1st PartWind speeds and analysis of turbulent flow in a

boundary layer

October 2002Júlíus Sólnes

Wind effects on structuresStatic and dynamic response of structures to wind loading

U z t U z u x y z t

U zT

U z t dt T

U z ttU z s ds moving average

t

t T

t

( , ) ( ) ( , , ; )

( ) ( , ) , min

( , ) ( , ) ,

= +

= =

=

+

∫

∫

1 10

1

0

Standard wind speed in m/s ismeasured at 10 m height andaveraged over 10 minutes. Amoving average gives a smoothedslowly varying mean value

Wind speed measurementsKeilisnes, SW Iceland

Measurements and PhotoJónas Snæbjörnsson

zR=10 m

Wind speeds at Keilisnes, SW IcelandThe wind speed U(zR,t), the 10 min. average U(zR)

and the moving average U(zR,t)

0 50 100 150 200 250 300 350 400 450 5008

10

12

14

16

18

20

22

24

Time in seconds

U(zR)

P vref,0-basic value as presented by appropriatewind maps in a National Annex (NA)

PCDIR-direction factor taken as 1,0 unlessotherwise specified in the NA

PCTEMP-temporal (seasonal) factor taken as 1,0unless otherwise specified in the NA

PCALT-altitude factor taken as 1,0 unless otherwisespecified in the NA (will probably be dropped inthe final version of Eurocode 1)

Reference windThe 10 min. average wind speed at 10 metre height (U(zR) m/s)is the fundamental design parameter called vref,0 in Eurocode 1

vref=CDIRCTEMPCALTvref,0

V T V K TKref ref

e e

n

( ) (50) log [ log ( / )],

=− − −

+

1 1 1 11 3 902

1

1

PNational authorities have to prepare wind maps(zonation) with maximum wind speeds V m/s

PUse extreme value statistics< P[V#v]=exp(-exp(-a(v-U))) , the Gumbel distribution with

the attraction coefficients 1/a and U-equal to the “mode ofthe data”

< V(T)=U-1/a(loge(-loge(1-1/T))) is the maximum wind speedwith a return period of T years

< Usually V(50)=U+3,902/a-the 50 year wind< Eurocode 1 suggests the formula below for other periods T

Distribution of maximumwind speeds

For convenience vref,0 is given by the random variable V m/s

K1=0,2 and n=0,5 if not otherwise specified in the NA

Map showing the modeU m/s of the gradientwind speed VG m/s

European windregime

U x y z tV x y z tW x y z

U z t u x y z tv x y z tz x y z t

or

U x t U x t u x ti i i i i

( , , ; )( , , ; )( , , ; )

( , ) ( , , ; )( , , ; )( , , ; )

( ; ) ( , ) ( ; )

=

+

= +

00

3

[ ]

[ ]

E U x t U x t U xandE u x t

i i i i

i i

( ; ) ( ; ) ( )

( ; )

= ≈

=

3 3

0

Wind as a boundary layer air flowMean wind speed plus a turbulent component

Buildingfacade

The x-axis (1-axis) is themain direction of the wind

σσ τ ττ σ ττ τ σ

ττ τ ττ τ ττ τ τ

=

= =

x xy xz

yx y yz

yx yz z

ij

11 12 13

21 22 23

31 32 33

x xkk kkk

==∑

1

3

uuxj kj

k, =

∂∂

Flow stresses in tensornotation

In hydrodynamics, the stresses Jij often signify the stressvelocities Jij=MJij/Mt

z,x3,3

y,x2,2

JyxJxy

Fzz=Fz

x,x1,1

i,j,k0{1,2,3}

The summationconvention of Einstein.If an index is twofold itmeans a summation

Einstein used thisshort hand term forpartial derivatives

τ µ γ µ ∂∂

τ µ ρνzx zxuz

u u= ⋅ = ⋅ = = ⋅ = ⋅31 1 3 1 3, ,

Kinematic and dynamic viscosity

D=1,25 Kg/m3, air density:=1,81 g/(cmqs), dynamicviscosity of air<=:/D, kinematic viscosity ofair

z,x3,3

x,x1,1)z

)u

(zx

Jzx

{ }& , , ,, , ,U U U p i ki k i k i ik k+ ⋅ = − + ∈1 1 1 2 3ρ ρ

τ

τ τ∂∂

∂∂

∂∂

∂∂

τ ∂τ∂

∂τ∂

∂τ∂

ij xy

i ji

j

k i ki i i

ik ki i i

U Ux

U U U Ux

U Ux

U Ux

x x x

=

=

⋅ = + +

= + +

,

,

, ( )

11

22

33

1

1

2

2

3

3

Ui={U1,U2,U3}is the wind speed (m/s)p(xi) is the air pressureJij is the boundary flowshear stress(Jii=Fii=0)

Fundamental equationsgoverning viscous flow

The Navier-Stokes equations

Tensor Notation:

[ ]E U U U E pi k i k i ik k&

, , ,+ ⋅ = − −

1 1ρ ρ

τ

[ ] { }∂∂ ρ

∂∂

Ut

U U pE u ux

i kik i k i

i k

k

( )( )

, , ,, ,= + ⋅ = − +− ⋅

∈0 1 1 2 3

{ }& , , ,, , ,U U U p i ki k i k i ik k+ ⋅ = − + ∈1 1 1 2 3ρ ρ

τ

Taking the expectation of both sides gives theReynold’s equation:

The Reynold’s equationE[Ui(xi;t)]=Ui(xi) , E[ui(xi;t)]=0

and disregarding the viscos shear stress Jik

Compare with the original Navier-Stokes equation

Measurements show that:J12=-DE[u1u2].0J23=-DE[u2u3].0J13=-DE[u1u3]…0 , that is,only the turbulencecomponent in the direction ofthe mean wind speed u1 andthe vertical component u3seem to be correlated

The Reynold’s shear stressThe last term of the Reynold’s equation can beinterpreted as a shear stress induced by the mixingof the turbulence components: Jij=-DE[uiuj]

U1(x3)

x3

x1

x2

u1(xi;t)

Moreover: E[u1u3]#0 for higher wind speeds withoutmajor temperature influences

[ ]κτ

ρ= = −13

21 32

U

E u u

U

[ ]u E u u U* = = − = ⋅τρ

κ131 3

The surface roughness coefficient6 is interpreted as a dimensionlessReynold’s shear stress and can bemeasured through E[u1u3]

The roughness coefficient 6The Reynold’s stress -DE[u1u3] gives an indication of the

surface roughness; a rougher surface increases thecorrelation between the two components u1 and u3

The shear velocity orfrictional velocity isdirectly related to thesquare root of theroughness coefficienttimes the meanreference velocity

τ ρ∂∂xz KUz

=

K k zu k z Ua a= = ⋅* κ

τ ρκxz U=2

The shear stress of the turbubulent flow isgiven by

A simple model for turbulent flowDue to the surface roughness and the vortices produced by u1 andu3, the mean velocity decreases nearer the surface where it iszero.This condition can be described by the K-model turbulencetheory of von Karman. For convenience let x=x1 and z=x3

K is the eddy viscosity dependent on the size and velocity ofthe vortices

where ka is the von Karman coefficient. Measurementsshow it to be approximately constant equal to 0,4. Forcomparison, the Reynold’s shear stress is written as

[ ]

τ ρκ ρ κ τ

κ

κ

κ

xz R a R xz

a

R

R ar r

Rr

ra R

U k z U dUdz

dUkU dz

z

U zU k

zz

k zz

c z z z

z zk

kk

E u u

U

= = ⋅ =

=

=

=

= ≥

=

= =

−

2

0 00

01 31

0 4

( ) ln ln ( ) ,

exp ,,

Wind velocity profilesThe two different versions of the shear stress Jxz are put equal

10 min. averagewind speed m/s

Height in metresabove ground

UR

U(z)

z0 is a constant of integration, the roughness length. In Eurocode 1,kr is called the terrain factor and cr(z) the roughness coefficient.

Terrain category kr z0

[m]zmin

[m]g

I Rough open sea. Lakeshore with at least5 km fetch upwind and smooth flat countrywithout obstacles.

0,17 0,01 2 0,13

II Farmlands with boundary hedges,occasional small farm structures, housesor trees.

0,19 0,05 4 0,26

III Suburban or industrial areas andpermanent forests. 0,22 0,3 8 0,37

IV Urban areas in which at least 15% of thesurface is covered with buildings and theiraverage height exceeds 15m.

0,24 1 16 0,46

The above parameters are fitted to available wind measurement data. The coefficient g is used for specialanalysis of inline response of structures.

Eurocode 1: Terrain categories

Wind measurements: KeilisnesJónas Þór Snæbjörnsson

0 50 100 150 200 250 300 350 4000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

MEoALVINDÁTT (E)

Wind Measurements: ReykjavíkBústaðavegur: Jónas Þór Snæbjörnsson

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

100

200

300

400

500

600

700

kr

-14 -12 -10 -8 -6 -4 -2 0 20

100

200

300

400

500

600

700

ln(zo)

cfor

s fors for

t =<

+ ⋅ ⋅ < ≤+ ⋅ >

1 0 051 2 0 05 0 31 0 6 0 3

ΦΦ Φ

Φ

,, ,

, ,

The reference wind speedis multiplied by thetopography coefficient ctqvref

Topography Coefficient ctWind speed increases when blowing over isolated hills and

escarpments (not undulating and mountainous regions)

Wind

M

M

Situation I

Situation IIWind

M is the slope of thehill/escarpments is the hill parameter to beobtained from graphs

LL forH fore =

< <

≥

0 05 0 3

0 30 3

, ,

,,

Φ

Φ

The hill factor sSituation I: cliffs and escarpments

H

LM

Downwind slope <0,05

-x +x

-x

z

z/Le

2,01,51,0

0,5

0,2

0,1

0,60

0,80

up wind down wind x/Le

0,0 0,5 1,0 1,5 2,0-0,5-1,0-1,5 2,5The effective slope length:

Building on an upwind slope: The factor s

LL forH fore =

< <

≥

0 05 0 3

0 30 3

, ,

,,

Φ

Φ

The hill factor sSituation II: hills and ridges

H

LM Downwind slope >0,05

L

-x +x

z

x

Building on a hill

The effective slope length:

z/Le

2,01,51,0

0,5

0,2

0,1

0,60

0,8

up wind down wind x/Le

0,0 0,5 1,0 1,5 2,0-0,5-1,0-1,5 2,5

The factor s

c zk z

zfor z z m

c z for z zr

r

r

( )ln min

min min

=

≤ ≤

<

0

200

After applying the topography coefficient ct andcorrecting for the appropriate height aboveground z m, the mean wind speed is given by

Modified reference wind speed10 min. average wind speed: Eurocode 1

vm(z)=cr(z)ctvref

where cr(z) is the roughness coefficient

End