When a surface wave is superimposed over turbulent current, a periodic velocity fluctuation is observed and the conventional Reynolds decomposition of the velocity field is not satisfactory and phase-averaging of the velocity signal needs to perform why?
Why phase averaging is necessary for such type of flow?
I am not sure if you are talking about the validity of decomposition, rather I suppose you are talking about the validity of the statistical time-averaging for non-stationary process.
The Reynolds averaging, when defined in terms of the time-averaging, produces a statistical variable that is steady. For statistically non-stationary turbulence this averaginh has no sense and an ensemble averaging is considered. As a result the statistically averaged variable is still time-dependent and in the Reynolds decomposition the terms like
Dear Professor Denaro
Thank you very much for your explanation. Yes, I am not talking about decomposition's validity, and I am fully convinced of your explanation. However, I have one more issue on my question, for e.g., the decomposition of velocity, u = u1+u2+u3, where u1, u2, and u3 are the averaged-velocity, turbulent fluctuation, and periodic wave velocity respectively. For the calculation of turbulence statistics of such flow, we have to remove wave velocity from the turbulence by ensemble phase-averaging. My concern is why this is necessary to do the phase-averaging. We can easily compute the time-averaged turbulence statistics with this wave component.
I try to better detail the topic. Consider the standard decomposition based on time averaging
u(x,t) =(x) + u'(x,t)
where
(x)= lim T->Inf (1/T) Int[0,T] u(x,t) dt
you can also use it if your periodic wave velocity is such to produce a statistically steady field. A typical example would be the periodic motion of a piston in a cylinder. But make it really sense to compute the global time averaging for a process depending on an external time-dependent forcing? Of course you want to describe the statistical behavior over the period of this forcing.
To do that the decomposition should be performed according to
u(x,t) =(x,t) + u'(x,t)
where now an ensemble averaging makes sense for a time-depended averaged velocity
(x,t)= lim N->Inf (1/N) Sum[n] un(x,t) dt
In conclusion, at any time t you compute the result of the averaging of N different realizations.
As a consequence,
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