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\begin{document}
\title{\Large{PIV Post-Processing Equations}}
\author{\normalsize{MA Wei}}
\date{\today}
\maketitle
%\begin{abstract}
%\end{abstract}
\section{Velocity}
$u_n; v_n; w_n; \vert{v}\vert_n= \sqrt{u_n^2+v_n^2+w_n^2}$
\section{Ensemble average of velocity}
velocity = mean velocity + fluctuation velocity
\begin{equation}
u_n=U+{u_n}^{'};
v_n=V+{v_n}^{'};
w_n=W+{w_n}^{'};
\vert{v}\vert_n =\vert{V}\vert+\vert{v}\vert_n^{'}
\end{equation}
\begin{equation}
U=\dfrac{1}{N}\displaystyle \sum_{n=1}^N u_n;
V=\dfrac{1}{N}\displaystyle \sum_{n=1}^N v_n;
W=\dfrac{1}{N}\displaystyle \sum_{n=1}^N w_n
\end{equation}
\begin{equation}
\begin{array}{rcl}
\vert{V}\vert & = & \dfrac{1}{N}\displaystyle \sum_{n=1}^N\overrightarrow{v}_n
= \dfrac{1}{N}\displaystyle \sum_{n=1}^N\left( \sqrt{u_n^2+v_n^2+w_n^2}\right)
= \sqrt{ \left({\dfrac{1}{N}\displaystyle \sum_{n=1}^N u_n}\right)^2
+\left({\dfrac{1}{N}\displaystyle \sum_{n=1}^N v_n}\right)^2
+\left({\dfrac{1}{N}\displaystyle \sum_{n=1}^N w_n}\right)^2 }\\
& = & \sqrt{U^2+V^2+W^2}
\end{array}
\end{equation}
\section{RMS (Root-Mean-Square) of fluctuation velocity}
\begin{equation}
\begin{array}{rcl}
u^{'}_{rms}&=&\sqrt{\dfrac{1}{N}\displaystyle{\sum_{n=1}^N}(u_n^{'})^2}=\sqrt{\dfrac{1}{N}\displaystyle{\sum_{n=1}^N}(u_n-U)^2}
= \sqrt{\dfrac{1}{N}\displaystyle{\sum_{n=1}^N}(u_n^2-2u_iU+U^2)}
= \sqrt{\dfrac{1}{N}\displaystyle{\sum_{n=1}^N}(u_n^2-U^2)} \\
v^{'}_{rms}&=&\sqrt{\dfrac{1}{N}\displaystyle{\sum_{n=1}^N}(v_n^{'})^2}=\sqrt{\dfrac{1}{N}\displaystyle{\sum_{n=1}^N}(v_n-V)^2}
= \sqrt{\dfrac{1}{N}\displaystyle{\sum_{n=1}^N}(v_n^2-2v_iV+V^2)}
= \sqrt{\dfrac{1}{N}\displaystyle{\sum_{n=1}^N}(v_n^2-V^2)} \\
w^{'}_{rms}&=&\sqrt{\dfrac{1}{N}\displaystyle{\sum_{n=1}^N}(w_n^{'})^2}=\sqrt{\dfrac{1}{N}\displaystyle{\sum_{n=1}^N}(w_n-W)^2}
= \sqrt{\dfrac{1}{N}\displaystyle{\sum_{n=1}^N}(w_n^2-2w_iW+W^2)}
= \sqrt{\dfrac{1}{N}\displaystyle{\sum_{n=1}^N}(w_n^2-W^2)} \\
\vert{v}\vert^{'}_{rms}&=&\sqrt{\dfrac{1}{N}\displaystyle{\sum_{n=1}^N}(\vert{v}\vert_n^{'})^2}
= \sqrt{\dfrac{1}{N}\displaystyle{\sum_{n=1}^N}(\vert{v}\vert_n-\vert{V}\vert)^2}
= \sqrt{\dfrac{1}{N}\displaystyle{\sum_{n=1}^N}(\vert{v}\vert_n^2-\vert{V}\vert^2)}\\
&=&\sqrt{\dfrac{1}{N}\displaystyle{\sum_{n=1}^N}\biggr((u_n^2-U^2)+(v_n^2-V^2)+(w_n^2-W^2)\biggr)}\\
&=&\sqrt{(u^{'}_{rms})^2+(v^{'}_{rms})^2+(w^{'}_{rms})^2}
\end{array}
\end{equation}
\section{Reynolds stress}
\begin{equation}
\begin{array}{rcl}
\tau_{xy}&=&\dfrac{1}{N}\displaystyle{\sum_{n=1}^N\biggr( (u_n-U)(v_n-V)\biggr) }
=\dfrac{1}{N}\displaystyle{\sum_{n=1}^N\left( u_n^{'}v_n^{'}\right)}\\
\tau_{xz}&=&\dfrac{1}{N}\displaystyle{\sum_{n=1}^N\biggr( (u_n-U)(w_n-W)\biggr) }
=\dfrac{1}{N}\displaystyle{\sum_{n=1}^N\left( u_n^{'}w_n^{'}\right)}\\
\tau_{yz}&=&\dfrac{1}{N}\displaystyle{\sum_{n=1}^N\biggr( (v_n-V)(w_n-W)\biggr) }
=\dfrac{1}{N}\displaystyle{\sum_{n=1}^N\left( v_n^{'}w_n^{'}\right)}\\
\tau_{xx}&=&\dfrac{1}{N}\displaystyle{\sum_{n=1}^N\biggr( (u_n-U)(u_n-U)\biggr) }
=\dfrac{1}{N}\displaystyle{\sum_{n=1}^N ( u_n^{'})^2}
=(u^{'}_{rms})^2 \\
\tau_{yy}&=&\dfrac{1}{N}\displaystyle{\sum_{n=1}^N\biggr( (v_n-V)(v_n-V)\biggr) }
=\dfrac{1}{N}\displaystyle{\sum_{n=1}^N ( v_n^{'})^2}
=(v^{'}_{rms})^2\\
\tau_{zz}&=&\dfrac{1}{N}\displaystyle{\sum_{n=1}^N\biggr( (w_n-W)(w_n-W)\biggr) }
=\dfrac{1}{N}\displaystyle{\sum_{n=1}^N ( w_n^{'})^2}
=(w^{'}_{rms})^2
\end{array}
\end{equation}
\section{Kinetic energy}
Average kinetic energy:
\begin{equation}
E_{ake}=\vert{V}\vert^2=U^2+V^2+W^2
\end{equation}
Turbulent kinetic energy:
\begin{equation}
\begin{array}{rcl}
k&=&\dfrac{1}{2}\biggr( \dfrac{1}{N}\displaystyle{\sum_{n=1}^N}(u_n^{'})^2 +
\dfrac{1}{N}\displaystyle{\sum_{n=1}^N}(v_n^{'})^2 +
\dfrac{1}{N}\displaystyle{\sum_{n=1}^N}(w_n^{'})^2 \biggr)\\
&=&\dfrac{1}{2}\biggr( \tau_{xx}+\tau_{yy}+\tau_{zz} \biggr) \\
&=&\dfrac{1}{2}\biggr( (u^{'}_{rms})^2+(v^{'}_{rms})^2+(w^{'}_{rms})^2 \biggr) \\
&=&\dfrac{1}{2}(\vert{v}\vert^{'}_{rms} )^2
\end{array}
\end{equation}
\section{Turbulence intensity $Tu$}
\begin{equation}
\begin{array}{rcl}
Tu &=& \dfrac{\sqrt{\dfrac{1}{3} \biggr( \dfrac{1}{N}\displaystyle{\sum_{n=1}^N}(u_n^{'})^2 +
\dfrac{1}{N}\displaystyle{\sum_{n=1}^N}(v_n^{'})^2 +
\dfrac{1}{N}\displaystyle{\sum_{n=1}^N}(w_n^{'})^2 \biggr) }}
{\vert{V}\vert_{\infty}}\\
&=& \dfrac{\sqrt{\dfrac{2}{3} k }}{\vert{V}\vert}
= \dfrac{\sqrt{\dfrac{2}{3} k }}{\sqrt{U^2+V^2+W^2}}
\end{array}
\end{equation}
\section{Vorticity } %$\overrightarrow{\omega}$
\begin{equation}
\overrightarrow{\omega}={\overrightarrow{\nabla}}\times{\overrightarrow{v}}=
\left\vert \begin{array}{ccc}
\overrightarrow{i} & \overrightarrow{j} & \overrightarrow{n} \\
\dfrac{\partial}{\partial x} & \dfrac{\partial}{\partial y} & \dfrac{\partial}{\partial z} \\
u & v & w
\end{array} \right\vert
= \overrightarrow{i}(\dfrac{\partial{w}}{\partial{y}}-\dfrac{\partial{v}}{\partial{z}})
+\overrightarrow{j}(\dfrac{\partial{u}}{\partial{z}}-\dfrac{\partial{w}}{\partial{x}})
+\overrightarrow{k}(\dfrac{\partial{v}}{\partial{x}}-\dfrac{\partial{u}}{\partial{y}})
\end{equation}
\begin{equation}
\begin{array}{rcl}
\omega_x=\dfrac{\partial{w}}{\partial{y}}-\dfrac{\partial{v}}{\partial{z}};
\omega_y=\dfrac{\partial{u}}{\partial{z}}-\dfrac{\partial{w}}{\partial{x}};
\omega_z=\dfrac{\partial{v}}{\partial{x}}-\dfrac{\partial{u}}{\partial{y}}
\end{array}
\end{equation}
\section{Q criterion}
\begin{equation}
Q=\dfrac{1}{2}\left(\vert \vert \mathbf{\Omega} \vert \vert ^2 -\vert\vert
\mathbf{S} \vert\vert^2\right)=\dfrac{1}{2}\left( {\Omega_{ij}\Omega_{ij}} -
{S_{ij}S_{ij}} \right) > 0
\end{equation}
Where $\vert \vert \mathbf{\Omega} \vert \vert=tr[\mathbf{\Omega}\mathbf{\Omega}^t]^{1/2}$ and $\vert \vert \mathbf{S} \vert
\vert=tr[\mathbf{S}\mathbf{S}^t]^{1/2}$; $S$ and $\Omega$ are the synnetric and antisymmetric components of $\nabla u$ defined as
$\mathbf{S}=1/2[\nabla u+(\nabla u)^t]$ and $\mathbf{\Omega}= 1/2[\nabla u-(\nabla u)^t]$ respectively. Usually $\mathbf{S}$ is called the
rate-of-strain tensor, while $\mathbf{\Omega}$ is called the vorticity tensor.
\begin{equation}
\mathbf{S}=S_{ij}=\dfrac{1}{2}\left( \dfrac{\partial{u_i}}{\partial{x_j}}+\dfrac{\partial{u_j}}{\partial{x_i}} \right)
= \left[ \begin{array}{ccc}
\dfrac{\partial{u}}{\partial{x}} &
\dfrac{1}{2} \left( \dfrac{\partial{u}}{\partial{y}} +\dfrac{\partial{v}}{\partial{x}} \right) &
\dfrac{1}{2} \left( \dfrac{\partial{u}}{\partial{z}} +\dfrac{\partial{w}}{\partial{x}} \right) \\
\dfrac{1}{2} \left( \dfrac{\partial{u}}{\partial{y}} +\dfrac{\partial{v}}{\partial{x}} \right) &
\dfrac{\partial{v}}{\partial{y}} &
\dfrac{1}{2} \left( \dfrac{\partial{v}}{\partial{z}} +\dfrac{\partial{w}}{\partial{y}} \right) \\
\dfrac{1}{2} \left( \dfrac{\partial{u}}{\partial{z}} +\dfrac{\partial{w}}{\partial{x}} \right) &
\dfrac{1}{2} \left( \dfrac{\partial{v}}{\partial{z}} +\dfrac{\partial{w}}{\partial{y}} \right) &
\dfrac{\partial{w}}{\partial{z}}
\end{array} \right]
\end{equation}
\begin{equation}
\mathbf{\Omega}=\Omega_{ij}=\dfrac{1}{2}\left( \dfrac{\partial{u_i}}{\partial{x_j}}-\dfrac{\partial{u_j}}{\partial{x_i}} \right)
= \left[ \begin{array}{ccc}
0 &
\dfrac{1}{2} \left( \dfrac{\partial{u}}{\partial{y}} -\dfrac{\partial{v}}{\partial{x}} \right) &
\dfrac{1}{2} \left( \dfrac{\partial{u}}{\partial{z}} -\dfrac{\partial{w}}{\partial{x}} \right) \\
\dfrac{1}{2} \left( \dfrac{\partial{u}}{\partial{y}} -\dfrac{\partial{v}}{\partial{x}} \right) &
0 &
\dfrac{1}{2} \left( \dfrac{\partial{v}}{\partial{z}} -\dfrac{\partial{w}}{\partial{y}} \right) \\
\dfrac{1}{2} \left( \dfrac{\partial{u}}{\partial{z}} -\dfrac{\partial{w}}{\partial{x}} \right) &
\dfrac{1}{2} \left( \dfrac{\partial{v}}{\partial{z}} -\dfrac{\partial{w}}{\partial{y}} \right) &
0
\end{array} \right]
\end{equation}
\section{$\Delta$ criterion}
\begin{equation}
\Delta=\left(\dfrac{Q}{3}\right)^3+\left(\dfrac{det(\nabla u)}{2}\right)^2 > 0
\end{equation}
\section{$\lambda$ criterion}
\begin{equation}
\lambda_2 \left( \mathbf{S}^2+\mathbf{\Omega}^2 \right) <>
\end{equation}
where $\lambda_2(\mathbf{A})$ denotes the intermediate eigenvalue of a symmetric tensor $\mathbf{A}$.
\section{Turbulence dissipation rate $\epsilon$}
\begin{equation}
\epsilon=2\nu\overline{s_{ij}^{'}s_{ij}^{'}}
\end{equation}
where $\nu$ is kinematic viscosity.
\begin{equation}
s_{ij}^{'}=\dfrac{1}{2}\left( \dfrac{\partial{u_i^{'}}}{\partial{x_j}}+\dfrac{\partial{u_j^{'}}}{\partial{x_i}} \right)
= \left[ \begin{array}{ccc}
\dfrac{\partial{u^{'}}}{\partial{x}} &
\dfrac{1}{2} \left( \dfrac{\partial{u^{'}}}{\partial{y}} +\dfrac{\partial{v^{'}}}{\partial{x}} \right) &
\dfrac{1}{2} \left( \dfrac{\partial{u^{'}}}{\partial{z}} +\dfrac{\partial{w^{'}}}{\partial{x}} \right) \\
\dfrac{1}{2} \left( \dfrac{\partial{u^{'}}}{\partial{y}} +\dfrac{\partial{v^{'}}}{\partial{x}} \right) &
\dfrac{\partial{v^{'}}}{\partial{y}} &
\dfrac{1}{2} \left( \dfrac{\partial{v^{'}}}{\partial{z}} +\dfrac{\partial{w^{'}}}{\partial{y}} \right) \\
\dfrac{1}{2} \left( \dfrac{\partial{u^{'}}}{\partial{z}} +\dfrac{\partial{w^{'}}}{\partial{x}} \right) &
\dfrac{1}{2} \left( \dfrac{\partial{v^{'}}}{\partial{z}} +\dfrac{\partial{w^{'}}}{\partial{y}} \right) &
\dfrac{\partial{w^{'}}}{\partial{z}}
\end{array} \right]
\end{equation}
\section{Kolmogorov scale (dissipative scale) $\eta$, energy-containing scale $L$ and Taylor scale $\lambda$, $Re_\lambda$, $Re_L$}
\begin{equation}
\eta \sim \left(\dfrac{\nu^3}{\epsilon}\right)^{1/4}
\end{equation}
\begin{equation}
L \sim \dfrac{k^{3/2}}{\epsilon}
\end{equation}
\begin{equation}
\lambda \sim 15^{1/2}L^{1/3}\eta^{2/3}
\end{equation}
\begin{equation}
Re_\lambda=15^{1/2}\left( \dfrac{L}{\eta} ^{2/3}\right)
\end{equation}
\begin{equation}
Re_L=\dfrac{1}{15}Re_\lambda^2
\end{equation}
\section*{Equation}
\begin{equation}
S_{ij}S{ij}=\displaystyle{\sum_{j=1}^3}\displaystyle{\sum_{i=1}^3}S_{ij}S_{ij}=
S_{11}S_{11}+S_{12}S_{12}+S_{13}S_{13}+S_{21}S_{21}+S_{22}S_{22}+S_{23}S_{23}+S_{31}S_{31}+S_{32}S_{32}+S_{33}S_{33}
\end{equation}
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