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Technical documentation update on lower stages

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Sampo Niskanen 2013-01-30 23:38:38 +02:00
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138
core/doc/techdoc/chapter-aerodynamic-properties.tex
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@ -2233,17 +2233,28 @@ attack, this approximation provides a sufficiently accurate estimate
for the purposes of this thesis.
\section{Lower stage aerodynamics}
\section{Tumbling bodies}
% Renaming of test vs. models here:
%
% #1 -> test 2
% #2 -> test 3
% #3 -> test 5
% #4 -> test 4
% #5 -> test 6
%
% Test 1 failed to produce a reliable result. Dimensions:
% n=3, Cr=50, Ct=25, s=50, l0=10, d=18, l=74, m=8.1
In staged rockets the lower stages of the rocket separate from the
main rocket body and descend to the ground on their own. While large
rockets have parachutes also in lower stages, most model rockets rely
on the stages falling to the ground without any recovery device. As
the lower stages typically are not aerodynamically stable, they tumble
during descent, significantly reducing their speed.
rockets typically have parachutes also in lower stages, most model
rockets rely on the stages falling to the ground without any recovery
device. As the lower stages normally are not aerodynamically stable,
they tumble during descent, significantly reducing their speed.
This kind of tumbling is difficult if not impossible to model in
6-DOF, and the orientation is typically not of interest anyway.
6-DOF, and the orientation is not of interest anyway.
For simulating the descent of aerodynamically unstable stages, it is
therefore sufficient to compute the average aerodynamic drag of
the tumbling lower stage.
@ -2252,32 +2263,29 @@ While model rockets are built in very peculiar forms, staged rockets
are typically much more conservative in their design. The lower
stages are most often formed of just a body tube and fins. Five such
models were constructed for testing their descent aerodynamic drag.
The physical properties of the models are listed in
Table~\ref{tab-lower-stages}.
% # fins
% root chord
% tip chord
% fin height
% diameter
% mass
Models \#1 and \#2 are identical except for the number of fins. \#3
represents a large, high-power booster stage. \#4 is a body tube
without fins, and \#5 fins without a body tube.
\begin{table}
\caption{Physical properties and drop results of the lower stage models}
\label{tab-lower-stages}
\caption{Physical properties of the lower stage models}
\begin{center}
\parbox{85mm}{
\parbox{80mm}{
\begin{tabular}{cccccc}
Model & \#1 & \#2 & \#3 & \#4 & \#5 \\
\hline
No. fins & 3 & 3 & 4 & 0 & 3 \\
$C_r$ / mm & 50 & 70 & 70 & - & 200 \\
$C_t$ / mm & 25 & 40 & 40 & - & 140 \\
$s$ / mm & 50 & 60 & 60 & - & 130 \\
$l_0$ / mm & 10 & 10 & 10 & - & 25 \\
$d$ / mm & 18 & 44 & 44 & 44 & 103 \\
$l$ / mm & 74 & 108 & 108 & 100 & 290 \\
$m$ / g & 8.1 & 18.0& 21.8& 6.8 & \\
No. fins & 3 & 4 & 3 & 0 & 4 \\
$C_r$ / mm & 70 & 70 & 200 & - & 85 \\
$C_t$ / mm & 40 & 40 & 140 & - & 85 \\
$s$ / mm & 60 & 60 & 130 & - & 50 \\
$l_0$ / mm & 10 & 10 & 25 & - & - \\
$d$ / mm & 44 & 44 & 103 & 44 & 0 \\
$l$ / mm & 108 & 108 & 290 & 100 & - \\
$m$ / g & 18.0& 22.0& 160 & 6.8 & 11.5 \\
\hline
$v_0$ / m/s & 5.6 & 6.3 & 6.6 & 5.4 & 5.0 \\
\end{tabular}
}
\parbox{50mm}{
@ -2286,20 +2294,82 @@ $m$ / g & 8.1 & 18.0& 21.8& 6.8 & \\
\end{center}
\end{table}
The drop tests were performed from a height of XX meters and the drop
was recorded on Full HD video. From the video frames the position of
the component was calculated XX times per second. The resulting graph
is presented in Figure~XX. The terminal velocity was determined for
all models.
The models were dropped from a height of 22 meters and the drop
was recorded on video. From the video frames the position of
the component was determined and the terminal velocity $v_0$
calculated with an accuracy of approximately $\pm 0.3\;\rm m/s$.
During the drop test the temperature was -5$^\circ$C, relative
humidity was 80\% and the dew point -7$^\circ$C. Together these yield
an air density of $\rho = 1.31\rm\;kg/m^3$. The physical properties
of the models and their terminal descent velocities are listed in
Table~\ref{tab-lower-stages}.
For a tumbling rocket, it is reasonable to assume that the drag force
is relative to the profile area of the rocket. For body tubes the
profile area is straightforward to calculate. For three and four fin
configurations the minimum profile area is taken instead.
Based on the results of models \#4 and \#5 it is clear that the
aerodynamic drag coefficient (relative to the profile area) is
significantly different for the body tube and fins. Thus we assume
the drag to consist of two independent components, one for the fins
and one for the body tube.
At terminal velocity the drag force is equal to that of gravity:
%
\begin{equation}
C_{D*} \cdot \frac{1}{2}\rho v_0^2 A_* = mg
\frac{1}{2}\rho v_0^2\; (C_{D,f}A_f + C_{D,bt}A_{bt}) = mg
\end{equation}
%
From this it is easy to determine the drag coefficient $C_{D*}$ for a
particular reference area $A_*$.
The values for $C_{D,f}$ and $C_{D,bt}$ were varied to optimize the
relative mean square error of the $v_0$ prediction, yielding a result
of $C_{D,f} = 1.42$ and $C_{D,bt} = 0.56$. Using these values, the
predicted terminal velocities varied between $3\%\ldots14\%$ from the
measured values.
During optimization it was noted that changing the error function
being optimized had a significant effect on the resulting fin drag
coefficient, but very little on the body tube drag coefficient. It is
assumed that the fin tumbling model has greater inaccuracy in this
aspect.
It is noteworthy that the body tube drag coefficient 0.56 is exactly
half of that of a circular cylinder perpendicular to the
airflow~\cite[p.~3-11]{hoerner}. This is expected of a cylinder that
is falling at a random angle of attack. The fin drag coefficient 1.42
is also similar to that of a flat plate 1.17 or an open hemispherical
cup 1.42 \cite[p.~3-17]{hoerner}.
The total drag coefficient $C_D$ of a tumbling lower stage is obtained
by combining and scaling the two drag coefficient components:
%
\begin{equation}
C_D = \frac{C_{D,f}A_f + C_{D,bt}A_{bt}}{\Aref}
\end{equation}
%
Here $A_{bt}$ is the profile area of the body, and $A_f$ the effective
fin profile area, which is the area of a single fin multiplied by the
efficiency factor. The estimated efficiency factors for various
numbers of fins are listed in Table~\ref{tab-lower-stage-fins}.
\begin{table}
\caption{Estimated fin efficiency factors for tumblig lower stages}
\label{tab-lower-stage-fins}
\begin{center}
\begin{tabular}{cc}
Number & Efficiency \\
of fins & factor \\
\hline
1 & 0.50 \\
2 & 1.00 \\
3 & 1.50 \\
4 & 1.41 \\
5 & 1.81 \\
6 & 1.73 \\
7 & 1.90 \\
8 & 1.85 \\
\hline
\end{tabular}
\end{center}
\end{table}
For a tumbling rocket, it is reasonable to assume that the drag force
is relative to the profile area of the rocket.

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