Welcome to the second lecture on Atmospheric Turbulence where we learn about some of the important concepts used in wind energy. Some of the learning objectives are, after this lecture, you will be able to explain some statistical concepts used to quantify turbulence. Use frequently used keywords and explain some key aspects related to data analysis. Before we go through all the learning objectives, a reiteration of why turbulence is relevant for wind energy. Again, if we remember the wind turbine, when the wind blows where it has random variations in the wind velocity, it bends the blade, which causes stresses at the root of the blade and ultimately leads to fatigue damage. This is what we want to avoid, and hence, try to quantify turbulence and understand how it influences the wind turbine loads. How do we study turbulence? A very fundamental way and a theoretical way is to learn something about what is called as the Navier-Stokes Equation. It is an evolution equation which tells you about how the wind speed evolves with time as well as in space. We can then perform some mathematical tricks on the Navier-Stokes Equation with which we can obtain other evolution equations. But another way to understand and to study turbulence is statically, so using data as an example, we have here time series where the fluctuations of the air parcels with time are clearly shown. On the X axis, we have the time going from zero to 1,000 seconds. On the Y axis we have the wind speed. The first thing that we do is separate the fluctuations from the mean. That is, we take the mean of the wind speed for the recorded time and then we get this red line, and the remaining values are simply the fluctuation. A very important statistical concept important for wind energy is turbulence spectra. Which simply tells us about the distribution of the kinetic energy of wind with respect to the frequency and we will learn a little bit more about spectra the next slides. What are the important key words used in wind energy? First and foremost, we need to understand that when we study turbulence, some understanding of the coordinate system, the base coordinate system, the transformed coordinate system and vector and matrix computations is very important. Wind vector, as we all call it, is composed of three components. Here, you can see u, v and w. And simply, this wind vector has a magnitude and a length, a magnitude and a direction, so the direction is measured with respect to a certain reference axis which we call as the wind direction. Another important key word, and probably the most frequently used key word in wind energy, is turbulence intensity. So this is a statistic, which is used to quantify turbulence, and what information is required to estimate sample turbulence intensity is the mean wind speed and the standard deviation. The standard deviation is simply computed using the fluctuations of the time series. The next very important key word is the turbulence scales. Again as an example of this time series where you see random fluctuations, we can see that there are fluctuations which are occurring on a very large time scale and fluctuations, which are happening on a very small time scale. That is rapid fluctuations. Now when we decompose this random time series using a concept called a Fourier series, then we simply get the mean component out and then every, then the whole fluctuating time series can be decomposed into waves with a certain frequency and a wavelength. For example here, there are shown three waves with a particular frequency and a wavelength and every wavelength can be thought as a length scale or the inverse of it. Another important keyword is integral scale. Physically, what integral scale means, is that if we take a look at this time series, and if we simply take a look at the wind speed at 200 seconds, what can we say about the wind speed at 250 or 300 seconds and 100 seconds? Just by looking at the time series, so it's important to understand that there is something that is called the memory that the time series has, which is nothing but stored or quantified as integrals scale. So from the time series, we compute the auto-correlation function as a function of the time difference between the two observations, and then we get a certain curve. And the area inside this curve gives us the integral scale, which is given by this mathematical expression. Turbulence spectra is also a very important keyword that is frequently used. As mentioned before, it gives us the distribution of the kinetic energy and the wind with respect to the frequency. So on the x axis, you normally have the frequency or the wave number, which is the inverse of the wavelength. On the y axis, we normally have the pre-multiply spectrum, multiplied by the wave number, or the frequency. The pre-multiplication is done in order to get the peak in the spectre very clearly, otherwise it's very difficult to see where the peak energy lies on the spectrum. The red line show the spectrum of the u component, the blue line is for the v, and the green is for the w. As we can clearly see, these scales which is the turbulence land scale, which is given by the wave length or the wave number corresponding to the maximum energy in the spectra, is very different for different components. Now we talk a little bit about the data analysis. So things to remember, averaging periods is very important. Ten minute period is routinely used in wind energy, but it is recommended the 30 minute period be used for turbulence studies. And what are the considerations? It mainly depends on the Integral Time Scale. One thing to remember is larger the Integral Time Scale, larger should be the averaging period. Coming to sampling frequencies, what are the considerations? It also depends on the integral time scale. So normally, the sampling frequency should be much smaller than the Integral Time Scale. Otherwise the random errors in the estimated turbulence statistics is quite large, which normally we don't like. And now you can see a connection between the concepts that were introduced previously and the concepts used also in routine data analysi Why these concepts are important to understand. Then, removal of spikes. This is a very typical time series where, you not only see fluctuations, but huge jumps in the wind velocities on a very shorter time scale. Which normally can be thought about as spikes and this, we need to remove, otherwise we artificially increase the amount of energy at higher frequencies. There's something called as de-trending the time series, as you can see here, if the mean is not constant here, the fluctuations simply, the magnitude of the fluctuations or the magnitude of the wind speed increases with time and there is some kind of a trend here. Which is non-stationarity while analyzing data. And this is also something that we need to remove. So in this lecture, we have learned about some statistical concepts used in understanding turbulence. Some keywords used in wind energy, and some key concepts related to data analysis.