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3.3 Rapid distortion theory The incompressible Navier-Stokes equation may be written as Du Dt +u ·∇ U = −1∇ p+non-lin. and viscous terms, (31) ρ where p is the pressure, and D/Dt ≡ ∂/∂t+U ·∇ is the ‘average Lagrangian derivative.’ Assuming a linear shear (∇ U constant), taking the curl, and dropping the non-linear and viscous terms we get Dω = Ω·∇ u +ω ·∇ U, (32) Dt where Ω and ω are the mean and the fluctuating part of the vorticity. It is not at all clear that this linearization is permissible. For example, it can be shown that if the curl of Eq. (31) is used to estimate the change in mean square vorticity the non-linear terms will dominate the linear. However, Hunt and Carruthers (1990) argue that when used for the calculation of the response of velocity fluctuations (u or Rij) to a sudden application of a large scale shearing or straining motion the linearization Eq. (32) is valid. Figure 30: Interpretation of the interplay of shear and turbulence: Two differently oriented eddies are followed over three successive times. Shear stretches (along the axis of rotation) and speeds up the upper eddy while the lower eddy is compressed and slowed down. Physically, the last term on the right hand side of Eq. (32) may be interpreted as the stretching of vorticity by the mean shear (see Figure 30). The first term is a distortion of the mean vorticity by velocity fluctuations. In order to solve Eq. (32) we have to Fourier transform the equation. In order to do so, it is important to notice that wave fronts are advected by the mean flow i.e. dk = −(∇ U)k. (33) dt The solution to this wave front advection equation is k(t) = exp(−∇ Ut)k0 where exp means the matrix exponential. For a general linear U Eq. (32) does not have analytic solution. However, for many simple situations such as unidirectional shear, non-rotational stretching or compression, etc. such solutions exists (Townsend, 1980). 54 <strong>DTU</strong> Wind Energy-E-Report-0029(EN) (34)
To get the velocity field from the vorticity we shall express dZ in terms of dΩ , which is the Fourier transform of ω defined in parallel to Eq. (26): ω = ∇ ×u ⇒ dΩ = ik ×dZ ⇒ −ik ×dΩ = k ×(k ×dZ). (35) Because of the general identity A×(B×C) = B(A·C)−C(A·B) and that k·dZ = 0 we get −ik ×dΩ = −k 2 k ×dΩ dZ ⇒ dZ = i k2 . We shall re-derive (3.11) in Mann (1994), i.e. set up the equations of motion for ⎛ ⎞ 0 0 0 (36) ∇ U = ⎝ 0 0 0 ⎠. (37) dU dz 0 0 In this case ⎛ 1 ⎞ 0 0 k(t) = exp(−∇ Ut)k0 = ⎝ 0 1 0 ⎠k0, (38) t 0 1 − dU dz in accordance with (3.13) of Mann (1994), and Ω = (0,dU/dz,0). The equations of motion Eq. (32) becomes ⎛ ⎞ dΩ3 Dk ×dZ = k2dZ + ⎝ 0 ⎠. Dβ 0 (39) Taking the cross product with k and adding ˙ k ×(k ×dZ) on both sides we get − Dk2 dZ Dβ Writing this more explicitly we get Dk2dZ Dβ = ⎛ ⎝ = Dk Dk ×dZ ×(k ×dZ)+k × Dβ Dβ = Dk Dβ ×(k ×dZ)+k2k ⎛ ×dZ + ⎝ (k 2 1 −k2 2 −k2 3 )dZ3 −2k1k3dZ1 2k1(k2dZ3 −k3dZ2) 0 0 k3 −k2 ⎞ ⎞ ⎠dΩ3. (40) ⎠ (41) and using Dk2 /Dβ = −2k1k3 from Eq. (38) this can be shown to be equivalent to (3.11) in Mann (1994). The differential equations Eq. (41) are easily solved given the initial conditions k(0) = k0 = (k1,k2,k30) and dZ(k0,0). Instead of time, t, we shall use the non-dimensional time, β, defined as β = dU t. (42) dz The solution to Eq. (41) is where with and ⎡ dZ(k,β) = ⎣ ζ1 = C2 = C1 − k2 C2 k1 1 0 ζ1 0 1 ζ2 0 0 k 2 0 /k2 , ζ2 = ⎤ ⎦dZ(k0,0), (43) k2 C1 +C2 k1 C1 = βk2 1(k 2 0 −2k 2 30 +βk1k30) k 2 (k 2 1 +k2 2 ) k2k 2 0 (k2 1 +k2 2 )32 arctan (44) (45) βk1(k2 1 +k2 2) 1 2 k2 0 −k30k1β . (46) Eqs. (38) and (43) give the temporal evolution of individual Fourier modes. <strong>DTU</strong> Wind Energy-E-Report-0029(EN) 55
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Remote Sensing for Wind Energy DTU
Page 3 and 4: Author: Alfredo Peña, Charlotte B.
Page 5 and 6: 4 Introduction to continuous-wave D
Page 7 and 8: 8 Nacelle-based lidar systems 157 8
Page 9 and 10: 12 Complex terrain and lidars 231 1
Page 11 and 12: 1 Remote sensing of wind Torben Mik
Page 13 and 14: Figure 2: Calibration, laboratory w
Page 15 and 16: Figure 3: Example of scatter plots
Page 17 and 18: 1.2.3 Summary of sodars Most of tod
Page 19 and 20: 1.3.3 Wind lidars Measuring wind wi
Page 21 and 22: Figure 6: CW wind lidars (ZephIRs)
Page 23 and 24: Further developments Furthermore, n
Page 25 and 26: 2 The atmospheric boundary layer S
Page 27 and 28: Figure 9: Large spatial scale varia
Page 29 and 30: Du3 Dt Du1 Dt Du2 Dt The three mome
Page 31 and 32: Figure 13: Consensus relations betw
Page 33 and 34: ψ z L ∼ − 5 L . For unstable c
Page 35 and 36: Figure 15: Behavior of the turbulen
Page 37 and 38: Figure 17: Newly developed models t
Page 39 and 40: The value of q0 at the surface is d
Page 41 and 42: the spray is the source of icing on
Page 43 and 44: u∗2 u∗1 u1(h) = u∗1 k ln h
Page 45 and 46: Figure 26:Land-seabreeze system,whe
Page 47 and 48: Figure28:Three dimensionalpicture o
Page 49 and 50: sufficient information. Finally, we
Page 51 and 52: Mann, J. (1998) Wind field simulati
Page 53: (2010) and comparison under differe
Page 57 and 58: τ(k) [Arbitrary units] 10 3 10 2 1
Page 59 and 60: (Koopmans, 1974; Bendat and Piersol
Page 61 and 62: anemometer was installed at each en
Page 63 and 64: and fSw(f) u 2 ∗ = 1.05n 1+5.3n 5
Page 65 and 66: and n = 0.468. This spectrum implie
Page 67 and 68: to be calculated. We do that on a m
Page 69 and 70: Notation A Charnock constant neutra
Page 71 and 72: Maxey M. R. (1982) Distortion of tu
Page 73 and 74: 4.2 Basic principles of lidar opera
Page 75 and 76: 4.2.5 Wind profiling in conical sca
Page 77 and 78: 4.3.1 Behaviour of scattering parti
Page 79 and 80: the beam radius at the output lens.
Page 81 and 82: from which the value of VLOS is der
Page 83 and 84: the atmosphere. The SNR 4 for a win
Page 85 and 86: individual line-of-sight wind speed
Page 87 and 88: A general approach to mitigating th
Page 89 and 90: from ±VH sinδ (if the tilt is tow
Page 91 and 92: as a down draught (of the same abso
Page 93 and 94: Table 7: Combined results from 28 Z
Page 95 and 96: in Eastern Jutland between January
Page 97 and 98: Figure 56: Normalized power curves
Page 99 and 100: the concept. Developments include i
Page 101 and 102: References x horizontal position in
Page 103 and 104: Wagner R., Mikkelsen T., and Courtn
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5.2 End-to-end description of pulse
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Scanner Coherent lidar measure the
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Figure 62: Radial wind velocity ret
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transform in order to use data obta
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This wavelength is also the most fa
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Eq.(132)isadaptedforcollimatedsyste
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5.3.6 Existing systems and actual p
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(Gottshall et al., 2010; Albers et
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References Albers A., Janssen A. W.
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derived from fluctuations of the wi
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Acoustic received echo (ARE) method
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Figure 71: Sample time-height cross
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A gradient minimum is characterized
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Figure73:Bragg-relatedacoustic(belo
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stability (inversion strength) can
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Figure 76: Combined soundingwith a
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Figure 79: Favorite regions (shaded
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Direct detection of MLH from acoust
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Engelbart D.A.M.and Bange J. (2002)
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7 What can remote sensing contribut
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uyms 9.0 8.5 8.0 7.5 7.0 120 140 16
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Bottom of rotor Φ rotation r w u
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Height Height Hub 1.6 1.4 1.2 1.0 0
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PP rated PP rated 1.0 0.8 0.6 0.4 0
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KEprofileKEhub 1.2 1.1 1.0 0.9 0.8
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PP rated WS Lidarms 1.0 0.8 0.6 0.4
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8 Nacelle-based lidar systems Andre
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• Flexibletrajectories.Dependingo
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Figure 104: Sketch of simultaneous
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The normal wind direction vector nw
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Figure 109: Test site at DTU Wind E
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Figure 112: Power curve met mast an
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Notation C number of sent photons C
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9 Lidars and wind turbine control -
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for three unknowns, it is impossibl
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model of the blade pitch actuator,
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|GRL| [-] 1 0.8 0.6 0.4 0.2 10 k [r
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PSD(Ωg) [(rpm) 2 /Hz] PSD(Ωg) [
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0.04 0.03 ˆk [ rad m ] 0.02 0.63 0
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[%] 10 0 −10 −20 −30 MyT Moop
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PSD(θ1) [rad 2 /Hz] PSD(Moop1) [Nm
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Pel/Pel,max [-] 1 0.98 0.96 0.94 0.
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fL weighting function GRL transfer
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E. Hau, Windkraftanlagen, 4th ed. S
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¨¦¦§©¡§ ¥§¨¦¦§£ ¡¥
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vertical (m) 150 100 50 R d 0
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With feedback only, on the other ha
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Figure 136: Estimated preview requi
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Normalized C r = C r P Q 2 W (r) b
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Normalized C r = C r P Q 2 W (r) b
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Coherence 1 0.8 0.6 0.4 0.2 0 10
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Coherence 1 0.8 0.6 0.4 0.2 0 10
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Magnitude Squared 10 8 10 7 10 6 10
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Figure 148: During simulation, FAST
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Figure 149: Collective flap respons
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Magnitude (abs) blade pitch gen spe
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• Measurement coherence, which ca
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Jonkman, B. (2009) TurbSim user’s
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11 Lidars and wind profiles Alfredo
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z [m] 160 100 80 60 40 20 10 15 20
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z [−] zo 1 κ ln 40 38 36 34 32 3
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z [m] z [m] 1000 900 800 700 600 50
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the growth of the length scale, agr
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12 Complex terrain and lidars Ferha
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Figure 158: The ZephIR models which
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Uconst wΑx l h Φ h.tanΦ Figure 1
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Figure 162: Lavrio: The scatter plo
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Figure 163: Panahaiko: The scatter
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References Albers A. and Janssen A.
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Wexler (1968), where the limitation
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13.2.1 Systematic turbulence errors
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where x is the center of the scanni
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The theoretical systematic errors a
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Height (m) 160 140 120 100 80 60 40
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Height (m) Height (m) 160 140 120 1
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RMSPE (%) 70 60 50 40 30 20 10 0 vu
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involves interaction of all compone
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Lindelöw P. (2007) Fibre Based Coh
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plications, including meteorologica
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Figure 173: Rayleigh-Jeans’ appro
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Figure 176: Brightness temperature
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with rain are shown in Fig. 179. A
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This method gives a good accuracy (
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Figure 183: A recent maintenance in
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Figure 185: Temperature profiles in
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References s point in space Sν(s)
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Also the Doppler Centroid anomaly c
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Christiansen et al., 2006). The res
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educe speckle noise, a random noise
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Figure 188: Envisat ASAR wind field
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Figure 190: Envisat ASAR wind field
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Satellites in sun-synchronous polar
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500 overlapping scenes for wind res
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(with the fewest samples). The unce
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Badger M., Hasager C. B., Thompson
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Ren Y. Z., Lehner S., Brusch S., Li
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tracking, climate studies, air-sea
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The modified logarithmic wind profi
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sea ice mask was applied and σ0 me
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QuikSCAT 25 20 15 10 5 Horns Rev Fi
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Figure 203:Example of an ASCAT coas
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References Bourassa M.A., Legler D.
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DTU Wind Energy Technical Universit
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