# 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 typical 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.