Lecture 3 Design Wind Speed Tokyo Polytechnic University The 21st Century Center of Excellence Program Yukio Tamura Wind Climates Temperature Gradient due to Differential Solar Heating Density Difference Pressure Gradient (Conservation of mass/momentum) - Global Circulations (Hadley, 1973) - Monsoons - Frontal Depressions - Tropical Cyclones (Hurricanes, Typhoons, Cyclones) - Thunderstorms (Down Bursts) - Tornadoes - Devils - Gravity Winds (Katabatic( Winds) - Lee Waves etc. 1
Tropical Cyclones (Hurricanes, Typhoons) 1000 hpa 1000 hpa 100 2000 km EYE OF HURRICANE Temporal Variation of Wind Speed (Hurricane Celia, 1970) Maximum Peak Gust 19:00 20:00 21:00 22:00 23:00 24:00 1:00 Cook, 1983 2
Wind Speed Distribution in Tropical Cyclones Wind Speed r 1 / r 0 r m 100 Distance from Center r Wind Speed Distribution in Tropical Cyclones Moving Direction 10m/s 20m/s 30m/s 40m/s 50m/s Dangerous Semicircle 3
Temporal Variation of Wind Speed (Typhoon) 13:28 50.6m/s SSE 240m 180m 120m Cumulonimbus Downbursts Down Draft Down Draft 60m 600m Stagnation Cone Gust Front 1200m 1800m 2400m 3000m 3600m 4km Intense Blowoff 4
Tornadoes Cumulonimbus Geostrophic Wind Geostrophic Wind Speed U G dp + ρ a U G f = 0 dn ρ a Low Pressure High Pressure Pressure Force U G Coriolis Force : Air Density f : Coriolis Parameter = 2Ω sinφ = 1.454 10 10-4 sinφ (rad/s) Ω : Earth s s Rotational Speed φ : Latitude Isobars 5
Gradient Wind Gradient Wind Speed U g (without consideration of ground surface friction ) Centrifugal Force Low Pressure dp U 2 g + ρ a U g f + ρ a = 0 dn r r : Radius of Curvature Isobars Pressure Force U g Coriolis Force High Pressure Atmospheric Boundary Layer Gradient Height Z g U g Friction Free Wind A.B.L. < Z g / 10 d Outer Layer 200m Surface Layer (τ : constant) Interfacial Layer d : Zero-plane displacement Average building height 6
Gradient Wind (Z( >Z g ) Pressure Gradient Force Coriolis Force + Centrifugal Force Outer Layer ( Z( g /10 < Z < Z g ) - Momentum transfer from Gradient Wind - Momentum delivered to lower altitude due to Friction Effects (Reynolds Stress : Turbulent Transfer) Strong Wind Condition - Sufficient mixing of air - Thermally neutral condition ( θ/ Z 0, θ :Temperature) - Effects of ground roughness are predominant. - Shear force (Reynolds stress: τ = ρ a uw ) increases from zero at the gradient height Z g to a maximum at the zero-plane displacement. - Surface shear stress: τ 0 = ρ a uw max = ρ a u 2 cf. Boussinesq: Kinetic Eddy Viscosity * du u * : Friction velocity ρ a uw = ρ a ε : Kinetic Eddy Viscosity ε dz 7
Ekman Layer Geostrophic Balance With Friction Low Pressure Pressure Force U Isobars Friction Force Coriolis Force High Pressure Lower Altitude U U G Ekman Spiral Wind Speed Fluctuation U(t ) = U + u(t u ) Fluctuating Component u(t) Mean Wind Speed U 8
Power Spectrum of Wind Speed (van der Hoven for Brookhaven, 1957) Power Spectral Density fs U (f) (m/s) 2 4days 1day Macrometeorological peak 1hour 10min 1min 5s Spectral Gap Micrometeorological peak 0.001 0.01 0.1 1 10 100 1000 Frequency f (cycle/hour) Long Term and Short Term Fluctuations Averaging Time of Wind Speed T = 10min (Japan, ISO4354, etc.) 1hour Long Term Fluctuation T > 10min 1hour --- Variation of Mean Wind Speed (Design Wind Speed) --- Static Effects on Buildings Short Term Fluctuation T < 10min 1hour --- Turbulence or Gust --- Dynamic Effects on Buildings 9
Mean Wind Speed Structure Geographical Location Ground Roughness and Height Topographic Effects Direction Seasonal Variation Return Period etc. Wind Speed Profile U g U g U g 10
Wind Speed Profile (Theoretical Approach) Surface Layer u * Z d U = { ln( ) ) + A } k z 0 Law of Wall (Prandtl( Prandtl,, 1932) Near Gradient Height u * u U * g = { ln( ) B } k f z 0 Velocity Defect Law (Karman, 1930) z 0 : Roughness length, u * : Friction velocity k : Karman constant ( ( 0.4) f : Coriolis parameter, A, B : Empirical constants Wind Speed Profile in Codes Log Law u * Z d U = ln( ) k z 0 - Uniform roughness & Constant τ Power Law (fully empirical) Z d U = U r ( ) α Z r z 0 : Roughness length, u * : Friction velocity k : Karman constant ( ( 0.4) d : Zero-plane displacement (0 0.75H) U r : Wind speed at reference height Z r Z r : Reference height, α : Power-law index Deaves & Harris Model (log + polynomial) 11
Wind Speed Profile (Field Data) Terrain z 0 α sea, mudflats, snow covered flat land, etc. flat open countryside, fields with crops, fences and few trees, etc. (Meteorological Standard) dense woodland, domestic housing, suburban area city large city center 0.0005m - 0.003m 0.003m - 0.2m 0.2m -1m 1m -2m 2m - 4m 0.1-0.13 0.14-0.2 0.2-0.25 0.25-0.3 0.3-0.5 Wind speed difference between two sites with different roughnesses Site A Site B 12
Wind Speed Profiles at Two Sites with Different Roughnesses N Seaside Urban 12k 1 Urban 2 m 500 200 100 50 Urban 2 Seaside Urban 2 Seaside Tokyo Bay 20 10 Wind Speed Profile (AIJ Rec.1993) Large City Center Seaside AIJ 1993 13
Yearly Variation of Annual Maximum Wind Speed and Building Volume in Japan Return Period : R Annual V Maximum Wind Speed U U R Vr r 1 N Rr = lim t i N N i = 1 0 t t 1 t (year) t 2 t 3 t 4 t 5 Return Period R = Mean Time Between Failure (MTBF) 14
Wind Speed U R and Return Period R Wind Speed U R (m/s) Return Period R (years) 26 28 34 37 42 52 5 10 50 100 500 1000 Annual probability of exceedence of wind speed U R : Q(U 1 R ) = R Probability of non-exceedence (CDF) of wind speed U R : 1 P(U R ) = 1 R Tokyo R-year Recurrence Wind Speed U R Probability of exceedence of U R during Building s Life Time, e.g. T L = 50 years T 1 Q(U R : T L ) = 1 P(U R ) L = 1 (1 ) R T L R (years) Q(U R : 50) 50 63.6% 100 39.5% 500 9.5% 1000 4.9% (T L = 50 years) 15
Extreme Value Analysis (Asymptotic Form: Cumulative Distribution Function) Fisher-Tippett Type I (Gumbel( Distribution) CDF: P(U) ) = exp[ exp{ a(u b)}] 1/a : dispersion, b : mode Fisher-Tippett Type II (Frechet( Distribution) c γ CDF: P(U) ) = exp{ ( ) } U ε - having lower limit : ε Fisher-Tippett Type III (Weibull( Distribution) U w γ CDF: P(U) ) = exp{ ( ) } v w - having upper limit : w Gumbel Distribution CDF: P(U R ) = exp[ exp{ a(u R b)}] 1/a : dispersion, b : mode Cumulative Distribution Function of R-year Recurrence Wind Speed U R 1 P(U R ) = 1 R R-year Recurrence Wind Speed Based on Gumbel Distribution 1 1 U R = ln{ ln ( 1 ) } + b a R Modified Jensen-Franck Method (Cook, 1982) (Individual Storms, Gumbel,, Threshold) 16
Basic Design Wind Speed in Japan (AIJ Rec.1993) 100-year Recurrence 10-min Mean Z =10m, α = 0.15 (Category II) 30 30 40 40 40 30 40 30 40 30 30 40 (m/s) 40 48 40 Okinawa: 50m/s 48 Topographic Effects Escarpments Cliffs Ridges Hills Valleys - Speed-up Ratio (Mean Wind Speed) - Turbulence Speed-up Ratio : AIJ Rec.1993 17
Topographic Effects Fluctuating Wind Speed Structure Gust Factor / Peak Factor Power Spectrum Spatial- / Temporal- Correlation Taylor s s Hypothesis of Frozen Turbulence Turbulence Intensity Turbulence Scale 18
Wind Speed Components z w U Mean Wind Speed Mean Wind Direction v u y x Temporal Variation of Wind Speed U(t ) = U + u(t u ) U max = U + u max Wind Speed Fluctuating Component u(t) Mean Wind Speed U σ u σ u 0 1 2 3 4 5 Time (min) Turbulence Intensity I u σ u I u =, σ 2 u = S u (f)df U 19
Gust Factor and Peak factor Gust Factor: G U U max U + u G max U = = U U Peak Factor: g U u max σ uumax g U = max σ u G U = 1 + = 1 + g U = 1 + g U I U U U Turbulence Intensity (AIJ Rec.1993) 400 200 Height Z (m) 100 50 20 10 Terrain Category I II III V IV 0.1 0.2 I 0.5 u 20
Wind Speed Measurement Spatial & Temporal Variations of Wind Speed Height Z (m) 150 115 80 45 0 500 Horizontal Distance tu (m) 21
Peak Factor of Wind Speed Peak Factor g U Ave. 3.50 Mean Wind Speed U (m/s) Power Spectrum of Wind Speed Power Spectral Density fs U (f) / σ 2 u Typhoon Frontal Depression 22
Schematic view of turbulence Turbulence Spectrum Inertial Range (Energy cascade) f S u (f) Production (Instability of mean flow) Dissipation (Viscosity) Large Eddies log f Small Eddies Cook, 1983 23
Power Spectrum of Wind Speed (Fichtl-McVehill Model) f S u (f)) 4 f* = σ 2 u (1 + a f* f β ) 5 3β Gamma Function f L ux 4 β 1.5 (1/β ) β Γ(5/3β) f* * =, a = 1.5, b = U b β Γ(1/β) Γ(2/3β) β = 1 (Kaimal( Spectrum) β = 5/3 (Panofsky( Spectrum) β = 2 (Karman Spectrum) 5 f S u (f) f 3 (Kolmogorov s Hypothesis of Local Isotropy, Power of 5/3 Law in Inertial Subrange) Power Spectrum of Wind Speed Power Spectral Density fs U (f) / σ 2 u 24
Power Spectrum of Wind Speed (TTU) log{ f S u (f) /σ u 2 } log f (Tieleman, 1995) Power Spectrum of Wind Speed (Tieleman,1995) f S u (f) u * 2 = A f* γ (C + B f* f α ) β f Z f* * = U Z = Height α = 1, β = 5/3, γ = 1 (Blunt Model) α = 5/3, β =1, γ = 1 (Pointed Model) 25
Power Spectrum of Wind Speed (Typhoon 9119) Power Spectral Density fs U (f) / σ 2 u Before eye passing over : 90min After eye passing over : 110min Turbulence Scale Average Scale of Fluctuation L ux ux R uu (x)dx R uu = U R uu (τ )dτ uu (x)) : Spatial Correlation Coefficient of u(t) = u(r,t )u(r r + x,t x ) /σ 2 u uu (τ ) : Auto-correlation Coefficient of u(t) = u(r,t )u(r,t t + +τ ) /σ 2 u R uu 0 0 } Taylor s Hypothesis of Frozen Turbulence u(r, t ) u(r+ x, t ) x 26
Turbulence Scale Average Scale of Fluctuation Spatial Correlation Coefficient R uu (x) u(r,t ) u(r + x, t) x 1 L ux = 0 0 x L ux R uu (x)dx Turbulence Scale Wiener-Khintchine Relation S u (f) /σ 2 u = R uu (τ )exp( i 2π fτ )dτ S u (0) /σ 2 u = 2 2 R uu (τ )dτ = 2L ux / U 0 Turbulence Scale L ux = US u (0) / 2σ 2 u Isotropic Turbulence Field L ux = 2L2 uy = 2L2 uz (Near Ground: L ux 3L uz ) 27
Turbulence Scale (TTU) Garg,, et al., 1995 Turbulence Scale (AIJ Rec.1993) 200 Height Z (m) 100 50 20 10 Z ux = 100( ) α 30 L ux 20 50 100 200 500 L ux (m) 28
Correlation of Wind Speed (Root Coherence) Coherence f Z Reduced Frequency U missing!! Correlation of Wind Speed (Root Coherence) k 2 y y 2 +k 2 z z 2 Coherence exp ( f ) U 1 U 2 z U 1 (t)) = U 1 +u 1 (t) y U 2 (t)) = U 2 +u 2 (t) 29