2377 lines
87 KiB
TeX
2377 lines
87 KiB
TeX
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\chapter{Aerodynamic properties of model rockets~~}
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\label{chap-aerodynamics}
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A model rocket encounters three basic forces during its flight:
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thrust from the motors, gravity, and aerodynamical forces. Thrust is
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generated by the motors by exhausting high-velocity gases in the
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opposite direction. The thrust of a motor is directly proportional to
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the velocity of the escaping gas and the mass per time unit that is
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exhausted. The thrust of commercial model rocket motors as a function
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of time have been measured in static motor tests and are readily
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available online~\cite{thrust-curve-database}. Normally the thrust of
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a rocket motor is aligned on the center axis of the rocket, so that it
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produces no angular moment to the rocket.
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Every component of the rocket is also affected by gravitational
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force. When the forces and moments generated are summed up, the
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gravitational force can be seen as a single force originating from the
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{\it center of gravity} (CG). A homogeneous gravitational field does
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not generate any angular moment on a body relative to the CG.
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Calculating the gravitational force is therefore a simple matter of
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determining the total mass and CG of the rocket.
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Aerodynamic forces, on the other hand, produce both net forces and
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angular moments. To determine the effect of the aerodynamic
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forces on the rocket, the total force and moment must be calculated
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relative to some reference point. In this chapter, a method for
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determining these forces and moments will be presented.
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\section{General aerodynamical properties}
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\label{sec-general-aerodynamics}
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The aerodynamic forces acting on a rocket are usually split into
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components for further examination. The two most important
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aerodynamic force components of interest in a typical model rocket are
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the {\it normal force} and {\it drag}. The aerodynamical normal
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force is the force component that generates the corrective moment
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around the CG and provides stabilization of the rocket. The
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drag of a rocket is defined as the force component parallel to the
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velocity of the rocket. This is the aerodynamical force that opposes
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the movement of the rocket through air.
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Figure~\ref{fig-aerodynamic-forces}(a) shows the thrust, gravity,
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normal force and drag of a rocket in free flight. It should be noted
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that if the rocket is flying at an angle of attack $\alpha>0$, then
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the normal force and drag are not perpendicular. In order to have
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independent force components, it is necessary to define component
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pairs that are always perpendicular to one another. Two such pairs
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are the normal force and axial drag, or side force and drag, shown in
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Figure~\ref{fig-aerodynamic-forces}(b). The two pairs coincide if the
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angle of attack is zero. The component pair that will be used as a
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basis for the flight simulations is the normal force and axial drag.
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\begin{figure}
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\centering
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\parbox{35mm}{\centering
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\epsfig{file=figures/aerodynamics/free-flight-forces,width=35mm} \\ (a)}
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\hfill
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\parbox{35mm}{\centering
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\epsfig{file=figures/aerodynamics/aero-force-components,width=35mm} \\ (b)}
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\hfill
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\parbox{35mm}{\centering
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\epsfig{file=figures/aerodynamics/pitch-yaw-roll,width=35mm} \\ (c)}
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\caption{(a) Forces acting on a rocket in free flight: gravity $G$,
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motor thrust $T$, drag $D$ and normal force $N$. (b) Perpendicular
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component pairs of the total aerodynamical force: normal force $N$
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and axial drag $D_A$; side force $S$ and drag $D$. (c) The pitch,
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yaw and roll directions of a model rocket.}
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\label{fig-aerodynamic-forces}
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\end{figure}
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The three moments around the different axis are called the
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{\it pitch}, {\it yaw} and {\it roll moments}, as depicted in
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Figure~\ref{fig-aerodynamic-forces}(c). Since a typical rocket has no
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``natural'' roll angle of flight as an aircraft does, we may choose
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the pitch angle to be in the same plane as the angle of attack, \ie
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the plane defined by the velocity vector and the centerline of the
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rocket. Thus, the normal force generates the pitching moment and no
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other moments.
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\subsection{Aerodynamic force coefficients}
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When studying rocket configurations, the absolute force values are
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often difficult to interpret, since many factors affect them. In
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order to get a value better suited for comparison, the forces are
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normalized by the current dynamic pressure $q=\frac{1}{2}\rho v_0^2$
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and some characteristic area \Aref\ to get a non-dimensional force
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coefficient. Similarly, the moments are normalized by the dynamic
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pressure, characteristic area and characteristic length $d$. Thus,
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the normal force coefficient corresponding to the normal force $N$ is
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defined as
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%
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\begin{equation}
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C_N = \frac{N}{\frac{1}{2}\rho v_0^2 \, \Aref}
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\label{eq-CN-def}
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\end{equation}
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%
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and the pitch moment coefficient for a pitch moment $m$ as
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%
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\begin{equation}
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C_m = \frac{m}{\frac{1}{2}\rho v_0^2 \, \Aref\, d}.
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\label{eq-Cm-def}
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\end{equation}
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%
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A typical choice of reference area is the base of the rocket's nose
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cone and the reference length is its diameter.
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The pitch moment is always calculated around some reference point,
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while the normal force stays constant regardless of the point of
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origin. If the moment coefficient $C_m$ is known for some reference
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point, the moment coefficient at another point $C_m'$ can be
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calculated from
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%
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\begin{equation}
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C_m'd = C_md - C_N\Delta x
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\label{eq-moment-reference}
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\end{equation}
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%
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where $\Delta x$ is the distance along the rocket centerline.
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Therefore it is sufficient to calculate the moment coefficient only at
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some constant point along the rocket body. In this thesis the
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reference point is chosen to be the tip of the nose cone.
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The {\it center of pressure} (CP) is defined as the position from
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which the total normal force alone produces the current pitching
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moment. Therefore the total normal force produces no moment around
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the CP itself, and an equation for the location of the CP
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can be obtained from (\ref{eq-moment-reference}) by selecting setting
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$C_m'=0$:
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%
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\begin{equation}
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X = \frac{C_m}{C_N}\,d
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\end{equation}
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%
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Here $X$ is the position of the CP along the rocket centerline from
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the nose cone tip. This equation is valid when $\alpha>0$. As
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$\alpha$ approaches zero, both $C_m$ and $C_N$ approach zero. The CP
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is then obtained as a continuous extension using l'H<>pital's rule
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%
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\begin{equation}
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X = \left.\frac{\;\frac{\partial C_m}{\partial\alpha}\;}
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{\;\frac{\partial C_N}{\partial\alpha}\;}\,d\right|_{\alpha=0}
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= \frac{\Cma}{\CNa}\,d
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\label{eq-CP-position}
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\end{equation}
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%
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where the normal force coefficient and pitch moment coefficient
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derivatives have been defined as
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%
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\begin{equation}
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\CNa = \left.\frac{\partial C_N}{\partial\alpha}\right|_{\alpha=0}
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\hspace{5mm}\mbox{and}\hspace{5mm}
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\Cma = \left.\frac{\partial C_m}{\partial\alpha}\right|_{\alpha=0}.
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\label{eq-CNa-derivative}
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\end{equation}
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At very small angles of attack we may approximate $C_N$ and $C_m$ to
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be linear with $\alpha$, so to a first approximation
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%
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\begin{equation}
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C_N \approx \CNa\,\alpha
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\hspace{5mm}\mbox{and}\hspace{5mm}
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C_m \approx \Cma\,\alpha.
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\label{eq-CNa-approx}
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\end{equation}
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%
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The Barrowman method uses the coefficient derivatives to determine the
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CP position using equation~(\ref{eq-CP-position}). However, there are
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some significant nonlinearities in the variation of $C_N$ as a
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function of $\alpha$. These will be accounted for by holding the
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approximation of equation~(\ref{eq-CNa-approx}) exact and letting
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\CNa\ and \Cma\ be a function of $\alpha$. Therefore, for the
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purposes of this thesis we define
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%
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\begin{equation}
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\CNa = \frac{C_N}{\alpha}
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\hspace{5mm}\mbox{and}\hspace{5mm}
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\Cma = \frac{C_m}{\alpha}
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\label{eq-CNa-definition}
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\end{equation}
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%
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for $\alpha>0$ and by equation~(\ref{eq-CNa-derivative}) for
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$\alpha=0$. These definitions are compatible, since
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equation~(\ref{eq-CNa-definition}) simplifies to the partial
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derivative~(\ref{eq-CNa-derivative}) at the limit
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$\alpha\rightarrow0$. This definition also allows us to stay true to
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Barrowman's original method which is familiar to many rocketeers.
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Similar to the normal force coefficient, the drag coefficient is
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defined as
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%
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\begin{equation}
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C_D = \frac{D}{\frac{1}{2}\rho v_0^2 \, \Aref}.
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\label{eq-CD-def}
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\end{equation}
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%
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Since the size of the rocket has been factored out, the drag
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coefficient at zero angle of attack $C_{D0}$ allows a straightforward
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method of comparing the effect of different rocket shapes on drag.
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However, this coefficient is not constant and will vary with \eg the
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speed of the rocket and its angle of attack.
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If each of the fins of a rocket are canted at some angle $\delta>0$
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with respect to the rocket centerline, the fins will produce a roll
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moment on the rocket. Contrary to the normal force and pitching
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moment, canting the fins will produce a non-zero rolling moment but no
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corresponding net force. Therefore the only quantity computed is the
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roll moment coefficient, defined by
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%
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\begin{equation}
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C_l = \frac{l}{\frac{1}{2}\rho v_0^2 \, \Aref\, d}
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\label{eq-Cl-def}
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\end{equation}
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%
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where $l$ is the roll moment.
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It shall be shown later that rockets with axially-symmetrical fin
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configurations experience no forces that would produce net yawing
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moments. However, a single fin may produce all six types of forces and
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moments. The equations for the forces and moments of a single fin will
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not be explicitly written out, and they can be computed from the
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geometry in question.
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\subsection{Velocity regions}
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Most of the aerodynamic properties of rockets vary with the velocity
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of the rocket. The important parameter is the {\it Mach number},
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which is the free-stream velocity of the rocket divided by the local
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speed of sound
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%
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\begin{equation}
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M = \frac{v_0}{c}.
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\end{equation}
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%
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The velocity range encountered by rockets is divided into regions with
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different impacts on the aerodynamical properties, listed in
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Table~\ref{tab-sonics}.
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In {\it subsonic flight} all of the airflow around the rocket occurs
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below the speed of sound. This is the case for approximately $M<0.8$.
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At very low Mach numbers air can be effectively treated as an
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incompressible fluid, but already above $M\approx 0.3$ some
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compressibility issues may have to be considered.
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In {\it transonic flight} some of the air flowing around the rocket
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accelerates above the speed of sound, while at other places it remains
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subsonic. Some local shock waves are generated and hard-to-predict
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interference effects may occur. The drag of a rocket has a sharp
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increase in the transonic region, making it hard to pass into the
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supersonic region. Transonic flight occurs at Mach numbers of
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approximately 0.8--1.2.
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In {\it supersonic flight} all of the airflow is faster than the
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speed of sound (with the exception of \eg the nose cone tip). A shock
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wave is generated by the nose cone and fins. In supersonic flight
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the drag reduces from that of transonic flight, but is generally
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greater than that of subsonic flight. Above approximately Mach 5 new
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phenomena begin to emerge that are not encountered at lower supersonic
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speeds. This region is called {\it hypersonic flight}.
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\begin{table}
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\caption{Velocity regions of rocket flight}
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\label{tab-sonics}
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\begin{center}
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\begin{tabular}{cr@{ -- }l}
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Region & \multicolumn{2}{c}{Mach number ($M$)} \\
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\hline
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Subsonic & \hspace{10mm} 0 & 0.8 \\
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Transonic & 0.8 & 1.2 \\
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Supersonic & 1.2 & $\sim5$ \\
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Hypersonic & $\sim5$ & \\
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\hline
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\end{tabular}
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\end{center}
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\end{table}
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Methods for predicting the aerodynamic properties of subsonic flight
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and some extensions to supersonic flight will be presented. Since the
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analytical prediction of aerodynamic properties in the transonic
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region is quite difficult, this region will be accounted for by using
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some suitable interpolation function that corresponds reasonably to
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actual measurements. Hypersonic flight will not be considered, since
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practically no model or high power rockets ever achieve such speeds.
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\subsection{Flow and geometry parameters}
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There exist many different parameters that characterize aspects of
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flow or a rocket's geometry. One of the most important flow
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parameters is the {\it Reynolds number} $R$. It is a dimensionless
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quantity that characterizes the ratio of inertial forces and viscous
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forces of flow. Many aerodynamic properties depend on the Reynolds
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number, defined as
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%
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\begin{equation}
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R = \frac{v_0\; L}{\nu}.
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\end{equation}
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%
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Here $v_0$ is the free-stream velocity of the rocket, $L$ is a
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characteristic length and $\nu$ is the kinematic viscosity of air. It
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is notable that the Reynolds number is dependent on a characteristic
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length of the object in question. In most cases, the length used is
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the length of the rocket. A typical 30~cm sport model flying at
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50~m/s has a corresponding Reynolds number of approximately
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1\s000\s000.
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Another term that is frequently encountered in aerodynamical equations
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has been defined its own parameter $\beta$, which characterizes the
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flow speed both in subsonic and supersonic flow:
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%
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\begin{equation}
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\beta = \sqrt{\envert{M^2-1}} =
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\left\{
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\begin{array}{ll}
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\sqrt{1-M^2}, & {\rm if\ } M<1 \\
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\sqrt{M^2-1}, & {\rm if\ } M>1
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\end{array}
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\right.
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\end{equation}
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%
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As the flow speed approaches the transonic region $\beta$ approaches
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zero. This term appears for example in the {\it Prandtl factor} $P$
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which corrects subsonic force coefficients for compressible flow:
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%
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\begin{equation}
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P = \frac{1}{\beta} = \frac{1}{\sqrt{1-M^2}}
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\label{eq-prandtl-factor}
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\end{equation}
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It is also often useful to define parameters characterizing general
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properties of a rocket. One such parameter is the {\it caliber},
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defined as the maximum body diameter. The caliber is often used to
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indicate relative distances on the body of a rocket, such as the
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stability margin. Another common parameter characterizes the
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``slenderness'' of a rocket. It is the {\it fineness ratio} of a
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rocket $f_B$, defined as the length of the rocket body divided by the
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maximum body diameter. Typical model rockets have a fineness ratio in
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the range of 10--20, but extreme models may have a fineness ratio as
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low as 5 or as large as 50.
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\subsection{Coordinate systems}
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During calculation of the aerodynamic properties a coordinate system
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fixed to the rocket will be used. The origin of the coordinates is at
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the nose cone tip with the positive $x$-axis directed along the rocket
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centerline. This convention is also followed internally in the
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produced software. In the following sections the position of the $y$-
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and $z$-axes are arbitrary; the parameter $y$ is used as a general
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spanwise coordinate when discussing the fins. During simulation,
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however, the $y$- and $z$-axes are fixed in relation to the rocket,
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and do not necessarily align with the plane of the pitching moments.
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\clearpage
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\section{Normal forces and pitching moments}
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Barrowman's method~\cite{barrowman-thesis} for determining the total
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normal force coefficient derivative \CNa, the pitch moment
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coefficient derivative \Cma\ and the CP location at subsonic speeds
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first splits the rocket into simple separate components, then
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calculates the CP location and \CNa\ for each component separately and
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then combines these to get the desired coefficients and CP
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location. The general assumptions made by the derivation are:
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%
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\begin{enumerate}
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\item The angle of attack is very close to zero.
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\item The flow around the body is steady and non-rotational.
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\item The rocket is a rigid body.
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\item The nose tip is a sharp point.
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\item The fins are flat plates.
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\item The rocket body is axially symmetric.
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\end{enumerate}
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The components that will be discussed are nose cones, cylindrical body
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tube sections, shoulders, boattails and fins, in an arbitrary
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order. The interference effect between the body and fins will be
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taken into account by a separate correction term. Extensions to
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account for body lift and arbitrary fin shapes will also be derived.
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\subsection{Axially symmetric body components}
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The body of the rocket is assumed to be an axially symmetric body of
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rotation. The entire body could be considered to be a single
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component, but in practice it is divided into nose cones, shoulders,
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boattails and cylindrical body tube sections. The geometry of typical
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nose cones, shoulders and boattails are described in
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Appendix~\ref{app-nosecone-geometry}.
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The method presented by Barrowman for calculating the normal force and
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pitch moment coefficients at supersonic speeds is based on a
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second-order shock expansion method. However, this assumes that the
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body of the rocket is very streamlined, and it cannot handle areas
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with a slope larger than than $\sim30^\circ$. Since the software
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allows basically any body shape, applying this method would be
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difficult.
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Since the emphasis is on subsonic flow, for the purposes of this
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thesis the normal force and pitching moments produced by the body are
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assumed to be equal at subsonic and supersonic speeds. The assumption
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is that the CP location is primarily affected by the fins. The effect
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of supersonic flight on the drag of the body will be accounted for in
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Section~\ref{sec-drag}.
|
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\subsubsection{\CNa\ of body components at subsonic speeds}
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The normal force for an axially symmetric body at position $x$ in
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subsonic flow is given by
|
||
%
|
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\begin{equation}
|
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N(x) = \rho v_0 \; \frac{\partial}{\partial x}[A(x)w(x)]
|
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\label{eq-normal-force}
|
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\end{equation}
|
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%
|
||
where $A(x)$ is the cross-sectional area of the body, and the $w(x)$
|
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is the local downwash, given as a function of the angle of attack as
|
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%
|
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\begin{equation}
|
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w(x) = v_0 \sin\alpha.
|
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\end{equation}
|
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%
|
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For angles of attack very close to zero $\sin\alpha\approx\alpha$, but
|
||
contrary to the original derivation, we shall not make this
|
||
simplification. From the definition of the normal force
|
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coefficient~(\ref{eq-CN-def}) and equation~(\ref{eq-normal-force}) we
|
||
obtain
|
||
%
|
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\begin{equation}
|
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C_N(x) = \frac{N(x)}{\frac{1}{2}\rho v_0^2\;\Aref}
|
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= \frac{2\; \sin\alpha}{\Aref}\; \frac{\dif A(x)}{\dif x}.
|
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\label{eq-CNx}
|
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\end{equation}
|
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%
|
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Assuming that the derivative $\frac{\dif A(x)}{\dif x}$ is
|
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well-defined, we can integrate over the component length $l$ to obtain
|
||
%
|
||
\begin{equation}
|
||
C_N = \frac{2\; \sin\alpha}{\Aref}\;
|
||
\int_0^l \frac{\dif A(x)}{\dif x}\dif x
|
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= \frac{2\; \sin\alpha}{\Aref}\; [A(l)-A(0)].
|
||
\end{equation}
|
||
%
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||
We then have
|
||
%
|
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\begin{equation}
|
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\CNa = \frac{C_N}{\alpha}
|
||
= \frac{2}{\Aref}\; [A(l)-A(0)]\;
|
||
\underbrace{\frac{\sin\alpha}{\alpha}}_
|
||
{\parbox{10mm}{\scriptsize\centering
|
||
$\rightarrow 1$ as \\ $\alpha\rightarrow0$}}.
|
||
\label{eq-body-CNa}
|
||
\end{equation}
|
||
%
|
||
This is the same equation as derived by Barrowman with the exception
|
||
of the correction term $\sin\alpha/\alpha$.
|
||
|
||
Equation~(\ref{eq-body-CNa}) shows that as long as the cross-sectional
|
||
area of the component changes smoothly, the normal force coefficient
|
||
derivative does not depend on the component shape, only the difference
|
||
of the cross-sectional area at the beginning and end. As a
|
||
consequence, according to Barrowman's theory, a cylindrical body tube
|
||
has no effect on the normal force coefficient or CP location.
|
||
However, the lift due to cylindrical body tube sections has been noted
|
||
to be significant for long, slender rockets even at angles of attack
|
||
of only a few degrees~\cite{galejs}. An extension
|
||
for the effect of body lift will be given shortly.
|
||
|
||
|
||
|
||
|
||
\subsubsection{\Cma\ of body components at subsonic speeds}
|
||
|
||
A normal force $N(x)$ at position $x$ produces a pitching moment
|
||
%
|
||
\begin{equation}
|
||
m(x) = xN(x).
|
||
\end{equation}
|
||
%
|
||
at the nose cone tip. Therefore the pitching moment coefficient is
|
||
%
|
||
\begin{equation}
|
||
C_m(x) = \frac{m(x)}{\frac{1}{2}\rho v_0^2\;\Aref\, d}
|
||
= \frac{xN(x)}{\frac{1}{2}\rho v_0^2\;\Aref\, d}.
|
||
\end{equation}
|
||
%
|
||
Substituting equation~(\ref{eq-CNx}) we obtain
|
||
%
|
||
\begin{equation}
|
||
C_m(x) = \frac{x\;C_N(x)}{d}
|
||
= \frac{2\; \sin\alpha\; x}{\Aref\, d}\; \frac{\dif A(x)}{\dif x}.
|
||
\end{equation}
|
||
%
|
||
This can be integrated over the length of the body to obtain
|
||
%
|
||
\begin{equation}
|
||
C_m = \frac{2\;\sin\alpha}{\Aref\,d}
|
||
\int_0^l x \left(\od{A(x)}{x}\right) \dif x
|
||
= \frac{2\;\sin\alpha}{\Aref\,d}
|
||
\sbr{ lA(l)-\int_0^l A(x) \dif x }.
|
||
\end{equation}
|
||
%
|
||
The resulting integral is simply the volume of the body $V$.
|
||
Therefore we have
|
||
%
|
||
\begin{equation}
|
||
C_m = \frac{2\;\sin\alpha}{\Aref\,d} \sbr{ lA(l)-V }
|
||
\end{equation}
|
||
%
|
||
and
|
||
%
|
||
\begin{equation}
|
||
\Cma = \frac{2}{\Aref\,d}\sbr{ lA(l)-V }\; \frac{\sin\alpha}{\alpha}.
|
||
\label{eq-body-Cma}
|
||
\end{equation}
|
||
%
|
||
This is, again, the result derived by Barrowman with the additional
|
||
correction term $\sin\alpha/\alpha$.
|
||
|
||
|
||
\subsubsection{Effect of body lift}
|
||
\label{sec-body-lift}
|
||
|
||
The analysis thus far has neglected the effect of body lift as
|
||
negligible at small angles of attack. However, in the flight of long,
|
||
slender rockets the lift may be quite significant at angles of attack
|
||
of only a few degrees, which may occur at moderate wind
|
||
speeds~\cite{galejs}.
|
||
|
||
Robert Galejs suggested adding a correction term to the body
|
||
component \CNa\ to account for body lift~\cite{galejs}. The normal
|
||
force exerted on a cylindrical body at an angle of attack $\alpha$
|
||
is~\cite[p.~3-11]{hoerner}
|
||
%
|
||
\begin{equation}
|
||
C_N = K\; \frac{A_{\rm plan}}{\Aref}\; \sin^2\alpha
|
||
\end{equation}
|
||
%
|
||
where $A_{\rm plan} = d\cdot l$ is the planform area of the cylinder
|
||
and K is a constant $K\approx 1.1$. Galejs had simplified the
|
||
equation with $\sin^2\alpha\approx\alpha^2$, but this shall not be
|
||
performed here. At small angles of attack, when the approximation is
|
||
valid, this yields a linear correction to the value of \CNa.
|
||
|
||
It is assumed that the lift on non-cylindrical components can be
|
||
approximated reasonably well with the same equation. The CP location
|
||
is assumed to be the center of the planform area, that is
|
||
%
|
||
\begin{equation}
|
||
X_{\rm lift} = \frac{\int_0^l x\; 2r(x)\dif x}{A_{\rm plan}}.
|
||
\end{equation}
|
||
%
|
||
This is reminiscent of the CP of a rocket flying at an angle of attack
|
||
of $90^\circ$. For a cylinder the CP location is at the center of the
|
||
body, which is also the CP location obtained at the limit with
|
||
equation~(\ref{eq-body-CP-position}). However, for nose cones,
|
||
shoulders and boattails it yields a slightly different position than
|
||
equation~(\ref{eq-body-CP-position}).
|
||
|
||
%The value of $K$ has been experimentally fitted to experimental data
|
||
%from wind tunnels.
|
||
|
||
|
||
|
||
|
||
|
||
|
||
\subsubsection{Center of pressure of body components}
|
||
|
||
The CP location of the body components can be calculated by
|
||
inserting equations~(\ref{eq-body-CNa}) and (\ref{eq-body-Cma}) into
|
||
equation~(\ref{eq-CP-position}):
|
||
%
|
||
\begin{equation}
|
||
X_B = \frac{(\Cma)_B}{(\CNa)_B}\;d
|
||
= \frac{lA(l)-V}{A(l)-A(0)}
|
||
\label{eq-body-CP-position}
|
||
\end{equation}
|
||
%
|
||
It is worth noting that the correction term $\sin\alpha/\alpha$
|
||
cancels out in the division, however, it is still present in the
|
||
value of \CNa\ and is therefore significant at large angles of attack.
|
||
|
||
The whole rocket body could be numerically integrated and the
|
||
properties of the whole body computed. However, it is often more
|
||
descriptive to split the body into components and calculate the
|
||
parameters separately. The total CP location can be calculated from
|
||
the separate CP locations $X_i$ and normal force coefficient
|
||
derivatives $(\CNa)_i$ by the moment sum
|
||
%
|
||
\begin{equation}
|
||
X = \frac{\sum_{i=1}^n X_i(\CNa)_i}{\sum_{i=1}^n (\CNa)_i}.
|
||
\label{eq-moment-sum}
|
||
\end{equation}
|
||
%
|
||
In this manner the effect of the separate components can be more
|
||
easily analyzed.
|
||
|
||
|
||
|
||
\subsection{Planar fins}
|
||
\label{sec-planar-fins}
|
||
|
||
The fins of the rocket are considered separately from the body. Their
|
||
CP location and normal force coefficient are determined and added to
|
||
the total moment sum~(\ref{eq-moment-sum}). The interference between
|
||
the fins and the body is taken into account by a separate correction
|
||
term.
|
||
|
||
In addition to the corrective normal force, the fins can induce a roll
|
||
rate if each of the fins are canted at an angle $\delta$. The roll
|
||
moment coefficient will be derived separately in
|
||
Section~\ref{sec-roll-dynamics}.
|
||
|
||
Barrowman's original report and thesis derived the equations for
|
||
trapezoidal fins, where the tip chord is parallel to the body
|
||
(Figure~\ref{fig-fin-geometry}(a)). The equations can be extended to
|
||
\eg elliptical fins~\cite{barrowman-elliptical-fins}
|
||
(Figure~\ref{fig-fin-geometry}(b)), but many model rocket fin
|
||
designs depart from these basic shapes. Therefore an
|
||
extension is presented that approximates the aerodynamical
|
||
properties for a free-form fin defined by a list of $(x,y)$
|
||
coordinates (Figure~\ref{fig-fin-geometry}(c)).
|
||
|
||
|
||
\begin{figure}
|
||
\centering
|
||
\parbox{35mm}{\centering
|
||
\epsfig{file=figures/fin-geometry/fin-trapezoidal,scale=0.5} \\ (a)}
|
||
\hfill
|
||
\parbox{35mm}{\centering
|
||
\epsfig{file=figures/fin-geometry/fin-elliptical,scale=0.5} \\ (b)}
|
||
\hfill
|
||
\parbox{35mm}{\centering
|
||
\epsfig{file=figures/fin-geometry/fin-free,scale=0.5} \\ (c)}
|
||
\caption{Fin geometry of (a) a trapezoidal fin, (b) an elliptical fin
|
||
and (c) a free-form fin.}
|
||
\label{fig-fin-geometry}
|
||
\end{figure}
|
||
|
||
|
||
|
||
Additionally, Barrowman considered only cases with three or four
|
||
fins. This shall be extended to allow for any reasonable number of
|
||
fins, even single fins.
|
||
|
||
|
||
\subsubsection{Center of pressure of fins at subsonic and supersonic
|
||
speeds}
|
||
|
||
Barrowman argued that since the CP of a fin is located along its mean
|
||
aerodynamic chord (MAC) and on the other hand at low subsonic speeds
|
||
on its quarter chord, then the CP must be located at the intersection
|
||
of these two (depicted in Figure~\ref{fig-fin-geometry}(a)). He
|
||
proceeded to calculate this intersection point analytically from the
|
||
fin geometry of a trapezoidal fin.
|
||
|
||
Instead of following the derivation Barrowman used, an alternative
|
||
method will be presented that allows simpler extension to free-form
|
||
fins. The two methods yield identical results for trapezoidal fins.
|
||
The length of the MAC $\bar c$, its spanwise position $y_{\rm MAC}$,
|
||
and the effective leading edge location $x_{\rm MAC,LE}$ are given
|
||
by~\cite{appl-comp-aero-fins}
|
||
%
|
||
\begin{align}
|
||
\bar c &= \frac{1}{\Afin} \int_0^s c^2(y) \dif y
|
||
\label{eq-MAC-length} \\
|
||
y_{\rm MAC} &= \frac{1}{\Afin} \int_0^s yc(y) \dif y
|
||
\label{eq-MAC-ypos} \\
|
||
x_{\rm MAC,LE} &= \frac{1}{\Afin} \int_0^s x_{\rm LE}(y)c(y) \dif y
|
||
\label{eq-MAC-xpos}
|
||
\end{align}
|
||
%
|
||
where $\Afin$ is the one-sided area of a single fin, $s$ is the span of
|
||
one fin, and $c(y)$ is the length of the fin chord and $x_{\rm LE}(y)$
|
||
the leading edge position at spanwise position $y$.
|
||
|
||
When these equations are applied to trapezoidal fins and the
|
||
lengthwise position of the CP is selected at the quarter chord,
|
||
$X_f=x_{\rm MAC,LE}+0.25\,\bar c$,
|
||
one recovers exactly the results derived by Barrowman:
|
||
%
|
||
\begin{align}
|
||
y_{\rm MAC} &= \frac{s}{3}\,\frac{C_r+2C_t}{C_r+C_t} \\
|
||
X_f &= \frac{X_t}{3}\,\frac{C_r+2C_t}{C_r+C_t} +
|
||
\frac{1}{6}\,\frac{C_r^2+C_t^2+C_rC_t}{C_r+C_t}
|
||
\end{align}
|
||
%
|
||
However, equations~(\ref{eq-MAC-length})--(\ref{eq-MAC-xpos}) may also
|
||
be directly applied to elliptical or free-form fins.
|
||
|
||
Barrowman's method assumes that the lengthwise position of the CP
|
||
stays at a constant 25\% of the MAC at subsonic speeds. However, the
|
||
position starts moving rearward above approximately Mach 0.5. For
|
||
$M>2$ the relative lengthwise position of the CP is given by an empirical
|
||
formula~\cite[p.~33]{fleeman}
|
||
%
|
||
\begin{equation}
|
||
\frac{X_f}{\bar c} = \frac{\AR\beta - 0.67}{2\AR\beta-1}
|
||
\label{eq-fin-CP-mach2}
|
||
\end{equation}
|
||
%
|
||
where $\beta=\sqrt{M^2-1}$ for $M>1$ and \AR\ is the aspect ratio of
|
||
the fin defined using the span $s$ as $\AR=2s^2/\Afin$.
|
||
%
|
||
Between Mach 0.5 and 2 the lengthwise position of the CP is
|
||
interpolated. A suitable function that gives a curve similar to that
|
||
of Figure~2.18 of reference~\cite[p.~33]{fleeman} was found to be a
|
||
fifth order polynomial $p(M)$ with the constraints
|
||
%
|
||
\begin{equation}
|
||
\begin{split}
|
||
p(0.5) & = 0.25 \\
|
||
p'(0.5) & = 0 \\
|
||
p(2) & = f(2) \\
|
||
p'(2) & = f'(2) \\
|
||
p''(2) & = 0 \\
|
||
p'''(2) & = 0
|
||
\end{split}
|
||
\end{equation}
|
||
%
|
||
where $f(M)$ is the function of equation~(\ref{eq-fin-CP-mach2}).
|
||
|
||
|
||
|
||
The method presented here can be used to estimate the CP location of
|
||
an arbitrary thin fin. However, problems arise with the method if the
|
||
fin shape has a jagged edge as shown in
|
||
Figure~\ref{fig-fin-jagged}(a). If $c(y)$ would include only the sum
|
||
of the two separate chords in the area containing the gap, then the
|
||
equations would yield the same result as for a fin shown in
|
||
Figure~\ref{fig-fin-jagged}(b). This clearly would be incorrect,
|
||
since the position of the latter fin portion would be neglected. To
|
||
overcome this problem, $c(y)$ is chosen as the length from the leading
|
||
edge to the trailing edge of the fin, effectively adding the portion
|
||
marked by the dotted line to the fin. This corrects the CP position
|
||
slightly rearwards. The fin area used in
|
||
equations~(\ref{eq-MAC-length})--(\ref{eq-MAC-xpos}) must in this case
|
||
also be calculated including this extra fin area, but the extra area
|
||
must not be included when calculating the normal force coefficient.
|
||
|
||
This correction is also approximate, since in reality such a jagged
|
||
edge would cause some unknown interference factor between the two fin
|
||
portions. Simulating such jagged edges using these methods should
|
||
therefore be avoided.
|
||
|
||
|
||
\begin{figure}
|
||
\centering
|
||
\parbox{35mm}{\centering
|
||
\epsfig{file=figures/fin-geometry/fin-jagged,scale=0.5} \\ (a)}
|
||
\hspace{5mm}
|
||
\parbox{35mm}{\centering
|
||
\epsfig{file=figures/fin-geometry/fin-jagged-equivalent,scale=0.5} \\ (b)}
|
||
\caption{(a) A jagged fin edge, and (b) an equivalent fin if $c(y)$ is
|
||
chosen to include only the actual fin area.}
|
||
\label{fig-fin-jagged}
|
||
\end{figure}
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
\subsubsection{Single fin \CNa\ at subsonic speeds}
|
||
\label{sec-average-angle}
|
||
|
||
Barrowman derived the normal force coefficient derivative value based
|
||
on Diederich's semi-empirical method~\cite{diederich}, which states that
|
||
for one fin
|
||
%
|
||
\begin{equation}
|
||
\del{\CNa}_1 = \frac{\CNa_0\; F_D \left(\frac{\Afin}{\Aref}\right)
|
||
\cos\Gamma_c}
|
||
{2+F_D\sqrt{1+\frac{4}{F_D^2}}},
|
||
\label{eq-fin-CNa-base}
|
||
\end{equation}
|
||
%
|
||
where
|
||
%
|
||
\begin{itemize}
|
||
\item[$\CNa_0$] = normal force coefficient derivative of a 2D airfoil
|
||
\item[$F_D$] = Diederich's planform correlation parameter
|
||
\item[$\Afin$] = area of one fin
|
||
\item[$\Gamma_c$] = midchord sweep angle (depicted in
|
||
Figure~\ref{fig-fin-geometry}(a)).
|
||
\end{itemize}
|
||
%
|
||
Based on thin airfoil theory of potential flow corrected for
|
||
compressible flow
|
||
%
|
||
\begin{equation}
|
||
\CNa_0 = \frac{2\pi}{\beta}
|
||
\label{eq-fin-CNa0}
|
||
\end{equation}
|
||
%
|
||
where $\beta=\sqrt{1-M^2}$ for $M<1$. $F_D$ is a parameter that
|
||
corrects the normal force coefficient for the sweep of the fin.
|
||
According to Diederich, $F_D$ is given by
|
||
%
|
||
\begin{equation}
|
||
F_D=\frac{\AR}{\frac{1}{2\pi}\CNa_0\cos\Gamma_c}.
|
||
\label{eq-fin-FD}
|
||
\end{equation}
|
||
%
|
||
Substituting equations~(\ref{eq-fin-CNa0}), (\ref{eq-fin-FD}) and
|
||
$\AR=2s^2/\Afin$ into (\ref{eq-fin-CNa-base}) and simplifying one
|
||
obtains
|
||
%
|
||
\begin{equation}
|
||
%\del{\CNa}_1 = \frac{2\pi\; \AR \del{\frac{\Afin}{\Aref}}}
|
||
% {2+\sqrt{4 + \del{\frac{\beta \AR}{\cos\Gamma_c}}^2}}.
|
||
\del{\CNa}_1 = \frac{2\pi\; \frac{s^2}{\Aref}}
|
||
{1+\sqrt{1 + \del{\frac{\beta s^2}{\Afin\cos\Gamma_c}}^2}}.
|
||
\label{eq-CNa1}
|
||
\end{equation}
|
||
%
|
||
This is the normal force coefficient derivative for one fin, where the
|
||
angle of attack is between the airflow and fin surface.
|
||
|
||
The value of equation~(\ref{eq-CNa1}) can be calculated directly for
|
||
trapezoidal and elliptical fins. However, in the case of free-form
|
||
fins, the question arises of how to define the mid-chord angle
|
||
$\Gamma_c$. If the angle $\Gamma_c$ is taken as the angle from the
|
||
middle of the root chord to the tip of the fin, the result may not be
|
||
representative of the actual shape, as shown by angle $\Gamma_{c1}$ in
|
||
Figure~\ref{fig-midchord-angle}.
|
||
|
||
\begin{figure}
|
||
\centering
|
||
\epsfig{file=figures/fin-geometry/fin-midchord-angle,scale=0.7}
|
||
\caption{A free-form fin shape and two possibilities for the midchord
|
||
angle $\Gamma_c$.}
|
||
\label{fig-midchord-angle}
|
||
\end{figure}
|
||
|
||
Instead the fin planform is divided into a large number of chords, and
|
||
the angle between the midpoints of each two consecutive chords is
|
||
calculated. The midchord angle used in equation~(\ref{eq-CNa1}) is
|
||
then the average of all these angles. This produces an angle better
|
||
representing the actual shape of the fin, as angle $\Gamma_{c2}$ in
|
||
Figure~\ref{fig-midchord-angle}. The angle calculated by this method
|
||
is also equal to the natural midchord angles for trapezoidal and
|
||
elliptical fins.
|
||
|
||
|
||
|
||
|
||
\subsubsection{Single fin \CNa\ at supersonic speeds}
|
||
\label{sec-single-fin-CNa-supersonic}
|
||
|
||
The method for calculating the normal force coefficient of fins at
|
||
supersonic speed presented by Barrowman is based on a third-order
|
||
expansion according to Busemann theory~\cite{barrowman-fin}. The
|
||
method divides the fin into narrow streamwise strips, the normal force
|
||
of which are calculated separately. In this presentation the method
|
||
is further simplified by assuming the fins to be flat plates and by
|
||
ignoring a third-order term that corrects for fin-tip Mach cone
|
||
effects.
|
||
|
||
%
|
||
% Angle of Inclination = between ray and surface
|
||
% Angle of Incidence = between ray and normal of surface
|
||
%
|
||
|
||
|
||
The local pressure coefficient of strip $i$ is calculated by
|
||
%
|
||
\begin{equation}
|
||
C_{P_i} = K_1 \,\eta_i + K_2 \,\eta_i^2 + K_3 \,\eta_i^3
|
||
\label{eq-local-pressure-coefficient}
|
||
\end{equation}
|
||
%
|
||
where $\eta_i$ is the inclination of the flow at the surface and the
|
||
coefficients are
|
||
%
|
||
\begin{align}
|
||
K_1 &= \frac{2}{\beta} \\
|
||
K_2 &= \frac{(\gamma+1)M^4 - 4\,\beta^2}{4\,\beta^4} \\
|
||
K_3 &= \frac{(\gamma+1)M^8 + (2\gamma^2-7\gamma-5)M^6 +
|
||
10(\gamma+1)M^4 + 8}{6\,\beta^7}
|
||
\end{align}
|
||
%
|
||
It is noteworthy that the coefficients $K_1$, $K_2$ and $K_3$ can be
|
||
pre-calculated for various Mach numbers, which makes the pressure
|
||
coefficient of a single strip very fast to compute. At small
|
||
angles of inclination the pressure coefficient is nearly linear, as
|
||
presented in Figure~\ref{fig-fin-strip-pressure-coefficient}.
|
||
|
||
|
||
\begin{figure}
|
||
\centering
|
||
\epsfig{file=figures/fin-geometry/Cp-supersonic,scale=0.6}
|
||
\caption{The local pressure coefficient as a function of the
|
||
strip inclination angle at various Mach numbers. The dotted line
|
||
depicts the linear component of
|
||
equation~(\ref{eq-local-pressure-coefficient}).}
|
||
\label{fig-fin-strip-pressure-coefficient}
|
||
\end{figure}
|
||
|
||
|
||
%If the rocket is not rolling, the inclinations $\eta_i$ are the
|
||
%same for all strips of a fin and only one pressure coefficient needs
|
||
%to be computed. However, presence of a roll velocity generates
|
||
%varying inclinations for all strips. Therefore the current
|
||
%examination is performed with separate inclinations and later
|
||
%simplified for non-rolling conditions. The effects of roll are
|
||
%further discussed in Section~\ref{sec-roll-dynamics}.
|
||
|
||
The lift force of strip $i$ is equal to
|
||
%
|
||
\begin{equation}
|
||
F_i = C_{P_i} \cdot \frac{1}{2} \rho v_0^2 \cdot
|
||
\underbrace{c_i \Delta y}_{\rm area}.
|
||
\label{eq-supersonic-strip-lift-force}
|
||
\end{equation}
|
||
%
|
||
The total lift force of the fin is obtained by summing up the
|
||
contributions of all fin strips. The normal force coefficient is then
|
||
calculated in the usual manner as
|
||
%
|
||
\begin{align}
|
||
C_N &= \frac{\sum_i F_i}{\frac{1}{2}\rho v_0^2\; \Aref} \\
|
||
&= \frac{1}{\Aref}\sum_i C_{P_i} \cdot c_i\Delta y.
|
||
\end{align}
|
||
|
||
When computing the corrective normal force coefficient of the fins the
|
||
effect of roll is not taken into account. In this case, and
|
||
assuming that the fins are flat plates, the inclination angles
|
||
$\eta_i$ of all strips are the same, and the pressure coefficient is
|
||
constant over the entire fin. Therefore the normal force coefficient
|
||
is simply
|
||
%
|
||
\begin{equation}
|
||
(C_N)_1 = \frac{\Afin}{\Aref} \;C_P.
|
||
\end{equation}
|
||
%
|
||
Since the pressure coefficient is not linear with the angle of attack,
|
||
the normal force coefficient slope is defined using
|
||
equation~(\ref{eq-CNa-definition}) as
|
||
%
|
||
\begin{equation}
|
||
(\CNa)_1 = \frac{(C_N)_1}{\alpha} =
|
||
\frac{\Afin}{\Aref} \; \del{K_1 + K_2\,\alpha + K_3\,\alpha^2}.
|
||
\end{equation}
|
||
|
||
|
||
|
||
|
||
\subsubsection{Multiple fin \CNa}
|
||
\label{update-roll-angle}
|
||
|
||
In his thesis, Barrowman considered only configurations with three and
|
||
four fins, one of which was parallel to the lateral airflow. For
|
||
simulation purposes, it is necessary to lift these restrictions to
|
||
allow for any direction of lateral airflow and for any number of
|
||
fins.
|
||
|
||
The lift force of a fin is perpendicular to the fin and originates
|
||
from its CP. Therefore a single fin may cause a rolling and yawing
|
||
moment in addition to a pitching moment. In this case all of the
|
||
forces and moments must be computed from the geometry. If there are
|
||
two or more fins placed symmetrically around the body then the yawing
|
||
moments cancel, and if additionally there is no fin cant then the
|
||
total rolling moment is also zero, and these moments need not be
|
||
computed.
|
||
|
||
The geometry of an uncanted fin configuration is depicted in
|
||
Figure~\ref{fig-dihedral-angle}. The dihedral angle between each of
|
||
the fins and the airflow direction is denoted $\Lambda_i$. The fin
|
||
$i$ encounters a local angle of attack of
|
||
%
|
||
\begin{equation}
|
||
\alpha_i = \alpha \sin\Lambda_i
|
||
\end{equation}
|
||
%
|
||
for which the normal force component (the component parallel to the
|
||
lateral airflow) is then
|
||
%
|
||
\begin{equation}
|
||
\del{\CNa}_{\Lambda_i} = \del{\CNa}_1 \sin^2 \Lambda_i.
|
||
\end{equation}
|
||
|
||
|
||
\begin{figure}
|
||
\centering
|
||
\epsfig{file=figures/fin-geometry/dihedral-angle,scale=1}
|
||
\caption{The geometry of an uncanted three-fin configuration (viewed
|
||
from rear).}
|
||
\label{fig-dihedral-angle}
|
||
\end{figure}
|
||
|
||
|
||
The sum of the coefficients for $N$ fins then yields
|
||
%
|
||
\begin{equation}
|
||
\sum_{k=1}^N \del{\CNa}_{\Lambda_k} =
|
||
\del{\CNa}_1 \sum_{k=1}^N \sin^2\Lambda_k.
|
||
\label{N-fin-equation}
|
||
\end{equation}
|
||
%
|
||
However, when $N\geq 3$ and the fins are spaced equally around the
|
||
body of the rocket, the sum simplifies to a constant
|
||
%
|
||
\begin{equation}
|
||
\sum_{k=1}^N \sin^2 (2\pi k/N + \theta) = \frac{N}{2}.
|
||
\label{N-fin-simplification}
|
||
\end{equation}
|
||
%
|
||
This equation predicts that the normal force produced by three or more
|
||
fins is independent of the roll angle $\theta$ of the vehicle.
|
||
Investigation by Pettis~\cite{pettis} showed that the normal force
|
||
coefficient derivative of a four-finned rocket at Mach~1.48 decreased
|
||
by approximately 6\% at a roll angle of $45^\circ$, and the roll angle
|
||
had negligible effect on an eight-finned rocket. Experimental data of
|
||
a four-finned sounding rocket at Mach speeds from 0.60 to 1.20
|
||
supports the 6\% estimate~\cite{experimental-transonic}.
|
||
|
||
The only experimental data available to the author of three-fin
|
||
configurations was of a rocket with a rounded triangular body cross
|
||
section~\cite{triform-fin-data}. This data suggests an effect of
|
||
approximately 15\% on the normal force coefficient derivative
|
||
depending on the roll angle. However, it is unknown how much of this
|
||
effect is due to the triangular body shape and how much from the fin
|
||
positioning.
|
||
|
||
It is also hard to predict such an effect when examining singular
|
||
fins. If three identical or very similar singular fins are placed on
|
||
a rocket body, the effect should be the same as when the fins belong
|
||
to the same three-fin configuration. Due to these facts the effect of
|
||
the roll angle on the normal force coefficient derivative is ignored
|
||
when a fin configuration has three or more fins.
|
||
%
|
||
\footnote{In OpenRocket versions prior to 0.9.6 a sinusoidal reduction
|
||
of 15\% and 6\% was applied to three- and four-fin configurations,
|
||
respectively. However, this sometimes caused a significantly
|
||
different predicted CP location compared to the pure Barrowman method,
|
||
and also caused a discrepancy when such a fin configuration was
|
||
decomposed into singular fins. It was deemed better to follow the
|
||
tested and tried Barrowman method instead of introducing additional
|
||
terms to the equation.}
|
||
|
||
However, in configurations with many fins the fin--fin
|
||
interference may cause the normal force to be less than that estimated
|
||
directly by equation~(\ref{N-fin-equation}). According to
|
||
reference~\cite[p.~5-24]{MIL-HDBK}, the normal force coefficients
|
||
for six and eight-fin configurations are 1.37 and 1.62 times that of
|
||
the corresponding four-fin configuration, respectively. The values
|
||
for five and seven-fin configurations are interpolated between these
|
||
values.
|
||
|
||
\pagebreak[4]
|
||
Altogether, the normal force coefficient derivative $(\CNa)_N$ is
|
||
calculated by:
|
||
%
|
||
\begin{equation}
|
||
(\CNa)_N = \del{\sum_{k=1}^N \sin^2\Lambda_k} \del{\CNa}_1 \cdot
|
||
\left\{
|
||
\begin{array}{ll}
|
||
1.000\; & N_{\rm tot}=1,2,3,4 \vspace{1mm}\\
|
||
0.948\; & N_{\rm tot}=5 \vspace{1mm}\\
|
||
0.913\; & N_{\rm tot}=6 \vspace{1mm}\\
|
||
0.854\; & N_{\rm tot}=7 \vspace{1mm}\\
|
||
0.810\; & N_{\rm tot}=8 \vspace{1mm}\\
|
||
0.750\; & N_{\rm tot}>8
|
||
\end{array}
|
||
\right.
|
||
\end{equation}
|
||
%
|
||
%\begin{equation}
|
||
%(\CNa)_N =
|
||
% \left\{
|
||
%\begin{array}{ll}
|
||
% \del{\sum_{k=1}^N \sin^2\Lambda_k} \del{\CNa}_1 & N=1,2 \vspace{1mm}\\
|
||
% 1.50\; \del{\CNa}_1 & N=3 \vspace{1mm}\\
|
||
% 2.00\; \del{\CNa}_1 & N=4 \vspace{1mm}\\
|
||
% 2.37\; \del{\CNa}_1 & N=5 \vspace{1mm}\\
|
||
% 2.74\; \del{\CNa}_1 & N=6 \vspace{1mm}\\
|
||
% 2.99\; \del{\CNa}_1 & N=7 \vspace{1mm}\\
|
||
% 3.24\; \del{\CNa}_1 & N=8
|
||
%\end{array}
|
||
% \right.
|
||
%\end{equation}
|
||
%
|
||
Here $N$ is the number of fins in this fin set, while $N_{\rm tot}$ is
|
||
the total number of parallel fins that have an interference effect.
|
||
The sum term simplifies to $N/2$ for $N\geq3$ according to
|
||
equation~(\ref{N-fin-simplification}). The interference effect for
|
||
$N_{\rm tot}>8$ is assumed at 25\%, as data for such configurations is
|
||
not available and such configurations are rare and eccentric in any
|
||
case.
|
||
|
||
\subsubsection{Fin--body interference}
|
||
|
||
The normal force coefficient must still be corrected for fin--body
|
||
interference, which increases the overall produced normal force. Here
|
||
two distinct effects can be identified: the normal force on the fins
|
||
due to the presence of the body and the normal force on the body due
|
||
to the presence of fins. Of these the former is significantly larger;
|
||
the latter is therefore ignored. The effect of the extra fin lift is
|
||
taken into account using a correction term
|
||
%
|
||
\begin{equation}
|
||
\del{\CNa}_{T(B)} = K_{T(B)}\;\del{\CNa}_N
|
||
\end{equation}
|
||
%
|
||
where $\del{\CNa}_{T(B)}$ is the normal force coefficient derivative
|
||
of the tail in the presence of the body. The term $K_{T(B)}$ can be
|
||
approximated by~\cite{barrowman-rd}
|
||
%
|
||
\begin{equation}
|
||
K_{T(B)} = 1 + \frac{r_t}{s + r_t},
|
||
\end{equation}
|
||
%
|
||
where $s$ is the fin span from root to tip and $r_t$ is the body
|
||
radius at the fin position. The value $\del{\CNa}_{T(B)}$ is then
|
||
used as the final normal force coefficient derivative of the fins.
|
||
|
||
|
||
%The normal force coefficient must still be corrected for fin--body
|
||
%interference, which increases the produced normal force. The effect
|
||
%of the interference can be split into two components, the normal force
|
||
%of the fins in the presence of the body $\del{\CNa}_{T(B)}$ and of the
|
||
%body in the presence of the fins $\del{\CNa}_{B(T)}$. (The subscript
|
||
%$T$ refers to {\it tail}.) The interference is taken into account
|
||
%using the correction factors
|
||
%%
|
||
%\begin{align}
|
||
%\del{\CNa}_{T(B)} &= K_{T(B)}\;\del{\CNa}_N \\
|
||
%\del{\CNa}_{B(T)} &= K_{B(T)}\;\del{\CNa}_N.
|
||
%\end{align}
|
||
%%
|
||
%In his original report, Barrowman simplified the factor $K_{T(B)}$ to
|
||
%%
|
||
%\begin{equation}
|
||
%K_{T(B)} = 1 + \frac{r_t}{s + r_t},
|
||
%\end{equation}
|
||
%%
|
||
%where $s$ is the fin span from root to tip and $r_t$ is the body
|
||
%radius at the fin position, and ignored the effect of $K_{B(T)}$ as
|
||
%small compared to $K_{T(B)}$ for typical fin dimensions. However,
|
||
%$K_{T(B)}$ may be significant for fins with a span short compared to
|
||
%the body radius. In his thesis, a more accurate equation for
|
||
%$K_{T(B)}$ is provided, and $K_{B(T)}$ is given
|
||
%as~\cite[p.~31]{barrowman-thesis}
|
||
%%
|
||
%\begin{equation}
|
||
%K_{B(T)} = \del{1 + \frac{r_t}{s + r_t}}^2 - K_{T(B)}.
|
||
%\end{equation}
|
||
%%
|
||
%Therefore the total interference effect can be accounted for by a
|
||
%factor
|
||
%%
|
||
%\begin{equation}
|
||
%K_T = K_{T(B)} + K_{B(T)} = \del{1 + \frac{r_t}{s + r_t}}^2
|
||
%\end{equation}
|
||
%%
|
||
%and
|
||
%%
|
||
%\begin{equation}
|
||
%\del{\CNa}_{\rm fins} = K_T\; (\CNa)_N.
|
||
%\end{equation}
|
||
|
||
%This equation takes the increase on body lift and adds it as an
|
||
%additional force on the fins. The equation holds at subsonic speeds
|
||
%and also at supersonic speeds until the fin tip Mach cone intersects
|
||
%the body. In the latter case more complex interference factors would
|
||
%be required, which have been ignored in the current software.
|
||
|
||
% TODO: FUTURE: supersonic interference effects, MIL-HDBK page 5-25 or B'man
|
||
|
||
|
||
|
||
\pagebreak[4]
|
||
\subsection{Pitch damping moment}
|
||
|
||
So far the effect of the current pitch angular velocity has been
|
||
ignored as marginal. This is the case during the upward flight of a
|
||
stable rocket. However, if a rocket is launched nearly vertically in
|
||
still air, the rocket flips over rather rapidly at apogee. In
|
||
some cases it was observed that the rocket was left wildly oscillating
|
||
during descent. The pitch damping moment opposes the fast rotation of
|
||
the rocket thus damping the oscillation.
|
||
|
||
Since the pitch damping moment is notable only at apogee, and
|
||
therefore does not contribute to the overall flight characteristics,
|
||
only a rough estimate of its magnitude is required. A cylinder in
|
||
perpendicular flow has a drag coefficient of approximately $C_D=1.1$,
|
||
with the reference area being the planform area of the
|
||
cylinder~\cite[p.~3-11]{hoerner}. Therefore a short piece of cylinder
|
||
$\dif\xi$ at a distance $\xi$ from a rotation axis, as shown in
|
||
Figure~\ref{fig-pitch-velocity}, produces a force
|
||
%
|
||
\begin{equation}
|
||
\dif F = 1.1 \cdot \frac{1}{2}\rho(\omega\xi)^2 \cdot
|
||
\underbrace{2r_t\,\dif\xi}_{\rm ref.area}
|
||
\end{equation}
|
||
%
|
||
when the cylinder is rotating at an angular velocity $\omega$. The
|
||
produced moment is correspondingly $\dif m = \xi\dif F$. Integrating
|
||
this over $0\ldots l$ yields the total pitch moment
|
||
%
|
||
\begin{equation}
|
||
m = 0.275 \cdot \rho\, r_t\, l^4 \omega^2
|
||
\end{equation}
|
||
%
|
||
and thus the moment damping coefficient is
|
||
%
|
||
\begin{equation}
|
||
C_{\rm damp} =
|
||
0.55 \cdot \frac{l^4\; r_t}{\Aref\, d}\cdot\frac{\omega^2}{v_0^2}.
|
||
\end{equation}
|
||
%
|
||
This value is computed separately for the portions of the rocket body
|
||
fore and aft of the CG using an average body radius as $r_t$.
|
||
|
||
\begin{figure}
|
||
\centering
|
||
\epsfig{file=figures/components/body-pitch-rate,scale=0.8}
|
||
\caption{Pitch damping moment due to a pitching body component.}
|
||
\label{fig-pitch-velocity}
|
||
\end{figure}
|
||
|
||
|
||
|
||
Similarly, a fin with area $\Afin$ at a distance $\xi$ from the CG
|
||
produces a moment of approximately
|
||
%
|
||
\begin{equation}
|
||
C_{\rm damp} =
|
||
0.6\cdot \frac{N\,\Afin\;\xi^3}{\Aref\,d}\cdot\frac{\omega^2}{v_0^2}
|
||
\end{equation}
|
||
%
|
||
where the effective area of the fins is assumed to be
|
||
$\Afin\cdot N/2$. For $N>4$ the value $N=4$ is used, since the other
|
||
fins are not exposed to any direct airflow.
|
||
|
||
The damping moments are applied to the total pitch moment in the
|
||
opposite direction of the current pitch rate. It is noteworthy
|
||
that the damping moment coefficients are proportional to $\omega^2/v_0^2$,
|
||
confirming that the damping moments are insignificant during most of
|
||
the rocket flight, where the angles of deflection are small and the
|
||
velocity of the rocket large. Through roll coupling the yaw rate may also
|
||
momentarily become significant, and therefore the same correction is
|
||
also applied to the yaw moment.
|
||
|
||
|
||
|
||
|
||
|
||
\clearpage
|
||
\section{Roll dynamics}
|
||
\label{sec-roll-dynamics}
|
||
|
||
When the fins of a rocket are canted at some angle $\delta>0$, the
|
||
fins induce a rolling moment on the rocket. On the other hand, when a
|
||
rocket has a specific roll velocity, the portions of the fin far from
|
||
the rocket centerline encounter notable tangential velocities
|
||
which oppose the roll. Therefore a steady-state roll velocity,
|
||
dependent on the current velocity of the rocket, will result.
|
||
|
||
The effect of roll on a fin can be examined by dividing the fin into
|
||
narrow streamwise strips and later integrating over the strips. A
|
||
strip $i$ at distance $\xi_i$ from the rocket centerline encounters a
|
||
radial velocity
|
||
%
|
||
\begin{equation}
|
||
u_i = \omega \xi_i
|
||
\end{equation}
|
||
%
|
||
where $\omega$ is the angular roll velocity, as shown in
|
||
Figure~\ref{fig-roll-velocity}. The radial velocity induces an angle
|
||
of attack
|
||
%
|
||
\begin{equation}
|
||
\eta_i = \tan^{-1} \del{\frac{u_i}{v_0}} =
|
||
\tan^{-1}\del{\frac{\omega\xi_i}{v_0}}
|
||
\approx \frac{\omega\xi_i}{v_0}
|
||
\label{eq-tan-approx}
|
||
\end{equation}
|
||
%
|
||
to the strip. The approximation $\tan^{-1} \eta \approx \eta$ is
|
||
valid for $u_i\ll v_0$, that is, when the velocity of the rocket is
|
||
large compared to the radial velocity. The approximation is
|
||
reasonable up to angles of $\eta \approx 20^\circ$, above which angle
|
||
most fins stall, which limits the validity of the equation in any
|
||
case.
|
||
|
||
When a fin is canted at an angle $\delta$, the total
|
||
inclination of the strip to the airflow is
|
||
%
|
||
\begin{equation}
|
||
\alpha_i = \delta - \eta_i.
|
||
\label{eq-roll-aoa-variation}
|
||
\end{equation}
|
||
|
||
Assuming that the force produced by a strip is directly proportional
|
||
to the local angle of attack, the force on strip $i$ is
|
||
%
|
||
\begin{equation}
|
||
F_i = k_i \alpha_i = k_i (\delta - \eta_i)
|
||
\end{equation}
|
||
%
|
||
for some $k_i$. The total moment produced by the fin is then
|
||
%
|
||
\begin{equation}
|
||
%l = \int_0^s (r+y) k (\delta-\eta(y))\dif y
|
||
% = \int_0^s (r+y) k \delta \dif y - \int_0^s (r+y) k \eta(y) \dif y
|
||
l = \sum_i \xi_i F_i = \sum_i \xi_i k_i (\delta - \eta_i)
|
||
= \sum_i \xi_i k_i \delta - \sum_i \xi_i k_i \eta_i.
|
||
\end{equation}
|
||
%
|
||
This shows that the effect of roll can be split into two components:
|
||
the first term $\sum_i \xi_i k_i \delta$ is the roll moment induced by
|
||
a fin canted at the angle $\delta$ when flying at zero roll rate
|
||
($\omega=0$), while the second term $\sum_i \xi_i k_i \eta_i$ is the
|
||
opposing moment generated by an uncanted fin ($\delta=0$) when flying
|
||
at a roll rate $\omega$. These two moments are called the roll
|
||
forcing moment and roll damping moment, respectively. These
|
||
components will be analyzed separately.
|
||
|
||
|
||
\begin{figure}
|
||
\centering
|
||
\epsfig{file=figures/fin-geometry/roll-velocity,scale=0.8}
|
||
\caption{Radial velocity at different positions of a fin. Viewed from
|
||
the rear of the rocket.}
|
||
\label{fig-roll-velocity}
|
||
\end{figure}
|
||
|
||
|
||
|
||
\subsection{Roll forcing coefficient}
|
||
|
||
As shown previously, the roll forcing coefficient can be computed by
|
||
examining a rocket with fins canted at an angle $\delta$ flying at
|
||
zero roll rate ($\omega=0$). In this case, the cant angle $\delta$ acts
|
||
simply as an angle of attack for each of the fins. Therefore, the
|
||
methods computed in the previous section can be directly applied.
|
||
Because the lift force of a fin originates from the mean aerodynamic
|
||
chord, the roll forcing coefficient of $N$ fins is equal to
|
||
%
|
||
\begin{equation}
|
||
C_{lf} = \frac{N (y_{\rm MAC}+r_t) \del{\CNa}_1 \delta}{d}
|
||
\label{eq-roll-forcing-moment}
|
||
\end{equation}
|
||
%
|
||
where $y_{\rm MAC}$ and $\del{\CNa}_1$ are computed using the methods
|
||
described in Section~\ref{sec-planar-fins} and $r_t$ is the radius of
|
||
the body tube at the fin position. This result is applicable
|
||
for both subsonic and supersonic speeds.
|
||
|
||
|
||
|
||
\subsection{Roll damping coefficient}
|
||
|
||
The roll damping coefficient is computed by examining a rocket with
|
||
uncanted fins ($\delta=0$) flying at a roll rate $\omega$. Since
|
||
different portions of the fin encounter different local angles of
|
||
attack, the damping moment must be computed from the separate
|
||
streamwise airfoil strips.
|
||
|
||
At subsonic speeds the force generated by strip $i$ is equal to
|
||
%
|
||
\begin{equation}
|
||
F_i = \CNa_0 \; \frac{1}{2}\rho v_0^2 \;
|
||
\underbrace{c_i \Delta\xi_i}_{\rm area} \; \eta_i.
|
||
\end{equation}
|
||
%
|
||
Here $\CNa_0$ is calculated by equation~(\ref{eq-fin-CNa0}) and
|
||
$c_i \Delta\xi_i$ is the area of the strip. The roll damping moment
|
||
generated by the strip is then
|
||
%
|
||
\begin{equation}
|
||
\del{C_{ld}}_i
|
||
= \frac{F_i\,\xi_i}{\frac{1}{2}\rho v_0^2\,\Aref\, d}
|
||
= \frac{\CNa_0}{\Aref\,d} \; \xi_i c_i\Delta\xi_i \; \eta_i.
|
||
\end{equation}
|
||
%
|
||
By applying the approximation~(\ref{eq-tan-approx}) and summing
|
||
(integrating) the airfoil strips the total roll damping moment for $N$
|
||
fins is obtained as:
|
||
%
|
||
\begin{align}
|
||
C_{ld} & = N \sum_i (C_{ld})_i \nonumber \\
|
||
& = \frac{N \;\CNa_0\;\omega}{\Aref\,d\,v_0} \sum_i c_i\xi_i^2\Delta\xi_i.
|
||
\label{eq-roll-damping-moment}
|
||
\end{align}
|
||
%
|
||
The sum term is a constant for a specific fin shape. It can be
|
||
computed numerically from the strips or analytically for specific
|
||
shapes. For trapezoidal fins the term can be integrated as
|
||
%
|
||
\begin{equation}
|
||
\sum_i c_i\xi_i^2\Delta\xi_i =
|
||
\frac{C_r+C_t}{2}\;r_t^2s + \frac{C_r+2C_t}{3}\;r_ts^2 +
|
||
\frac{C_r+3C_t}{12}\;s^3
|
||
\end{equation}
|
||
%
|
||
and for elliptical fins
|
||
%
|
||
\begin{equation}
|
||
\sum_i c_i\xi_i^2\Delta\xi_i =
|
||
C_r \del{ \frac{\pi}{4}\;r_t^2s + \frac{2}{3}\;r_ts^2 +
|
||
\frac{\pi}{16}\;s^3 }.
|
||
\end{equation}
|
||
|
||
|
||
The roll damping moment at supersonic speeds is calculated
|
||
analogously, starting from the supersonic strip lift force,
|
||
equation~(\ref{eq-supersonic-strip-lift-force}), where the angle of
|
||
inclination of each strip is calculated using
|
||
equation~(\ref{eq-tan-approx}). The roll moment at supersonic speeds
|
||
is thus
|
||
%
|
||
\begin{equation}
|
||
C_{ld} = \frac{N}{\Aref\,d} \sum_i C_{P_i}\, c_i \xi_i \Delta \xi_i.
|
||
\end{equation}
|
||
%
|
||
The dependence on the incidence angle $\eta_i$ is embedded within the
|
||
local pressure coefficient $C_{P_i}$,
|
||
equation~(\ref{eq-local-pressure-coefficient}). Since the dependence
|
||
is non-linear, the sum term is a function of the Mach number as well
|
||
as the fin shape.
|
||
|
||
|
||
|
||
|
||
\subsection{Equilibrium roll frequency}
|
||
|
||
One quantity of interest when examining rockets with canted fins
|
||
is the steady-state roll frequency that the fins induce on a rocket
|
||
flying at a specific velocity. This is obtained by equating the roll
|
||
forcing moment~(\ref{eq-roll-forcing-moment}) and roll damping
|
||
moment~(\ref{eq-roll-damping-moment}) and solving for the roll rate
|
||
$\omega$. The equilibrium roll frequency at subsonic speeds is
|
||
therefore
|
||
%
|
||
\begin{equation}
|
||
f_{\rm eq} = \frac{\omega_{\rm eq}}{2\pi} =
|
||
\frac{\Aref\; \beta v_0 \; y_{\rm MAC} \; (\CNa)_1 \; \delta}
|
||
{4\pi^2\; \sum_i c_i\xi_i^2\Delta\xi_i}
|
||
\label{eq-subsonic-roll-rate}
|
||
\end{equation}
|
||
%
|
||
It is worth noting that the arbitrary reference area \Aref\ is
|
||
cancelled out by the reference area appearing within $(\CNa)_1$,
|
||
as is to be expected.
|
||
|
||
At supersonic speeds the dependence on the incidence angle is
|
||
non-linear and therefore the equilibrium roll frequency must be solved
|
||
numerically. Alternatively, the second and third-order terms of the
|
||
local pressure coefficient of
|
||
equation~(\ref{eq-local-pressure-coefficient}) may be ignored, in
|
||
which case an approximation for the equilibrium roll frequency nearly
|
||
identical to the subsonic case is obtained:
|
||
%
|
||
\begin{equation}
|
||
f_{\rm eq} = \frac{\omega_{\rm eq}}{2\pi} =
|
||
\frac{\Aref\; \beta v_0 \; y_{\rm MAC} \; (\CNa)_1 \; \delta}
|
||
{4\pi\; \sum_i c_i\xi_i^2\Delta\xi_i}
|
||
\label{eq-supersonic-roll-rate}
|
||
\end{equation}
|
||
%
|
||
The value of $(\CNa)_1$ must, of course, be computed using different
|
||
methods in the subsonic and supersonic cases.
|
||
|
||
|
||
|
||
|
||
%\subsection{Roll sensitivity}
|
||
%
|
||
%The vast majority of model rockets have uncanted fins and in
|
||
%principle no roll is induced to these rockets. However, in practice
|
||
%imprecision in the attachment of the fins and other protrusions always
|
||
%cause some roll to the rocket during flight. In some applications,
|
||
%such as launching rockets with onboard video cameras, it is desirable
|
||
%to design the rockets so as to minimize the roll rate. To assist this
|
||
%design, a quantity called the {\it roll sensitivity} of the rocket is
|
||
%defined as
|
||
%%
|
||
%\begin{equation}
|
||
%f_{\rm sens} = \frac{1}{N}\eval{\pd{f_{\rm eq}}{\delta}}_{\delta=0}.
|
||
%\end{equation}
|
||
%%
|
||
%This is the slope of the equilibrium roll frequency at $\delta=0$,
|
||
%divided by the number of fins. If measured in units of
|
||
%$\rm Hz/^\circ$, the quantity indicates the number of rotations per
|
||
%second induced by one fin being attached at an angle of $1^\circ$. By
|
||
%minimizing the roll sensitivity of a rocket, the effect of
|
||
%construction imperfections on the roll rate can be minimized. From
|
||
%equation~(\ref{eq-roll-rate}) the subsonic roll sensitivity is
|
||
%obtained as
|
||
%%
|
||
%\begin{equation}
|
||
%f_{\rm sens} =
|
||
%\frac{\Aref\; \beta v_0 \; y_{\rm MAC} \; (\CNa)_1}
|
||
%{N\; 4\pi^2\; \sum_i c_i\xi_i^2\Delta\xi_i}
|
||
%\end{equation}
|
||
%%
|
||
%or conversely,
|
||
%%
|
||
%\begin{equation}
|
||
%f_{\rm eq} = N\; f_{\rm sens}\,\delta.
|
||
%\end{equation}
|
||
%%
|
||
%Similarly, in supersonic flight the roll sensitivity may either be
|
||
%solved numerically, or computed using the linear approximation for
|
||
%$C_{P_i}$ yielding
|
||
%%
|
||
%\begin{equation}
|
||
%f_{\rm sens} =
|
||
%\frac{\Aref\; \beta v_0 \; y_{\rm MAC} \; (\CNa)_1}
|
||
%{N\; 4\pi\; \sum_i c_i\xi_i^2\Delta\xi_i}.
|
||
%\end{equation}
|
||
%
|
||
%
|
||
%When the fins are canted by design, the roll sensitivity loses its
|
||
%significance. Therefore if all the fins on a rocket are uncanted, the
|
||
%quantity of intrest is the roll sensitivity, while for a rocket with
|
||
%canted fins it is the equilibrium roll frequency.
|
||
|
||
|
||
|
||
|
||
|
||
|
||
\clearpage
|
||
\section{Drag forces}
|
||
\label{sec-drag}
|
||
|
||
Air flowing around a solid body causes drag, which resists the
|
||
movement of the object relative to the air. Drag forces arise from
|
||
two basic mechanisms, the air pressure distribution around the rocket
|
||
and skin friction. The pressure distribution is further divided into
|
||
body pressure drag (including shock waves generated as supersonic
|
||
speeds), parasitic pressure drag due to protrusions such as launch
|
||
lugs and base drag. Additional sources of drag include interference
|
||
between the fins and body and vortices generated at fin tips when
|
||
flying at an angle of attack. The different drag sources are depicted
|
||
in Figure~\ref{fig-drag-components}. Each drag source will be analyzed
|
||
separately; the interference drag and fin-tip vortices will be
|
||
ignored as small compared to the other sources.
|
||
|
||
As described in Section~\ref{sec-general-aerodynamics}, two different
|
||
drag coefficients can be defined: the (total) drag coefficient $C_D$
|
||
and the axial drag coefficient $C_A$. At zero angle of attack these
|
||
two coincide, $C_{D_0} = C_{A_0}$, but at other angles a distinction
|
||
between the two must be made. The value of significance in the
|
||
simulation is the axial drag coefficient $C_A$ based on the choice of
|
||
force components. However, the drag coefficient $C_D$ describes the
|
||
deceleration force on the rocket, and is a more commonly known value
|
||
in the rocketry community, so it is informational to calculate its
|
||
value as well.
|
||
|
||
In this section the zero angle-of-attack drag coefficient
|
||
$C_{D_0} = C_{A_0}$ will be computed first. Then, in
|
||
Section~\ref{sec-axial-drag} this will be extended for angles of
|
||
attack and $C_A$ and $C_D$ will be computed. Since the drag force of
|
||
each component is proportional to its particular size, the subscript
|
||
$\bullet$ will be used for coefficients that are computed using the
|
||
reference area of the specific component. This reference area is the
|
||
frontal area of the component unless otherwise noted. Conversion to
|
||
the global reference area is performed by
|
||
%
|
||
\begin{equation}
|
||
C_{D_0} = \frac{A_{\rm component}}{\Aref} \cdot C_{D\bullet}.
|
||
\end{equation}
|
||
|
||
|
||
\begin{figure}
|
||
\centering
|
||
\epsfig{file=figures/aerodynamics/drag-components,width=13.5cm}
|
||
\caption{Types of model rocket drag at subsonic speeds.}
|
||
\label{fig-drag-components}
|
||
\end{figure}
|
||
|
||
|
||
\subsection{Laminar and turbulent boundary layers}
|
||
|
||
At the front of a streamlined body, air flows smoothly around the
|
||
body in layers, each of which has a different velocity. The layer
|
||
closest to the surface ``sticks'' to the object having zero velocity.
|
||
Each layer gradually increases the speed until the free-stream
|
||
velocity is reached. This type of flow is said to be {\it laminar}
|
||
and to have a {\it laminar boundary layer}. The thickness of the
|
||
boundary layer increases with the distance the air has flowed along
|
||
the surface. At some point a transition occurs and the layers of air
|
||
begin to mix. The boundary layer becomes {\it turbulent} and thickens
|
||
rapidly. This transition is depicted in
|
||
Figure~\ref{fig-drag-components}.
|
||
|
||
A turbulent boundary layer induces a notably larger skin friction drag
|
||
than a laminar boundary layer. It is therefore necessary to consider
|
||
how large a portion of a rocket is in laminar flow and at what point
|
||
the flow becomes turbulent. The point at which the flow becomes
|
||
turbulent is the point that has a {\it local critical Reynolds number}
|
||
%
|
||
\begin{equation}
|
||
R_{\rm crit} = \frac{v_0 \; x}{\nu},
|
||
\label{eq-transition-Re}
|
||
\end{equation}
|
||
%
|
||
where $v_0$ is the free-stream air velocity, $x$ is the distance along
|
||
the body from the nose cone tip and
|
||
$\nu\approx 1.5\cdot10^{-5}\;\rm m^2/s$ is the kinematic viscosity of
|
||
air. The critical Reynolds number is approximately
|
||
$R_{\rm crit} = 5\cdot10^5$~\cite[p.~43]{barrowman-thesis}. Therefore,
|
||
at a velocity of 100~m/s the transition therefore occurs approximately
|
||
7~cm from the nose cone tip.
|
||
|
||
% Air viscosity:
|
||
% http://www.engineeringtoolbox.com/air-absolute-kinematic-viscosity-d_601.html
|
||
|
||
|
||
%Since the drag force is approximately proportional to the square of
|
||
%the free-stream velocity, the value of $C_D$ is most critical at high
|
||
%velocities. Equation~(\ref{eq-transition-Re}) shows that at a velocity
|
||
%of 100~m/s the transition to turbulent flow occurs about 7~cm
|
||
%from the nose cone tip. Therefore at these speeds most of the wetted
|
||
%area of a typical model rocket is in turbulent flow.
|
||
|
||
Surface roughness or even slight protrusions may also trigger the
|
||
transition to occur prematurely. At a velocity of 60~m/s the critical
|
||
height for a cylindrical protrusion all around the body is of the
|
||
order of 0.05~mm~\cite[p.~348]{advanced-model-rocketry}. The
|
||
body-to-nosecone joint, a severed paintbrush hair or some other
|
||
imperfection on the surface may easily exceed this limit and cause
|
||
premature transition to occur.
|
||
|
||
Barrowman presents methods for computing the drag of both fully
|
||
turbulent boundary layers as well as partially-laminar layers. Both
|
||
methods were implemented and tested, but the difference in apogee
|
||
altitude was less than 5\% in with all tested designs. Therefore,
|
||
the boundary layer is assumed to be fully turbulent in all cases.
|
||
|
||
%A typical model rocket may therefore be assumed to have a fully
|
||
%turbulent boundary layer. Only sport models which have been
|
||
%finished to fine precision may benefit from a partial laminar
|
||
%flow around the rocket. These different types of rockets will be
|
||
%taken into account by having two modes of calculation, one for typical
|
||
%model rockets that assumes a fully turbulent boundary layer, and
|
||
%another one which assumes very fine precision finish.
|
||
|
||
|
||
|
||
|
||
|
||
|
||
\subsection{Skin friction drag}
|
||
|
||
Skin friction is one of the most notable sources of model rocket
|
||
drag. It is caused by the friction of the viscous flow of air
|
||
around the rocket. In his thesis Barrowman presented formulae for
|
||
estimating the skin friction coefficient for both laminar and
|
||
turbulent boundary layers as well as the transition between the
|
||
two~\cite[pp.~43--47]{barrowman-thesis}. As discussed above, a fully
|
||
turbulent boundary layer will be assumed in this thesis.
|
||
|
||
The skin friction coefficient $C_f$ is defined as the drag coefficient
|
||
due to friction with the reference area being the total wetted area
|
||
of the rocket, that is, the body and fin area in contact with the
|
||
airflow:
|
||
%
|
||
\begin{equation}
|
||
C_f = \frac{D_{\rm friction}}{\frac{1}{2} \rho v_0^2\;A_{\rm wet}}
|
||
\end{equation}
|
||
%
|
||
The coefficient is a function of the rocket's Reynolds number $R$ and
|
||
the surface roughness. The aim is to first calculate the skin
|
||
friction coefficient, then apply corrections due to compressibility
|
||
and geometry effects, and finally to convert the coefficient to the
|
||
proper reference area.
|
||
|
||
|
||
\subsubsection{Skin friction coefficients}
|
||
\label{sec-skin-friction-coefficient}
|
||
|
||
The values for $C_f$ are given by different formulae depending on the
|
||
Reynolds number. For fully turbulent flow the coefficient is given by
|
||
%
|
||
\begin{equation}
|
||
C_f = \frac{1}{(1.50\; \ln R - 5.6)^2}.
|
||
\label{eq-turbulent-friction}
|
||
\end{equation}
|
||
|
||
The above formula assumes that the surface is ``smooth'' and the
|
||
surface roughness is completely submerged in a thin, laminar sublayer.
|
||
At sufficient speeds even slight roughness may have an effect on the
|
||
skin friction. The critical Reynolds number corresponding to the
|
||
roughness is given by
|
||
%
|
||
\begin{equation}
|
||
R_{\rm crit} = 51\left(\frac{R_s}{L}\right)^{-1.039},
|
||
\end{equation}
|
||
%
|
||
where $R_s$ is an approximate roughness height of the surface. A few
|
||
typical roughness heights are presented in Table~\ref{tab-roughnesses}.
|
||
For Reynolds numbers above the critical value, the skin friction
|
||
coefficient can be considered independent of Reynolds number, and has
|
||
a value of
|
||
%
|
||
\begin{equation}
|
||
C_f = 0.032\left(\frac{R_s}{L}\right)^{0.2}.
|
||
\label{eq-critical-friction}
|
||
\end{equation}
|
||
%
|
||
|
||
|
||
\begin{table}
|
||
\caption{Approximate roughness heights of different
|
||
surfaces~\cite[p.~5-3]{hoerner}}
|
||
\label{tab-roughnesses}
|
||
\begin{center}
|
||
\begin{tabular}{lc}
|
||
Type of surface & Height / \um \\
|
||
\hline
|
||
Average glass & 0.1 \\
|
||
Finished and polished surface & 0.5 \\
|
||
Optimum paint-sprayed surface & 5 \\
|
||
Planed wooden boards & 15 \\
|
||
% planed = h<>yl<79>tty ???
|
||
Paint in aircraft mass production & 20 \\
|
||
Smooth cement surface & 50 \\
|
||
Dip-galvanized metal surface & 150 \\
|
||
Incorrectly sprayed aircraft paint & 200 \\
|
||
Raw wooden boards & 500 \\
|
||
Average concrete surface & 1000 \\
|
||
\hline
|
||
\end{tabular}
|
||
\end{center}
|
||
\end{table}
|
||
|
||
|
||
Finally, a correction must be made for very low Reynolds numbers. The
|
||
experimental formulae are applicable above approximately
|
||
$R\approx10^4$. This corresponds to velocities typically below 1~m/s,
|
||
which therefore have negligible effect on simulations. Below this
|
||
Reynolds number, the skin friction coefficient is assumed to be equal
|
||
as for $R=10^4$.
|
||
|
||
Altogether, the skin friction coefficient for turbulent flow is
|
||
calculated by
|
||
%
|
||
\begin{equation}
|
||
C_f = \left\{
|
||
\begin{array}{ll}
|
||
1.48\cdot10^{-2}, & \mbox{if $R<10^4$} \\
|
||
\mbox{Eq.~(\ref{eq-turbulent-friction})}, & \mbox{if $10^4<R<R_{\rm crit}$} \\
|
||
\mbox{Eq.~(\ref{eq-critical-friction})}, & \mbox{if $R>R_{\rm crit}$}
|
||
\end{array}
|
||
\right. .
|
||
\end{equation}
|
||
%
|
||
These formulae are plotted with a few different surface roughnesses in
|
||
Figure~\ref{fig-skinfriction-plot}. Included also is the laminar and
|
||
transitional skin friction values for comparison.
|
||
|
||
\begin{figure}
|
||
\centering
|
||
\epsfig{file=figures/drag/skin-friction-coefficient,width=11cm}
|
||
\caption{Skin friction coefficient of turbulent, laminar and
|
||
roughness-limited boundary layers.}
|
||
\label{fig-skinfriction-plot}
|
||
\end{figure}
|
||
|
||
|
||
\subsubsection{Compressibility corrections}
|
||
|
||
A subsonic speeds the skin friction coefficient turbulent and
|
||
roughness-limited boundary layers need to be corrected for
|
||
compressibility with the factor
|
||
%
|
||
\begin{equation}
|
||
{C_f}_c = C_f\; (1-0.1\, M^2).
|
||
\end{equation}
|
||
%
|
||
In supersonic flow, the turbulent skin friction coefficient must be
|
||
corrected with
|
||
%
|
||
\begin{equation}
|
||
{C_f}_c = \frac{C_f}{(1+0.15\, M^2)^{0.58}}
|
||
\end{equation}
|
||
%
|
||
and the roughness-limited value with
|
||
%
|
||
\begin{equation}
|
||
{C_f}_c = \frac{C_f}{1 + 0.18\, M^2}.
|
||
\end{equation}
|
||
%
|
||
However, the corrected roughness-limited value should not be used if
|
||
it would yield a value smaller than the corresponding turbulent
|
||
value.
|
||
|
||
|
||
\subsubsection{Skin friction drag coefficient}
|
||
\label{sec-skin-friction-drag}
|
||
|
||
After correcting the skin friction coefficient for compressibility
|
||
effects, the coefficient can be converted into the actual drag
|
||
coefficient. This is performed by scaling it to the correct reference
|
||
area. The body wetted area is corrected for its cylindrical geometry,
|
||
and the fins for their finite thickness.
|
||
%effect of finite fin thickness which Barrowman handled
|
||
%separately is also included~\cite[p.~55]{barrowman-thesis}.
|
||
The total friction drag coefficient is then
|
||
%
|
||
\begin{equation}
|
||
(C_D)_{\rm friction} = {C_f}_c \; \frac{
|
||
\del{1 + \frac{1}{2f_B}} \cdot A_{\rm wet,body} +
|
||
\del{1 + \frac{2t}{\bar c}} \cdot A_{\rm wet,fins}}
|
||
{\Aref}
|
||
\label{eq-friction-drag-scale}
|
||
\end{equation}
|
||
%
|
||
where $f_B$ is the fineness ratio of the rocket, and $t$ the thickness
|
||
and $\bar c$ the mean aerodynamic chord length of the fins. The
|
||
wetted area of the fins $A_{\rm wet,fins}$ includes both sides of the
|
||
fins.
|
||
|
||
|
||
|
||
|
||
|
||
\subsection{Body pressure drag}
|
||
|
||
Pressure drag is caused by the air being forced around the rocket. A
|
||
special case of pressure drag are shock waves generated at supersonic
|
||
speeds. In this section methods for estimating the pressure drag of
|
||
nose cones will be presented and reasonable estimates also for
|
||
shoulders and boattails.
|
||
|
||
|
||
\subsubsection{Nose cone pressure drag}
|
||
|
||
At subsonic speeds the pressure drag of streamlined nose cones is
|
||
significantly smaller than the skin friction drag. In fact, suitable
|
||
shapes may even yield negative pressure drag coefficients, producing a
|
||
slight reduction in drag. Figure~\ref{fig-nosecone-cd} presents
|
||
various nose cone shapes and their respective measured pressure drag
|
||
coefficients.~\cite[p.~3-12]{hoerner}
|
||
|
||
It is notable that even a slight rounding at the joint between the nose
|
||
cone and body reduces the drag coefficient dramatically. Rounding the
|
||
edges of an otherwise flat head reduces the drag coefficient from 0.8
|
||
to 0.2, while a spherical nose cone has a coefficient of only 0.01.
|
||
The only cases where an appreciable pressure drag is present is when
|
||
the joint between the nose cone and body is not smooth, which may
|
||
cause slight flow separation.
|
||
|
||
The nose pressure drag is approximately
|
||
proportional to the square of the sine of the joint angle $\phi$
|
||
(shown in
|
||
Figure~\ref{fig-nosecone-cd})~\cite[p.~237]{handbook-supersonic-aerodynamics}:
|
||
%
|
||
\begin{equation}
|
||
(C_{D\bullet,M=0})_p = 0.8 \cdot \sin^2\phi.
|
||
\label{eq-nosecone-pressure-drag}
|
||
\end{equation}
|
||
%
|
||
This yields a zero pressure drag for all nose cone shapes that have a
|
||
smooth transition to the body. The equation does not take into
|
||
account the effect of extremely blunt nose cones (length less than
|
||
half of the diameter). Since the main drag cause is slight flow
|
||
separation, the coefficient cannot be corrected for compressibility
|
||
effects using the Prandtl coefficient, and the value is applicable
|
||
only at low subsonic velocities.
|
||
|
||
\begin{figure}
|
||
\centering
|
||
\epsfig{file=figures/nose-geometry/nosecone-cd-top,width=11cm}
|
||
\caption{Pressure drag of various nose cone
|
||
shapes~\cite[p.~3-12]{hoerner}.}
|
||
\label{fig-nosecone-cd}
|
||
\end{figure}
|
||
|
||
|
||
At supersonic velocities shock waves increase the pressure drag
|
||
dramatically. In his report Barrowman uses a second-order
|
||
shock-expansion method that allows determining the pressure
|
||
distribution along an arbitrary slender rotationally symmetrical
|
||
body~\cite{second-order-shock-expansion-method}. However,
|
||
the method has some problematic limitations. The method cannot handle
|
||
body areas that have a slope larger than approximately $30^\circ$,
|
||
present in several typical nose cone shapes. The local airflow in
|
||
such areas may decrease below the speed of sound, and the method
|
||
cannot handle transonic effects. Drag in the transonic
|
||
region is of special interest for rocketeers wishing to build rockets
|
||
capable of penetrating the sound barrier.
|
||
|
||
Instead of a general piecewise computation of the air pressure around
|
||
the nose cone, a simpler semi-empirical method for estimating the
|
||
transonic and supersonic pressure drag of nose cones is used. The
|
||
method, described in detail in
|
||
Appendix~\ref{app-nosecone-drag-method}, combines theoretical and
|
||
empirical data of different nose cone shapes to allow estimating the
|
||
pressure drag of all the nose cone shapes described in
|
||
Appendix~\ref{app-nosecone-geometry}.
|
||
|
||
The semi-empirical method is used at Mach numbers above 0.8.
|
||
At high subsonic velocities the pressure drag is interpolated between
|
||
that predicted by equation~(\ref{eq-nosecone-pressure-drag}) and the
|
||
transonic method. The pressure drag is assumed to be non-decreasing
|
||
in the subsonic region and to have zero derivative at $M=0$. A
|
||
suitable interpolation function that resembles the shape of the
|
||
Prandtl factor is
|
||
%
|
||
\begin{equation}
|
||
(C_{D\bullet})_{\rm pressure} = a\cdot M^b + (C_{D\bullet,M=0})_p
|
||
\label{eq-nosecone-pressure-interpolator}
|
||
\end{equation}
|
||
%
|
||
where $a$ and $b$ are computed to fit the drag coefficient and its
|
||
derivative at the lower bound of the transonic method.
|
||
|
||
|
||
|
||
\subsubsection{Shoulder pressure drag}
|
||
|
||
Neither Barrowman nor Hoerner present theoretical or experimental
|
||
data on the pressure drag of transitions at subsonic velocities. In
|
||
the case of shoulders, the pressure drag coefficient is assumed to be
|
||
the same as that of a nose cone, except that the reference area is the
|
||
difference between the aft and fore ends of the transition. The
|
||
effect of a non-smooth transition at the beginning of the shoulder is
|
||
ignored, since this causes an increase in pressure and thus cannot
|
||
cause flow separation.
|
||
|
||
While this assumption is reasonable at subsonic velocities, it is
|
||
somewhat dubious at supersonic velocities. However, no comprehensive
|
||
data set of shoulder pressure drag at supersonic velocities was
|
||
found. Therefore the same assumption is made for supersonic
|
||
velocities and a warning is generated during such simulations (see
|
||
Section~\ref{sec-warnings}). The refinement of the supersonic
|
||
shoulder pressure drag estimation is left as a future enhancement.
|
||
|
||
|
||
|
||
\subsubsection{Boattail pressure drag}
|
||
|
||
The estimate for boattail pressure drag is based on the body base
|
||
drag estimate, which will be presented in Section~\ref{sec-base-drag}.
|
||
At one extreme, the transition length is zero, in which case the
|
||
boattail pressure drag will be equal to the total base drag. On the
|
||
other hand, a gentle slope will allow a gradual pressure change
|
||
causing approximately zero pressure drag. Hoerner has presented
|
||
pressure drag data for wedges, which suggests that at a
|
||
length-to-height ratio below 1 has a constant pressure drag
|
||
corresponding to the base drag and above a ratio of 3 the pressure
|
||
drag is negligible. Based on this and the base drag
|
||
equation~(\ref{eq-base-drag}), an approximation for the pressure drag
|
||
of a boattail is given as
|
||
%
|
||
\begin{equation}
|
||
(C_{D\bullet})_{\rm pressure} =
|
||
\frac{A_{\rm base}}{A_{\rm boattail}} \cdot (C_{D\bullet})_{\rm base}
|
||
\cdot
|
||
\left\{
|
||
\begin{array}{cl}
|
||
1 & \mbox{if\ } \gamma < 1 \\
|
||
\frac{3-\gamma}{2} & \mbox{if\ } 1 < \gamma < 3 \\
|
||
0 & \mbox{if\ } \gamma > 3
|
||
\end{array}
|
||
\right.
|
||
\end{equation}
|
||
%
|
||
where the length-to-height ratio $\gamma = l/(d_1-d_2)$ is calculated
|
||
from the length and fore and aft diameters of the boattail. The
|
||
ratios 1 and 3 correspond to reduction angles of $27^\circ$ and
|
||
$9^\circ$, respectively, for a conical boattail. The base drag
|
||
$(C_{D\bullet})_{\rm base}$ is calculated using
|
||
equation~(\ref{eq-base-drag}).
|
||
|
||
Again, this approximation is made primarily based on subsonic data.
|
||
At supersonic velocities expansion fans exist, the counterpart of
|
||
shock waves in expanding flow. However, the same equation is used for
|
||
subsonic and supersonic flow and a warning is generated during
|
||
transonic simulation of boattails.
|
||
|
||
|
||
|
||
\subsection{Fin pressure drag}
|
||
|
||
The fin pressure drag is highly dependent on the fin profile shape.
|
||
Three typical shapes are considered, a rectangular profile, rounded
|
||
leading and trailing edges, and an airfoil shape with rounded leading
|
||
edge and tapering trailing edge. Barrowman estimates the fin pressure
|
||
drag by dividing the drag further into components of a finite
|
||
thickness leading edge, thick trailing edge and overall fin
|
||
thickness~\cite[p.~48--57]{barrowman-thesis}. In this report the fin
|
||
thickness was already taken into account as a correction to the skin
|
||
friction drag in Section~\ref{sec-skin-friction-drag}. The division
|
||
to leading and trailing edges also allows simple extension to the
|
||
different profile shapes.
|
||
|
||
The drag of a rounded leading edge can be considered as a circular
|
||
cylinder in cross flow with no base drag. Barrowman derived
|
||
an empirical formula for the leading edge pressure drag as
|
||
%
|
||
\begin{equation}
|
||
(C_{D\bullet})_{LE\perp} =
|
||
\left\{
|
||
\begin{array}{ll}
|
||
(1-M^2)^{-0.417} - 1 & \mbox{for $M<0.9$} \\
|
||
1-1.785(M-0.9) & \mbox{for $0.9 < M < 1$} \\
|
||
1.214 - \frac{0.502}{M^2} + \frac{0.1095}{M^4} & \mbox{for $M>1$}
|
||
\end{array}
|
||
\right. .
|
||
\end{equation}
|
||
%
|
||
The subscript $\perp$ signifies the the flow is perpendicular to the
|
||
leading edge.
|
||
|
||
In the case of a rectangular fin profile the leading edge pressure
|
||
drag is equal to the stagnation pressure drag as derived in
|
||
equation~\ref{eq-blunt-cylinder-drag} of
|
||
Appendix~\ref{app-blunt-cylinder-drag}:
|
||
\begin{equation}
|
||
(C_{D\bullet})_{LE\perp} = (C_{D\bullet})_{\rm stag}
|
||
\end{equation}
|
||
|
||
The leading edge pressure drag of a slanted fin is obtained from the
|
||
cross-flow principle~\cite[p.~3-11]{hoerner} as
|
||
%
|
||
\begin{equation}
|
||
(C_{D\bullet})_{LE} = (C_{D\bullet})_{LE\perp} \cdot \cos^2\Gamma_L
|
||
\end{equation}
|
||
%
|
||
where $\Gamma_L$ is the leading edge angle. Note that in the equation
|
||
both coefficients are relative to the frontal area of the cylinder, so
|
||
the ratio of their reference areas is also $\cos\Gamma_L$. In the
|
||
case of a free-form fin the angle $\Gamma_L$ is the average leading
|
||
edge angle, as described in Section~\ref{sec-average-angle}.
|
||
|
||
The fin base drag coefficient of a square profile fin is the same as
|
||
the body base drag coefficient in equation~\ref{eq-base-drag}:
|
||
%
|
||
\begin{equation}
|
||
(C_{D\bullet})_{TE} = (C_{D\bullet})_{\rm base}
|
||
\end{equation}
|
||
%
|
||
For fins with rounded edges the value is taken as half of the total
|
||
base drag, and for fins with tapering trailing edges the base
|
||
drag is assumed to be zero.
|
||
|
||
The total fin pressure drag is the sum of the leading and trailing
|
||
edge drags
|
||
%
|
||
\begin{equation}
|
||
(C_{D\bullet})_{\rm pressure} =
|
||
(C_{D\bullet})_{LE} + (C_{D\bullet})_{TE}.
|
||
\end{equation}
|
||
%
|
||
The reference area is the fin frontal area $N\cdot ts$.
|
||
|
||
% TODO: FUTURE: supersonic shock wave drag???
|
||
|
||
|
||
|
||
|
||
|
||
\subsection{Base drag}
|
||
\label{sec-base-drag}
|
||
|
||
Base drag is caused by a low-pressure area created at the base of the
|
||
rocket or in any place where the body radius diminishes rapidly
|
||
enough. The magnitude of the base drag can be estimated using the
|
||
empirical formula~\cite[p.~23]{fleeman}
|
||
%
|
||
\begin{equation}
|
||
(C_{D\bullet})_{\rm base} =
|
||
\left\{
|
||
\begin{array}{ll}
|
||
0.12+0.13M^2, & \mbox{if $M<1$} \\
|
||
0.25/M, & \mbox{if $M>1$}
|
||
\end{array}
|
||
\right. .
|
||
\label{eq-base-drag}
|
||
\end{equation}
|
||
%
|
||
The base drag is disrupted when a motor exhausts into the area. A
|
||
full examination of the process would need much more detailed
|
||
information about the motor and would be unnecessarily complicated. A
|
||
reasonable approximation is achieved by subtracting the area of the
|
||
thrusting motors from the base reference area~\cite[p.~23]{fleeman}.
|
||
Thus, if the base is the same size as the motor itself, no base drag
|
||
is generated. On the other hand, if the base is large with only a
|
||
small motor in the center, the base drag is approximately the same as
|
||
when coasting.
|
||
|
||
The equation presented above ignores the effect that the rear body
|
||
slope angle has on the base pressure. A boattail at the end of the
|
||
rocket both diminishes the reference area of base drag, thus reducing
|
||
drag, but the slope also directs air better into the low pressure
|
||
area. This effect has been neglected as small compared to the effect
|
||
of reduced base area.
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
\subsection{Parasitic drag}
|
||
|
||
Parasitic drag refers to drag caused by imperfections and protrusions
|
||
on the rocket body. The most significant source of parasitic drag in
|
||
model rockets are the launch guides that protrude from the rocket
|
||
body. The most common type of launch guide is one or two launch lugs,
|
||
which are pieces of tube that hold the rocket on the launch rod during
|
||
takeoff. Alternatives to launch lugs include replacing the tube with
|
||
metal wire loops or attaching rail pins that hold the rocket on a
|
||
launch rail. These three guide types are depicted in
|
||
Figure~\ref{fig-launch-guides}. The effect of launch lugs on the
|
||
total drag of a model rocket is small, typically in the range of
|
||
0--10\%, due to their comparatively small size. However, studying
|
||
this effect may be of notable interest for model rocket designers.
|
||
|
||
|
||
\begin{figure}
|
||
\centering
|
||
\epsfig{file=figures/components/launch-guides,width=12cm}
|
||
\caption{Three types of common launch guides.}
|
||
\label{fig-launch-guides}
|
||
\end{figure}
|
||
|
||
|
||
A launch lug that is long enough that no appreciable airflow occurs
|
||
through the lug may be considered a solid cylinder next to the main
|
||
rocket body. A rectangular protrusion that has a length at least
|
||
twice its height has a drag coefficient of 0.74, with reference area
|
||
being its frontal area~\cite[p.~5-8]{hoerner}. The drag coefficient
|
||
varies proportional to the stagnation pressure as in the case of a
|
||
blunt cylinder in free airflow, presented in
|
||
Appendix~\ref{app-blunt-cylinder-drag}.
|
||
|
||
A wire held perpendicular to airflow has instead a drag coefficient of
|
||
1.1, where the reference area is the planform area of the
|
||
wire~\cite[p.~3-11]{hoerner}. A wire loop may be thought of as a
|
||
launch lug with length and wall thickness equal to the thickness of
|
||
the wire. However, in this view of a launch lug the reference area
|
||
must not include the inside of the tube, since air is free to flow
|
||
within the loop.
|
||
|
||
These two cases may be unified by changing the used reference area as
|
||
a function of the length of the tube $l$. At the limit $l=0$ the
|
||
reference area is the simple planform area of the loop, and when the
|
||
length is greater than the diameter $l>d$ the reference area includes
|
||
the inside of the tube as well. The slightly larger drag coefficient
|
||
of the wire may be taken into account as a multiplier to the blunt
|
||
cylinder drag coefficient.
|
||
|
||
Therefore the drag coefficient of a launch guide can be approximately
|
||
calculated by
|
||
%
|
||
\begin{equation}
|
||
(C_{D\bullet})_{\rm parasitic} =
|
||
\max\{1.3-0.3\;l/d, 1\} \cdot (C_{D\bullet})_{\rm stag}
|
||
\end{equation}
|
||
%
|
||
where $(C_{D\bullet})_{\rm stag}$ is the stagnation pressure
|
||
coefficient calculated in equation~(\ref{eq-blunt-cylinder-drag}), and
|
||
the reference area is
|
||
%
|
||
\begin{equation}
|
||
A_{\rm parasitic} = \pi r_{ext}^2 - \pi r_{int}^2 \cdot
|
||
\max\{1-l/d,0\}.
|
||
\end{equation}
|
||
|
||
This approximation may also be used to estimate the drag of rail
|
||
pins. A circular pin protruding from a wall has a drag coefficient of
|
||
0.80~\cite[p.~5-8]{hoerner}. Therefore the drag of the pin is
|
||
approximately equal to that of a lug with the same frontal area. The
|
||
rail pins can be approximated in a natural manner as launch lugs with
|
||
the same frontal area as the pin and a length equal to their
|
||
diameter.
|
||
|
||
|
||
|
||
|
||
\subsection{Axial drag coefficient}
|
||
\label{sec-axial-drag}
|
||
|
||
The total drag coefficient may be calculated by simply scaling the
|
||
coefficients to a common reference area and adding them together:
|
||
%
|
||
\begin{equation}
|
||
C_{D_0} = \sum_T \frac{A_T}{\Aref}(C_{D\bullet})_T
|
||
+ (C_D)_{\rm friction}
|
||
\end{equation}
|
||
%
|
||
where the sum includes the pressure, base and parasitic drags. The
|
||
friction drag was scaled to the reference area \Aref\ already in
|
||
equation~(\ref{eq-friction-drag-scale}).
|
||
|
||
This yields the total drag coefficient at zero angle of attack. At an
|
||
angle of attack the several phenomena begin to affect the drag.
|
||
More frontal area is visible to the airflow, the pressure gradients
|
||
along the body change and fin-tip vortices emerge. On the other hand,
|
||
the drag force is no longer axial, so the axial drag force is less
|
||
than the total drag force.
|
||
|
||
Based on experimental data an empirical formula was produced for
|
||
calculating the axial drag coefficient at an angle of attach $\alpha$
|
||
from the zero-angle drag coefficient. The scaling function is a
|
||
two-part polynomial function that starts from 1 at $\alpha=0^\circ$,
|
||
increases to 1.3 at $\alpha=17^\circ$ and then decreases to zero at
|
||
$\alpha=90^\circ$; the derivative is also zero at these points. Since
|
||
the majority of the simulated flight is at very small angles of
|
||
attack, this approximation provides a sufficiently accurate estimate
|
||
for the purposes of this thesis.
|
||
|
||
|
||
\section{Tumbling bodies}
|
||
\label{sec-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 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 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.
|
||
|
||
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.
|
||
|
||
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}
|
||
\begin{center}
|
||
\parbox{80mm}{
|
||
\begin{tabular}{cccccc}
|
||
Model & \#1 & \#2 & \#3 & \#4 & \#5 \\
|
||
\hline
|
||
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}{
|
||
\epsfig{file=figures/lower-stage/lower-stage,width=50mm}
|
||
}
|
||
\end{center}
|
||
\end{table}
|
||
|
||
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}
|
||
\frac{1}{2}\rho v_0^2\; (C_{D,f}A_f + C_{D,bt}A_{bt}) = mg
|
||
\end{equation}
|
||
%
|
||
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}
|
||
|