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\newcommand{\nutp}{{\nu_t^+}}
\title{Transient Turbulent Simulations,  Conditional Averaging }
\author{Tariq Talha }
\institute{University of Warwick}
% \date{13 July, 2010}
%  \date{}

\begin{document}
\setlength{\unitlength}{\textwidth}  % measure in textwidths
\begin{frame}
  \titlepage
\end{frame}

\begin{frame}[compress]{Temporal Constant Acceleration Flow Simulations}

The Navier-Stokes equations for acceleration,
\begin{eqnarray}
\frac{\partial \overline u_i}{\partial x_i}\ = 0.
\end{eqnarray}
\begin{eqnarray}
\frac{\partial \overline u_i }{\partial t}\ + \frac{\partial}{\partial x_j}\
\left(\overline u_i \overline u_j \right) = - \frac{\partial \overline {p} }
{\partial x_i}\ + \frac{1}{Re}\ \frac{\partial^2 \overline u_i}
{\partial x_j \partial x_j}\ -    \frac{\partial}{\partial x_j}\ \tau_{ij} +  \delta_{i1} P_x  \pm \delta_{i1} f.
\end{eqnarray}

Where the over bar denotes a filtered variable in case of LES, $P_x$ is the mean-pressure gradient, $f$ is the acceleration parameter, and $\tau_{ij}$ is the sub-grid scale stress given as,
 
\begin{eqnarray}
\tau_{ij} \,-\, 1/3 \delta_{ij} \tau_{kk} \;=\; -2\nu_t S_{ij} \;=\; \overline {u_i u_j} - \overline u_i \overline u_j.
\end{eqnarray}

\end{frame}


\begin{frame}[compress]{Acceleration Parameter}
The non dimensional acceleration parameter as defined by He and Jackson (2001) for pipe flow  is given as

\begin{eqnarray}
\gamma\; =\;  \dfrac{D}{u_{\tau0}}\, \bigg (\dfrac{1}{U_{m0}}\, \dfrac{dU_m}{dt} \bigg ). 
\label {eq:ramp-up-1}
\end{eqnarray}

In present numerical simulations, Equation \ref{eq:ramp-up-1} can be  non-dimensionalised as:

\begin{eqnarray}
 \gamma\; =\;  2\dfrac{Re_{m0}}{Re_{\tau0}} \bigg (\dfrac{dU^*_m}{dt^*} \bigg ) \; =\; 2\dfrac{Re_{m0}}{Re_{\tau0}} f
\end{eqnarray}
Where $U^*_m$, $t^*$ and $f$ are the non-dimensional bulk-mean velocity, time and acceleration parameter respectively. $f=0.2 (5 s)$ is used for DNS simulations while $f=0.2(5s),0.33(3 s), 0.5(2 s) $ are used for LES.  
 \end{frame}

\begin{frame}{Temporal Acceleration}
\begin{table}[ht]
\begin{center}
%\begin{tabular}{cccccc}
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}ccccccc}  
%\hline SIMULATION&  Present&Abe \textit{et al.}(2004) \\[5 pt]
\hline \hline
DNS/LES &Initial $Re_b$ & Final $Re_b$ & Initial $Re_\tau$ &  Final $Re_\tau$ & Time \\ [2 pt]
 \hline \hline
DNS& 3500 & 15000 & 215 & 800 & 16.4 \\  
LES& 3500 & 22600 & 215 & 1100 & 27.3 \\ \hline
\end{tabular*}
\end{center}
 \caption{Simulations setup.}
\end{table}

\begin{table}[ht]
\begin{center}
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}ccccccc} \hline 
\hline   DNS/LES & $L_x  \times L_y \times L_z$ & $N_x \times N_y \times N_z$ & $\Delta x^+$  & $ \Delta y_{min}^+ $ & $ \Delta y_{max}^+ $& $\Delta z^+$  \\ [2pt] \hline \hline
         DNS     & $6h \times 2h \times 4h $ & $ 384 \times 384 \times 640$ & $12.5$  & $0.4$ & $9.7$ & $5.0$   \\
         LES     & $6h \times 2h \times 4h $ & $ 256 \times 192 \times 384$ & $25.7$  & $1$ & $27.5$ & $11.4$   \\

\hline
\end{tabular*}
\end{center}
 \caption{Simulations parameters based on $Re_{\tau} = 800 (Re_b = 15000)$ for DNS and  $Re_{\tau} = 1100 (Re_b = 22600)$ for LES.}
\label{table: sim-parmeters-les-ru-22600}
\end{table}
\end{frame}

\begin{frame}{$\lambda_2 $ Iso-surface Contours (DNS).}
 \vspace{-0.3 cm}
\begin{minipage}[b]{.4\linewidth}
\includegraphics[trim = 0.2cm 0.2cm  0.2cm 0.3cm, clip,width=1.3\columnwidth,clip]{lambdaQ-3d-st-3500-run11-0p0115-FD}
\end{minipage} \hfil
 \hspace {1cm}
\begin{minipage}[b]{.4\linewidth}
\includegraphics[trim = 0.2cm 0.2cm  0.2cm 0.3cm, clip,width=1.3\columnwidth,clip]{lambdaQ-3d-9000-sim5-0p0224-FD}
 \end{minipage} \hfil

\hspace {15 mm} a) $Re=3500$ \hspace{.35\textwidth}  b) $Re=9000$ 

% \vspace {1cm}

\begin{minipage}[b]{.4\linewidth}
\includegraphics[trim = 0.2cm 0.2cm  0.2cm 0.2cm, clip,width=1.3\columnwidth,clip]{lambdaQ-3d-12000-sim5-0p335-FD}
\end{minipage} \hfil
 \hspace {1cm}
\begin{minipage}[b]{.4\linewidth}
\includegraphics[trim = 0.2cm 0.2cm  0.2cm 0.2cm, clip, width=1.3\columnwidth,clip]{lambdaQ-3d-unst-15000-sim5-2p33-FD}
 \end{minipage} \hfil\

\hspace {15 mm} c)  $Re=12000$ \hspace{.35\textwidth}  d)  $Re=15000$ 

\vspace { 0.2 cm}
\hspace {30 mm}
 $\lambda_ 2$ iso-surfaces during transient (DNS).
\end{frame}

\begin{frame}{Active Area Quantification Procedure}

\textbf{Considerations:}
\vspace { 0.2 cm}
\begin{itemize}
\item Property used: $\lambda'_{2}$ for DNS, $\lambda'_{2}$, $v'$ or $v'^2$ for LES. 
\item Wall normal integration of $\lambda'_{2}$. Test window of $\Delta y^+ = 150$ used.
\item Local averaging in $xz$ plane for $\lambda'_{2}$ over a test window .
\item Window size of $150 \times 40$ as per Jeong {\it et al\/}. (1997).
\item Locally averaged  $\lambda'_{2}$ comparison with wall-normal integrated. PDF for $\lambda'_{2}$  at several steady Reynolds number are analysed to determine cut-off value.
\item Use of conditionally averaged $u_\tau$ and RMS $\lambda'_{2}$.
\end{itemize}
\end{frame}

\begin{frame}{Wall-normal Integration Window}
\vspace{-0.3 cm}
\begin{minipage}[b]{.4\linewidth}
\includegraphics[trim = 0.2cm 0.2cm  0.2cm 0.3cm, clip,width=1.3\columnwidth,clip]{lambdaQ-3d-12000-sim5-0p335-FD}
\end{minipage} \hfil
 \hspace {1cm}
\begin{minipage}[b]{.4\linewidth}
\includegraphics[trim = 0.2cm 0.2cm  0.2cm 0.3cm, clip,width=1.3\columnwidth,clip]{yp-dependence}
 \end{minipage} \hfil

\vspace { 0.2 cm}
\hspace {10 mm}

 Effect of $\Delta y^+$ on area quantification during transient at $Re=12000$.
\end{frame}


% \begin{frame}{$xz$ Plane Local Averaging Window}
% \begin{minipage}[b]{.4\linewidth}
% \includegraphics[trim = 0.2cm 0.2cm  0.2cm 0.3cm, clip,width=1.3\columnwidth,clip]{lambdaQ-3d-12000-sim5-0p335-FD}
% \end{minipage} \hfil
%  \hspace {1cm}
% \begin{minipage}[b]{.4\linewidth}
% \includegraphics[trim = 0.2cm 0.2cm  0.2cm 0.3cm, clip,width=1.3\columnwidth,clip]{yp-dependence}
%  \end{minipage} \hfil
% 
% \hspace {10 mm}
%  Effect of $\Delta y^+$ on area quantification during transient at $Re=12000$.
% \end{frame}


\begin{frame}{$xz$ Plane Local Averaging Window}
 \vspace{-0.3 cm}
\begin{figure}
\begin{center}
\includegraphics[width=\WidthTwo]{lambdaQ-3d-12000-sim5-0p335-FD}
% \label{fig:l2-3d-ru-dns15000}
\end{center}
\end{figure}

\begin{minipage}[b]{.4\linewidth}
\includegraphics[trim = 0.2cm 0.2cm  0.2cm 0.3cm, clip,width=1.3\columnwidth,clip]{ik-xz-single}
\end{minipage} \hfil
 \hspace {1cm}
\begin{minipage}[b]{.4\linewidth}
\includegraphics[trim = 0.2cm 0.2cm  0.2cm 0.3cm, clip,width=1.3\columnwidth,clip]{ik-xz-plane-avg}
 \end{minipage} \hfil

\hspace {15 mm} a) Without averaging \hspace{.25\textwidth}  b) With averaging 

% \vspace { 0.2 cm}
\hspace {10 mm}

 Effect of $xz$ plane averaging on area quantification during transient at $Re=12000$.
\end{frame}

\begin{frame}{Probability Density Functions for Steady Cases (PDF)}
\vspace{-0.3 cm}
\begin{minipage}[b]{.4\linewidth}
\includegraphics[trim = 0.2cm 0.2cm  0.2cm 0.3cm, clip,width=1.3\columnwidth,clip]{pdf-l2-3500-st-new.png}
\end{minipage} \hfil
 \hspace {1cm}
\begin{minipage}[b]{.4\linewidth}
\includegraphics[trim = 0.2cm 0.2cm  0.2cm 0.3cm, clip,width=1.3\columnwidth,clip]{pdf-l2-11000-st-new.png}
 \end{minipage} \hfil

\hspace {15 mm} a) $Re=3500$ \hspace{.35\textwidth}  b) $Re=11000$ 

\hspace {15 mm}
PDF of $\lambda'_2 $ at several steady turbulent channel flows.

\begin{table}[ht]
\begin{center}
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}cccccc} \hline  \hline 
$Re$          & RMS $\lambda'_2 $              & cut-off $\lambda'_2$          & Ratio       \\ [2pt] \hline  \hline 
$3500$        & $ 8.0 \times 10^{-3}$         &  $ 1.1 \times 10^{-3}$       & $0.14$             \\
$6900$        & $ 8.5 \times 10^{-3}$         &  $ 1.7 \times 10^{-3}$       & $0.20$             \\
$1100$        & $ 1.0 \times 10^{-2}$         &  $ 1.5 \times 10^{-3}$       & $0.15$             \\
 \hline
\end{tabular*}
\end{center}
\caption{Ratio between volume integrated RMS $\lambda'_2$ and cut-off $\lambda'_2$ }
\label{tab:Ratio-l2-rms}
\end{table}
\end{frame}

\begin{frame}{Algorithm for Active Area Quantification Procedure}


\begin{itemize}
\item Make an initial guess for the $u_\tau$ value which is $xz$ plane averaged local $u_\tau$ value.
\item Determine the physical size of test window i.e. $ 300  \times 50$ in wall units based based on the $u_\tau$ value.
\item Determine local average (3D)  $\lambda'_{2}$ and (wall-normal average) RMS $\lambda'_{2}$ .
\item Determine the area having turbulence in each computational cell on $xz$ based on the cutt-off $\lambda_2$ criteria.
\item Calculate conditional-averaged value of $u_\tau$ and $\lambda'_{2}$ for active cells.
\item Compare the value of conditional $u_\tau$ with that of initial guess in step (1).
\item Use conditionally averaged $u_\tau$ and RMS $\lambda'_{2}$ and go to step (2).
\item Repeat the process to get convergence of $u_\tau$.
\end{itemize}
\end{frame}

\begin{frame}{Active Area (DNS)}
 \vspace{-0.3 cm}
\begin{figure}
\begin{center}
\includegraphics[trim = 0.2cm 0.2cm  0.2cm 0.3cm, clip, width=.70\linewidth]{area-filled-ru-dns}
\label{fig:l2-3d-ru-dns15000}
\end{center}
\caption{Percentage of active area at different stages of acceleration.}
\end{figure}



\end{frame}

\begin{frame}{Conditionally Averaged Profiles (DNS).}
 \vspace{-0.2 cm}
\begin{minipage}[b]{.3\linewidth}
\includegraphics[trim = 0.2cm 0.2cm  0.2cm 0.3cm, clip,width=1.3\columnwidth,clip]{lam2-con-unst-12000}
\end{minipage} \hfil
 \hspace {1cm}
\begin{minipage}[b]{.3\linewidth}
\includegraphics[trim = 0.2cm 0.2cm  0.2cm 0.3cm, clip,width=1.3\columnwidth,clip]{urms-con-unst-12000}
 \end{minipage} \hfil

\hspace {15 mm} a) RMS $\lambda'_{2}$ \hspace{.35\textwidth}  b) $u_{rms}$ 

% \vspace {1cm}

\begin{minipage}[b]{.3\linewidth}
\includegraphics[trim = 0.2cm 0.2cm  0.2cm 0.2cm, clip,width=1.3\columnwidth,clip]{vrms-con-unst-12000.png}
\end{minipage} \hfil
 \hspace {1cm}
\begin{minipage}[b]{.3\linewidth}
\includegraphics[trim = 0.2cm 0.2cm  0.2cm 0.2cm, clip, width=1.3\columnwidth,clip]{wrms-con-unst-12000.png}
 \end{minipage} \hfil\

\hspace {15 mm} c)  $v_{rms}$ \hspace{.35\textwidth}  d)  $w_{rms}$ 


\hspace {10 mm}
 $\lambda_ 2$ Conditionally averaged profiles during transient at $Re=12000$.
\end{frame}


\begin{frame}{Conditionally Averaged Profiles (DNS).}
 
\begin{minipage}[b]{.4\linewidth}
\includegraphics[trim = 0.2cm 0.2cm  0.2cm 0.3cm, clip,width=1.3\columnwidth,clip]{lam2-comp-11000}
\end{minipage} \hfil
 \hspace {1cm}
\begin{minipage}[b]{.4\linewidth}
\includegraphics[trim = 0.2cm 0.2cm  0.2cm 0.3cm, clip,width=1.3\columnwidth,clip]{lam2-comp-15000}
 \end{minipage} \hfil

 \hspace {15 mm} a) $Re=11000$ \hspace{.35\textwidth}  b) $Re=15000$

 \vspace {1cm}
\hspace {20 mm}
 $\lambda_ 2$ Conditionally averaged profiles during transient.
\end{frame}


\begin{frame}{$\lambda_2 $ Iso-surface Contours (LES).}
 \vspace{-0.3 cm}
\begin{minipage}[b]{.4\linewidth}
\includegraphics[trim = 0.2cm 0.2cm  0.2cm 0.3cm, clip,width=1.3\columnwidth,clip]{l2-rms-3d-sim1-3500-0p015.png}
\end{minipage} \hfil
 \hspace {1cm}
\begin{minipage}[b]{.4\linewidth}
\includegraphics[trim = 0.2cm 0.2cm  0.2cm 0.3cm, clip,width=1.3\columnwidth,clip]{l2-rms-3d-sim1-9800-0p065.png}
 \end{minipage} \hfil

\hspace {15 mm} a) $Re=3500$ \hspace{.35\textwidth}  b) $Re=10000$ 

% \vspace {1cm}

\begin{minipage}[b]{.4\linewidth}
\includegraphics[trim = 0.2cm 0.2cm  0.2cm 0.2cm, clip,width=1.3\columnwidth,clip]{l2-rms-3d-sim1-12000-0p56.png}
\end{minipage} \hfil
 \hspace {1cm}
\begin{minipage}[b]{.4\linewidth}
\includegraphics[trim = 0.2cm 0.2cm  0.2cm 0.2cm, clip, width=1.3\columnwidth,clip]{l2-rms-3d-sim1-16000-1p6.png}
 \end{minipage} \hfil\

\hspace {15 mm} c)  $Re=12000$ \hspace{.35\textwidth}  d)  $Re=16000$ 

\vspace { 0.2 cm}
\hspace {30 mm}
 $\lambda_ 2$ iso-surfaces during transient (LES).
\end{frame}

\begin{frame}{$\lambda_2 $ Iso-surface Contours (LES).}
 \vspace{-0.2 cm}
\begin{minipage}[b]{.4\linewidth}
\includegraphics[trim = 0.2cm 0.2cm  0.2cm 0.3cm, clip,width=1.3\columnwidth,clip]{l2-rms-3d-len-12-8900-0p03.png}
\end{minipage} \hfil
 \hspace {1cm}
\begin{minipage}[b]{.4\linewidth}
\includegraphics[trim = 0.2cm 0.2cm  0.2cm 0.3cm, clip,width=1.3\columnwidth,clip]{l2-rms-3d-len-12-11200-0p35.png}
 \end{minipage} \hfil

\hspace {15 mm} c)  $Re=9000$ \hspace{.35\textwidth}  d)  $Re=11000$ 

\end{frame}

\begin{frame}{$\lambda_2 $ Iso-surface Contours (LES).}
 \begin{figure}
\begin{center}
\includegraphics[trim = 0.2cm 0.2cm  0.2cm 0.3cm, clip,width=.7\linewidth]{l2-rms-3d-len-12-13500-1p1.png}
% \label{fig:l2-3d-ru-dns15000}
\end{center}
\caption{$Re=13500$.}
\end{figure}
\end{frame}

\end{document}
