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\def\plderivxy#1#2#3{\partial #1 / \partial #2 \partial #3}

\def\derivmat#1#2{\frac{D#1}{D#2}}

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\def\derivdel#1#2{\frac{\delta#1}{\delta#2}}

\def\lderivdel#1#2{\delta#1/\delta#2}

\def\derivcal#1#2{\frac{{\cal D}#1}{{\cal D}#2}}

\def\lderivcal#1#2{{\cal D}#1/{\cal D}#2}

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\newcommand {\eps}[0]{\varepsilon}

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{\newpage
\clearpage
\samepage \begin{equation}T\;\mbox{:}\;\mbox{\boldmath  $x$} = \mbox{\boldmath  $x$}(\mbox{\boldmath  $\xi$}) \mbox{, } 
   \quad \mbox{\boldmath  $x$} \in \Omega \mbox{, } \mbox{\boldmath  $\xi$} \in G
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\end{equation}
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{\newpage
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\samepage \begin{equation}% latex2html id marker 168
{\mbox{\boldmath  $a$}}_{(\alpha)} = \frac{\partial \mbox{\boldmath  $x$}}
                                  {\partial \xi^\alpha}
                             \mbox{,} \quad
  {\mbox{\boldmath  $a$}}^{(\alpha)} = \frac{\partial \xi^\alpha}
                                  {\partial \mbox{\boldmath  $x$}}
                             \mbox{,} \quad
  g_{\alpha \beta} = \mbox{\boldmath  $a$}_{(\alpha)} \cdot \mbox{\boldmath  $a$}_{(\beta)}
                     \mbox{,} \quad
  g^{\alpha \beta} = \mbox{\boldmath  $a$}^{(\alpha)} \cdot \mbox{\boldmath  $a$}^{(\beta)}
\end{equation}
}

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{\newpage
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\samepage \begin{eqnarray}% latex2html id marker 192
&&\phi_{,\alpha} = \frac{\partial \phi}{\partial \xi^\alpha} \label{covzero} \\ 
  &&U^{\alpha}_{,\beta} = \frac{\partial U^\alpha}{\partial \xi^\beta} + 
                          \{\!\!\!\begin{array}{c}
\begin{array}{c}\alpha \end{array}\\  
\gamma \beta \end{array}\!\!\!\} U^\gamma \label{covuab} \\ 
  &&U^{\alpha}_{,\alpha} = \frac{1}{\sqrt{g}} \frac{\partial \sqrt{g} U^\alpha}
                           {\partial \xi^\alpha} \label{covone} \\ 
  &&T^{\alpha \beta}_{,\beta} = \frac{1}{\sqrt{g}} \frac{\partial \sqrt{g}
                           T^{\alpha \beta}}{\partial \xi^\beta} +
                           \{\!\!\!\begin{array}{c}
\begin{array}{c}\alpha \end{array}\\  
\gamma \beta \end{array}\!\!\!\} T^{\gamma \beta}
     \label{covtwo}
\end{eqnarray}
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{\newpage
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\samepage \begin{equation}% latex2html id marker 225
\{\!\!\!\begin{array}{c}
\begin{array}{c}\alpha \end{array}\\  
\gamma \beta \end{array}\!\!\!\} = \mbox{\boldmath  $a$}^{(\alpha)} \cdot
                                  \frac{\partial \mbox{\boldmath  $a$}_{(\beta)}}
                                       {\partial \xi^\gamma} = 
                                  \frac{\partial \xi^\alpha}{\partial
                                        x^\delta} \cdot
                                  \frac{\partial^2 x^\delta}
                                       {\partial x^\gamma \partial
                                        \xi^\beta} = 
                                  \{\!\!\!\begin{array}{c}
\begin{array}{c}\alpha \end{array}\\  
\beta \gamma \end{array}\!\!\!\}              
\end{equation}
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{\newpage
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\samepage \begin{equation}% latex2html id marker 242
U^{\alpha}_{,\alpha} = 0 \label{conteq}
\end{equation}
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{\newpage
\clearpage
\samepage \begin{equation}% latex2html id marker 247
\frac{\partial U^\alpha}{\partial t} + (U^\alpha U^\beta)_{,\beta} + 
  (g^{\alpha \beta} p)_{,\beta} - ((\nu + \nu_t) (g^{\alpha \gamma}
  U^\beta_{,\gamma} + g^{\beta \gamma}U^\alpha_{,\gamma} ))_{,\beta}
   = F^\alpha \label{NavSteq}
\end{equation}
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{\newpage
\clearpage
\samepage \begin{equation}\nu_t =  c_\mu \frac{k^2}{\varepsilon } \label{eddy}
\end{equation}
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\samepage \begin{equation}% latex2html id marker 265
\frac{\partial k}{\partial t} + (U^\alpha k)_{,\alpha} -
       (\frac{\nu_t}{\sigma_k} g^{\alpha \beta} k_{,\beta} )_{,\alpha} =
       P_k - \varepsilon  \label{equak} 
\end{equation}
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{\newpage
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\samepage \begin{equation}% latex2html id marker 276
\frac{\partial \varepsilon }{\partial t} + (U^\alpha \varepsilon )_{,\alpha} -
       (\frac{\nu_t}{\sigma_\varepsilon } g^{\alpha \beta} \eps_{,\beta} )_{,\alpha} =
       \frac{\varepsilon }{k}( c_{\varepsilon  1} P_k - c_{\varepsilon  2} \varepsilon  ) \label{equaeps} 
\end{equation}
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{\newpage
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\samepage \begin{equation}% latex2html id marker 291
P_k = \nu_t g_{\alpha \delta} (g^{\alpha \gamma} U^\beta_{,\gamma} + 
       g^{\beta \gamma} U^\alpha_{,\gamma}) U^\delta_{,\beta} \label{prod}
\end{equation}
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\samepage \begin{equation}\tau_w = 
           \left\{ 
             \begin{array}{ll}
                \nu \frac{\mbox{\boldmath  $u \cdot t$}}{Y},  & Y^+ < 11.3 \\ 
                \\ 
                \frac{c^{1/4}_\mu \kappa \sqrt{k}}
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{\newpage
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\samepage \begin{equation}Y^+ = \frac{ c_\mu^{1/4} Y \sqrt{k}}{\nu}
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\samepage \begin{equation}P_k = \tau_w \frac{\mbox{\boldmath  $u \cdot t$}}{Y}
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\samepage \begin{equation}\varepsilon  = 
         \left\{
           \begin{array}{ll}
             c_\mu^{3/4} k^{3/2} \frac{Y^+}{Y},               & Y^+ < 11.3 \\ 
             \\ 
             c_\mu^{3/4} k^{3/2} \frac{ \ln(E Y^+)}{\kappa Y},& Y^+ \geq 11.3
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{\newpage
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\samepage \begin{figure}[htb]
\centerline{\psfig{figure=fig2.eps,height=4cm}}

\label{compgrid}
\end{figure}
}

{\newpage
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{\newpage
\clearpage
\samepage \begin{equation}% latex2html id marker 376
T^{\alpha \beta}_{,\beta} = F^\alpha - \frac{\partial U^\alpha}{\partial t} \label{newmomeq}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}% latex2html id marker 383
T^{\alpha \beta} = U^\alpha U^\beta + g^{\alpha \beta}p - 
                     (\nu + \nu_t) (g^{\alpha \gamma}
                     U^\beta_{,\gamma} + g^{\beta \gamma}U^\alpha_{,\gamma} )
  \label{momtensor}
\end{equation}
}

{\newpage
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\samepage \begin{equation}% latex2html id marker 392
Q^{\alpha}_{,\alpha} = S_\phi - \frac{\partial \phi}{\partial t} \label{newtrnspeq}
\end{equation}
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{\newpage
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\samepage \begin{equation}% latex2html id marker 399
Q^\alpha = U^\alpha \phi - \frac{\nu_t}{\sigma_\phi} g^{\alpha \beta}
              \phi_{,\beta} \label{trnsptensor}
\end{equation}
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{\newpage
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{\newpage
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\samepage \begin{figure}[htb]
\centerline{\psfig{figure=fig1.eps,height=5cm}}

\label{loccoor}
\end{figure}
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{\newpage
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\samepage \begin{equation}% latex2html id marker 414
\int_{\Omega} U^{\alpha}_{,\alpha} d\Omega = \int_{\Omega} \frac{\partial \sqrt{g}
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{\newpage
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{\newpage
\clearpage
\samepage \begin{eqnarray}\int_{\Omega} T^{1 \beta}_{,\beta} d\Omega & = & \int_{\Omega} \frac{\partial \sqrt{g}
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\begin{array}{c}1 \end{array}\\  
\gamma \beta \end{array}\!\!\!\}
   T^{\gamma \beta} \sqrt{g} d\xi^1 d\xi^2 \nonumber \\ 
   & \approx &
   \sqrt{g} T^{11} |^{(2,0)}_{(0,0)} + \sqrt{g} T^{12} |^{(1,1)}_{(1,-1)} +
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\begin{array}{c}1 \end{array}\\  
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\samepage \begin{equation}% latex2html id marker 460
\int_{\Omega} Q^{\alpha}_{,\alpha} d\Omega \approx \sqrt{g} Q^1 
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{\newpage
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\int_{\Omega} (F^\alpha - \frac{\partial U^\alpha}{\partial t}) d\Omega \approx 
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{\newpage
\clearpage
\samepage \begin{equation}\int_{\Omega} (S_\phi - \frac{\partial \phi}{\partial t}) d\Omega \approx {\sqrt{g}}_{(0,0)}
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{\newpage
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{\newpage
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{\newpage
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Pe^\alpha_{(k,l)} = \frac{\sigma_\phi V^\alpha_{(k,l)}}
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{\newpage
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\samepage \begin{equation}s(Pe) = 
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{\newpage
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\samepage \begin{figure}[htb]
\begin{minipage}[t]{7cm}
\begin{center}
\hspace*{0,7cm}\fbox{\hspace{2mm} } \hspace{0,5cm} + \hspace{0,5cm}
\fbox{\hspace{2mm} } \hspace{0,5cm} + \hspace{0,5cm}
\fbox{\hspace{2mm} }\\ [2ex]
\hspace*{0,9cm}\fbox{\rule[1mm]{0mm}{3mm}}\hspace{2,1cm}\fbox{\rule[1mm]{0mm}{3mm}}\\ [2ex]
\hspace*{0,7cm}\fbox{\hspace{2mm} } \hspace{0,5cm} + \hspace{0,5cm}
\rule{6mm}{2mm} \hspace{0,5cm} + \hspace{0,5cm}
\fbox{\hspace{2mm} }\\ [2ex]
\hspace*{0,9cm}\fbox{\rule[1mm]{0mm}{3mm}}\hspace{2,1cm}\fbox{\rule[1mm]{0mm}{3mm}}\\ [2ex]
\hspace*{0,7cm}\fbox{\hspace{2mm} } \hspace{0,5cm} + \hspace{0,5cm}
\fbox{\hspace{2mm} } \hspace{0,5cm} + \hspace{0,5cm}
\fbox{\hspace{2mm} }
\end{center}
\end{minipage}
\hfill
\begin{minipage}[t]{5cm}
\fbox{\hspace{2mm} } - $V^{1}$ unknown\\ [2ex]
\hspace*{0,3cm}\fbox{\rule{0mm}{3mm}}\hspace{0,1cm} - $V^{2}$ unknown\\ [2ex]
\hspace*{0,3cm}+\hspace{0,1cm} - $p$ unknown
\end{minipage}

\label{stenmom}
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\end{equation}
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{\newpage
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{\newpage
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\samepage \begin{figure}[htb]
  \centerline{
       \psfig{file=drivcav.eps,height=4cm}
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  \label{drivcav1}
\end{figure}
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{\newpage
\clearpage
\samepage \begin{equation}k = \frac{u_{\tau_s}^2}{\sqrt{c_\mu}}, \qquad 
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{\newpage
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\samepage \begin{figure}[htb]
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  \label{drivcav2}
\end{figure}
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  \vspace{-0.3cm}
  
  \label{drivcav3}
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{\newpage
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  \label{bfs11}
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{\newpage
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\samepage \begin{figure}[htb]
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{\newpage
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\samepage \begin{figure}[htb]
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\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{figure}[htb]
  \centerline{
       \psfig{file=contube.eps,height=3cm,width=10cm}
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  \label{ctube1}
\end{figure}
}

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{\newpage
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{\newpage
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\samepage \begin{equation}u_{in} = u_0 (1 + \frac{y}{R_0})^{1/6.4}
\end{equation}
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{\newpage
\clearpage
\samepage \begin{equation}k_{in} = 1.5 I_{\mbox{\tiny T}}^2 u_{in}^2, \qquad \eps_{in} = 
  \frac{c_\mu^{3/4} k^{3/2}_{in}}{l} \label{inturb}
\end{equation}
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{\newpage
\clearpage
\samepage \begin{equation}l = \min(\kappa y, 0.1 R_0)
\end{equation}
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{\newpage
\clearpage
\samepage \begin{figure}[htb]
  \centerline{
       \psfig{file=ctubegrid.eps,height=3cm,angle=-90}
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  \label{ctube2}
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}

{\newpage
\clearpage
\samepage \begin{table}[htb]
\begin{center}
\begin{tabular}{||l|c|c||}
\hline
      &Separation $x_s/R_0$ &Reattachment $x_r/R_0$\\ 
\hline
Experiment                &0.5  &4.0 \\ 
                          &     &    \\ 
Rastogi, $ 41 \times  21$ &1.2  &2.4 \\ 
Melaaen, $ 52 \times  22$ &0.89 &3.02\\ 
                          &     &    \\ 
Present, $ 50 \times  20$ &1.06 &3.07\\ 
Present, $ 75 \times  30$ &0.93 &3.10\\ 
Present, $100 \times  60$ &0.64 &3.36\\ 
Present, $150 \times 100$ &0.56 &3.50\\ 
\hline
\end{tabular}
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\label{griddep}
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