\documentclass[a4paper,9pt]{beamer}
\usepackage{beamerthemeshadow}
\usepackage[english]{babel}
\usepackage{amsbsy,amscd,amsfonts,amsmath,amsopn,amssymb,amsxtra}
\usepackage{mathptm,mathptmx}
\usepackage{pgf,pgfarrows,pgfnodes,pgfautomata,pgfheaps,pgfshade,pgfpages}
\usepackage{pdfpages}
%\usepackage{times}

\newcommand{\undertilde}[1]{\underset{\widetilde{}}{#1}}
\newcommand{\vect}[1]{\undertilde{\mbox{$#1$}}\hskip 0mm}
\graphicspath{{./}{../}{./FIGURES/}{./Talha/}{../Talha/FIGURES/}{../Talha/FIGURES_DNS/}{../Talha/FIGURES_25Feb2011/}{../Talha/FIGURES_LES_24Feb2011/}{../Talha/FIGURES_10Feb2011/}{../Talha/FIGURES_04Feb2011/}{../Talha/FIGURES_02Feb2011/}{../Talha/FIGURES_22Jan2011/}{../Talha/FIGURES_21Jan2011/}{../Talha/FIGURES_11Jan2011/}}
%{../Talha/FIGURES_26Nov2010/}{../Talha/FIGURES_22Nov2010/}{../Talha/FIGURES_15Nov2010/}}

\DeclareGraphicsExtensions{.pdf,.PDF,.jpg,.JPG,.png,.PNG,.bmp,.eps,.EPS}
\definecolor{aguamarina}{cmyk}{0.80,0.30,0.50,0}

\newlength{\WidthOne}
\setlength{\WidthOne}{.70\textwidth}
\newlength{\WidthTwoOne}
\setlength{\WidthTwoOne}{.60\textwidth}
\newlength{\WidthTwo}
\setlength{\WidthTwo}{.50\textwidth}
\newlength{\WidthThree}
\setlength{\WidthThree}{.32\textwidth}

\newcommand{\ENU}{\begin{enumerate}}
\newcommand{\ENUD}{\end{enumerate}}
\newcommand{\ITE}{\begin{itemize}}
\newcommand{\ITED}{\end{itemize}}
\newcommand{\etal}{{\it et al\/}.\ }
\newcommand{\etall}{{\it et al\/}.,\ }
\newcommand{\eg}{{\it e.g.}, }
\newcommand{\ie}{{\it i.e.}, }
\newcommand{\Cf}{C_f}
\newcommand{\Cs}{C_s}
\newcommand{\degree}{^\circ\mathrm{C}}
\newcommand{\DR}{\mbox{DR}}
\newcommand{\ReD}{Re_D}
\newcommand{\ReDo}{Re_{D0}}
\newcommand{\ReDn}{Re_{D1}}
\newcommand{\Reh}{Re}
%\newcommand{\Reh}{Re_h}
\newcommand{\Rem}{{Re_m}}
\newcommand{\Remo}{Re_{m 0}}
\newcommand{\Ren}{Re_{1}}
\newcommand{\Reo}{Re_{0}}
\newcommand{\Retau}{Re_\tau}
\newcommand{\Retauo}{Re_{\tau 0}}
\newcommand{\rms}{{rms}}
\newcommand{\Sij}{S_{ij}}
\newcommand{\tauij}{\tau_{ij}}
\newcommand{\tauw}{\tau_w}
\newcommand{\tauwo}{\tau_{w0}}
\newcommand{\tauwno}{\tau_{w_{no}}}
\newcommand{\td}{t^*}
\newcommand{\tNon}{t\Umo/h}
%\newcommand{\tNon}{t\Umo/R}
\newcommand{\tRef}{h/\Umo}
%\newcommand{\tRef}{R/\Umo}
\newcommand{\Te}{T_e}
\newcommand{\Um}{U_m}
\newcommand{\Umo}{U_{m0}}
\newcommand{\umax}{u'_{max}}
\newcommand{\vp}{u^+}
\newcommand{\urms}{u_{rms}}
\newcommand{\uirms}{u_{i,rms}}
%\newcommand{\urms}{u'}
%\newcommand{\uirms}{u'_i}
\newcommand{\uirmsp}{u'^+_i}
\newcommand{\urmsp}{u'^+}
\newcommand{\utau}{u_\tau}
\newcommand{\utausq}{u_\tau^2}
\newcommand{\utauo}{u_{\tau0}}
\newcommand{\utauno}{u_{\tau_{no}}}
\newcommand{\uu}{\overline{uu}}
\newcommand{\uv}{-\overline{uv}}
\newcommand{\vf}{\overline{v^2}-f}
\newcommand{\vmax}{v'_{max}}
\newcommand{\vpmax}{v'^+_{max}}
%\newcommand{\vrms}{v'}
\newcommand{\vrms}{v_{rms}}
\newcommand{\vrmsp}{v'^+}
\newcommand{\vv}{\overline{vv}}
\newcommand{\vwall}{v'_w}
\newcommand{\vwallp}{v'^+_w}
\newcommand{\wmax}{w'_{max}}
%\newcommand{\wrms}{w'}
\newcommand{\wrms}{w_{rms}}
\newcommand{\wrmsp}{w'^+}
\newcommand{\ww}{\overline{ww}}
\newcommand{\wx}{\omega'_x}
\newcommand{\wxmax}{\omega'_{x, max}}
\newcommand{\wxmin}{\omega'_{x, min}}
\newcommand{\wxwall}{\omega'_{x, wall}}
\newcommand{\wxp}{\omega'^+_x}
\newcommand{\wy}{\omega'_y}
\newcommand{\wymax}{\omega'_{y, max}}
\newcommand{\wypmax}{\omega'^+_{y, max}}
\newcommand{\wyp}{\omega'^+_y}
\newcommand{\wz}{\omega'_z}
\newcommand{\wzwall}{\omega'_{z, wall}}
\newcommand{\xp}{x^+}
\newcommand{\yd}{y_d}
\newcommand{\ydo}{y_d^o}
\newcommand{\ydop}{y_{d,op}}
\newcommand{\ydp}{y_d^+}
\newcommand{\ydpl}{\tilde{y}_d^+}
\newcommand{\ydpo}{{y_d^+}_0}
\newcommand{\ydpop}{y_{d,op}^+}
\newcommand{\ydpno}{{y_d^+}_{no}}
\newcommand{\yp}{y^+}
\newcommand{\ypmax}{y_{max}^+}
\newcommand{\ypmin}{y_{min}^+}
\newcommand{\ypA}{y^+_A}
\newcommand{\ypo}{y^+_0}
\newcommand{\zp}{z^+}

\mode<presentation>
{% .--------------------------------------------------------------------
  %%FONT THEME%%
  %\usefonttheme{default}
  %\usefonttheme{serif}
  \usefonttheme{professionalfonts}
} % .--------------------------------------------------------------------
\setbeamertemplate{bibliography item}[text]
\AtBeginSection[] {
  \begin{frame}<beamer>%[shrink=4]
    \frametitle{Contents}
    \tableofcontents[currentsection]
  \end{frame}
}
% -----------------------------------------------------------------
\title[{\centering Unsteady Turbulent Flow Simulations
\hspace{10em} \insertframenumber/\inserttotalframenumber}]{%
Numerical Simulations of Unsteady Turbulent Flows: \\Temporal Acceleration}
% -----------------------------------------------------------------
\author[Y M Chung]{Dr Yongmann Chung}
% -----------------------------------------------------------------
\institute[The University of Warwick]
{School of Engineering and Centre for Scientific Computing\\University of Warwick, United Kingdom\\
\vspace{2mm}
Acknowledgement: T Talha, E Hurst, M Jones, Dr S Y Jung
}
\date{}

\logo{\href{http://www2.warwick.ac.uk/}{\includegraphics[height=6mm]{the_warwick_uni_blue}}\hskip 1.5mm}
% -----------------------------------------------------------------
% put second_logo logo in bottom left
\setbeamertemplate{sidebar left}{%
   \vfill%
   \rlap{\hskip0.1cm%
         \href{http://www.warwick.ac.uk/go/csc/}%
         {\includegraphics[height=1cm]{csc_csc_rgb_screen}}}%
   \vskip2pt%
   \llap{\usebeamertemplate***{navigation symbols}\hskip0.1cm}%
   \vskip2pt%
}
\begin{document}
\begin{frame}
 \maketitle
\end{frame}


\section[Introduction]{Introduction }
\subsection[Governing equations]{Governing equations }
\begin{frame}{Governing equations}
\begin{block}{The Navier-Stokes equations}
\begin{eqnarray}
\frac{\partial u_i }{\partial t} + \frac{\partial}{\partial x_j} u_i u_j
= - \frac{\partial p}{\partial x_i}
+ \frac{1}{Re} \frac{\partial^2 u_i}{\partial x_j \partial x_j} + f \delta_{i1},
\end{eqnarray}
\begin{eqnarray}
\frac{\partial u_i}{\partial x_i} = 0.
\end{eqnarray}
where $Re$ is the Reynolds number, $\displaystyle Re = \frac{\Umo h}{\nu}$.
\end{block}
\begin{block}{Non-dimensionalisation}
\ITE
\item $h$: the half channel height,
\item $\Umo$: the initial bulk mean velocity.
\ITED
\end{block}
\end{frame}


\subsection[Temporal acceleration]{Acceleration parameter}
\begin{frame}{Temporal acceleration}
\begin{block}{Non-dimensional acceleration parameter, f}
\ITE
\item $f$ is the  non-dimensional acceleration parameter, $\displaystyle f = \frac{dU^*_m}{dt^*}$.
\ITED
\begin{align*}
f &= \frac{h}{\Umo^2} \frac{d\Um}{dt},\\
\gamma &= \frac{D}{\utauo \Umo} \frac{d\Um}{dt} = 2\dfrac{\Remo}{\Retauo} f.
\end{align*}
\end{block}
\begin{block}{The effect of $f$}
\begin{center}
\begin{tabular}{cccc} \hline
$f$ & 0.2 & 0.35 & 0.5 \\
& DNS/LES & LES & LES \\ \hline
\end{tabular}
\end{center}
\end{block}
\end{frame}


\subsection[Numerical methods]{Numerical methods }
\begin{frame}{Numerical methods}
\begin{block}{Finite volume method}
\ITE
\item Implicit, fractional step method (FSM) with Crank-Nicolson method,
    \ITE
    \item Kim \etal (2001)
    \ITED
\item Second-order accurate, finite volume scheme,
\item Periodic boundary conditions in the streamwise and spanwise directions,
\item Ensemble average with 15 realisations.
\ITED
\end{block}
\begin{block}{Validation: Steady DNS/LES comparison}
\begin{table}
\begin{center}
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}cccccccccc} \hline
$Re$ & 2800 & 3500 & 7000 & 11000 & 1200 & 15000 & 17000 & 20000 & 22600 \\
$\Retau$ & 180 & 210 & 395 & 590 & 640 & 780 & 860 & 1000 & 1110 \\
 & DNS & DNS & DNS & DNS & LES & DNS & LES & LES & LES \\ \hline
\end{tabular*}
\end{center}
\end{table}
\end{block}
\end{frame}


\begin{frame}{Temporal acceleration}
\begin{block}{Simulation parameters for LES}
\begin{table}[ht]
\begin{center}
\begin{tabular}{cccc} \hline
 & initial & final (DNS/LES) & ratios \\
$Re$ & 3500 & 15000/22600 & 4.3/6.4 \\
$Re_\tau$ & 210 & 800/1110 & 3.8/5.2 \\
$\tauw$ & & & 14.5/27.4 \\ \hline
\end{tabular}
\end{center}
\end{table}
\begin{figure}
\begin{center}
\includegraphics[width=\WidthTwoOne]{mass-history-RU}
\end{center}
\end{figure}
\end{block}
\end{frame}


\begin{frame}{Low speed streaks}
\begin{block}{Challenges in grid resolution}
\ITE
\item $Re=3500$ and $15000$
\ITED
\end{block}
\begin{figure}
\begin{center}
\includegraphics[width=\WidthTwoOne]{streaks-3500-steady-1-n}\\
\includegraphics[width=\WidthTwoOne]{streaks-15000-steady-1-n}
\end{center}
\end{figure}
\end{frame}


\begin{frame}{Simulations parameters}
\begin{block}{DNS based on $\Retau=800$}
\begin{table}[ht]
\begin{center}
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}cccccc} \hline
$L_x \times L_y \times L_z$ & $N_x \times N_y \times N_z$ & $\Delta \xp$ & $ \Delta \ypmin $ & $ \Delta \ypmax $ & $\Delta \zp$ \\
$6h \times 2h \times 4h$ & $384 \times 384 \times 640$ & 12.5 & 0.4 & 9.7 & 5.0 \\ \hline
\end{tabular*}
\end{center}
\end{table}
\ITE
\item Acceleration time: $T=16.4$
\ITED
\end{block}

\begin{block}{LES based on $\Retau=1110$}
\begin{table}[ht]
\begin{center}
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}cccccc} \hline
$L_x^+ \times L_y^+ \times L_z^+$ & $N_x \times N_y \times N_z$ & $\Delta \xp$ & $ \Delta \ypmin $ & $ \Delta \ypmax $ & $\Delta \zp$ \\
$6780 \times 2260 \times4520$ & $256\times192\times384$ & 26.2 & 1.0 & 28.0 & 11.6\\ \hline
\end{tabular*}
\end{center}
\end{table}
\ITE
\item Acceleration time: $T=27.3$, $16.3$ and $10.9$.
\ITED
\end{block}
\end{frame}


\section[Results]{Results}
\begin{frame}{Temporal acceleration}
\begin{block}{Mean velocity profile variation}
\ITE
\item Three $f$ values ($f=0.2$, $0.35$ and $0.5$) with steady results.
\ITED
\end{block}
\begin{figure}
\begin{center}
\includegraphics[width=\WidthOne]{umean-comp-RU-tot}
\end{center}
\end{figure}
\end{frame}


\subsection[Wall shear stress]{Wall shear stress}
\begin{frame}{Wall shear stress}
\begin{block}{Four distinctive stages}
\ITE
\item IT: initial transient ($3500 < \Reh < 4200$, or $0 < \tNon < 1$)
\item WT: weak transient ($4200 < \Reh < 12000$, or $1 < \tNon < 12$)
\item ST: strong transient ($12000 < \Reh < 16000$, or $12 < \tNon < 17$)
\item PS: pseudo-steady stage ($\Reh > 16000$, or $\tNon > 17$)
\ITED
\end{block}
\begin{figure}
\begin{center}
\includegraphics[width=\WidthOne]{tw-comp-RU-tot-new}
\end{center}
\end{figure}
\end{frame}


\begin{frame}{Wall shear stress}
\begin{block}{Wall shear stress variation}
\ITE
\item $\Retau$ (left) and the rate of change of $\Retau$ (right)
\item Good agreement with Dean and Bradshaw (1976)
\ITED
\end{block}
\begin{figure}
\begin{center}
\includegraphics[width=\WidthTwo]{retau}
\includegraphics[width=\WidthTwo]{Retau_t}
\end{center}
\end{figure}
\end{frame}


\subsection[Mean velocity profile]{Mean velocity profile}
\begin{frame}{Mean velocity profile}
\begin{block}{Mean velocity profile variation with $f=0.2$ and $0.5$}
\ITE
\item WT: Uniform increase in velocity in the early stage of transient.
\item PS: It reaches the pseudo-steady velocity at the end of acceleration.
\ITED
\end{block}
\begin{figure}
\begin{center}
\includegraphics[width=\WidthTwo]{tp5-umean-comp-y}
\includegraphics[width=\WidthTwo]{tp2-umean-comp-y}\\
$f=0.2$ \hspace{.40\textwidth} $f=0.5$
\end{center}
\end{figure}
\end{frame}


\begin{frame}{Log-law profile}
\begin{block}{Mean velocity profile variation}
\ITE
\item A downward shift at the IT stage due to a higher $\utau$ value.
\item An upward shift at the WT stage due to delay in new turbulence generation.
\ITED
\end{block}
\begin{figure}
\begin{center}
\includegraphics[width=\WidthTwo]{tp5-ru-umeanp-y}
\includegraphics[width=\WidthTwo]{tp2-ru-umeanp-y}\\
$f=0.2$ \hspace{.40\textwidth} $f=0.5$
\end{center}
\end{figure}
\end{frame}


\begin{frame}{Mean velocity}
\begin{block}{Velocity change in time}
\ITE
\item Near-wall region: smaller acceleration due to the no-slip condition
\item Centre region: almost constant acceleration.
\ITED
\end{block}
\begin{figure}
\begin{center}
\includegraphics[width=\WidthTwo]{tp5-ru-umean-time-his}
\includegraphics[width=\WidthTwo]{tp2-ru-umean-time-his}\\
$f=0.2$ \hspace{.40\textwidth} $f=0.5$
\end{center}
\end{figure}
\end{frame}


\subsection[Fluctuations]{Fluctuations}
\begin{frame}{Velocity fluctuations}
\begin{block}{$\urms$}
\ITE
\item Near-wall turbulence ($y<0.2$) responds first to acceleration.
\item Turbulence intensities in the core region are smaller than steady values.
\ITED
\end{block}
\begin{figure}
\begin{center}
\includegraphics[width=\WidthTwo]{urms-y-RU-tp5}
\includegraphics[width=\WidthTwo]{urms-y-RU-tp2}\\
$f=0.2$ \hspace{.40\textwidth} $f=0.5$
\end{center}
\end{figure}
\end{frame}


\begin{frame}{Velocity fluctuations}
\begin{block}{$\vrms$}
\ITE
\item Slower response than $\urms$
\item $\vrms$ in $y<0.5$ region responded first to acceleration.
\ITED
\end{block}
\begin{figure}
\begin{center}
\includegraphics[width=\WidthTwo]{vrms-y-RU-tp5}
\includegraphics[width=\WidthTwo]{vrms-y-RU-tp2}\\
$f=0.2$ \hspace{.40\textwidth} $f=0.5$
\end{center}
\end{figure}
\end{frame}


\begin{frame}{Velocity fluctuations}
\begin{block}{$\wrms$}
\ITE
\item A similar trend to the response of $\vrms$.
\item Slower response than $\urms$
\ITED
\end{block}
\begin{figure}
\begin{center}
\includegraphics[width=\WidthTwo]{wrms-y-RU-tp5}
\includegraphics[width=\WidthTwo]{wrms-y-RU-tp2}\\
$f=0.2$ \hspace{.40\textwidth} $f=0.5$
\end{center}
\end{figure}
\end{frame}


\begin{frame}{Velocity fluctuations}
\begin{block}{Velocity fluctuations in wall units}
\ITE
\item Fluctuations are normalised by local $\utau$.
\item Near-wall turbulence responds to the acceleration first.
\item Turbulence is transported to the core region.
\ITED
\end{block}
\begin{figure}
\begin{center}
\includegraphics[width=\WidthOne]{urmsp-unst-2d-rey-ru-les-t5-new}\\
\includegraphics[width=\WidthOne]{vrmsp-unst-2d-rey-ru-les-t5-new}\\
\includegraphics[width=\WidthOne]{wrmsp-unst-2d-rey-ru-les-t5-new}
\end{center}
\end{figure}
\end{frame}


\begin{frame}{$\urms$ velocity fluctuations with three $f$ values ($f=0.2$, $0.35$ and $0.5$)}
\begin{figure}
\begin{center}
\includegraphics[width=\WidthTwoOne]{urmsp-yplus-2d-les-ru-tp5}\\
\includegraphics[width=\WidthTwoOne]{urmsp-yplus-2d-les-ru-tp3}\\
\includegraphics[width=\WidthTwoOne]{urmsp-yplus-2d-les-ru-tp2}
\end{center}
\end{figure}
\end{frame}


\begin{frame}{$\urms$ velocity fluctuations}
\begin{block}{Maximum $\urms$ location}
\ITE
\item For steady, $\yp=15$.
\item At $Re=3500$, $y=0.07$; and $Re=22600$, $y=0.015$.
\ITED
\end{block}
\begin{figure}
\begin{center}
\includegraphics[width=\WidthTwo]{urms-max-loc-re}
\includegraphics[width=\WidthTwo]{urms-max-loc-time}\\
$Re$ \hspace{.40\textwidth} time
\end{center}
\end{figure}
\end{frame}


\begin{frame}{$\urms$ velocity fluctuations}
\begin{block}{Maximum $\urms$ location in wall units}
\ITE
\item For steady, $\yp=15$.
\ITED
\end{block}
\begin{figure}
\begin{center}
\includegraphics[width=\WidthOne]{urms-max-loc-re-yplus}
\end{center}
\end{figure}
\end{frame}


\begin{frame}{$\urms$ and $\vrms$ velocity fluctuations}
\begin{block}{Maximum $\urms$ and $\vrms$}
\ITE
\item $\urms(t)-\urms(0)$ normalised by the steady values.
\item An overshoot for $f=0.5$ case.
\ITED
\end{block}
\begin{figure}
\begin{center}
\includegraphics[width=\WidthTwo]{urmsfluc-max-norm-st-comp-ru-les}
\includegraphics[width=\WidthTwo]{vrmsfluc-max-norm-st-comp-ru-les}\\
$\urms$ \hspace{.40\textwidth} $\vrms$
\end{center}
\end{figure}
\end{frame}


\begin{frame}{Vorticity fluctuations, $\wx$ with three $f$ values ($f=0.2$, $0.35$ and $0.5$)}
\begin{figure}
\begin{center}
\includegraphics[width=\WidthTwoOne]{wxp-re-yp-2d-les-ru-tp5}\\
\includegraphics[width=\WidthTwoOne]{wxp-re-yp-2d-les-ru-tp3}\\
\includegraphics[width=\WidthTwoOne]{wxp-re-yp-2d-les-ru-tp2}
\end{center}
\end{figure}
\end{frame}


\begin{frame}{Streamwise vorticity fluctuations, $\wx$}
\begin{block}{Wall $\wx$ value}
\ITE
\item A delay followed by a sudden increase.
\item A longer delay for a large $f$ value.
\ITED
\end{block}
\begin{figure}
\begin{center}
\includegraphics[width=\WidthTwo]{wx-wall-val-comp-re-ru-les}
\includegraphics[width=\WidthTwo]{wx-wall-val-comp-time-ru-les}\\
$Re$ \hspace{.40\textwidth} time
\end{center}
\end{figure}
\end{frame}


\begin{frame}{Streamwise vorticity fluctuations, $\wx$}
\begin{block}{Maximum $\wx$ value}
\ITE
\item A similar trend to the wall $\wx$ value.
\item Weaker than the steady state value.
\ITED
\end{block}
\begin{figure}
\begin{center}
\includegraphics[width=\WidthTwo]{wx-max-val-comp-re-ru-les}
\includegraphics[width=\WidthTwo]{wx-max-val-comp-time-ru-les}\\
$Re$ \hspace{.40\textwidth} time
\end{center}
\end{figure}
\end{frame}


\begin{frame}{Streamwise vorticity fluctuations, $\wx$}
\begin{block}{Minimum $\wx$ location}
\ITE
\item Two sudden decreases.
\item Located further away from the wall than the steady state case.
\ITED
\end{block}
\begin{figure}
\begin{center}
\includegraphics[width=\WidthTwo]{wx-min-loc-comp-re-ru-les}
\includegraphics[width=\WidthTwo]{wx-min-loc-comp-time-ru-les}\\
$Re$ \hspace{.40\textwidth} time
\end{center}
\end{figure}
\end{frame}


\begin{frame}{Wall-normal vorticity fluctuations, $\wy$}
\begin{block}{Maximum $\wy$ value}
\ITE
\item A similar trend to the wall $\wx$ value.
\item Weaker than the steady state value.
\ITED
\end{block}
\begin{figure}
\begin{center}
\includegraphics[width=\WidthTwo]{wy-max-val-comp-re-ru-les}
\includegraphics[width=\WidthTwo]{wy-max-val-comp-time-ru-les}\\
$Re$ \hspace{.40\textwidth} time
\end{center}
\end{figure}
\end{frame}


\begin{frame}{Wall-normal vorticity fluctuations, $\wy$}
\begin{block}{Maximum $\wy$ location}
\ITE
\item Similar to minimum $\wy$ location: two sudden decreases.
\item Located further away from the wall than the steady state case.
\ITED
\end{block}
\begin{figure}
\begin{center}
\includegraphics[width=\WidthTwo]{wy-max-loc-comp-re-ru-les}
\includegraphics[width=\WidthTwo]{wy-max-loc-comp-time-ru-les}\\
$Re$ \hspace{.40\textwidth} time
\end{center}
\end{figure}
\end{frame}


\begin{frame}{Wall-normal vorticity fluctuations, $\wy$}
\begin{block}{Maximum $\wx$ and $\wy$ location in wall units}
\ITE
\item A similar trend to the wall $\wx$ value.
\item Weaker than the steady state value.
\ITED
\end{block}
\begin{figure}
\begin{center}
\includegraphics[width=\WidthTwo]{wx-max-loc-comp-re-ru-les-yplus}
\includegraphics[width=\WidthTwo]{wy-max-loc-comp-re-ru-les-yplus}\\
$\wx$ \hspace{.40\textwidth} $\wy$
\end{center}
\end{figure}
\end{frame}


\subsection[Flow structures]{Flow structures}
\begin{frame}{Low speed streaks}
\begin{figure}
\begin{center}
\includegraphics[width=\WidthTwo]{streaks-3500-steady-1-n}
\includegraphics[width=\WidthTwo]{streaks-8000-unsteady-1-n}\\
$Re=3500$ \hspace{.40\textwidth} $Re=8000$
\includegraphics[width=\WidthTwo]{streaks-12000-unsteady-1-n}
\includegraphics[width=\WidthTwo]{streaks-15000-unsteady-2-n}\\
$Re=12000$ \hspace{.40\textwidth} $Re=15000$
\end{center}
\end{figure}
\end{frame}


\begin{frame}{$\lambda_2$ iso-surface contours}
\begin{figure}
\begin{center}
\includegraphics[width=\WidthTwo]{lambdaQ-3d-st-3500-run11-0p0115-FD}
\includegraphics[width=\WidthTwo]{lambdaQ-3d-9000-sim5-0p0224-FD}\\
$Re=3500$ \hspace{.40\textwidth} $Re=9000$
\includegraphics[width=\WidthTwo]{lambdaQ-3d-12000-sim5-0p335-FD}
\includegraphics[width=\WidthTwo]{lambdaQ-3d-unst-15000-sim5-2p33-FD}\\
$Re=12000$ \hspace{.40\textwidth} $Re=15000$
\end{center}
\end{figure}
\end{frame}


\begin{frame}{Domain size test}
\begin{block}{Destruction of turbulence at $Re=9000$}
\end{block}
\begin{figure}
\begin{center}
\includegraphics[width=0.9\WidthTwo]{lambdaQ-3d-9000-sim5-0p0224-FD}
\includegraphics[width=1.35\WidthTwo]{l2-rms-3d-len-12-8900-0p03}\\
$6 \times 2 \times 4$ \hspace{.40\textwidth} $12 \times 2 \times 4$
\end{center}
\end{figure}
\end{frame}


\begin{frame}{Domain size test}
\begin{block}{Turbulence structures at $Re=11000$}
\end{block}
\begin{figure}
\begin{center}
\includegraphics[width=0.9\WidthTwo]{lambdaQ-3d-12000-sim7-0p121-FD}
\includegraphics[width=1.35\WidthTwo]{l2-rms-3d-len-12-11200-0p35}\\
$6 \times 2 \times 4$ \hspace{.40\textwidth} $12 \times 2 \times 4$
\end{center}
\end{figure}
\end{frame}


\begin{frame}{Active area}
\begin{block}{Percentage of active area during temporal acceleration}
\ITE
\item Destruction of \textbf{old} turbulence.
\item Generation of \textbf{new} turbulence.
\ITED
\end{block}
\begin{figure}
\begin{center}
\includegraphics[width=\WidthOne]{area-filled-ru-dns}
\end{center}
\end{figure}
\end{frame}


\begin{frame}{Conditional average at $Re=12000$}
\begin{block}{Conditional avarage}
\ITE
\item Area with/without active flow structures.
\item Location for the maximum $\urms$.
\ITED
\end{block}
\begin{figure}
\begin{center}
\includegraphics[width=\WidthTwo]{lam2-con-unst-12000}
\includegraphics[width=\WidthTwo]{urms-con-unst-12000}\\
$\lambda_2$ \hspace{.40\textwidth} $\urms$
\end{center}
\end{figure}
\end{frame}


\begin{frame}{Conditional average at $Re=12000$}
\begin{block}{Destruction of turbulence structure}
\ITE
\item Large changes.
\item New and old turbulence.
\ITED
\end{block}
\begin{figure}
\begin{center}
\includegraphics[width=\WidthTwo]{vrms-con-unst-12000}
\includegraphics[width=\WidthTwo]{wrms-con-unst-12000}\\
$\vrms$ \hspace{.40\textwidth} $\wrms$
\end{center}
\end{figure}
\end{frame}



\begin{frame}{Conditional average (DNS)}
\begin{block}{Destruction of turbulence structure}
\ITE
\item New turbulence is much weaker at $Re=11000$ than the steady flow.
\item Generation of new turbulence is almost completed at $Re=15000$.
\ITED
\end{block}
\begin{figure}
\begin{center}
\includegraphics[width=\WidthTwo]{lam2-comp-11000}
\includegraphics[width=\WidthTwo]{lam2-comp-15000}
\hspace {15mm} $Re=11000$ \hspace{.30\textwidth}  $Re=15000$
\end{center}
\end{figure}
\end{frame}


\section[Conclusions]{Conclusions}
\begin{frame}{Conclusions}
\begin{block}{Turbulent channel flow with temporal acceleration}
\ITE
\item Turbulence responds rapidly to the temporal acceleration after a certain delay until pseudo-steady equilibrium is achieved.
\item $\urms$ velocity component responds to the acceleration first, followed by $\vrms$ and $\wrms$ velocity components.
\item Three delays: production, redistribution, and transport.
\item Destruction of the initial turbulence and generation of new turbulence.
\item Effect of the acceleration parameter.
\item Effect of the initial flow condition.
\ITED
\end{block}
\end{frame}


\begin{frame}{The End}
\begin{center}
\Huge Thank You!
\end{center}
\vspace{10mm}
\end{frame}


\begin{frame}{Wall shear stress}
\begin{block}{Wall shear stress variation}
\ITE
\item $\Retau$ (left) and the rate of change of $\Retau$ (right)
\item Good agreement with Dean and Bradshaw (1976)
\ITED
\end{block}
\begin{figure}
\begin{center}
\includegraphics[width=\WidthTwo]{retau-comp-total-les}
\includegraphics[width=\WidthTwo]{dret-comp-total-les}
\end{center}
\end{figure}
\end{frame}


%\bibliographystyle{plain}	% (uses file "plain.bst")
\bibliographystyle{acm}	% (uses file "plain.bst")
%\bibliographystyle{abbrv}	% (uses file "plain.bst")
%\bibliographystyle{siam}	% (uses file "plain.bst")
\bibliography{myrefs}		% expects file "myrefs.bib"
\end{document} 