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Wind loading and structural response Lecture 6 Dr. J.D. Holmes Atmospheric boundary layers and turbulence I Atmospheric boundary layers and turbulence 153 metres 64 metres 12 metres 35 30 Wind speed (m/s) 25 20 15 10 5 0 0 1 2 3 4 5 Time (minutes) Wind speeds from 3 different levels recorded from a synoptic gale Atmospheric boundary layers and turbulence Features of the wind speed variation : • Increase in mean (average) speed with height • Turbulence (gustiness) at each height level • Broad range of frequencies in the fluctuations • Similarity in gust patterns at lower frequencies Atmospheric boundary layers and turbulence • Mean wind speed profiles : • Logarithmic law dU is a function of (z, ρ a τ 0 ) dz 0 - surface shear stress a - air density dU u constant . dz z u = friction velocity = (0/a) integrating w.r.t. z : U (1/ k ) . u log e z constant Atmospheric boundary layers and turbulence • Logarithmic law u U(z) log e (z/z0 ) k • k = von Karman’s constant (constant for all surfaces) • zo = roughness length (constant for a given ground surface) logarithmic law - only valid for z >zo and z < about 100 m Atmospheric boundary layers and turbulence • Modified logarithmic law for very rough surfaces (forests, urban) u z - zh z U(z) log e k o • zh= zero-plane displacement zh is about 0.75 times the average height of the roughness Atmospheric boundary layers and turbulence • logarithmic law applied to two different heights U(z 1 ) log e z1/z o U(z 2 ) log e z 2 /z o • or with zero-plane displacement : U(z 1 ) log e (z 1 z h )/z o U(z 2 ) log e (z 2 z h )/z o Atmospheric boundary layers and turbulence • Surface drag coefficient : Non-dimensional surface shear stress : 0 2 u 2 U10 U10 2 u 10 from logarithmic law : U10 log e z k o 2 k 10 log e z o Atmospheric boundary layers and turbulence • Terrain types : Terrain Type Roughness Surface Drag Length (m) Coefficient Very flat terrain (snow, desert) 0.001 - 0.005 0.002 – 0.003 Open terrain (grassland, few trees) 0.01 – 0.05 0.003 – 0.006 Suburban terrain (buildings 3-5 m) 0.1 – 0.5 0.0075 – 0.02 Dense urban (buildings 10-30 m) 1–5 0.03 – 0.3 Atmospheric boundary layers and turbulence • Power law z U ( z ) U10 10 • = changes with terrain roughness and height range 1 log ( z / z ) e ref 0 zref = reference height Atmospheric boundary layers and turbulence • Matching of power and logarithmic laws : zo = 0.02 m = 0.128 zref = 50 metres Logarithmic law Power law 100 80 Height, z (m) 60 40 20 0 0.0 0.5 1.0 1.5 Atmospheric boundary layers and turbulence • Mean wind speed profiles over the ocean: • Surface drag coefficient () and roughness length (zo) vary with mean wind speed 2 au 2* aκU10 zo (Charnock, 1955) g g g - gravitational constant a - empirical constant a lies between 0.01 and 0.02 2 2 a kU10 κ k substituting : zo 10 g log e 10/zo log e z o Implicit relationship between zo and U10 Atmospheric boundary layers and turbulence • Mean wind speed profiles over the ocean: Assume g = 9.81 m/s2 ; a = 0.0144 (Garratt) ; k =0.41 U10 (m/s) Roughness Length (mm) 10 0.21 15 0.59 20 1.22 25 2.17 30 3.51 Applicable to non-hurricane conditions Atmospheric boundary layers and turbulence • Relationship between upper level and surface winds : • Geostrophic drag coefficient Cg u* Ug Ug Rossby Number : Ro fzo balloon measurements : Cg = 0.16 Ro-0.09 (Lettau, 1959) Can be used to determine wind speed near ground level over different terrains : Log law Lettau Lettau Log law U10, terrain 1 u*,terrain 1 Ug u*,terrain 2 U10, terrain 2 Atmospheric boundary layers and turbulence • Mean wind profiles in hurricanes : • Aircraft flights down to 200 metres • Drop-sonde (probe dropped from aircraft - tracked by satellite) : recently started • Sonic radar (SODAR) measurements in Okinawa • Tower measurements • not enough • usually in outer radius of hurricane and/or higher latitudes Atmospheric boundary layers and turbulence • Mean wind profiles in hurricanes : North US Navy West Cape antennas Exmouth EXMOUTH GULF 100 km • Northern coastline of Western Australia • Profiles from 390 m mast in late nineteen-seventies Atmospheric boundary layers and turbulence • Mean wind profiles in hurricanes : • In region of maximum winds : steep logarithmic profile to 60-200 m • Nearly constant mean wind speed at greater heights 1000 log e ( z / 0.3) U z U10 Height z, (m) log e (10 / 0.3) for z < 100 m 100 Uz =U100 for z 100 m 10 0.0 1.0 2.0 U(z)/U(10) Atmospheric boundary layers and turbulence • Mean wind profiles in thunderstorms (downbursts) : • Doppler radar • Some tower measurements (not enough) • Horizontal wind profile shows peak at 50-100 m • Model of Oseguera and Bowles (stationary downburst): λR 2 U 2r 1 e r/R 2 e z/z e z/ε r - radial coordinate R - characteristic radius z* - characteristic height out of the boundary layer - characteristic height in the boundary layer - scaling factor Atmospheric boundary layers and turbulence • Mean wind profiles in thunderstorms (downbursts) : Model of Oseguera and Bowles (stationary downburst) : r/R = 1.121 R = 1000 m 600 r/R = 1.121 400 z* = 200 metres Height (m) = 30 metres 200 = 0.25 (1/sec) 0 0 20 40 60 Wind speed (m/s) Atmospheric boundary layers and turbulence • Mean wind profiles in thunderstorms (downbursts) : Add component constant with height (moving downburst) : R = 1000 m 600 r/R = 1.121 400 Height (m) z* = 60 metres = 50 metres 200 = 1.3 (1/sec) 0 0 20 40 60 80 100 Wind speed (m/s) Uconst = 35 m/s Atmospheric boundary layers and turbulence 153 metres 64 metres 12 metres 35 30 Wind speed (m/s) 25 20 15 10 5 0 0 1 2 3 4 5 Time (minutes) Turbulence represents the fluctuations (gusts) in the wind speed It can usually be represented as a stationary random process Atmospheric boundary layers and turbulence Components of turbulence : • u(t) - longitudinal - parallel to mean wind direction - parallel to ground (usually horizontal) • v(t) - parallel to ground - right angles to u(t) • w(t) - right angles to ground (usually vertical) w(t) v(t) U+u(t) ground Atmospheric boundary layers and turbulence Turbulence intensities : 2 u { U (t ) U dt} 2 T 1 1 • standard deviation of u(t) : T 0 Iu = u /U (longitudinal turbulence intensity) (non dimensional) Iv = v /U (lateral turbulence intensity) Iw = w /U (vertical turbulence intensity) Atmospheric boundary layers and turbulence Turbulence intensities : near the ground, u 2.5u* 2.5u 1 Iu = u /U u /0.4 log e z/z 0 log e z/z 0 from logarithmic law 0.88 v 2.2u* Iv log e z/z 0 0.55 w 1.37u* Iw log e z/z 0 Atmospheric boundary layers and turbulence Turbulence intensities : rural terrain, zo = 0.04 m : Height, z (m) Iu 2 0.26 5 0.21 10 0.18 20 0.16 50 0.14 100 0.13 Atmospheric boundary layers and turbulence Probability density : • The components of turbulence (constantU) can generally be represented quite well by the Gaussian, or normal, p.d.f. : 1 u U 2 f u u 1 for u(t) : exp σ σ u 2π 2 u 1 v 2 f v v 1 for v(t) : exp σ v 2π 2 σv 1 w 2 f w w 1 for w(t) : exp σ w 2π 2 σw End of Lecture 6 John Holmes 225-405-3789 JHolmes@lsu.edu