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\date{Nov. 8th, 2016}
\author[<NAME>]{<NAME>}
\institute{Department of Mechanical Engineering \\ The University of
Texas at Austin}
\title[Dissertation Defense]{%
    \mbox{Numerical Simulation of Synthetic,}
    \mbox{Buoyancy-Induced Columnar Vortices}
}

\begin{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%===============================================================================
% Title page
\begin{frame}
%
\titlepage{}


%\begin{centering}
\mbox{{\scriptsize Acknowledgment: This material is based upon work
 supported}}
\mbox{{\scriptsize by the Department of Energy [ARPA-E] under Award
 Number [DE-FOA-0000670]}} 
%\end{centering}
\raisebox{-0.2in}{\hbox{\includegraphics[width=4.5in]{figs/Footer}}\hfill}
%
\end{frame}

%===============================================================================

 \section{Motivation}
 %
 %

\begin{frame}
  \frametitle{The dust devil phenomenon}
  \begin{columns}[]
    \begin{column}{0.45\linewidth}
      \begin{center}
        \includegraphics[width=0.6\linewidth]{Dust_devil}
      \end{center}
      \begin{center}
        \includegraphics[width=0.8\linewidth]{martian_dust}
      \end{center}
    \end{column}
  %
    \begin{column}{0.55\linewidth}

      \begin{block}{Turbulent, unstable motions}
        \begin{itemize}
	 \item Incident solar energy absorbed into ground
	       \begin{itemize}
		\item peak solar insolation 1000~W/$\text{m}^2$ 
	       \end{itemize}
	 \item Ground heats air by conduction
	 \item Heated air expands, lowering density, 
	       driven upward by force of buoyancy
	 \item Arizona in Summer ($\Delta T= 30$ K) 
	 \item $\text{Ra} \approx 10^9 - 10^{11} \rarrow$ 
	       unstably stratified fluid
	 \item Velocities can exceed 33 m/s.
        \end{itemize}
      \end{block}
      How much energy is present in one of these objects? 
      \end{column}
    \end{columns}
  \end{frame}

%
%
%
\begin{frame}
\frametitle{Energy Estimate [Sinclair, Battan, etc. ]}

   \begin{center}
    \includegraphics[width=0.6\linewidth]{sinclair_cut}
   \end{center}

\begin{itemize}
  \item Estimate energy from Sinclair's measured velocities 
  \item R $\approx$ 5 meters, $V_{\theta} \approx V_{z} \approx 10$ m/s 
  \item Kinetic Energy flux $ = \frac{1}{2}\rho \int V_z (V_z^2 + V_{\theta}^2)
	\, dA \approx 45$ kW
 \begin{itemize}
  \item Exceeds PV per $\text{m}^2$
 \end{itemize}
\end{itemize}

\end{frame}

\begin{frame}
\frametitle{Dust Devil Structure [Sinclair,Kanak,Renno]}

\begin{columns}[]
  \begin{column}{0.45\linewidth}
   \begin{center}
    \includegraphics[width=0.9\linewidth]{ground}
   \end{center}
  \end{column}
  %
  \begin{column}{0.55\linewidth}
    \begin{block}{Dust devil structure}
    \begin{itemize}
     \item Driven by buoyancy of hot ground
     \item Near surface: radial inflow spinning region (core) 
     \item Downward flow in \myred{hot}, low-pressure  center (``eye'')
     \item Fluid from higher region entrained (inviscid potential flow region)
     \item Characterized by strong rotation
     \begin{itemize}
      \item What generates vorticity?
        \begin{itemize}
        \item vortex stretching
        \item vortex tilting
          \end{itemize}
     \end{itemize}
    \end{itemize}
   \end{block}
  \end{column}
\end{columns}

\end{frame}


%
%
%
\begin{frame}
\frametitle{Synthetic Dust Devils}

\begin{columns}[]
  \begin{column}{0.45\linewidth}
   \begin{center}
    \includegraphics[width=0.9\linewidth]{ground_vanes}
   \end{center}
  \end{column}
  %
  \begin{column}{0.55\linewidth}
   \begin{block}{Motivation}
    \begin{itemize}
     \item Engineer synthetic vortices for energy generation
       \begin{itemize}
	\item Two tier configuration mimics natural phenomenon
	\item Turbine at upper tier vanes to extract energy
       \end{itemize}

     \item Difficult to explore space experimentally
	   \begin{itemize}
	    \item Field tests in Arizona (GT)
	    \item expensive to alter configuration
	    \item difficult to measure everything
	   \end{itemize}
    \end{itemize}
   \end{block}

  \end{column}
\end{columns}
   
\end{frame}

%
%
%
\begin{frame}
\frametitle{Example configuration}

\begin{columns}[]
  \begin{column}{0.5\linewidth}

   \begin{center}
    \includegraphics[width=0.8\linewidth]{side_example}
   \end{center}


  \end{column}
  %
  \begin{column}{0.5\linewidth}

   \begin{center}
    \includegraphics[width=0.8\linewidth]{top_example}
   \end{center}

  \end{column}
\end{columns}

   \begin{block}{Turning Vanes}
    \begin{itemize}
     \item Several control surfaces used to turn the flow
     \item Top tier significantly taller than the bottom
     \item A cone may be added to the top to contract the flow
     \item A turbine is placed above the vanes to extract kinetic energy
    \end{itemize}
   \end{block}
   
\end{frame}



%
%
%
\begin{frame}
\frametitle{Project Objectives}

 \begin{block}{System Feasibility Assessment}
  \begin{itemize}
   \item Assess technological capability
  \end{itemize}
 \end{block}


 \begin{block}{Scenario Space}
  \begin{itemize}
   \item Atmosphere, buoyancy, winds, etc. 
  \end{itemize}
  \end{block}

 \begin{block}{Solar Vortex (SoV) System Configuration}
 \begin{itemize}
  \item Represent complex configuration with CFD
	\begin{itemize}
	 \item Turning vanes
	 \item Cone (and solid surfaces)
	 \item Turbine
	\end{itemize}
  \item Inform design for 2016 field test
	\begin{itemize}
	 \item Probe and optimize SoV configuration
	 \item Predict energy output
	\end{itemize}
 \end{itemize}
 \end{block}

\end{frame}


%===============================================================================

\section{Modeling}
%
%


%
%
%
\begin{frame}
\frametitle{Dynamical Equations}
The equations describing fluid flow with natural convection are,
\begin{align}
  \frac{\partial {\bf u}}{\partial t} + {\bf u} \cdot \nabla {\bf u} =& \,
  -\frac{1}{\rho}\nabla P + \nu \nabla^2 {\bf u} \textcolor{orange}{-
 {\bf g} \frac{T-T_0}{T_0}}\\
  \nabla \cdot {\bf u} =& \, 0 \\
  \rho c_p \frac{\partial T}{\partial t} + {\bf u} \cdot \nabla T =& \, \nabla
 \cdot ( K \nabla T)
\end{align} 

\begin{itemize}
  \item Incompressible N-S + \textcolor{orange}{Boussinesq buoyancy}
  \item Temperature variation small compared to mean temperature
  \item Low mach number
\end{itemize}
\begin{center}
  However, $Ra \approx 10^9 - 10^{11}$ -- \textcolor{blue}{Need turbulence model!}
\end{center}
\end{frame}

%
%
%
\begin{frame}
\frametitle{Diffusivity Modeling}
 
 \begin{columns}[]
  \begin{column}{0.45\linewidth}

 MO (spatially homogeneous)

  \end{column}
  \begin{column}{0.55\linewidth}

   \begin{center}
    \includegraphics[width=0.6\linewidth]{wind_measurements}
   \end{center}
   
  \end{column}
\end{columns}

\end{frame}


%
%
%
\begin{frame}
\frametitle{Turning Vane Representation}

\begin{columns}[]
  \begin{column}{0.5\linewidth}

   \begin{center}
    \includegraphics[width=0.65\linewidth]{gridded_region}
   \end{center}

  \end{column}

  \begin{column}{0.5\linewidth}
   \begin{center}
    \includegraphics[width=0.5\linewidth]{forcing_region}
   \end{center}

  \end{column}
\end{columns}


 \begin{block}{Virtual Vanes}

  \begin{itemize}
   \item Volumetric forcing: 
	 \begin{itemize}
	  \item Very general formulation, no re-meshing 
	  \item Vane surface not represented
	  \item Similar to actuator-disk
	 \end{itemize}	 
%  \end{itemize}
%  
%  \begin{itemize}
%   \item Unit normal to vane surface, 
%	 \begin{equation}
%	  {\bf n}({\bf x}) = \sin \left(\textcolor{red}{\phi(r)} \right) \hat{{\bf r}}+ \cos
%	   \left(\textcolor{red}{\phi(r)} \right) \hat{{\bf \theta}} 
%	 \end{equation}
   \item Apply ${\bf f_v}$ to force flow parallel to vanes,
	 \begin{equation}
	  {\bf f}_v= -\frac{1}{\textcolor{red}{\ell_v}}|{\bf u}|({\bf u}\cdot{\bf n_v}){\bf n_v}
	 \end{equation}
  \end{itemize}
 \end{block}

\end{frame}

%
%
%
\begin{frame}
\frametitle{Scenario Modeling}
 
 \begin{columns}[]
  \begin{column}{0.5\linewidth}

   \begin{center}
    \includegraphics[width=0.7\linewidth]{wind_measurements}
   \end{center}

  \end{column}
  \begin{column}{0.5\linewidth}
 
   \begin{center}
    \includegraphics[width=0.7\linewidth]{wind_direction}
   \end{center}
  
  \end{column}
\end{columns}


  \begin{block}{Ambient Conditions}
  \begin{itemize}
  \item	Thermal B.L. profile fit to field measurements
  \item Wind heading largely constant 
  \item Wind magnitude steady 
  \end{itemize}
  \end{block}	

\end{frame}


%
%
%
\begin{frame}
\frametitle{Wind Cases}

\begin{block}

   \begin{center}
    \includegraphics[width=0.7\linewidth]{wind_3d}
   \end{center}

   \begin{itemize}
   \item Specified Inflow (Dirichlet)
   \item Top and Outflow have special mixed B.C.
   \item Sponge layers on Top and Outflow
   \item Natural boundaries on the side (Neumann)
   \end{itemize}

\end{block}
\end{frame}


%
%
%
\begin{frame}
  \frametitle{Numerics}
  \begin{block}{Formulation}
    \begin{itemize}
    \item Using Galerkin FEM discretization (N-S in weak form)
    \item Stabilized with SUPG stabilization \textit{al la} Hughes/Becker+Braack
      \begin{itemize}
      \item Adjoint stabilization scheme ($\tau$)
      \end{itemize}
    \item Time discretization via backward Euler \textcolor{blue}{(if not steady)}
    \item Newton method used to solve resulting non-linear implicit system
    \end{itemize}
  \end{block}

  \begin{block}{Software}
    \begin{itemize}
    \item Using GRINS(+Libmesh) library developed by Bauman and Stogner
    \end{itemize}
  \end{block}

  \begin{block}{Hardware}
    \begin{itemize}
    \item Runs performed on LS4,LS5, Stampede at TACC
    \item 264-528 processing cores
    \item Several million degrees of freedom for each run, locally $O(10^4)$ 
    \end{itemize}
  \end{block}

\end{frame}


%
%
%
\begin{frame}
  \frametitle{Modeling Hierarchy}

    \begin{center}
    \includegraphics[width=0.5\linewidth]{validation_hierarchy}
   \end{center}

\end{frame}

%
%
%
\begin{frame}
 \frametitle{Validation}

 \begin{columns}[]
  \begin{column}{0.55\linewidth}

   \begin{block}{Scenarios and Configurations}
    \begin{itemize}
     \item Tabletop Laboratory
     \item Cold Wind Tunnel
     \item Field Tests
     \item Hierarchy of CFD models 
	   \begin{itemize}
	    \item Gridded vs. virtual
	    \item Steady vs. unsteady
	    \item Sensitivities to QoI
	   \end{itemize}
    \end{itemize}
    \end{block}

  \end{column}

  \begin{column}{0.45\linewidth}
  \begin{center}
   \includegraphics[width=.85\linewidth]{sim_vs_exp_30_vt}
  \end{center}
  \end{column}
  \end{columns}

 %
 % 2nd column
 
 \begin{columns}[]
  \begin{column}{0.55\linewidth}
   
   \begin{center}
    \includegraphics[width=.55\linewidth]{gridded_wind}
   \end{center}
\end{column}
 \begin{column}{0.55\linewidth}
   \begin{center}
    \includegraphics[width=.55\linewidth]{virtual_wind}
   \end{center}
\end{column}
 \end{columns}


\end{frame}


%===============================================================================

%\section{Steady vs. Transient Virtual Vanes}
%
%
\begin{frame}
 \frametitle{Steady vs. Transient Virtual Vanes}

 \begin{columns}[]
  \begin{column}{0.5\linewidth}

    \begin{figure}[htb]
     \centering
     \includegraphics[width=.7\linewidth]{rom_iterate}
    \end{figure}

    \end{column}
  \begin{column}{0.5\linewidth}

    \begin{figure}[htb]
     \centering
     \includegraphics[width=.7\linewidth]{rom_radial}
    \end{figure}

  \end{column}
 \end{columns}

 \begin{block}{Unsteady vs. Steady}
   \begin{itemize}
   \item Transient vs. no $\frac{\partial ()}{\partial t}$
   %\item Solutions temporally averaged
   \item Run-time 12 hrs vs. 2 minutes
   \item Steady is principle design tool
   \end{itemize}
 \end{block}
 
\end{frame}


\section{Optimization Heuristics}
%
%
\begin{frame}
 \frametitle{Optimization}

    \begin{figure}[htb]
     \centering
     \includegraphics[width=.53\linewidth]{turbine_opt}
    \end{figure}
 
\end{frame}


%===============================================================================

\section{2016 Field Test}
%
%

%\begin{columns}
%  \begin{column}{0.5\linewidth}

\begin{frame}
 \frametitle{Top Tier Vanes}
    \begin{figure}[htb]
     \centering
     \includegraphics[width=.53\linewidth]{interp_entire_top}
    \end{figure}

%   \end{column}

%   \begin{column}{0.5\linewidth}
%    \begin{tabular}{l|l|l}
%     Name                        & Value & Meaning                    \\
%     \hline
%     $r^{\text{cyl}}$            &  3.0 meters & Radius of rigid cylinder
% 	    \\ 
%     $r^t_{\text{min}}$          &  3.0 meters & Smallest radius of top
% 	    tier vanes, relative to ground \\ 
%     $L_x$                       &  12 meters  & Distance upstream
% 	    of vanes relative to center\\
%     $\phi^{t,l}$ &  $0^{\circ}$   & Outer angle for the top tier, left
% 	    side vanes \\ 
%     $\phi^{t,r}$   &  $0^{\circ}$   & Outer angle for the top tier right
% 	    side \\ 
%     $\theta^{t,l}$ &   $30^{\circ} +\frac{\alpha}{3}$  & Inner angle for
% 	    the top tier left side vanes\\ 
%     $\theta^{t,r}$   &   $75^{\circ} +\frac{\alpha}{6}$   & Inner angle
% 	    for the top tier right side \\ 
%     $L^{t,r}$                   &  12 meters  & Width of vane in front of
% 	    cylinder \\ 
%     $L^{t,l}$                   &  10 meters  & Width of vane to the side
% 	    of cylinder \\ 
%    \end{tabular}
   
% yar  
%   \end{column}
% \end{columns}
 
 \begin{block}{}
  \begin{itemize}
   \item Vanes aligned with streamwise (wind) velocity
   \item Vanes contract flow into central region
   \item Impermeable half cylinder arc
   \item 6 meter inner diameter
  \end{itemize}
 \end{block}
\end{frame}

%
%
%
\begin{frame}
 \frametitle{Effect of the Wind}

  \begin{itemize}
    \item Energy flux more than doubles with $\Delta T$
    \item Kinetic energy flux only 100 Watts in thermal-only
    \item Higher wind velocities increases energy flux
    \item Implications for natural dust devils? 
  \end{itemize}

 \begin{figure}[htb]
  \centering
  \includegraphics[width=.55\linewidth]{wind_impact}
 \end{figure}
 
\end{frame}

%
%
%
\begin{frame}
 \frametitle{Bottom Tier Vanes}
    \begin{figure}[htb]
     \centering
     \includegraphics[width=.55\linewidth]{interp_entire_bottom}
    \end{figure}
 
 \begin{block}{}
  \begin{itemize}
   \item Bottom tier vanes not symmetric along upstream/downstream
	 sides:
	 \begin{itemize}
	  \item Different heights
	  \item Different final angles
	 \end{itemize}
   \item Significant inflow from downstream
  \end{itemize}
 \end{block}
\end{frame}

%
%
%
\begin{frame}
 \frametitle{Vertical View}
    \begin{figure}[htb]
     \centering
     \includegraphics[width=.8\linewidth]{vertical_design}
    \end{figure}
 
 \begin{block}{}
  \begin{itemize}
   \item Cone visible
   \item Height asymmetry in bottom tier vanes visible
   \item Turbine placed at top of cone
   \item Vertical partition on top of 2nd tier vanes 
  \end{itemize}
 \end{block}
\end{frame}

%
%
%
\begin{frame}
 \frametitle{CAD drawings}

 \begin{columns}[]
  \begin{column}{0.55\linewidth}

    \begin{figure}[htb]
     \centering
     \includegraphics[width=.9\linewidth]{top_down_cad_notop}
    \end{figure}

  \end{column}
   \begin{column}{0.45\linewidth}

    \begin{figure}[htb]
     \centering
     \includegraphics[width=1.0\linewidth]{skewed_cad}
    \end{figure}

    \begin{block}{Images}
     \begin{itemize}
      \item Cone clearly visible
      \item Horizontal Partitions
      \item Turning Vane Posts (drag)
      \item Consistent with field test 
      \item CAD credit: <NAME>, GT
     \end{itemize}
     \end{block}

   \end{column}
 \end{columns}

\end{frame}

%
%
%
\begin{frame}
 \frametitle{Turbine Model}

 \begin{center}
  \includegraphics[width=0.58\linewidth]{actuator_disk}
 \end{center}
 
 \begin{block}{Actuator Disk (Blade Element Momentum)}
  \begin{itemize}
  \item Represents turbine blade geometry as spinning disk
  \item Assumes axisymmetric turbine, neglects unsteady effects
  \item Not validated
  \item Common in wind turbine industry
  \end{itemize}
 \end{block}

\end{frame}

%
%
%
\begin{frame}
 \frametitle{Turbine Model}

 common model
 math behind actuator disk equations ($F=?$)
 interpolate
 flat plate, 90, 180, etc.

   \begin{center}
    \includegraphics[width=0.5\linewidth]{90}
   \end{center}

 
\end{frame}


%
%
\begin{frame}
 \frametitle{Turbine Optimization Method}

 \begin{columns}[]
  \begin{column}{0.45\linewidth}
   \begin{center}
    \includegraphics[width=1.05\linewidth]{utrc_plot}
   \end{center}

  \end{column}
    \begin{column}{0.55\linewidth}

     \begin{block}{Coupled Effort}
     \begin{itemize}
      \item Optimization effort based on UTRC (<NAME>) ``frozen flow'' 
	    and full CFD
      \item Both use actuator disk model
      \item Frozen flow used CFD as input
      \item Iterative procedure
	    \begin{itemize}
	     \item Frozen flow prediction 
	     \item ``Confirmed'' by CFD
	     \item Not automated!
	    \end{itemize}
      \item Results broadly agree
	    \begin{itemize}
	     \item inflow updated after several iterations
	    \end{itemize}
     \end{itemize}

      \end{block}
    \end{column}
 \end{columns}
\end{frame}

%
%
\begin{frame}
 \frametitle{Final Design}

\begin{columns}[]
  \begin{column}{0.45\linewidth}
 
    \begin{figure}[htb]
     \centering
     \includegraphics[width=.9\linewidth]{rotor_assem_oblique_2}
    \end{figure}

    \begin{figure}[htb]
     \centering
     \includegraphics[width=.5\linewidth]{turbine_built}
    \end{figure}

  \end{column}
  \begin{column}{0.45\linewidth}
 
 turbine energy?

 turbine placed at top of vanes

 gap in turbine 

 turbine efficiency?

 \end{column}
 \end{columns}

\end{frame}



%
%
\begin{frame}
 \frametitle{Solution Structure}


\begin{figure}[!htb]
  \centering
  \includegraphics[width=.47\linewidth]{u_field_vert}
  \hfill
  \includegraphics[width=.47\linewidth]{w_field_vert}
  \\
  \includegraphics[width=.47\linewidth]{T_field_vert}
  \hfill
  \includegraphics[width=.47\linewidth]{v_field_vert}
\end{figure}

\end{frame}

%
%
%
\begin{frame}
 \frametitle{Solution Structure}


\begin{figure}[!htb]
  \centering
  \includegraphics[width=.67\linewidth]{field_streamlines}
 \end{figure}

\begin{block}{title}
 \begin{itemize}
  \item Entraining fluid from wide region in front of device
  \item Substantial portion of backflow also included
\end{itemize}
\end{block}

\end{frame}


%
%
%
\begin{frame}
 \frametitle{2016 Field Test}

\begin{block}{What we know}
\begin{itemize}
 \item Little or no flow through device
 \item Configuration largely consistent with CFD
\end{itemize}
\end{block}

\begin{block}{Possible Explainations}
\begin{itemize}
 \item Shading
       \begin{itemize}
	\item ground temperature inside the vane region set to the ambient air 
       \end{itemize}
 \item Closed back vanes
       \begin{itemize}
	\item reduced but non-zero power
       \end{itemize}
 \item Poor field conditions
       \begin{itemize}
	\item intermittent winds, $100^{\circ}$ F, etc. 
       \end{itemize}
\end{itemize}
\end{block}

\bigbreak
\centering 
Several other possible model corrections were investigated

\end{frame}

%
%
%
\begin{frame}
 \frametitle{Drag addition}
\begin{itemize}
 \item Initial analysis indicated drag not significant
 \item Revisited after 2016 Field test
 \item Model introduced skin-friction effects
 \item Drag computed based on smallest distance between vanes
       \begin{itemize}
	\item ``worst-case'' scenario
       \end{itemize}
 \item Kinetic energy flux never reduced by more than 12\%.
\end{itemize}


\end{frame}


\begin{frame}
\frametitle{Drag Addition}

\end{frame}

%
%
\begin{frame}
 \frametitle{Solidity Modification}

\end{frame}

%
%
\begin{frame}
 \frametitle{Solidity Modification}


 \begin{columns}[]
  \begin{column}{0.55\linewidth}


   \begin{center}
    \includegraphics[width=1.1\linewidth]{solidity}
   \end{center}

  \end{column}
  \begin{column}{0.45\linewidth}
   
   \begin{block}{Correction to actuator disk}
    \begin{itemize}
     \item Model assumes flow separates from turbine blade
	   \begin{itemize}
	    \item ``worst case'' estimate
	   \end{itemize}
     \item Six and eight turbine blades greatly impacted
     \item Need calibration/validation data for more detailed assessment  
	   \begin{itemize}
	    \item Data needed from field or gridded simulation
	   \end{itemize}
     \item Even with correction, model appears inadequate
    \end{itemize}

   \end{block}
  \end{column}
 \end{columns}

\end{frame}



%
%
\begin{frame}
 \frametitle{Idealized Turbine Drag Polars}

in contrast to previous efforts, 
we now optimize power extraction for a generic turbine

\end{frame}


%
%
\section{Conclusions}
\begin{frame}
 \frametitle{System Feasibility Assessment}

 \begin{block}{Not competitive with other renewable sources}
 \begin{itemize}
 \item Order of magnitude less power for same 6 meter diameter
 \begin{itemize}
   \item Assume 1000~W/$\text{m}^2$, efficiency of 15\%
 \end{itemize}
 \end{itemize}
 \end{block}


 \begin{block}{Additional Risks}
 \begin{itemize}
 \item No validation
 \item Solutions are ``Fragile''
 \end{itemize}
 \end{block}

 \begin{block}{Additional Questions}
 \begin{itemize}
 \item Independent of economic assessment (e.g. $\text{kW}/\$$?)
 \item Why is power generated by SoV lower than Natural?
 \end{itemize}
 \end{block}


\end{frame}

%
%
\begin{frame}
 \frametitle{Conclusions and Future Work}

 \begin{block}{Contributions}
  \begin{itemize}
  \item SoV not promising
  \item Turning vane formulation (and separation, drag models)
  \item Steady models and optimization heuristic useful tools for exploring configurations
  \item Turbine design using actuator disk for non-traditional flows
  \item Simulations indicate that experiments generating dust devil-like structures
  \end{itemize}
 \end{block}


 \begin{block}{Future Work}
  \begin{itemize}
  \item Application in different scenarios (waste heat, etc.)?
  \item Turbine solidity implications for actuator disk?
  \item Tool to simulate dust devils?
  \end{itemize}
 \end{block}
 


\end{frame}

%
%
%
 \begin{frame}
   \frametitle{The End}

   \begin{figure}[htb]
     \centering
     \includegraphics[width=.55\linewidth]{arrows}
   \end{figure}

   \begin{block}{Thank you!}
    \center{<EMAIL>}
    \end{block}
 \end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% BONUS ROUND
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\appendix
\section{Extra Slides}

%
% scaling
%
\begin{frame}
  \frametitle{Estimate of Energy Scaling}
    \begin{itemize}
    \item Medium size dust devil: $3\text{m} = \text{D}$, $U =  5 \text{
	  m/s}$, $\Delta T = 30$ K, $3$ meters tall
    \item Assume a turbulent boundary layer ($\delta = 10$ cm),
      \begin{equation*}
        u(z) = U \text{ min }\left(\left(\frac{z}{\delta}\right)^7,1\right)
      \end{equation*}        
    \end{itemize}

  \begin{columns}[]
    \begin{column}{0.45\linewidth}
      Boussinesq potential energy flux over the upstream flow [Renno]:
      \begin{align*}
        E_p & = \int u(z) (\rho(z)-\rho_\infty) g z \, dA \\
        E_p & = g  \pi R \beta \rho_0 U \Delta T \left[ \frac{z_\text{max}^2}{2} - \frac{7 \delta^2}{18} \right] \\
       & = 217 \text{ Watts}
      \end{align*}

    \end{column}
  %
    \begin{column}{0.55\linewidth}
      The KE flux:  surface integral over upstream face of dust devil: 
      \begin{align*}
        \text{KE} & = \int \frac{\vec V^2}{2} \rho \vec V \cdot \hat n \, dA \\
        \text{KE} & = R \rho U^3 \left[ z_{\text{max}} - \frac{10}{11}\delta \right] \\
        & = 1144 \text{ Watts}
      \end{align*}
    \end{column}
  \end{columns}
\end{frame}


%
%
%
\begin{frame}
\frametitle{Calibration of Virtual Vanes}

    \begin{figure}[htb]
     \centering
     \includegraphics[width=.85\linewidth]{annealing}
    \end{figure}
\end{frame}

%
% mesh
%
\begin{frame}
 \frametitle{Mesh}
 
 \begin{figure}[htb]
  \centering
  \includegraphics[width=.55\linewidth]{meshing}
 \end{figure}
 
 \begin{block}{Discretization}
  \begin{itemize}
   \item Single refinement in vane region
  \end{itemize}
\begin{equation*}
	  \text{Re}_\text{cell} = \frac{\text{max}(\Delta x,\Delta y)
	   \, u}{\nu_T}
\end{equation*}	
  \begin{itemize}
   \item Boundary layer mesh visible
  \end{itemize}
 \end{block}
\end{frame}

%
% SUPG Stab
%
\begin{frame}
\frametitle{Stabilization Outline}

\begin{itemize}
 \item Cast Navier Stokes + Boussinesq equations into weak form
 \item Prepare as operator $Lc=f$
 \item Calculate Fr\'echet derivative (to calculate operator adjoint)
 \item Separate into differential (P) and constant (Z) components,
       $L'[c] = P + Z$
 \item Choose stabilization operator such that $S = -P^*$
 \item Then stabilization has form, $a_h(c,\phi) = a(c,\phi) + \langle
       Lc,S\phi \rangle_\tau$
\end{itemize}
\end{frame}

%
%
%
\begin{frame}
\frametitle{Simulation Geometry and Boundary Conditions}

\begin{block}{Specified Inflow}

   \begin{itemize}
   \item 7th order turbulent boundary layer, 
     \begin{equation*}
       u_{\text{in}}(z) = U \text{ min }\left(\left(\frac{z}{\delta}\right)^7,1\right)
       \label{eq:bl_u}
     \end{equation*}
   \item The thermal inflow is then,
     \begin{equation*}
       T_{\text{in}}(z) = \Delta T \left(1- \text{ min
       }\left(\left(\frac{z}{\delta}\right)^7,1\right)\right)
       + T_0 \textcolor{blue}{- \beta}.  
     \end{equation*}
   \end{itemize}
\end{block}

\begin{block}{Ground}
  \begin{itemize}
    \item modeled with a Dirichlet boundary condition such that, 
      \begin{align*}
        {\bf u} &= 0 \quad \text{ on } \Gamma_G \\
        T &= T_g
      \end{align*}
  \end{itemize}
  \end{block}

\end{frame}

%
%
%
\begin{frame}
\frametitle{2015 Field Validation Study}
 
\begin{columns}[]
  \begin{column}{0.52\linewidth}
   \begin{block}{Scenario parameter uncertainty}
    \begin{itemize}
     \item Wind speed: 2-5 m/s
     \item Wind direction($\approx20^{\circ}$)
     \item $T_{\text{s}} = 50-60^{\circ}$ C ($\approx 121-140^{\circ}$ F)
     \item No boundary layer temperature data 
	   (DAQ equipment malfunctioned)
    \end{itemize}
   \end{block}
  \end{column}

  \begin{column}{0.49\linewidth}

   \begin{center}
    \includegraphics[width=0.99\linewidth]{validate_field}
   \end{center}
  \end{column}
 \end{columns}

 \begin{center}
  \begin{itemize}
   \item CFD broadly consistent with field observations
	 \begin{itemize}
	  \item Experimental data extremely limited
	 \end{itemize}
   \item Kinetic energy fluxes match to within 10\%
  \end{itemize}
  \end{center}

\end{frame}


%
%
%
\begin{frame}
\frametitle{Thermal-Only Structure}

\begin{figure}[!htb]
  \centering
  \includegraphics[width=.38\linewidth]{vt}
  \hfill
  \includegraphics[width=.38\linewidth]{vz}
  \\
  \includegraphics[width=.38\linewidth]{t}
  \hfill
  \includegraphics[width=.38\linewidth]{p}
\end{figure}


\end{frame}

%
%
%
\begin{frame}
\frametitle{Thermal-Only Structure}

    \begin{figure}[htb]
     \centering
     \includegraphics[width=.85\linewidth]{rankene}
    \end{figure}
\end{frame}

%
%
%
\begin{frame}
\frametitle{Proposal Timeline}
\small
\begin{center}
\begin{table}
\centering
\begin{minipage}[t]{1.6\linewidth}
\color{black}
 \rule{\linewidth}{1pt}
 \yto{April 2016}{Conclude parameter sweeps and optimization of apparatus}
 \yto{April 2016}{Turbine actuator-disk verification, validation and prediction}
 \ytl{May 2016}{Proposed configuration and predictions for experimental
 team}
 \ytb{July 2016}{Comparisons between synthetic and natural dust devil physics}
 \ytl{Aug 2016}{Validation against 2016 field data}
 \ytl{Sept 2016}{Optimal drag polar prediction}
\ytl{Nov. 2016}{Dissertation Defense}
 \bigskip
 \rule{\linewidth}{1pt}%
\end{minipage}%
\end{table}
 \label{tab:prop}
\end{center}

\end{frame}


%
%
%
\begin{frame}
 \frametitle{Course list}
 \footnotesize
 \begin{table}[h]
  \centering
  \begin{tabular}{llll}
   \hline \hline
   Course \# & Semester & Course Name & Instructor \\ 
   \hline 
   ME381P   & 2014 Spr.   & Validation \& Uncertainty Quantification &
	       Prof. <NAME> \\
   ME 382R5 & 2014 Spr.   & Advanced Combustion & Prof. <NAME> \\
   CSE 397  & 2014 Spr.   &  Comp. \& Var. Methods for
	   Inv. Problems & Prof. <NAME> \\
   SDS 384  & 2014 Fall   & Bayesian Statistical Methods & Prof. S.
	       Walker \\
   SDS 394  & 2014 Fall   & Scientific \& Technical Computing & Dr. <NAME> \\
   ME 382P  & 2015 Fall   & Adv. Exp. Methods in Thermal/Fluids
	   & Prof. <NAME> \\  
   \hline \hline
  \end{tabular} 
 \end{table}

\end{frame}


\end{document}
<commit_msg>adding steady notes, done with this slide<commit_after>\documentclass[mathserif]{beamer}
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#2.\\[7pt]}\\[-3pt]} 


\date{Nov. 8th, 2016}
\author[<NAME>]{<NAME>}
\institute{Department of Mechanical Engineering \\ The University of
Texas at Austin}
\title[Dissertation Defense]{%
    \mbox{Numerical Simulation of Synthetic,}
    \mbox{Buoyancy-Induced Columnar Vortices}
}

\begin{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%===============================================================================
% Title page
\begin{frame}
%
\titlepage{}


%\begin{centering}
\mbox{{\scriptsize Acknowledgment: This material is based upon work
 supported}}
\mbox{{\scriptsize by the Department of Energy [ARPA-E] under Award
 Number [DE-FOA-0000670]}} 
%\end{centering}
\raisebox{-0.2in}{\hbox{\includegraphics[width=4.5in]{figs/Footer}}\hfill}
%
\end{frame}

%===============================================================================

 \section{Motivation}
 %
 %

\begin{frame}
  \frametitle{The dust devil phenomenon}
  \begin{columns}[]
    \begin{column}{0.45\linewidth}
      \begin{center}
        \includegraphics[width=0.6\linewidth]{Dust_devil}
      \end{center}
      \begin{center}
        \includegraphics[width=0.8\linewidth]{martian_dust}
      \end{center}
    \end{column}
  %
    \begin{column}{0.55\linewidth}

      \begin{block}{Turbulent, unstable motions}
        \begin{itemize}
	 \item Incident solar energy absorbed into ground
	       \begin{itemize}
		\item peak solar insolation 1000~W/$\text{m}^2$ 
	       \end{itemize}
	 \item Ground heats air by conduction
	 \item Heated air expands, lowering density, 
	       driven upward by force of buoyancy
	 \item Arizona in Summer ($\Delta T= 30$ K) 
	 \item $\text{Ra} \approx 10^9 - 10^{11} \rarrow$ 
	       unstably stratified fluid
	 \item Velocities can exceed 33 m/s.
        \end{itemize}
      \end{block}
      How much energy is present in one of these objects? 
      \end{column}
    \end{columns}
  \end{frame}

%
%
%
\begin{frame}
\frametitle{Energy Estimate [Sinclair, Battan, etc. ]}

   \begin{center}
    \includegraphics[width=0.6\linewidth]{sinclair_cut}
   \end{center}

\begin{itemize}
  \item Estimate energy from Sinclair's measured velocities 
  \item R $\approx$ 5 meters, $V_{\theta} \approx V_{z} \approx 10$ m/s 
  \item Kinetic Energy flux $ = \frac{1}{2}\rho \int V_z (V_z^2 + V_{\theta}^2)
	\, dA \approx 45$ kW
 \begin{itemize}
  \item Exceeds PV per $\text{m}^2$
 \end{itemize}
\end{itemize}

\end{frame}

\begin{frame}
\frametitle{Dust Devil Structure [Sinclair,Kanak,Renno]}

\begin{columns}[]
  \begin{column}{0.45\linewidth}
   \begin{center}
    \includegraphics[width=0.9\linewidth]{ground}
   \end{center}
  \end{column}
  %
  \begin{column}{0.55\linewidth}
    \begin{block}{Dust devil structure}
    \begin{itemize}
     \item Driven by buoyancy of hot ground
     \item Near surface: radial inflow spinning region (core) 
     \item Downward flow in \myred{hot}, low-pressure  center (``eye'')
     \item Fluid from higher region entrained (inviscid potential flow region)
     \item Characterized by strong rotation
     \begin{itemize}
      \item What generates vorticity?
        \begin{itemize}
        \item vortex stretching
        \item vortex tilting
          \end{itemize}
     \end{itemize}
    \end{itemize}
   \end{block}
  \end{column}
\end{columns}

\end{frame}


%
%
%
\begin{frame}
\frametitle{Synthetic Dust Devils}

\begin{columns}[]
  \begin{column}{0.45\linewidth}
   \begin{center}
    \includegraphics[width=0.9\linewidth]{ground_vanes}
   \end{center}
  \end{column}
  %
  \begin{column}{0.55\linewidth}
   \begin{block}{Motivation}
    \begin{itemize}
     \item Engineer synthetic vortices for energy generation
       \begin{itemize}
	\item Two tier configuration mimics natural phenomenon
	\item Turbine at upper tier vanes to extract energy
       \end{itemize}

     \item Difficult to explore space experimentally
	   \begin{itemize}
	    \item Field tests in Arizona (GT)
	    \item expensive to alter configuration
	    \item difficult to measure everything
	   \end{itemize}
    \end{itemize}
   \end{block}

  \end{column}
\end{columns}
   
\end{frame}

%
%
%
\begin{frame}
\frametitle{Example configuration}

\begin{columns}[]
  \begin{column}{0.5\linewidth}

   \begin{center}
    \includegraphics[width=0.8\linewidth]{side_example}
   \end{center}


  \end{column}
  %
  \begin{column}{0.5\linewidth}

   \begin{center}
    \includegraphics[width=0.8\linewidth]{top_example}
   \end{center}

  \end{column}
\end{columns}

   \begin{block}{Turning Vanes}
    \begin{itemize}
     \item Several control surfaces used to turn the flow
     \item Top tier significantly taller than the bottom
     \item A cone may be added to the top to contract the flow
     \item A turbine is placed above the vanes to extract kinetic energy
    \end{itemize}
   \end{block}
   
\end{frame}



%
%
%
\begin{frame}
\frametitle{Project Objectives}

 \begin{block}{System Feasibility Assessment}
  \begin{itemize}
   \item Assess technological capability
  \end{itemize}
 \end{block}


 \begin{block}{Scenario Space}
  \begin{itemize}
   \item Atmosphere, buoyancy, winds, etc. 
  \end{itemize}
  \end{block}

 \begin{block}{Solar Vortex (SoV) System Configuration}
 \begin{itemize}
  \item Represent complex configuration with CFD
	\begin{itemize}
	 \item Turning vanes
	 \item Cone (and solid surfaces)
	 \item Turbine
	\end{itemize}
  \item Inform design for 2016 field test
	\begin{itemize}
	 \item Probe and optimize SoV configuration
	 \item Predict energy output
	\end{itemize}
 \end{itemize}
 \end{block}

\end{frame}


%===============================================================================

\section{Modeling}
%
%


%
%
%
\begin{frame}
\frametitle{Dynamical Equations}
The equations describing fluid flow with natural convection are,
\begin{align}
  \frac{\partial {\bf u}}{\partial t} + {\bf u} \cdot \nabla {\bf u} =& \,
  -\frac{1}{\rho}\nabla P + \nu \nabla^2 {\bf u} \textcolor{orange}{-
 {\bf g} \frac{T-T_0}{T_0}}\\
  \nabla \cdot {\bf u} =& \, 0 \\
  \rho c_p \frac{\partial T}{\partial t} + {\bf u} \cdot \nabla T =& \, \nabla
 \cdot ( K \nabla T)
\end{align} 

\begin{itemize}
  \item Incompressible N-S + \textcolor{orange}{Boussinesq buoyancy}
  \item Temperature variation small compared to mean temperature
  \item Low mach number
\end{itemize}
\begin{center}
  However, $Ra \approx 10^9 - 10^{11}$ -- \textcolor{blue}{Need turbulence model!}
\end{center}
\end{frame}

%
%
%
\begin{frame}
\frametitle{Diffusivity Modeling}
 
 \begin{columns}[]
  \begin{column}{0.45\linewidth}

 MO (spatially homogeneous)

  \end{column}
  \begin{column}{0.55\linewidth}

   \begin{center}
    \includegraphics[width=0.6\linewidth]{wind_measurements}
   \end{center}
   
  \end{column}
\end{columns}

\end{frame}


%
%
%
\begin{frame}
\frametitle{Turning Vane Representation}

\begin{columns}[]
  \begin{column}{0.5\linewidth}

   \begin{center}
    \includegraphics[width=0.65\linewidth]{gridded_region}
   \end{center}

  \end{column}

  \begin{column}{0.5\linewidth}
   \begin{center}
    \includegraphics[width=0.5\linewidth]{forcing_region}
   \end{center}

  \end{column}
\end{columns}


 \begin{block}{Virtual Vanes}

  \begin{itemize}
   \item Volumetric forcing: 
	 \begin{itemize}
	  \item Very general formulation, no re-meshing 
	  \item Vane surface not represented
	  \item Similar to actuator-disk
	 \end{itemize}	 
%  \end{itemize}
%  
%  \begin{itemize}
%   \item Unit normal to vane surface, 
%	 \begin{equation}
%	  {\bf n}({\bf x}) = \sin \left(\textcolor{red}{\phi(r)} \right) \hat{{\bf r}}+ \cos
%	   \left(\textcolor{red}{\phi(r)} \right) \hat{{\bf \theta}} 
%	 \end{equation}
   \item Apply ${\bf f_v}$ to force flow parallel to vanes,
	 \begin{equation}
	  {\bf f}_v= -\frac{1}{\textcolor{red}{\ell_v}}|{\bf u}|({\bf u}\cdot{\bf n_v}){\bf n_v}
	 \end{equation}
  \end{itemize}
 \end{block}

\end{frame}

%
%
%
\begin{frame}
\frametitle{Scenario Modeling}
 
 \begin{columns}[]
  \begin{column}{0.5\linewidth}

   \begin{center}
    \includegraphics[width=0.7\linewidth]{wind_measurements}
   \end{center}

  \end{column}
  \begin{column}{0.5\linewidth}
 
   \begin{center}
    \includegraphics[width=0.7\linewidth]{wind_direction}
   \end{center}
  
  \end{column}
\end{columns}


  \begin{block}{Ambient Conditions}
  \begin{itemize}
  \item	Thermal B.L. profile fit to field measurements
  \item Wind heading largely constant 
  \item Wind magnitude steady 
  \end{itemize}
  \end{block}	

\end{frame}


%
%
%
\begin{frame}
\frametitle{Wind Cases}

\begin{block}

   \begin{center}
    \includegraphics[width=0.7\linewidth]{wind_3d}
   \end{center}

   \begin{itemize}
   \item Specified Inflow (Dirichlet)
   \item Top and Outflow have special mixed B.C.
   \item Sponge layers on Top and Outflow
   \item Natural boundaries on the side (Neumann)
   \end{itemize}

\end{block}
\end{frame}


%
%
%
\begin{frame}
  \frametitle{Numerics}
  \begin{block}{Formulation}
    \begin{itemize}
    \item Using Galerkin FEM discretization (N-S in weak form)
    \item Stabilized with SUPG stabilization \textit{al la} Hughes/Becker+Braack
      \begin{itemize}
      \item Adjoint stabilization scheme ($\tau$)
      \end{itemize}
    \item Time discretization via backward Euler \textcolor{blue}{(if not steady)}
    \item Newton method used to solve resulting non-linear implicit system
    \end{itemize}
  \end{block}

  \begin{block}{Software}
    \begin{itemize}
    \item Using GRINS(+Libmesh) library developed by Bauman and Stogner
    \end{itemize}
  \end{block}

  \begin{block}{Hardware}
    \begin{itemize}
    \item Runs performed on LS4,LS5, Stampede at TACC
    \item 264-528 processing cores
    \item Several million degrees of freedom for each run, locally $O(10^4)$ 
    \end{itemize}
  \end{block}

\end{frame}


%
%
%
\begin{frame}
  \frametitle{Modeling Hierarchy}

    \begin{center}
    \includegraphics[width=0.5\linewidth]{validation_hierarchy}
   \end{center}

\end{frame}

%
%
%
\begin{frame}
 \frametitle{Validation}

 \begin{columns}[]
  \begin{column}{0.55\linewidth}

   \begin{block}{Scenarios and Configurations}
    \begin{itemize}
     \item Tabletop Laboratory
     \item Cold Wind Tunnel
     \item Field Tests
     \item Hierarchy of CFD models 
	   \begin{itemize}
	    \item Gridded vs. virtual
	    \item Steady vs. unsteady
	    \item Sensitivities to QoI
	   \end{itemize}
    \end{itemize}
    \end{block}

  \end{column}

  \begin{column}{0.45\linewidth}
  \begin{center}
   \includegraphics[width=.85\linewidth]{sim_vs_exp_30_vt}
  \end{center}
  \end{column}
  \end{columns}

 %
 % 2nd column
 
 \begin{columns}[]
  \begin{column}{0.55\linewidth}
   
   \begin{center}
    \includegraphics[width=.55\linewidth]{gridded_wind}
   \end{center}
\end{column}
 \begin{column}{0.55\linewidth}
   \begin{center}
    \includegraphics[width=.55\linewidth]{virtual_wind}
   \end{center}
\end{column}
 \end{columns}


\end{frame}


%===============================================================================

%\section{Steady vs. Transient Virtual Vanes}
%
%
\begin{frame}
 \frametitle{Steady vs. Transient Virtual Vanes}

 \begin{columns}[]
  \begin{column}{0.5\linewidth}

    \begin{figure}[htb]
     \centering
     \includegraphics[width=.9\linewidth]{rom_radial}
    \end{figure}

    \end{column}
  \begin{column}{0.5\linewidth}

    \begin{figure}[htb]
     \centering
     \includegraphics[width=.9\linewidth]{rom_iterate}
    \end{figure}

  \end{column}
 \end{columns}

 \begin{block}{Unsteady vs. Steady}
   \begin{itemize}
   \item Transient vs. no $\frac{\partial ()}{\partial t}$
   %\item Solutions temporally averaged
   \item Run-time 12 hrs vs. 2 minutes
   \item Steady is principle design tool
   \end{itemize}
 \end{block}
 
\end{frame}


\section{Optimization Heuristics}
%
%
\begin{frame}
 \frametitle{Optimization}

    \begin{figure}[htb]
     \centering
     \includegraphics[width=.53\linewidth]{turbine_opt}
    \end{figure}
 
\end{frame}


%===============================================================================

\section{2016 Field Test}
%
%

%\begin{columns}
%  \begin{column}{0.5\linewidth}

\begin{frame}
 \frametitle{Top Tier Vanes}
    \begin{figure}[htb]
     \centering
     \includegraphics[width=.53\linewidth]{interp_entire_top}
    \end{figure}

%   \end{column}

%   \begin{column}{0.5\linewidth}
%    \begin{tabular}{l|l|l}
%     Name                        & Value & Meaning                    \\
%     \hline
%     $r^{\text{cyl}}$            &  3.0 meters & Radius of rigid cylinder
% 	    \\ 
%     $r^t_{\text{min}}$          &  3.0 meters & Smallest radius of top
% 	    tier vanes, relative to ground \\ 
%     $L_x$                       &  12 meters  & Distance upstream
% 	    of vanes relative to center\\
%     $\phi^{t,l}$ &  $0^{\circ}$   & Outer angle for the top tier, left
% 	    side vanes \\ 
%     $\phi^{t,r}$   &  $0^{\circ}$   & Outer angle for the top tier right
% 	    side \\ 
%     $\theta^{t,l}$ &   $30^{\circ} +\frac{\alpha}{3}$  & Inner angle for
% 	    the top tier left side vanes\\ 
%     $\theta^{t,r}$   &   $75^{\circ} +\frac{\alpha}{6}$   & Inner angle
% 	    for the top tier right side \\ 
%     $L^{t,r}$                   &  12 meters  & Width of vane in front of
% 	    cylinder \\ 
%     $L^{t,l}$                   &  10 meters  & Width of vane to the side
% 	    of cylinder \\ 
%    \end{tabular}
   
% yar  
%   \end{column}
% \end{columns}
 
 \begin{block}{}
  \begin{itemize}
   \item Vanes aligned with streamwise (wind) velocity
   \item Vanes contract flow into central region
   \item Impermeable half cylinder arc
   \item 6 meter inner diameter
  \end{itemize}
 \end{block}
\end{frame}

%
%
%
\begin{frame}
 \frametitle{Effect of the Wind}

  \begin{itemize}
    \item Energy flux more than doubles with $\Delta T$
    \item Kinetic energy flux only 100 Watts in thermal-only
    \item Higher wind velocities increases energy flux
    \item Implications for natural dust devils? 
  \end{itemize}

 \begin{figure}[htb]
  \centering
  \includegraphics[width=.55\linewidth]{wind_impact}
 \end{figure}
 
\end{frame}

%
%
%
\begin{frame}
 \frametitle{Bottom Tier Vanes}
    \begin{figure}[htb]
     \centering
     \includegraphics[width=.55\linewidth]{interp_entire_bottom}
    \end{figure}
 
 \begin{block}{}
  \begin{itemize}
   \item Bottom tier vanes not symmetric along upstream/downstream
	 sides:
	 \begin{itemize}
	  \item Different heights
	  \item Different final angles
	 \end{itemize}
   \item Significant inflow from downstream
  \end{itemize}
 \end{block}
\end{frame}

%
%
%
\begin{frame}
 \frametitle{Vertical View}
    \begin{figure}[htb]
     \centering
     \includegraphics[width=.8\linewidth]{vertical_design}
    \end{figure}
 
 \begin{block}{}
  \begin{itemize}
   \item Cone visible
   \item Height asymmetry in bottom tier vanes visible
   \item Turbine placed at top of cone
   \item Vertical partition on top of 2nd tier vanes 
  \end{itemize}
 \end{block}
\end{frame}

%
%
%
\begin{frame}
 \frametitle{CAD drawings}

 \begin{columns}[]
  \begin{column}{0.55\linewidth}

    \begin{figure}[htb]
     \centering
     \includegraphics[width=.9\linewidth]{top_down_cad_notop}
    \end{figure}

  \end{column}
   \begin{column}{0.45\linewidth}

    \begin{figure}[htb]
     \centering
     \includegraphics[width=1.0\linewidth]{skewed_cad}
    \end{figure}

    \begin{block}{Images}
     \begin{itemize}
      \item Cone clearly visible
      \item Horizontal Partitions
      \item Turning Vane Posts (drag)
      \item Consistent with field test 
      \item CAD credit: <NAME>, GT
     \end{itemize}
     \end{block}

   \end{column}
 \end{columns}

\end{frame}

%
%
%
\begin{frame}
 \frametitle{Turbine Model}

 \begin{center}
  \includegraphics[width=0.58\linewidth]{actuator_disk}
 \end{center}
 
 \begin{block}{Actuator Disk (Blade Element Momentum)}
  \begin{itemize}
  \item Represents turbine blade geometry as spinning disk
  \item Assumes axisymmetric turbine, neglects unsteady effects
  \item Not validated
  \item Common in wind turbine industry
  \end{itemize}
 \end{block}

\end{frame}

%
%
%
\begin{frame}
 \frametitle{Turbine Model}

 common model
 math behind actuator disk equations ($F=?$)
 interpolate
 flat plate, 90, 180, etc.

   \begin{center}
    \includegraphics[width=0.5\linewidth]{90}
   \end{center}

 
\end{frame}


%
%
\begin{frame}
 \frametitle{Turbine Optimization Method}

 \begin{columns}[]
  \begin{column}{0.45\linewidth}
   \begin{center}
    \includegraphics[width=1.05\linewidth]{utrc_plot}
   \end{center}

  \end{column}
    \begin{column}{0.55\linewidth}

     \begin{block}{Coupled Effort}
     \begin{itemize}
      \item Optimization effort based on UTRC (Duane McCormick) ``frozen flow'' 
	    and full CFD
      \item Both use actuator disk model
      \item Frozen flow used CFD as input
      \item Iterative procedure
	    \begin{itemize}
	     \item Frozen flow prediction 
	     \item ``Confirmed'' by CFD
	     \item Not automated!
	    \end{itemize}
      \item Results broadly agree
	    \begin{itemize}
	     \item inflow updated after several iterations
	    \end{itemize}
     \end{itemize}

      \end{block}
    \end{column}
 \end{columns}
\end{frame}

%
%
\begin{frame}
 \frametitle{Final Design}

\begin{columns}[]
  \begin{column}{0.45\linewidth}
 
    \begin{figure}[htb]
     \centering
     \includegraphics[width=.9\linewidth]{rotor_assem_oblique_2}
    \end{figure}

    \begin{figure}[htb]
     \centering
     \includegraphics[width=.5\linewidth]{turbine_built}
    \end{figure}

  \end{column}
  \begin{column}{0.45\linewidth}
 
 turbine energy?

 turbine placed at top of vanes

 gap in turbine 

 turbine efficiency?

 \end{column}
 \end{columns}

\end{frame}



%
%
\begin{frame}
 \frametitle{Solution Structure}


\begin{figure}[!htb]
  \centering
  \includegraphics[width=.47\linewidth]{u_field_vert}
  \hfill
  \includegraphics[width=.47\linewidth]{w_field_vert}
  \\
  \includegraphics[width=.47\linewidth]{T_field_vert}
  \hfill
  \includegraphics[width=.47\linewidth]{v_field_vert}
\end{figure}

\end{frame}

%
%
%
\begin{frame}
 \frametitle{Solution Structure}


\begin{figure}[!htb]
  \centering
  \includegraphics[width=.67\linewidth]{field_streamlines}
 \end{figure}

\begin{block}{title}
 \begin{itemize}
  \item Entraining fluid from wide region in front of device
  \item Substantial portion of backflow also included
\end{itemize}
\end{block}

\end{frame}


%
%
%
\begin{frame}
 \frametitle{2016 Field Test}

\begin{block}{What we know}
\begin{itemize}
 \item Little or no flow through device
 \item Configuration largely consistent with CFD
\end{itemize}
\end{block}

\begin{block}{Possible Explainations}
\begin{itemize}
 \item Shading
       \begin{itemize}
	\item ground temperature inside the vane region set to the ambient air 
       \end{itemize}
 \item Closed back vanes
       \begin{itemize}
	\item reduced but non-zero power
       \end{itemize}
 \item Poor field conditions
       \begin{itemize}
	\item intermittent winds, $100^{\circ}$ F, etc. 
       \end{itemize}
\end{itemize}
\end{block}

\bigbreak
\centering 
Several other possible model corrections were investigated

\end{frame}

%
%
%
\begin{frame}
 \frametitle{Drag addition}
\begin{itemize}
 \item Initial analysis indicated drag not significant
 \item Revisited after 2016 Field test
 \item Model introduced skin-friction effects
 \item Drag computed based on smallest distance between vanes
       \begin{itemize}
	\item ``worst-case'' scenario
       \end{itemize}
 \item Kinetic energy flux never reduced by more than 12\%.
\end{itemize}


\end{frame}


\begin{frame}
\frametitle{Drag Addition}

\end{frame}

%
%
\begin{frame}
 \frametitle{Solidity Modification}

\end{frame}

%
%
\begin{frame}
 \frametitle{Solidity Modification}


 \begin{columns}[]
  \begin{column}{0.55\linewidth}


   \begin{center}
    \includegraphics[width=1.1\linewidth]{solidity}
   \end{center}

  \end{column}
  \begin{column}{0.45\linewidth}
   
   \begin{block}{Correction to actuator disk}
    \begin{itemize}
     \item Model assumes flow separates from turbine blade
	   \begin{itemize}
	    \item ``worst case'' estimate
	   \end{itemize}
     \item Six and eight turbine blades greatly impacted
     \item Need calibration/validation data for more detailed assessment  
	   \begin{itemize}
	    \item Data needed from field or gridded simulation
	   \end{itemize}
     \item Even with correction, model appears inadequate
    \end{itemize}

   \end{block}
  \end{column}
 \end{columns}

\end{frame}



%
%
\begin{frame}
 \frametitle{Idealized Turbine Drag Polars}

in contrast to previous efforts, 
we now optimize power extraction for a generic turbine

\end{frame}


%
%
\section{Conclusions}
\begin{frame}
 \frametitle{System Feasibility Assessment}

 \begin{block}{Not competitive with other renewable sources}
 \begin{itemize}
 \item Order of magnitude less power for same 6 meter diameter
 \begin{itemize}
   \item Assume 1000~W/$\text{m}^2$, efficiency of 15\%
 \end{itemize}
 \end{itemize}
 \end{block}


 \begin{block}{Additional Risks}
 \begin{itemize}
 \item No validation
 \item Solutions are ``Fragile''
 \end{itemize}
 \end{block}

 \begin{block}{Additional Questions}
 \begin{itemize}
 \item Independent of economic assessment (e.g. $\text{kW}/\$$?)
 \item Why is power generated by SoV lower than Natural?
 \end{itemize}
 \end{block}


\end{frame}

%
%
\begin{frame}
 \frametitle{Conclusions and Future Work}

 \begin{block}{Contributions}
  \begin{itemize}
  \item SoV not promising
  \item Turning vane formulation (and separation, drag models)
  \item Steady models and optimization heuristic useful tools for exploring configurations
  \item Turbine design using actuator disk for non-traditional flows
  \item Simulations indicate that experiments generating dust devil-like structures
  \end{itemize}
 \end{block}


 \begin{block}{Future Work}
  \begin{itemize}
  \item Application in different scenarios (waste heat, etc.)?
  \item Turbine solidity implications for actuator disk?
  \item Tool to simulate dust devils?
  \end{itemize}
 \end{block}
 


\end{frame}

%
%
%
 \begin{frame}
   \frametitle{The End}

   \begin{figure}[htb]
     \centering
     \includegraphics[width=.55\linewidth]{arrows}
   \end{figure}

   \begin{block}{Thank you!}
    \center{<EMAIL>}
    \end{block}
 \end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% BONUS ROUND
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\appendix
\section{Extra Slides}

%
% scaling
%
\begin{frame}
  \frametitle{Estimate of Energy Scaling}
    \begin{itemize}
    \item Medium size dust devil: $3\text{m} = \text{D}$, $U =  5 \text{
	  m/s}$, $\Delta T = 30$ K, $3$ meters tall
    \item Assume a turbulent boundary layer ($\delta = 10$ cm),
      \begin{equation*}
        u(z) = U \text{ min }\left(\left(\frac{z}{\delta}\right)^7,1\right)
      \end{equation*}        
    \end{itemize}

  \begin{columns}[]
    \begin{column}{0.45\linewidth}
      Boussinesq potential energy flux over the upstream flow [Renno]:
      \begin{align*}
        E_p & = \int u(z) (\rho(z)-\rho_\infty) g z \, dA \\
        E_p & = g  \pi R \beta \rho_0 U \Delta T \left[ \frac{z_\text{max}^2}{2} - \frac{7 \delta^2}{18} \right] \\
       & = 217 \text{ Watts}
      \end{align*}

    \end{column}
  %
    \begin{column}{0.55\linewidth}
      The KE flux:  surface integral over upstream face of dust devil: 
      \begin{align*}
        \text{KE} & = \int \frac{\vec V^2}{2} \rho \vec V \cdot \hat n \, dA \\
        \text{KE} & = R \rho U^3 \left[ z_{\text{max}} - \frac{10}{11}\delta \right] \\
        & = 1144 \text{ Watts}
      \end{align*}
    \end{column}
  \end{columns}
\end{frame}


%
%
%
\begin{frame}
\frametitle{Calibration of Virtual Vanes}

    \begin{figure}[htb]
     \centering
     \includegraphics[width=.85\linewidth]{annealing}
    \end{figure}
\end{frame}

%
% mesh
%
\begin{frame}
 \frametitle{Mesh}
 
 \begin{figure}[htb]
  \centering
  \includegraphics[width=.55\linewidth]{meshing}
 \end{figure}
 
 \begin{block}{Discretization}
  \begin{itemize}
   \item Single refinement in vane region
  \end{itemize}
\begin{equation*}
	  \text{Re}_\text{cell} = \frac{\text{max}(\Delta x,\Delta y)
	   \, u}{\nu_T}
\end{equation*}	
  \begin{itemize}
   \item Boundary layer mesh visible
  \end{itemize}
 \end{block}
\end{frame}

%
% SUPG Stab
%
\begin{frame}
\frametitle{Stabilization Outline}

\begin{itemize}
 \item Cast Navier Stokes + Boussinesq equations into weak form
 \item Prepare as operator $Lc=f$
 \item Calculate Fr\'echet derivative (to calculate operator adjoint)
 \item Separate into differential (P) and constant (Z) components,
       $L'[c] = P + Z$
 \item Choose stabilization operator such that $S = -P^*$
 \item Then stabilization has form, $a_h(c,\phi) = a(c,\phi) + \langle
       Lc,S\phi \rangle_\tau$
\end{itemize}
\end{frame}

%
%
%
\begin{frame}
\frametitle{Simulation Geometry and Boundary Conditions}

\begin{block}{Specified Inflow}

   \begin{itemize}
   \item 7th order turbulent boundary layer, 
     \begin{equation*}
       u_{\text{in}}(z) = U \text{ min }\left(\left(\frac{z}{\delta}\right)^7,1\right)
       \label{eq:bl_u}
     \end{equation*}
   \item The thermal inflow is then,
     \begin{equation*}
       T_{\text{in}}(z) = \Delta T \left(1- \text{ min
       }\left(\left(\frac{z}{\delta}\right)^7,1\right)\right)
       + T_0 \textcolor{blue}{- \beta}.  
     \end{equation*}
   \end{itemize}
\end{block}

\begin{block}{Ground}
  \begin{itemize}
    \item modeled with a Dirichlet boundary condition such that, 
      \begin{align*}
        {\bf u} &= 0 \quad \text{ on } \Gamma_G \\
        T &= T_g
      \end{align*}
  \end{itemize}
  \end{block}

\end{frame}

%
%
%
\begin{frame}
\frametitle{2015 Field Validation Study}
 
\begin{columns}[]
  \begin{column}{0.52\linewidth}
   \begin{block}{Scenario parameter uncertainty}
    \begin{itemize}
     \item Wind speed: 2-5 m/s
     \item Wind direction($\approx20^{\circ}$)
     \item $T_{\text{s}} = 50-60^{\circ}$ C ($\approx 121-140^{\circ}$ F)
     \item No boundary layer temperature data 
	   (DAQ equipment malfunctioned)
    \end{itemize}
   \end{block}
  \end{column}

  \begin{column}{0.49\linewidth}

   \begin{center}
    \includegraphics[width=0.99\linewidth]{validate_field}
   \end{center}
  \end{column}
 \end{columns}

 \begin{center}
  \begin{itemize}
   \item CFD broadly consistent with field observations
	 \begin{itemize}
	  \item Experimental data extremely limited
	 \end{itemize}
   \item Kinetic energy fluxes match to within 10\%
  \end{itemize}
  \end{center}

\end{frame}


%
%
%
\begin{frame}
\frametitle{Thermal-Only Structure}

\begin{figure}[!htb]
  \centering
  \includegraphics[width=.38\linewidth]{vt}
  \hfill
  \includegraphics[width=.38\linewidth]{vz}
  \\
  \includegraphics[width=.38\linewidth]{t}
  \hfill
  \includegraphics[width=.38\linewidth]{p}
\end{figure}


\end{frame}

%
%
%
\begin{frame}
\frametitle{Thermal-Only Structure}

    \begin{figure}[htb]
     \centering
     \includegraphics[width=.85\linewidth]{rankene}
    \end{figure}
\end{frame}

%
%
%
\begin{frame}
\frametitle{Proposal Timeline}
\small
\begin{center}
\begin{table}
\centering
\begin{minipage}[t]{1.6\linewidth}
\color{black}
 \rule{\linewidth}{1pt}
 \yto{April 2016}{Conclude parameter sweeps and optimization of apparatus}
 \yto{April 2016}{Turbine actuator-disk verification, validation and prediction}
 \ytl{May 2016}{Proposed configuration and predictions for experimental
 team}
 \ytb{July 2016}{Comparisons between synthetic and natural dust devil physics}
 \ytl{Aug 2016}{Validation against 2016 field data}
 \ytl{Sept 2016}{Optimal drag polar prediction}
\ytl{Nov. 2016}{Dissertation Defense}
 \bigskip
 \rule{\linewidth}{1pt}%
\end{minipage}%
\end{table}
 \label{tab:prop}
\end{center}

\end{frame}


%
%
%
\begin{frame}
 \frametitle{Course list}
 \footnotesize
 \begin{table}[h]
  \centering
  \begin{tabular}{llll}
   \hline \hline
   Course \# & Semester & Course Name & Instructor \\ 
   \hline 
   ME381P   & 2014 Spr.   & Validation \& Uncertainty Quantification &
	       Prof. <NAME> \\
   ME 382R5 & 2014 Spr.   & Advanced Combustion & Prof. <NAME> \\
   CSE 397  & 2014 Spr.   &  Comp. \& Var. Methods for
	   Inv. Problems & Prof. <NAME> \\
   SDS 384  & 2014 Fall   & Bayesian Statistical Methods & Prof. S.
	       Walker \\
   SDS 394  & 2014 Fall   & Scientific \& Technical Computing & Dr. <NAME> \\
   ME 382P  & 2015 Fall   & Adv. Exp. Methods in Thermal/Fluids
	   & Prof. <NAME> \\  
   \hline \hline
  \end{tabular} 
 \end{table}

\end{frame}


\end{document}

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