time averaging and Reynolds decomposition

time average generic variable $\varphi$

$$ \overline{\varphi}(\mathbf{x}) = \underset{T\rightarrow\infty}{\mathrm{lim}} \frac{1}{T}\int\limits_{t_0}^{t_0 + T} \varphi (\mathbf{x},t)\mathrm{d}t $$

only sensible for statistically stationary flows

Reynolds decomposition

$$ \varphi(\mathbf{x},t) = \overline{\varphi}(\mathbf{x}) + \varphi^\prime(\mathbf{x},t) $$

introduced by Osborne Reynolds in 1895

time average of Reynolds decomposition

\begin{aligned} \overline{\varphi}(\mathbf{x}) &= \underset{T\rightarrow\infty}{\mathrm{lim}} \frac{1}{T}\int\limits_{t_0}^{t_0 + T} \left[\overline{\varphi}(\mathbf{x}) + \varphi^\prime(\mathbf{x},t)\right]\mathrm{d}t\\ \overline{\varphi}(\mathbf{x}) &= \overline{\varphi}(\mathbf{x}) + \underset{T\rightarrow\infty}{\mathrm{lim}} \frac{1}{T}\int\limits_{t_0}^{t_0 + T} \varphi^\prime(\mathbf{x},t)\mathrm{d}t \end{aligned}

time averaged fluctuations are zero

$$ \overline{\varphi^\prime} = \underset{T\rightarrow\infty}{\mathrm{lim}} \frac{1}{T}\int\limits_{t_0}^{t_0 + T} \varphi^\prime(\mathbf{x},t)\mathrm{d}t = 0 $$

averaging twice has no effect $$ \overline{\overline{\varphi}} = \overline{\varphi} $$

time average of sum

\begin{aligned} \overline{\varphi + \psi} &= \overline{\overline{\varphi}+\varphi^\prime + \overline{\psi} + \psi^\prime}\\ &= \overline{\overline{\varphi}} + \overline{\varphi^\prime} + \overline{\overline{\psi}} + \overline{\psi^\prime}\\ &= \overline{\varphi} + \overline{\psi} \end{aligned}

equals the sum of time averages

time average of product $\overline{\varphi}\psi^\prime$

$$ \overline{\overline{\varphi}\psi^\prime} = \overline{\varphi} \overline{\psi^\prime} = 0 $$

time average of product $\varphi\psi$

\begin{aligned} \overline{\varphi \psi} &= \overline{(\overline{\varphi}+\varphi^\prime) (\overline{\psi} + \psi^\prime)}\\ &= \overline{\overline{\varphi}\overline{\psi}} + \overline{\varphi^\prime\overline{\psi}} + \overline{\overline{\varphi}\psi^\prime} + \overline{\varphi^\prime\psi^\prime}\\ &= \overline{\varphi}\overline{\psi} + \overline{\varphi^\prime\psi^\prime} \end{aligned}

time average and spatial derivative

\begin{aligned} \overline{\partial_s \varphi} &= \underset{T\rightarrow\infty}{\mathrm{lim}} \frac{1}{T}\int\limits_{t_0}^{t_0 + T} \partial_s\varphi(s,t)\mathrm{d}t\\ &= \partial_s \left(\underset{T\rightarrow\infty}{\mathrm{lim}} \frac{1}{T}\int\limits_{t_0}^{t_0 + T} \varphi(s,t)\mathrm{d}t\right)\\ &= \partial_s \overline{\varphi} \end{aligned}

are commutative

time average of time derivative

\begin{aligned} &\underset{T\rightarrow\infty}{\mathrm{lim}} \frac{1}{T}\int\limits_{t_0}^{t_0 + T} \partial_t\varphi (\mathbf{x},t)\mathrm{d}t =\dots\\ &\dots\underset{T\rightarrow\infty}{\mathrm{lim}} \frac{1}{T}\left(\varphi (\mathbf{x},t_0+T)-\varphi (\mathbf{x},t_0)\right) \\ &=0 \end{aligned}

for finite $\varphi$

vectorial notation

$$ \boldsymbol{\varphi}(\mathbf{x},t) = \overline{\boldsymbol{\varphi}}(\mathbf{x}) + \boldsymbol{\varphi}^\prime (\mathbf{x},t) $$

time average of divergence operator

\begin{aligned} \overline{\nabla\cdot\boldsymbol{\varphi}} &= \overline{\nabla\cdot(\overline{\boldsymbol{\varphi}} + \boldsymbol{\varphi}^\prime)}\\ &= \nabla\cdot\overline{(\overline{\boldsymbol{\varphi}} + \boldsymbol{\varphi}^\prime)}\\ &= \nabla\cdot\overline{\boldsymbol{\varphi}} \end{aligned}

equals the divergence of the time average

time average of Laplace operator

\begin{aligned} \overline{\nabla\cdot\nabla\boldsymbol{\varphi}} &= \overline{\nabla\cdot\nabla(\overline{\boldsymbol{\varphi}} + \boldsymbol{\varphi}^\prime)}\\ &= \nabla\cdot\nabla\overline{(\overline{\boldsymbol{\varphi}} + \boldsymbol{\varphi}^\prime)}\\ &= \nabla\cdot\nabla\overline{\boldsymbol{\varphi}} \end{aligned}

equals the Laplace of the time average

assumptions for non-stationary flows

separation of time scales $T_1\ll T \ll T_2$

assumption not always fulfilled

time average of time derivative

\begin{aligned} \frac{1}{T}\int\limits_{t_0}^{t_0 + T} \partial_t\varphi (\mathbf{x},t)\mathrm{d}t &= \frac{1}{T}\int\limits_{t_0}^{t_0 + T} \partial_t(\overline{\varphi} (\mathbf{x},t) + \varphi^\prime(\mathbf{x},t))\mathrm{d}t\\ &=\frac{1}{T}\left[\overline{\varphi} (\mathbf{x},t_0+T)-\overline{\varphi} (\mathbf{x},t_0) + \dots\right.\\ &\dots\left. \varphi^\prime (\mathbf{x},t_0+T)-\varphi^\prime (\mathbf{x},t_0)\right] \\ \end{aligned}

if $|\overline{\varphi}| \gg |\varphi^\prime|$ $\rightarrow$ $\overline{\partial_t \varphi} \approx \partial_t \overline{\varphi}$

correlation between mean and fluctuation

$$ \overline{\overline{\varphi}\varphi^\prime} := 0 $$

true only if $\overline{\varphi}$ and $\varphi^\prime$ uncorrelated

mean transport of a passive scalar

mean transport of mass and momentum

summary

• transport equations for mean quantities
• additional, unclosed turbulent diffusion
• time varying mean flow with multiple
  questionable assumptions
• some transport equations were not covered