Variances and co. (turbulence intensity offshore)

[post updated on 2022-02-28 with a sensitivity analysis om time-averaging (10, 20, 30 and 40 min)]

Turbulence Intensity (TI)… a recurrent topic in offshore wind! If you work in the field, you have surely come across it being discussed in journal- and conference articles, workshop proceedings and great deal of documentation. It is defined, for a 10-minute sample, as the ratio between the standard deviation and the mean of the horizontal wind speed U :

$$ TI = \frac{\sigma_U}{\overline{U}} $$

In effect, the freestream TI is a key input parameter for many analyses. For instance, for a given hub height wind speed bin, the upper 90-percent quantile of the TI is, combined to the wake-added turbulence, used for computing the microscale stochastic wind field used for computing aeroelastic loads on the turbine and the suppport structure. You can read about this in the following paper: “Modelling turbulence intensity within a large offshore wind farm” (Argyle et al., 2018).

If we recall that, in a simplified 2D model of the flow, the wind field is defined as:

$$U(z,t)=\overline{U(z)}+u'(z,t)$$

$$W(z,t)=w'(z,t)$$

Two covariance terms emerge:

$$\overline{u’u’}=\sigma_U^2$$

$$-\overline{u’w’}=u_*^2 \text{ with } u_* \text{ the friction velocity}$$

It turns out that the entire 3D microscale wind speed spectra can be almost entirely parametrised using and or . And, since is much easier to measure than (the former requires just a cup anemometer and some simple logger, while the later requires a sonic anemometer and more complex processing), TI became the go-to measurement for wind energy practitioners.

There are many sites offshore where there are no cup anemometer measurements, and in such situations practitioners usually rely on measurements from other, similar places. See for instance what we at C2Wind did for the Thor Offshore wind farm tender in Denmark, in Appendix B of this report.

That is though not what Section 6.4.3.3 of the IEC 61400-3-1 standard recommend at the first place. Instead, they propose to use the following method for estimating the mean TI (they later propose how to estimate the p90 based on this mean value):

First, assume a neutral wind profile:

$$\overline{U}(z)=\frac{u_{*0}}{\kappa}\ln(\frac{z}{z_0})$$

Where the surface friction velocity is linked to the aerodynamic roughness length by the Charnock relationship with a constant Charnock parameter :

$$z_0=\alpha_{Ch}\frac{u_{*0}^2}{g}$$

Then, assume that and that with .

And that’s it. The equations can be solved iteratively, using solely . But this doesn’t work well: it leads to TI values that are too large compared with what is observed. See for instance below at the IJmuiden met mast in the North Sea (which I have used in many of my previous posts), where I am also showing the effect of using alternative values for A and .

So, why is that? In this post, I will discuss three items which I think can be of interest to practitioners:

1. The dependency of and , and their proportionality, with the elevation;
2. The empirical evidence for the Charnock relationship and its relation to the log law wind profile;
3. A simple model for turbulence intensity offshore, for neutral conditions.

I have used the following measurement data sources:

• A range of air-sea interactions studies, reported in journal papers and technical reports, which I have compiled in a spreadsheet “Air-Sea experiments” at this LINK;
• The IJmuiden met mast data available via the “Wind op Zee” website: LINK;
• The M2 met mast data available via “www.winddata.com” LINK and documented in Section 3.1 of “Thor Offshore Wind Farm – Description of measurement datasets” LINK.
• The Vindeby SMW met mast data via “www.winddata.com” LINK (for the connoisseurs, these come from the RASEX campaign).

For each dataset, mast- and nearby wind farm disturbance has been accounted for.

The not-so-constant fluxes layer

A reccurent discussion topic in various papers and studies is the validity of the assumption that in the surface layer the momentum fluxes are constant. This because it is sufficient to make this assumption to derive the log law, via Prandtl’s mixing length hypothesis. See two recent publications:

• “An Evaluation of the Constant Flux Layer in the Atmospheric Flow Above the Wavy Air-Sea Interface” (Ortiz-Suslow et al., 2021)
• “Turbulence in a coastal environment: the case of Vindeby” (Putri et al.)

Yet, it is worth noting that this assumption is not necessary for deriving the log law. This is explained in “The Logarithmic Wind Profile” (Tennekes, 1973), and nicely discussed in (Ortiz-Suslow et al., 2021). It must admit that I did not know the theoretical reasons for this fact which all practitioners know from measurements; see onshore example:

• at Cabauw in Chapter 3 of “On Wind and Roughness over Land” (Verkaik, 2006),
• at Høvsøre (DK) in Figure 5-6 of “Measuring and modelling of the wind on the scale of tall wind turbines” (Floors, 2013).

See below what the data look like for a range of wind speed bins at IJmuiden: on the second plot in log-log scale a power law has been fitted to the data.

And here the same plot for the M2 met mast in the Danish North Sea:

The power-law exponent are similar for both datasets, see below.

And finally, see the results from the Vindeby (RASEX) dataset: it includes both momentum flux and variance.

The power-law exponent values are similar to the ones derived from IJmuiden and M2:

Yet another look at the Charnock relationship

RASEX, HEXMAX, ASGAMAGE, … over the past decades several field measurement campaigns have provided conccurent value of friction velocity, sea state and wind speed. I have compiled the results from most of them in the spreadsheet “Air-Sea experiments” at this LINK; see also below where the measurements of friction velocity are showed with blue markers, while the corresponding values derived from the Charnock relationship with a constant are showed in red. These measurement datasets were gathered at different elevations, for different averaging periods, and with different sea state conditions. Yet, they show somewhat consistent features:

• The values of decrease with the elevation: this is in particular true for the [TÜRK09] data, gathered at the FINO1 met mast (i.e. above ca 30 mMSL);
• For large seastates (wind speeds larger than ~15-20m/s), the measured values are larger than the ones obtained with .

Proportionality between friction velocity and standard deviation?

Looking a neutral atmospheric conditions, at the IJmuiden met mast, the proportionality factor A between the friction velocity and the standard deviation () is close to the canonical value of 2.5:

The same value of approx. 2.5 is found when looking at the, much lower, measurements at the SMW mast at Vindeby:

One thing to note is that the averaging period matters when looking at the variance. See below the measurements from IJmuiden: the comparison of TI values for 600s (10min) and 1800s (30min) averages makes a difference of about 1%! This is because of the mesoscale (slow-moving turbulence) described in papers such as “Full-Scale Spectrum of Boundary-Layer Winds” (Larsén et al., 2016). This is why, in the plot above, I have adjusted the measurement of from RASEX (30-minute averages).

Now if I take A =2.4, and try to derive (most use z = 10 m) as reference elevation, this is what I obtain for the three datasets: the values are showed with diamonds for the two bottom plots. The top plot shows the comparison with the measured friction velocities at Vindeby.

Clearly, something isn’t working for . The data should lie on the dashed lined (power law) we saw earlier. Instead, these 10m- seem too large. The differences are larger than the variations reported in the A parameter (typically between 2.3 and 2.5).

I am guessing that this is caused by the values of being recorded, in the Air-Sea experiments mentioned in the spreadsheet, for longer time intervals than 10 min. RASEX was 30min, HEXMAX 45min, etc. See below how the comparison looks like if I increase all TI values by 1%, to account for the change from 10 to 30min averages:

I am thinking that this difference in time averaging periods may be the reason for the difference between the HEXMAX (45min) and ASGAMAGE (18min) results: the later shows smaller values of friction velocities but both campaigns where carried out at the same place.

Another example of this time-averaging effect is shown below with the IJmuiden data (sonic anemometer): the longer the averaging period, the larger the flux.

Some more plots below with the illustration of the sensitivity of to the averaging period. It follows that the ratio between the two fluxes is constant only for large values of friction velocities, but is much larger for smaller values.

A simple model for turbulence intensity offshore in neutral conditions

In a first approach, and focusing on neutral conditions, we can use the learnings from the analyses presented below and assume that:

• the values of friction velocities from the Charnock relationship is taken valid at 10m;
• the longitudinal standard deviation is proportional to the friction velocity, and this proportionality factor is approximately 2.4 and is constant for large wind speeds, and with elevation (at least up to approx. 90 mASL);
• the friction velocity decreases according to a power law, which for wind speeds of interest to the practitioner (above 4-5 m/s) is constant and equal to ;
• The TI values are corrected by about 1% to account for the shorter time averaging (10min).

With all that, I can compute TI vs wind speed at several elevations using only one wind speed time series as input, see below. Note: in these plots I have added the results if I use the COARE 3.5 relationship from (Edson et al. 2013), where the Charnock parameter is made dependent on the sea state.

How about stable and convective conditions?

When looking at convective or stable conditions, things get trickier (like always), because the physics are just different (large scale motions in convective situations), and because the surface wind varies (in relation to the neutral wind), therefore this affects sea state too. Yet, the measurements show approximately the same thing, that is: decreasing fluxes with elevation, once a given wind-speed (sea state actually) threshold is passed.

Using simple Monin-Obukhov theory, it is therefore possible, only using one time series as input to compute the fluxes and derive the following, indicative model results:

Conclusions and wrap-up

Using the Monin-Obukhov theory and some simple Charnock relationships, combined with some pragmatic power-law scaling of the friction velocity with elevation and a simple proportionality relationship between variance and momentum flux, we have derived some simple models for assessing turbulence intensity offshore.

Questions remain:

• If the friction velocity varies with elevation, how to relate the value of friction velocity in the log law and the one from the air-sea experiments?
  • In particular, I notice that if I choose to set the reference elevation of to ~4-5 m (instead of 10m)m, then I do not need to “adjust” the TI. Most of these air-sea experiments weren’t carried out at 10m, often the sensor elevatio was lower (see the Air-Sea spreadsheet).
• What can be learned from the situations for which the variance increases with elevation? These seem to be present for small surface wind speeds. As showed in the figure below, this does not seem to be related to the relative importance of Wind-Sea and Swell (obtained here via a hindcast model).

Comments/questions are welcome,

Rémi