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Reynolds averaging

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Reynolds averaging refers to the process of averaging a variable or an equation in time. Let \Phi be any dependent variable that varies in time. This variable can be decomposed into a fluctuating part, \Phi' and an average part \overline{\Phi} in the following way:

\overline{\Phi} \equiv \frac{1}{T} \int_T \Phi(t) dt
(1)
\Phi' \equiv \Phi - \overline{\Phi}

Where T is a long enough time to average out the fluctuations in \Phi.

Reynolds averaging is often used in fluid dynamics to separate turbulent fluctuations from the mean-flow. The term Reynolds averaging originates from Osborne Reynolds, who was the first to use this type of averaging in fluid dynamics.

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