typical variance density spectrum

  33. 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

  34. 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

  35. 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.

  36. 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, themean-squareorvarianceofsurfaceelevation:

  37. 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

  38. 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 relationshipbetweenthepeakperiodandtheenergyperiod for thePierson-Mokowitzspectrum.

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

  40. irregular waves

  41. Waveclimate 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 heightand 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

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

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

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