What is Phase Averaged Turbulence Intensity ? How to calculate it from velocity(u(t),v(t)) data?

Suppose you have a periodic signal (for instance pressure signal). You can then define a period duration D. Then divide this duration into n equal intervals of time. At any given time t, you will be able to determine in which sub-interval i you are, i.e. :

A + B*D + (i-1)*D/n < t < A + B*D + i*D/n

with A a constant and B the number of periods since the beginning of the experiment or the calculation.

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For each sub-interval 1 <= i <= n :

*** compute classical statistics Mi(u)=U_i, Mi(v)=V_i, Mi(u^2) and Mi(v^2) (M means mean, _i for phase i)

*** phase fluctuations u'_i and v'_i are given by :

u'_i=sqrt(Mi(u^2)-U_i^2)

v'_i=sqrt(Mi(v^2)-V_i^2)

*** phase averaged turbulent intensity may be defined for instance by the ratio :

sqrt(u'_i^2+v'_i^2)/sqrt(U_i^2+V_i^2)

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In practice, you have to store arrays of size n for the sums of u, v, u^2 and v^2 and for the number of samples. At each time, identify the phase number i, and then add instantaneous u, v, u^2 and v^2 data to the i th element of the related arrays and increase by 1 the number of samples for the i th phase. At the end, for each i, divide the element i by the number of samples of the i th phase to obtain phase averaged statistics.

Hope this helps

Thank you for your explanation. I use almost the same technique as you described. My data is u,v,w,p from turbulent flow. Since 'u' is turbulent velocity, it is sum of periodic and fluctuating parts. To do the phase averaging of u,v,w, one of pressure signal is used as phase trigger. In this case, there is no problem. My problem is how to calculate the phase-averaged turbulence intensity mentioned at the following article.

'The falpping shear layer formed by flow separation from the forward corner of a square cylinder' by D.A.Lyn and W.Rodi, J.Fluid Mech. (1994) vol. 267, pp353-376. At pp355, "This average overall turbulence intensity excluding the periodic contribution may be considered in the context of the triple decomposition as the average over all phases of the phase-averaged turbulence intensity, denoted here as u0' and should be distinguished from the time-averaged turbulence intensity"