Wind turbulence

With the aim to characterize the wind action, Figure 3.2 reports the variability of wind power spectral density over dierent frequencies/timescales (Hoven 1957). From the gure two peaks can be noted in the low frequency zone and a third peak is evident in the high frequency area. The rst peak corresponds to about 4 days and represents the typical development of a storm. The second peak corresponds to the daily (day/night) wind period, representing variable breezes with a period of about 12 hours. The third peak at about 1-2 minutes is due to the atmospheric turbulence. In the center of the graph, the spectral power density displays a minimum within a time range of 10 minutes to 1 hour.

This area, called spectral gap, provides information for the experimental evaluation of the wind load: since the variance is small and constant, the speed of the wind can be calculated by averaging recordings over a period of 10 minutes to 1 hour. The gap allows to separate macro-meteorological (storm) and micro-meteorological eects (turbulence).

On the basis of the previous considerations it is possible to clearly distinguish the stationary components of the wind ¯Vu, ¯Vv and ¯Vw (which does not experience large changes from hour to hour) and the turbulent components vu, vv, vw. The directions u, v (horizontal) and w (vertical) are specied in Figure 3.3. Consequently it is possible to dene the wind velocity V in the three principals directions as the sum of the average value ( ¯V) obtained over a

MACRO METEROLOGY

MICRO METEROLOGY CYCLONIC

WEATHER

DIURNAL BREEZES

TURBULENCE

4 DAYS 12 HOURS 1 HOUR 2 MINUTES

SPECTRAL GAP

10 MINUTES

Fig. 3.2: Horizontal wind speed spectrum at Vrookhaven National Laboratory at about 100-m height in synoptic conditions (Hoven 1957).

u

v

w

Fig. 3.3: Principal directions u, v, w of wind velocity and of wind turbulence.

period of 10-60 and of the uctuating turbulent component:

V_u(z, t) = ¯V_u+ v_u(z, t) (3.1) Vv(z, t) = ¯Vv+ vv(z, t) (3.2)

V_w(z, t) = ¯V_w+ v_w(z, t) (3.3) The mean component characterizes the steady nature of wind. Typical prole of the mean-wind speed in the earth's atmospheric boundary layer is shown in Figure 3.4. The friction force derived from the earth's surface inuences the air movement with a consequent de-crease of the mean wind speed close to the ground. This eect is reduced with height up to a certain z called gradient height (δgr), known as the height of the atmospheric boundary layer. Above δgr the speed of the wind follows the isobars (Gradient velocity Vgr).

z

reference mean wind speed

x

mean-wind speed profile

V_ref

generic tall building gradient

wind velocity V_gr

gradient height

δ_gr

Fig. 3.4: a schematic representation of the mean wind speed of a generic tall building.

Two models can be used to describe the mean wind prole: the power law and the loga-rithmic law. For the sake of simplicity the two models are reported for the u component.

The rst model used is the power law, given by:

V¯_u(z) = ¯V_u(z_ref)( z

z_ref)^α (3.4)

where ¯V_u(z_ref) is the mean-wind speed at the reference height zref and α is the power law exponent which depends on the terrain type. The second model (logarithmic law) is described by:

V¯_u(z) = 1 κv∗lnz

z₀ (3.5)

where κ ≈ 0.4 is the Von Karman constant, z0 is the soil roughness and v∗=pτ₀/ρis the friction velocity depending on the surface shear stress τ0 and on the air density ρ.

Fluctuating components of wind velocity vu, vv and vw show gustiness of the wind. The turbulence is considered as a random process and needs to be studied with appropriate statistical models. In many cases it is justied the simplication that the wind turbulence is an ergodic and stationary Gaussian process and the rst two statistical moments allow to characterize it completely. However, many research studies are devoted the evaluation of the structural response by considering the wind non-Gaussianities (Blaise et al. 2016).

A physical description of the turbulence is given by the characterization of the following quantities (Borri and Pastò 2006):

1. turbulence intensities;

2. integral and time scales of turbulence;

3. power spectral density.

1. The turbulence intensities I, corresponding to the uctuating components of the wind velocity vu (longitudinal), vv (lateral), vw (vertical), can be computed as follows:

Iu(z) = σ_u(z)

vu (3.6)

I_v(z) = σv(z)

v_v (3.7)

I_w(z) = σ_w(z)

vw (3.8)

where σu(z), σv(z), σw(z) are the root mean square (RMS) of the uctuating com-ponents. The term σu(z) can be evaluated as (Holmes 1987):

σu(z) = E[u(z)²]^0.5 (3.9)

The lateral and vertical turbulence components are generally lower in magnitude than the corresponding longitudinal value and, for well-developed boundary-layer

winds, can be evaluated in a simple manner as a function of the friction velocity u∗

(Holmes 1987).

σv(z) ≈ 2.2u∗ (3.10)

σ_w(z) ≈ 1.3 − 1.4u∗ (3.11)

2. Obtaining the average sizes of eddies in a turbulent wind ow is important both for experimental and numerical simulations. The average sizes of eddies is described by a quantity known as the integral length scale (Simiu and Scanlan 1996). Corresponding to each of the uctuating components of the wind velocity (vu, vv, vw), there are three integral lengths scales corresponding to the coordinates (x, y, z) with a total of 9 scales. Considering the longitudinal component, the integral time scale can be computed as the area under the auto-correlation curve Rvu1,vu1 of the uctuating components of the velocity:

∆x, t1+ τ ) is the turbulence at the point 2. Considering the Taylor's hypothesis (Taylor 1938) that the ow disturbance vu travels with mean velocity ¯V, L^x_vu can be written:

Considering the wind ow in a certain time interval, the average time taken for the

uctuating components of the wind speed is described by a quantity known as the integral time scale:

T_vu(z) = Z ∞

0

R_vu(z, τ )dτ (3.14)

3. The turbulent nature of wind ow creates eddies of variable sizes which overlap with dierent frequencies and energies. The turbulence spectrum gives information about the vortex frequency content. Several empirical equations have also been suggested by dierent researchers for computing the longitudinal and transverse components of the wind velocity of which some examples are given (Borri and Pastò 2006):

ˆ Von Karman (longitudinal)

ˆ Lumley and Panofsky (transverse) nS_vw,vw(z, n)

In aerodynamics, a blu body is the one which has a length in the ow direction close or equal to that perpendicular to the ow direction. Most of the man-made structures are blu bodies. A principal feature of the ow around blu bodies structures is the formation of large vortices in their wakes, which strongly aect the wind loading and the structural response. The investigation, the analysis and the understanding of blu bodies aerodynamics are complex since the ow around blu bodies depends not only on the shape of the body, but on many factors, such as the Reynolds number and the characteristics of the turbulence (Irwin 2008). The Reynolds number is the ratio of inertial uid forces to viscous uid forces: