The comments above are spot on. You will need to first obtain a better understanding of what you would like to know, before coming back and readdressing your question. But to try and help in some way, I will attempt to throw some light on what I think you are asking.
Firstly the first question does not make too much sense but I will come back to this...
I will assume you have gathered some time-series data from your "acoustic Doppler current profiler" - whatever that may be. For the second part of your question I would firstly adopt structure function analysis. This technique provides a method of quantifying time variability without the problem of aliasing, or windowing, that are encountered using the traditional FFT technique. Potentially it is able to provide information on the nature of the process that causes variability in your flow. The method is mainly concerned with the categorization of underlying noise processes and the identification of correlation time-scales, that is the time-scale that is associated with actual flow dynamics and not turbulent noise.
Once you have established that there is in fact some flow variability (large scale variability of the flow) and the oscillatory time-series is not just down to noise alone, you can then look at adopting a FFT. Depending on the nature of the variability, this maybe sufficient to establish any underlying periods/quasi-periods. If you find that the flow is showing very non-linear behaviour, but that the structure function did reveal underlying flow you can then move to wavelet analysis.
Although FFTs can, to some extent detect and quantify such transient behaviour, it is far from ideal for such purposes. Wavelet analysis has been developed to overcome these difficulties. The wavelet transform is well suited to detect such transient periodic fluctuations as it can concentrate on windowed sections of the time-series data. It is powerful because it not only tells us which frequencies exist in the time-series, but also when they exist, enabling us to observe whether periodic behaviour varies in time.
Now, your first question. Do you mean angular velocity or the mean time-scale associated with the underlying turbulence? If it is the second case, the structure function will provide this information and is perfect in this case. You should also establish the turbulent length scales associated with this flow for your relevant Reynolds number, this will help you get a feel for the result provided by the structure function, and can be used to provide a good 'sanity test'. There are many books available that you will be able to look up the relevant formulas for pipe flow (another assumption!).
I hope this helps.