--- /dev/null
+
+
+\chapter{Nose cone and transition geometries}
+\label{app-nosecone-geometry}
+
+Model rocket nose cones are available in a wide variety of shapes and
+sizes. In this appendix the most common shapes and their defining
+parameters are presented.
+
+
+\section{Conical}
+
+The most simple nose cone shape is a right circular cone. They are
+easy to make from a round piece of cardboard. A conical nose cone is
+defined simply by its length and base diameter. An additional
+parameter is the opening angle $\phi$, shown in
+Figure~\ref{fig-nosecone-shapes}(a). The defining equation of a
+conical nose cone is
+%
+\begin{equation}
+r(x) = \frac{x}{L}\cdot R.
+\end{equation}
+
+
+\section{Ogival}
+
+Ogive nose cones have a profile which is an arc of a circle, as shown
+in Figure~\ref{fig-nosecone-shapes}(b). The most common ogive shape
+is the {\it tangent ogive} nose cone, which is formed when radius of
+curvature of the circle $\rho_t$ is selected such that the joint
+between the nose cone and body tube is smooth,
+%
+\begin{equation}
+\rho_t = \frac{R^2+L^2}{2R}.
+\end{equation}
+%
+If the radius of curvature $\rho$ is greater than this, then the
+resulting nose cone has an angle at the joint between the nose cone
+and body tube, and is called a {\it secant ogive}. The secant ogives
+can also be viewed as a larger tangent ogive with its base cropped.
+At the limit $\rho\rightarrow\infty$ the secant ogive becomes a
+conical nose cone.
+
+The parameter value $\kappa$ used for ogive nose cones is the ratio of
+the radius of curvature of a corresponding tangent ogive $\rho_t$ to
+the radius of curvature of the nose cone $\rho$:
+%
+\begin{equation}
+\kappa = \frac{\rho_t}{\rho}
+\end{equation}
+%
+$\kappa$ takes values from zero to one, where $\kappa=1$ produces a
+tangent ogive and $\kappa=0$ produces a conical nose cone (infinite
+radius of curvature).
+
+With a given length $L$, radius $R$ and parameter $\kappa$ the radius
+of curvature is computed by
+%
+\begin{equation}
+\rho^2 = \frac{ \del{L^2+R^2}\cdot
+ \del{ \del{\del{2-\kappa}L}^2 + \del{\kappa R}^2 }
+}{ 4\del{\kappa R}^2 }.
+\end{equation}
+%
+Using this the radius at position $x$ can be computed as
+%
+\begin{equation}
+r(x) = \sqrt{\rho^2 - \del{L/\kappa - x}^2} - \sqrt{\rho^2-\del{L/\kappa}^2}
+\end{equation}
+
+
+
+\section{Elliptical}
+
+Elliptical nose cones have the shape of an ellipsoid with one major
+radius is $L$ and the other two $R$. The profile has a shape of a
+half-ellipse with major axis $L$ and $R$,
+Figure~\ref{fig-nosecone-shapes}(c). It is a simple geometric shape
+common in model rocketry. The special case $R=L$ corresponds to a
+half-sphere.
+
+The equation for an elliptical nose cone is obtained by stretching the
+equation of a unit circle:
+\begin{equation}
+r(x) = R \cdot \sqrt{1-\del{1-\frac{x}{L}}^2}
+\end{equation}
+
+
+\section{Parabolic series}
+
+A parabolic nose cone is the shape generated by rotating a section of
+a parabola around a line perpendicular to its symmetry axis,
+Figure~\ref{fig-nosecone-shapes}(d). This is
+distinct from a paraboloid, which is rotated around this symmetry
+axis (see Appendix~\ref{app-power-series}).
+
+Similar to secant ogives, the base of a ``full'' parabolic nose cone
+can be cropped to produce nose cones which are not tangent with the
+body tube. The parameter $\kappa$ describes the portion of the larger
+nose cone to include, with values ranging from zero to one. The most
+common values are $\kappa=0$ for a conical nose cone, $\kappa=0.5$ for
+a 1/2~parabola, $\kappa=0.75$ for a 3/4~parabola and $\kappa=1$ for a
+full parabola. The equation of the shape is
+%
+\begin{equation}
+r(x) = R\cdot\frac{x}{L} \del{ \frac{2 - \kappa\frac{x}{L}}
+ {2-\kappa}}.
+\end{equation}
+
+
+
+\section{Power series}
+\label{app-power-series}
+
+The power series nose cones are generated by rotating the segment
+%
+\begin{equation}
+r(x) = R\del{\frac{x}{L}}^\kappa
+\end{equation}
+%
+around the x-axis, Figure~\ref{fig-nosecone-shapes}(e). The parameter
+value $\kappa$ can range from zero to one. Special cases are
+$\kappa=1$ for a conical nose cone, $\kappa=0.75$ for a 3/4~power nose
+cone and $\kappa=0.5$ for a 1/2~power nose cone or an ellipsoid.
+The limit $\kappa\rightarrow0$ forms a blunt cylinder.
+
+
+\section{Haack series}
+
+In contrast to the other shapes which are formed from rotating
+geometric shapes or simple formulae around an axis, the Haack series
+nose cones are mathematically derived to minimize the theoretical
+pressure drag. Even though they are defined as a series, two specific
+shapes are primarily used, the {\it LV-Haack}\ shape and the
+{\it LD-Haack}\ or {\it Von Kárman}\ shape. The letters LV and LD
+refer to length-volume and length-diameter, and they minimize the
+theoretical pressure drag of the nose cone for a specific length and
+volume or length and diameter, respectively. Since the parameters
+defining the dimensions of the nose cone are its length and radius,
+the Von Kárman nose cone (Figure~\ref{fig-nosecone-shapes}(f)) should,
+in principle, be the optimal nose cone shape.
+
+The equation for the series is
+%
+\begin{equation}
+r(x) = \frac{R}{\sqrt{\pi}} \;
+ \sqrt{\theta - \frac{1}{2}\sin(2\theta) + \kappa \sin^3\theta}
+\end{equation}
+%
+where
+%
+\begin{equation}
+\theta = \cos^{-1} \del{1-\frac{2x}{L}}.
+\end{equation}
+%
+The parameter value $\kappa=0$ produces the Von Kárman of LD-Haack
+shape and $\kappa=1/3$ produces the LV-Haack shape. In principle,
+values of $\kappa$ up to $2/3$ produce monotonic nose cone shapes.
+However, since there is no experimental data available for the
+estimation of nose cone pressure drag for $\kappa > 1/3$ (see
+Appendix~\ref{app-haack-series-pressure-drag}), the selection of the
+parameter value is limited in the software to the range
+$0 \ldots 1/3$.
+
+
+\section{Transitions}
+
+The vast majority of all model rocket transitions are conical.
+However, all of the nose cone shapes may be adapted as transition
+shapes as well. The transitions are parametrized with the fore and
+aft radii $R_1$ and $R_2$, length $L$ and the optional shape parameter
+$\kappa$.
+
+Two choices exist when adapting the nose cones as transition shapes.
+One is to take a nose cone with base radius $R_2$ and crop the tip of
+the nose at the radius position $R_1$. The length of the nose cone
+must be selected suitably that the length of the transition is $L$.
+Another choice is to have the profile of the transition resemble two
+nose cones with base radius $R_2-R_1$ and length $L$. These two
+adaptations are called {\it clipped} and {\it non-clipped} transitions,
+respectively. A clipped and non-clipped elliptical transition is
+depicted in Figure~\ref{fig-transition-clip}.
+
+For some transition shapes the clipped and non-clipped adaptations are
+the same. For example, the two possible ogive transitions have equal
+radii of curvature and are therefore the same. Specifically, the
+conical and ogival transitions are equal whether clipped or not, and
+the parabolic series are extremely close to each other.
+
+
+
+
+\begin{figure}[p]
+\centering
+\begin{tabular}{ccc}
+
+\epsfig{file=figures/nose-geometry/geometry-conical,scale=0.7} &&
+\epsfig{file=figures/nose-geometry/geometry-ogive,scale=0.7} \\
+(a) & \hspace{1cm} & (b) \\
+&& \\
+&& \\
+&& \\
+%&& \\
+
+\epsfig{file=figures/nose-geometry/geometry-elliptical,scale=0.7} &&
+\epsfig{file=figures/nose-geometry/geometry-parabolic,scale=1.0} \\
+(c) && (d) \\
+&& \\
+&& \\
+&& \\
+%&& \\
+
+\epsfig{file=figures/nose-geometry/geometry-power,scale=0.6} &&
+\epsfig{file=figures/nose-geometry/geometry-haack,scale=0.6} \\
+(e) && (f) \\
+&& \\
+%&& \\
+\end{tabular}
+\caption{Various nose cone geometries: (a)~conical, (b)~secant ogive,
+ (c)~elliptical, (d)~parabolic, (e)~1/2~power (ellipsoid) and
+ (f)~Haack series (Von Kárman).}
+\label{fig-nosecone-shapes}
+\end{figure}
+
+\begin{figure}[p]
+\vspace{5mm}
+\begin{center}
+\epsfig{file=figures/nose-geometry/geometry-transition,scale=0.7}
+\end{center}
+\caption{A clipped and non-clipped elliptical transition.}
+\label{fig-transition-clip}
+\end{figure}
+
+
+
+
+
+\chapter{Transonic wave drag of nose cones}
+\label{app-nosecone-drag-method}
+
+The wave drag of different types of nose cones vary largely in the
+transonic velocity region. Each cone shape has its distinct
+properties. In this appendix methods for calculating and
+interpolating the drag of various nose cone shapes at transonic and
+supersonic speeds are presented. A summary of the methods is
+presented in Appendix~\ref{app-transonic-nosecone-summary}.
+
+
+
+\section{Blunt cylinder}
+\label{app-blunt-cylinder-drag}
+
+A blunt cylinder is the limiting case for every nose cone shape at the
+limit $f_N\rightarrow 0$. Therefore it is useful to have a formula
+for the front pressure drag of a circular cylinder in longitudinal
+flow. As the object is not streamline, its drag coefficient does not
+vary according to the Prandtl factor~(\ref{eq-prandtl-factor}).
+Instead, the coefficient is approximately proportional to the
+{\it stagnation pressure}, or the pressure at areas perpendicular to
+the airflow. The stagnation pressure can be approximated by the
+function~\cite[pp.~15-2,~16-3]{hoerner}
+%
+\begin{equation}
+\frac{q_{\rm stag}}{q} =
+\left\{
+\begin{array}{ll}
+1 + \frac{M^2}{4} + \frac{M^4}{40}, & \mbox{for\ } M < 1 \\
+1.84 - \frac{0.76}{M^2} + \frac{0.166}{M^4} + \frac{0.035}{M^6}, &
+ \mbox{for\ } M > 1
+\end{array}
+\right. .
+\end{equation}
+%
+The pressure drag coefficient of a blunt circular cylinder as a
+function of the Mach number can then be written as
+%
+\begin{equation}
+(C_{D\bullet})_{\rm pressure} =
+(C_{D\bullet})_{\rm stag} = 0.85 \cdot \frac{q_{\rm stag}}{q}.
+\label{eq-blunt-cylinder-drag}
+\end{equation}
+
+
+
+
+
+\section{Conical nose cone}
+
+A conical nose cone is simple to construct and closely resembles many
+slender nose cones. The conical shape is also the limit of several
+parametrized nose cone shapes, in particular the secant ogive with
+parameter value 0.0 (infinite circle radius), the power series nose
+cone with parameter value 1.0 and the parabolic series with parameter
+value 0.0.
+
+Much experimental data is available on the wave drag of conical nose
+cones. Hoerner presents formulae for the value of $C_{D\bullet}$ at
+supersonic speeds, the derivative $\dif C_{D\bullet}/\dif M$ at
+$M=1$, and a figure of $C_{D\bullet}$ at
+$M=1$~\cite[pp.~16-18\ldots16-20]{hoerner}. Based on these and
+the low subsonic drag coefficient~(\ref{eq-nosecone-pressure-drag}), a
+good interpolation of the transonic region is possible.
+
+The equations presented by Hoerner are given as a function of the
+half-apex angle $\varepsilon$, that is, the angle between the conical
+body and the body centerline. The half-apex angle is related to the
+nose cone fineness ratio by
+%
+\begin{equation}
+\tan\varepsilon = \frac{d/2}{l} = \frac{1}{2f_N}.
+\end{equation}
+
+The pressure drag coefficient at supersonic speeds ($M\gtrsim1.3$) is
+given by
+%
+\begin{align}
+(C_{D\bullet})_{\rm pressure}
+& = 2.1\;\sin^2\varepsilon + 0.5\;
+ \frac{\sin\varepsilon}{\sqrt{M^2-1}} \nonumber\\
+& = \frac{2.1}{1+4f_N^2} + \frac{0.5}{\sqrt{(1+4f_N^2)\; (M^2-1)}} .
+\label{eq-conical-supersonic-drag}
+\end{align}
+%
+It is worth noting that as the Mach number increases, the drag
+coefficient tends to the constant value $2.1\sin^2\epsilon$. At $M=1$
+the slope of the pressure drag coefficient is equal to
+%
+\begin{equation}
+\eval{\frac{\partial (C_{D\bullet})_{\rm pressure}}{\partial M}}_{M=1} =
+ \frac{4}{\gamma+1} \cdot (1-0.5\;C_{D\bullet,M=1})
+\label{eq-conical-sonic-drag-derivative}
+\end{equation}
+%
+where $\gamma=1.4$ is the specific heat ratio of air and the drag
+coefficient at $M=1$ is approximately
+%
+\begin{equation}
+C_{D\bullet,M=1} = 1.0\; \sin\varepsilon.
+\label{eq-conical-sonic-drag}
+\end{equation}
+
+The pressure drag coefficient between Mach~0 and Mach~1 is
+interpolated using equation~(\ref{eq-nosecone-pressure-drag}).
+Between Mach~1 and Mach~1.3 the coefficient is calculated using
+polynomial interpolation with the boundary conditions from
+equations~(\ref{eq-conical-supersonic-drag}),
+(\ref{eq-conical-sonic-drag-derivative}) and
+(\ref{eq-conical-sonic-drag}).
+
+
+
+
+
+
+\section{Ellipsoidal, power, parabolic and Haack series nose cones}
+\label{app-haack-series-pressure-drag}
+
+A comprehensive data set of the pressure drag coefficient for all nose
+cone shapes at all fineness ratios at all Mach numbers is not
+available. However, Stoney has collected a compendium of nose cone
+drag data including data on the effect of the fineness ratio $f_N$ on
+the drag coefficient and an extensive study of drag coefficients of
+different nose cone shapes at fineness ratio
+3~\cite{nosecone-cd-data}. The same report suggests that the effects
+of fineness ratio and Mach number may be separated.
+
+The curves of the pressure drag coefficient as a function of the nose
+fineness ratio $f_N$ can be closely fitted with a function of the form
+%
+\begin{equation}
+(C_{D\bullet})_{\rm pressure} = \frac{a}{(f_N + 1)^b}.
+\label{eq-fineness-ratio-drag-interpolator}
+\end{equation}
+%
+The parameters $a$ and $b$ can be calculated from two data points
+corresponding to fineness ratios 0 (blunt cylinder,
+Appendix~\ref{app-blunt-cylinder-drag}) and ratio 3. Stoney includes
+experimental data of the pressure drag coefficient as a function of
+Mach number at fineness ratio 3 for power series $x^{1/4}$, $x^{1/2}$,
+$x^{3/4}$ shapes, $1/2$, $3/4$ and full parabolic shapes, ellipsoidal,
+L-V~Haack and Von Kárman nose cones. These curves are written into
+the software as data curve points. For parametrized nose cone shapes
+the necessary curve is interpolated if
+necessary. Typical nose cones of model rockets have fineness ratios
+in the region of 2--5, so the extrapolation from data of fineness
+ratio 3 is within reasonable bounds.
+
+
+
+
+
+\section{Ogive nose cones}
+
+One notable shape missing from the data in Stoney's report are secant
+and tangent ogives. These are common shapes for model rocket nose
+cones. However, no similar experimental data of the pressure drag as
+a function of Mach number was found for ogive nose cones.
+
+At supersonic velocities the drag of a tangent ogive is approximately
+the same as the drag of a conical nose cone with the same length and
+diameter, while secant ogives have a somewhat smaller
+drag~\cite[p.~239]{handbook-supersonic-aerodynamics}. The minimum
+drag is achieved when the secant ogive radius is approximately twice
+that of a corresponding tangent ogive, corresponding to the parameter
+value 0.5. The minimum drag is consistently 18\% less than that of a
+conical nose at Mach numbers in the range of 1.6--2.5 and for fineness
+ratios of 2.0--3.5. Since no better transonic data is available, it
+is assumed that ogives follow the conical drag profile through
+the transonic and supersonic region. The drag of the corresponding
+conical nose is diminished in a parabolic fashion with the ogive
+parameter, with a minimum of -18\% at a parameter value of 0.5.
+
+
+
+
+
+\section{Summary of nose cone drag calculation}
+\label{app-transonic-nosecone-summary}
+
+The low subsonic pressure drag of nose cones is calculated using
+equation~(\ref{eq-nosecone-pressure-drag}):
+%
+\begin{equation*}
+(C_{D\bullet,M=0})_p = 0.8 \cdot \sin^2\phi.
+\end{equation*}
+%
+The high subsonic region is interpolated using a function of the form
+presented in equation~(\ref{eq-nosecone-pressure-interpolator}):
+%
+\begin{equation*}
+(C_{D\bullet})_{\rm pressure} = a\cdot M^b + (C_{D\bullet,M=0})_p
+\end{equation*}
+%
+where $a$ and $b$ are selected according to the lower boundary of the
+transonic pressure drag and its derivative.
+
+The transonic and supersonic pressure drag is calculated depending on
+the nose cone shape as follows:
+%
+\begin{itemize}
+
+\item[\bf Conical:] At supersonic velocities ($M > 1.3$) the
+ pressure drag is calculated using
+ equation~(\ref{eq-conical-supersonic-drag}). Between Mach 1 and 1.3
+ the drag is interpolated using a polynomial with boundary conditions
+ given by equations~(\ref{eq-conical-supersonic-drag}),
+ (\ref{eq-conical-sonic-drag-derivative}) and
+ (\ref{eq-conical-sonic-drag}).
+\\
+
+
+
+\item[\bf Ogival:] The pressure drag at transonic and supersonic
+ velocities is equal to the pressure drag of a conical nose cone with
+ the same diameter and length corrected with a shape factor:
+%, multiplied by the shape factor
+%
+\begin{equation}
+(C_{D\bullet})_{\rm pressure} =
+\del{0.72 \cdot (\kappa - 0.5)^2 + 0.82} \cdot
+(C_{D\bullet})_{\rm cone}.
+\end{equation}
+%
+The shape factor is one at $\kappa = 0, 1$ and 0.82 at $\kappa=0.5$.
+\\
+
+
+
+\item[\bf Other shapes:] The pressure drag calculation is based on
+ experimental data curves:
+%
+\begin{enumerate}
+\item Determine the pressure drag $C_3$ of a similar nose cone
+ with fineness ratio $f_N=3$ from experimental data. If data for a
+ particular shape parameter is not available, interpolate the data
+ between parameter values.
+
+\item Calculate the pressure drag of a blunt cylinder $C_0$
+ using equation~(\ref{eq-blunt-cylinder-drag}).
+
+\item Interpolate the pressure drag of the nose cone using
+ equation~(\ref{eq-fineness-ratio-drag-interpolator}).
+ After parameter substitution the equation takes the form
+%
+\begin{equation}
+(C_{D\bullet})_{\rm pressure} \;=\;
+\frac{C_0}{(f_N+1)^{\log_4 C_0/C_3}} \;=\;
+C_0 \cdot \del{\frac{C_3}{C_0}}^{\log_4(f_N+1)}
+\end{equation}
+%
+ The last form is computationally more efficient since the exponent
+ $\log_4(f_N+1)$ is constant during a simulation.
+
+\end{enumerate}
+
+\end{itemize}
+
+
+
+
+
+
+
+
+
+
+
+\chapter{Streamer drag coefficient estimation}
+\label{app-streamers}
+
+
+A streamer is a typically rectangular strip of plastic or other
+material that is used as a recovery device especially in small model
+rockets. The deceleration is based on the material flapping in the
+passing air, thus causing air resistance. Streamer optimization has
+been a subject of much interest in the rocketry
+community~\cite{streamer-optimization}, and contests on streamer
+landing duration are organized regularly. In order to estimate the
+drag force of a streamer a series of experiments were performed and an
+empirical formula for the drag coefficient was developed.
+
+One aspect that is not taken into account in the present investigation
+is the fluctuation between the streamer and rocket. At one extreme a
+rocket with a very small streamer drops head first to the ground with
+almost no deceleration at all. At the other extreme there is a very
+large streamer creating significant drag, and the rocket falls below
+it tail-first. Between these two extremes is a point where the
+orientation is labile, and the rocket as a whole twirls
+around during descent. This kind of interaction between the rocket
+and streamer cannot be investigated in a wind tunnel and would require
+an extensive set of flight tests to measure. Therefore it is not
+taken into account, instead, the rocket is considered effectively a
+point mass at the end of the streamer, the second extreme mentioned
+above.
+
+
+\subsubsection*{Experimental methods}
+
+A series of experiments to measure the drag coefficients of streamers
+was performed using the $40\times40\times120$~cm wind tunnel of
+Pollux~\cite{pollux-wind-tunnel}. The experiments were performed
+using various materials, widths and lengths of streamers and at
+different wind speeds. The effect of the streamer size and shape was
+tested separately from the effect of the streamer material.
+
+A tube with a rounded $90^\circ$ angle at one end was installed in the
+center of the wind tunnel test section. A line was drawn through the
+tube so that one end of the line was attached to a the streamer and the
+other end to a weight which was placed on a digital scale. When the
+wind tunnel was active the force produced by the streamer was read
+from the scale. A metal wire was taped to the edge of the streamer to
+keep it rigid and the line attached to the midpoint of the wire.
+
+A few different positions within the test section and free line
+lengths were tried. All positions seemed to produce approximately
+equal results, but the variability was significantly lower when the
+streamer fit totally into the test section and had a only 10~cm length
+of free line between the tube and streamer. This configuration was
+used for the rest of the experiments.
+
+Each streamer was measured at three different velocities, 6~m/s, 9~m/s
+and 12~m/s. The results indicated that the force produced is
+approximately proportional to the square of the airspeed, signifying
+that the definition of a drag coefficient is valid also for streamers.
+
+The natural reference area for a streamer is the area of the strip.
+However, since in the simulation we are interested in the total drag
+produced by a streamer, it is better to first produce an equation for
+the drag coefficient normalized to unit area, $C_D \cdot \Aref$.
+These coefficient values were calculated separately for the different
+velocities and then averaged to obtain the final normalized drag
+coefficient of the streamer.
+
+
+\subsubsection*{Effect of streamer shape}
+
+\begin{figure}[p]
+\centering
+\hspace*{-7mm}
+\epsfig{file=figures/experimental/streamerCDvsWL,width=155mm}
+\caption{The normalized drag coefficient of a streamer as a function
+ of the width and length of the streamer. The points are the
+ measured values and the mesh is cubically interpolated between the
+ points.}
+\label{fig-streamer-CD-vs-shape}
+\end{figure}
+
+
+\begin{figure}[p]
+\centering
+\hspace*{-7mm}
+\epsfig{file=figures/experimental/streamerCDvsWLestimate,width=155mm}
+\caption{Estimated and measured normalized drag coefficients of a
+ streamer as a function of the width and length of the streamer. The
+ lines from the points lead to their respective estimate values.}
+\label{fig-streamer-shape-estimate}
+\end{figure}
+
+
+
+Figure~\ref{fig-streamer-CD-vs-shape} presents the normalized drag
+coefficient as a function of the streamer width and length for a fixed
+material of $\rm80~g/m^2$ polyethylene plastic. It was noticed that
+for a specific streamer length, the normalized drag coefficient was
+approximately linear with the width,
+%
+\begin{equation}
+C_D \cdot \Aref = k\cdot w,
+\label{eq-streamer-first-approx}
+\end{equation}
+%
+where $w$ is the width and $k$ is dependent on the streamer length.
+The slope $k$ was found to be approximately linear with
+the length of the streamer, with a linear regression of
+%
+\begin{equation}
+k = 0.034 \cdot (l+\rm 1~m).
+\label{eq-streamer-second-approx}
+\end{equation}
+%
+Substituting equation (\ref{eq-streamer-second-approx}) into
+(\ref{eq-streamer-first-approx}) yields
+%
+\begin{equation}
+C_D \cdot \Aref = 0.034 \cdot (l+{\rm 1~m})\cdot w
+\label{eq-streamer-estimate}
+\end{equation}
+%
+or using $\Aref = wl$
+%
+\begin{equation}
+C_D = 0.034 \cdot \frac{l+\rm 1~m}{l}.
+\label{eq-streamer-shape-estimate}
+\end{equation}
+
+
+The estimate as a function of the width and length is presented in
+Figure~\ref{fig-streamer-shape-estimate} along with the measured data
+points. The lines originating from the points lead to their
+respective estimate values. The average relative error produced by
+the estimate was 9.7\%.
+
+
+\subsubsection*{Effect of streamer material}
+
+
+
+The effect of the streamer material was studied by creating
+$4\times40$~cm and $8\times48$~cm streamers from various household
+materials commonly used in streamers. The tested materials were
+polyethylene plastic of various thicknesses, cellophane and crêpe
+paper. The properties of the materials are listed in
+Table~\ref{table-streamer-materials}.
+
+
+Figure~\ref{fig-streamer-material} presents the normalized drag
+coefficient as a function of the material thickness and surface
+density. It is evident that the thickness is not a good
+parameter to characterize the drag of a streamer. On the other hand,
+the drag coefficient as a function of surface density is nearly
+linear, even including the crêpe paper. While it is not as
+definitive, both lines seem to intersect with the $x$-axis at
+approximately $\rm-25~g/m^2$. Therefore the coefficient of the
+$\rm80~g/m^2$ polyethylene estimated by
+equation~(\ref{eq-streamer-shape-estimate}) is corrected for a
+material surface density $\rho_m$ with
+%
+\begin{equation}
+C_{D_m} = \left(\frac{\rho_m + \rm 25~g/m^2}{\rm 105~g/m^2}\right)
+ \cdot C_D.
+\end{equation}
+%
+Combining these two equations, one obtains the final empirical
+equation
+%
+\begin{equation}
+C_{D_m} = 0.034 \cdot
+ \left(\frac{\rho_m + \rm 25~g/m^2}{\rm 105~g/m^2}\right) \cdot
+ \left(\frac{l + 1~{\rm m}}{l}\right).
+\label{eq-streamer-CD-estimate}
+\end{equation}
+
+This equation is also reasonable since it produces positive and finite
+normalized drag coefficients for all values of $w$, $l$ and $\rho_m$.
+However, this equation does not obey the rule-of-thumb of rocketeers
+that the optimum width-to-length ratio for a streamer would be 1:10.
+According to equation~(\ref{eq-streamer-estimate}), the maximum drag
+for a fixed surface area is obtained at the limit $l\rightarrow0$,
+$w\rightarrow\infty$. In practice the rocket dimensions limit the
+practical dimensions of a streamer, from which the 1:10 rule-of-thumb
+may arise.
+
+
+\subsubsection*{Equation validation}
+
+To test the validity of the equation, several additional streamers
+were measured for their drag coefficients. These were of various
+materials and of dimensions that were not used in the fitting of the
+empirical formulae. These can therefore be used as an independent
+test set for validating equation~(\ref{eq-streamer-CD-estimate}).
+
+Table~\ref{table-streamer-validation} presents the tested streamers
+and their measured and estimated normalized drag coefficients. The
+results show relative errors in the range of 12--27\%. While rather
+high, they are considered a good result for estimating such a random
+and dynamic process as a streamer. Furthermore, due to the
+proportionality to the square of the velocity, a 25\% error in the
+normalized force coefficient translates to a 10--15\% error in the
+rocket's descent velocity. This still allows the rocket designer to
+get a good estimate on how fast a rocket will descend with a
+particular streamer.
+
+
+\begin{figure}[p]
+\centering
+\parbox{70mm}{\centering
+\epsfig{file=figures/experimental/streamerCDvsThickness2,width=70mm}
+\\ (a)
+}\parbox{70mm}{\centering
+\epsfig{file=figures/experimental/streamerCDvsDensity2,width=70mm} \\ (b)}
+\caption{The normalized drag coefficient of a streamer as a function
+ of (a) the material thickness and (b) the material surface density.}
+\label{fig-streamer-material}
+\end{figure}
+
+
+
+\begin{table}[p]
+\caption{Properties of the streamer materials experimented with.}
+\label{table-streamer-materials}
+\begin{center}
+\begin{tabular}{ccc}
+\hline
+Material & Thickness / \um & Density / $\rm g/m^2$ \\
+\hline
+Polyethylene & 21 & 19 \\
+Polyethylene & 22 & 10 \\
+Polyethylene & 42 & 41 \\
+Polyethylene & 86 & 80 \\
+Cellophane & 20 & 18 \\
+Crêpe paper & 110$\dagger$ & 24 \\
+\hline
+\end{tabular} \\
+{\footnotesize $\dagger$ Dependent on the amount of pressure applied.}
+\end{center}
+\end{table}
+
+
+
+\begin{table}[p]
+\caption{Streamers used in validation and their results.}
+\label{table-streamer-validation}
+\begin{center}
+\begin{tabular}{ccccccc}
+\hline
+Material & Width & Length & Density & Measured & Estimate & Error \\
+ & m & m & $\rm g/m^2$ &
+ \multicolumn{2}{c}{$10^{-3} (C_D\cdot\Aref)$} & \\
+\hline
+Polyethylene & 0.07 & 0.21 & 21 & 0.99 & 1.26 & 27\% \\
+Polyethylene & 0.07 & 0.49 & 41 & 1.81 & 2.23 & 23\% \\
+Polyethylene & 0.08 & 0.24 & 10 & 0.89 & 1.12 & 26\% \\
+Cellophane & 0.06 & 0.70 & 20 & 1.78 & 1.49 & 17\% \\
+Crêpe paper & 0.06 & 0.50 & 24 & 1.27 & 1.43 & 12\% \\
+\hline
+\end{tabular}
+\end{center}
+\end{table}
+
projects / debian / openrocket / blobdiff