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% latex2e adocument: smpaiaa.tex
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%  sample AIAA conference paper, journal article,
%  journal note, and journal submission
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%  -- bil kleb <10 jan 97>
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\documentclass[cover]{aiaa}% options: paper, article, note, or submit

% note: until we reach \begin{document}, we're in the `preamble'

% load the header and footer information for the various options:

% for journal submission:

\SubmitName{Kleb}

% for conference paper:

\PaperNumber{91--0825}

\CoverFigure{smpfig.eps}

\Conference{{\bfseries 31st AIAA Aerospace Sciences \\
            Meeting and Exhibit} \\
            January~6--9,~1997/Reno,~NV}

% for a journal simulation cover page:

\JournalName{Journal of Spacecraft and Rockets}
\JournalIssue{Volume~32, Number~6, Pages 715--717}

% journal article simulation:

\ArticleIssue{Vol.~32, No.~6, November--December 1995}% first page
\ArticleHeader{Kleb Et Al: Pitch-Over Manuever}%        subsequent pages

% journal note simulation:

\NoteHeader{J.Spacecraft, Vol.~32, No.~6: Engineering Notes}

% set copyright and other notices to appear
% as a footnote at the bottom of the first page:

\PaperNotice{\CopyrightC}

\JournalNotice{Presented as Paper~96--0825 at the AIAA
               34th Aerospace Sciences Meeting, Reno,~NV,
               Jan.~15--18,~1996; received Feb.~15,~1996;
               revision received Nov.~25,~1996;
               accepted for publication Jan.~18,~1997.
               \CopyrightD{1997}}

% load the title, author, and abstract for use with the \maketitle command

\title{Simulation of an Aerospace Vehicle \\
       Pitch-Over Maneuver}
                                
\author{
%
William L.~Kleb%
%
  \thanks{Research Engineer, Aerothermodynamics Branch,
          Aero- and Gas-Dynamics Division,
          Research Technology Group.} \\
%
         {\itshape NASA Langley Research Center,
                   Hampton,~VA~23681} \\
%
\and
%
A.~N.~Author%
%
  \thanks{Deligent worker,
          AIAA member.}
%
\ and Y.~F.~Anotherlongername%
%
  \thanks{The Very Fine Organization,
          Answering to This Organization,
          AIAA senior member.} \\
%
         {\itshape Someother Affliation,
         Atown,~ST~98293}
}

\abstract{
This abstract does not appear when a journal note simulation is
the chosen option.
The objective of the present work is to
summarize the application of unsteady computational fluid dynamic
methods to the problem of predicting verticle take-off/vertical
landing vehicle aerodynamics during an un-powered
pitch-over maneuver.
In addition to the time-dependent simulation of a pitch-over
maneuver, a series of steady solutions at discrete points
are also computed for comparison with wind-tunnel measurements
and as a means of quantifying unsteady effects.
As this application represents a new challenge to unsteady
computational fluid dynamics, observations concerning grid
resolution, far-field boundary placement, temporal resolution, and the
suitability of assuming flow-field symmetry are discussed.}

\begin{document}

\maketitle%  create the cover page, title block, and notices

\section{Nomenclature}

\begin{tabbing}
12345678 \= \kill
$c$     \> Sound speed, m/s \\
$M$     \> Mach number \\
$P$     \> Pressure, Pa \\
$Re_s$  \> Reynolds number based on length $s$ \\
$T$     \> Temperature, K \\
$V$     \> Velocity, m/s \\
$x,y,z$ \> Cartesian body axes, m \\
$\alpha$ \> Angle of attack, deg \\
$\eta$  \> Wall-normal distance, m \\
$\rho$  \> Density, kg/m$^3$
\end{tabbing}

\subsection{Subscripts}
\begin{tabbing}
12345678 \= \kill
$tran$          \> Transition \\
$w$             \> Wall \\
$\infty$        \> Freestream
\end{tabbing}

\subsection{Superscripts}
\begin{tabbing}
12345678 \= \kill
$0$          \> Fiduciary point \\
$n\!+\!1$        \> Time level
\end{tabbing}

\section{Introduction}

\dropword NASA's Access to Space Study\cite{bekey:94ja} recommends the
development of a {\em fully}\ reusable launch vehicle\cite{dornheim:94awst}
to replace the aged Space Shuttle.
A method of reaching this goal is to develop a vehicle which does not rely
on expendable boosters to reach orbit, a single-stage-to-orbit
vehicle\cite{austin:94ja}.  One such configuration being
investigated is a Vertical Take-off
and Vertical Landing (VTVL) concept\cite{austin:94ja}.  In one
scenario, the VTVL
vehicle, upon completion of its mission in low-Earth orbit, reenters
nose first, decelerates to subsonic speeds,
and then performs a rotation
maneuver\cite{dornheim:95awst,david:95sn,dornheim:95awst2}
to land vertically.

\Figure{f:trn2lndg} presents a schematic of the last portion
of a typical VTVL entry.
\begin{figure}
  \incfig{smpfig.eps}
  \caption{Transition to landing for a VTVL vehicle.}
  \label{f:trn2lndg}
\end{figure}
The pitch-over maneuver, which occurs near Mach 0.2, is characterized by
high angle of attack, unsteady, vortical flow.
Accurately predicting vehicle performance during this aerodynamic
pitch-over maneuver is quite challenging.
While ground-based facilities can readily
predict the vehicle's aerodynamics at discrete points during the
maneuver, simulating the transient motion
in a wind tunnel is difficult.\cite{oleary:94cp}
Time-dependent Computational Fluid Dynamics (CFD) offers another means of
analyzing the pitch-over maneuver. The majority of work in
unsteady CFD has, however, been restricted to small amplitude, harmonic
variations in angle of attack in support of aeroelastic flutter
predictions.\cite{edwards:92cp}

The objective of the present work is to summarize the application
of unsteady CFD methods to the problem of predicting VTVL vehicle
aerodynamics during an un-powered
pitch-over maneuver (further details are available in the
companion conference paper\cite{kleb:96cp}).
In addition to the time-dependent simulation of a pitch-over
maneuver, a series of steady solutions at discrete points
are also computed for comparison with wind-tunnel measurements
and as a means of quantifying unsteady effects.
As this application represents a new challenge to unsteady CFD,
observations concerning grid resolution, far-field boundary
placement, temporal resolution, and the
suitability of assuming flow-field symmetry are documented in
\Reference{kleb:96cp}.

\section{Geometry}

The vehicle's fore body is an 8 deg half-angle sphere cone with nose radius 
equal to 0.3 of the base radius.
The aft body, beginning at the 85 percent fuselage station,
is a cylinder with a partially squared-off cross-section
producing flat ``slices'' extending from the base of the vehicle to
approximately the 60 percent fuselage station.
The vehicle has a fineness ratio of 6.4.
A complete description of the vehicle geometry modeled has been given by
Woods in \Reference{woods:95cp}.

With the help of Karen Bibb,
we have a demonstration of a subfigure situation.
An early Lockheed-Martin X-33 configuration
was used in the remainder of the examples.  The full vehicle is shown in
\Figure{fig:both}.
\begin{figure}
  \centering
  \begin{tabular}{c}
    \subfigure[\bf First pretty picture.]
              {\incfig{smpfig.eps}\label{fig:first}}\\
    \subfigure[\bf Second pretty picture.]
              {\incfig{smpfig.eps}\label{fig:second}}
  \end{tabular}
  \caption{Temperature distribution comparisons at various wing
           semi-span stations as a function of chord.  Whuh?}
  \label{fig:both}
\end{figure}
This configuration (B1001A) was evaluated during
Phase I of the X-33 program.  It has twin vertical tails, fins,
and outboard body flaps.  The engines are modeled by the box-shaped
structure on the base.  With imbedded labels you can refer
to \Figure{fig:both} as a whole or specifically, things in
\Figure{fig:first} or \Figure{fig:second}.

We can also have a ``table'' of two, or four, or more figures
as in \Figure{fig:four}.
\begin{figure}
  \newlength{\sfigwidth}% compute the width that the subfigures
                        % should be
  \setlength{\sfigwidth}{.5\linewidth} %  half of the current line
                                       %  width
  \addtolength{\sfigwidth}{-\tabcolsep}% minus the column separation
  \centering
  \begin{tabular}{cc}
    \subfigure[\bf First pretty picture.]
    {\incfig[\sfigwidth]{smpfig.eps}}&
    \subfigure[\bf Second pretty picture.]
    {\incfig[\sfigwidth]{smpfig.eps}}\\
    \subfigure[\bf Second pretty picture, again.]
    {\incfig[\sfigwidth]{smpfig.eps}}&
    \subfigure[\bf First pretty picture, again.]
    {\incfig[\sfigwidth]{smpfig.eps}}
 \end{tabular}
  \caption{Four small figures in a table-like setting.}
  \label{fig:four}
\end{figure}

\section{Computational Mesh}

The underlying surface definition database was generated from
structured surface patches obtained using
GRIDGEN\cite{GRIDGEN3D_release}, GridTool\cite{GridTool}, and
simple analytical methods.
The unstructured surface and flow-field grids were then generated using
FELISA\cite{peraire:90cp} and TETMESH.\cite{kennon:92cp}
The coarsest mesh has 32,374 tetrahedra with 6,634 nodes.
Additional meshes are described in \Reference{kleb:96cp};
however, all the results shown here are the result of the coarsest meshes.

Since the flow about symmetric
configurations at high angles of attack,
even with zero side-slip, often involve asymmetric, vortex-dominated,
features,\cite{yoshinaga:94cp,cobleigh:94cp,dusing:93cp,fisher:94cp}
two different options for the computational domain were employed: one
modeling the complete vehicle and another modeling only half of the
vehicle, assuming symmetry across the pitch-plane.
This aspect of the study is covered in \Reference{kleb:96cp}.

\section{Numerical Method}

The 3D3U code of Batina\cite{batina:93aij} was used exclusively in this
study. The 3D3U code was originally developed to study harmonically
pitching wings and wing-bodies in transonic flow.  The code can
incorporate aeroelastic effects through assumed mode shapes, coupled
with a deforming mesh via the linear spring analogy.

The following defaults were used for the computed results in
this study: Roe's flux-difference splitting,
an eigenvalue limiter threshold value of 0.3,
second-order flux reconstruction using a $\kappa$ of 0.5,
and Gauss-Seidel implicit time integration with
a CFL number of one million.

\section{Maneuver Definition}

The pitch schedule chosen for this study is
the first half of a sine function.
Initially, the vehicle is in steady flight at 17.5 degrees
angle of attack, Mach 0.2.  At time zero, the vehicle begins
the pitch-over maneuver,
reaching a maximum pitch rate exactly half-way
through the maneuver and finishing
at a 180 deg angle of attack.  For this study,
the time to complete the maneuver was chosen as 90 seconds,
giving a maximum rotation rate of 3.1 deg per second
halfway through the maneuver.  For simplicity, it is
assumed that the free-stream Mach number remains constant
throughout the maneuver.

Sticking in a table for guidance (see Table~\ref{tab:sample}).
\begin{table}[htbp]
  \begin{center}
    \caption{A sample table}
    \begin{tabular}{ccc} \hline\hline
      \multicolumn{2}{c}{Header 1} & Header 2 \\ \hline
      a & b & c \\
      d & e & f \\ \hline\hline
    \end{tabular}
    \label{tab:sample}
  \end{center}
\end{table}
Normally there would be text following the table, so that it
is not left ``hanging'' into the next section.

\section{Results}

The main results are presented in two stages which are
followed by comments on flow asymmetries for both steady
and unsteady flows.
The first stage is computed steady data as it compares to experimental results.
while the second is a comparison of steady to unsteady data.

\subsection{Steady Flow}

As a baseline to examine the unsteady effects of the pitch-over
maneuver itself, steady flow at selected angles of attack were computed.
\Figure{f:expcomp} shows a comparison of the normal force coefficient
with the experimental data of Woods\cite{woods:95cp} for
angles of attack from 0 to 60 deg.
\begin{figure}
   \incfig{smpfig.eps}
   \caption{Comparison of steady normal force coefficient as a
     function of angle of attack with experimental data of
     Woods\protect{\cite{woods:95cp}}}
   \label{f:expcomp}
\end{figure}
The computed results agree well (within 20 percent) at small angles of attack,
and diverge from the experimental results as the angle
of attack increases largely due to the position of the lee-side
separation line---a viscous phenomenon.
Since the computed results are modeling inviscid
flow, the only mechanism for flow separation is the numerical
dissipation in the scheme.  Thus, the exact location of the computed
separation line is highly grid and scheme dependent. Compounding
this is the fact that the lee-side separation line nearly
encompasses the length of the vehicle; and thus, a slight
deviation can make a large difference in the integrated coefficients.

\subsection{Unsteady Flow}

\Figure{f:aerocomp} compares the normal force coefficient of
both the steady and unsteady calculations.
\begin{figure*}
  \incfig{smpfig.eps}
  \caption{Example of a figure that spans both columns.  The danger is
           that the figure numbering may be out of order.  You can correct
           this in the final version. Comparison of steady and unsteady normal force
    coefficients as a function of angle of attack.}
 \label{f:aerocomp}
\end{figure*}
The solid line represents the unsteady results and the
symbols are the steady-flow results.
Readily discernible is the fact that the steady and unsteady
results are significantly different.
The time-dependent results have the time lag behavior expected for
moderately unsteady flow: showing the same general qualitative trend throughout
the angle-of-attack range, but with the unsteady results lagging
behind the steady results. Other aerodynamic coefficients show
similar differences.\cite{kleb:96cp}

\subsection{Flow Asymmetry}

As documented in \Reference{kleb:96cp},
no appreciable asymmetries were found for the steady flow
cases computed although they were present in the experimental
data of Woods\cite{woods:95cp}.
However, for the unsteady case,
asymmetric flow is apparent as shown in Figure{f:uasym}.
\begin{figure}[t]
  \incfig{smpfig.eps}
  \caption{Unsteady flow asymmetry: side force coefficient as a
    function of angle of attack.}
  \label{f:uasym}
\end{figure}
This figure shows the appearance of a non-zero side-force
similar to that reported for {\em steady} flow by
Woods\cite{woods:95cp}.
The most significant
manifestations of the asymmetries occur in the 90 to 135 deg
angle-of-attack range.

\section{Concluding Remarks}

The objective of the present work was to focus an unsteady CFD method
on the prediction of VTVL vehicle aerodynamics during a
pitch-over maneuver.
This was accomplished through the use
of the inviscid 3D3U code\cite{batina:93aij}
A series of steady solutions at discrete points in the maneuver were
computed and it was shown that, even for the unrealistically slow
pitch-over rate studied, unsteady effects were large.
More importantly, the rotation maneuver creates flow asymmetries
which lead to side forces not apparent for the steady cases.

As this is an exploratory study, there is certainly room
for future work. The following is just a handful of extensions which
would be necessary to create an effective design tool:
incorporating viscous effects, allowing movable control surfaces,
coupling a six-degree-of-freedom
rigid body dynamics solver, adding control law, and
incorporating an adaptive grid capability.

\bibliography{smpbtx}
\bibliographystyle{aiaa}

\end{document}