\chapter{Aerodynamic properties of model rockets~~} \label{chap-aerodynamics} A model rocket encounters three basic forces during its flight: thrust from the motors, gravity, and aerodynamical forces. Thrust is generated by the motors by exhausting high-velocity gases in the opposite direction. The thrust of a motor is directly proportional to the velocity of the escaping gas and the mass per time unit that is exhausted. The thrust of commercial model rocket motors as a function of time have been measured in static motor tests and are readily available online~\cite{thrust-curve-database}. Normally the thrust of a rocket motor is aligned on the center axis of the rocket, so that it produces no angular moment to the rocket. Every component of the rocket is also affected by gravitational force. When the forces and moments generated are summed up, the gravitational force can be seen as a single force originating from the {\it center of gravity} (CG). A homogeneous gravitational field does not generate any angular moment on a body relative to the CG. Calculating the gravitational force is therefore a simple matter of determining the total mass and CG of the rocket. Aerodynamic forces, on the other hand, produce both net forces and angular moments. To determine the effect of the aerodynamic forces on the rocket, the total force and moment must be calculated relative to some reference point. In this chapter, a method for determining these forces and moments will be presented. \section{General aerodynamical properties} \label{sec-general-aerodynamics} The aerodynamic forces acting on a rocket are usually split into components for further examination. The two most important aerodynamic force components of interest in a typical model rocket are the {\it normal force} and {\it drag}. The aerodynamical normal force is the force component that generates the corrective moment around the CG and provides stabilization of the rocket. The drag of a rocket is defined as the force component parallel to the velocity of the rocket. This is the aerodynamical force that opposes the movement of the rocket through air. Figure~\ref{fig-aerodynamic-forces}(a) shows the thrust, gravity, normal force and drag of a rocket in free flight. It should be noted that if the rocket is flying at an angle of attack $\alpha>0$, then the normal force and drag are not perpendicular. In order to have independent force components, it is necessary to define component pairs that are always perpendicular to one another. Two such pairs are the normal force and axial drag, or side force and drag, shown in Figure~\ref{fig-aerodynamic-forces}(b). The two pairs coincide if the angle of attack is zero. The component pair that will be used as a basis for the flight simulations is the normal force and axial drag. \begin{figure} \centering \parbox{35mm}{\centering \epsfig{file=figures/aerodynamics/free-flight-forces,width=35mm} \\ (a)} \hfill \parbox{35mm}{\centering \epsfig{file=figures/aerodynamics/aero-force-components,width=35mm} \\ (b)} \hfill \parbox{35mm}{\centering \epsfig{file=figures/aerodynamics/pitch-yaw-roll,width=35mm} \\ (c)} \caption{(a) Forces acting on a rocket in free flight: gravity $G$, motor thrust $T$, drag $D$ and normal force $N$. (b) Perpendicular component pairs of the total aerodynamical force: normal force $N$ and axial drag $D_A$; side force $S$ and drag $D$. (c) The pitch, yaw and roll directions of a model rocket.} \label{fig-aerodynamic-forces} \end{figure} The three moments around the different axis are called the {\it pitch}, {\it yaw} and {\it roll moments}, as depicted in Figure~\ref{fig-aerodynamic-forces}(c). Since a typical rocket has no ``natural'' roll angle of flight as an aircraft does, we may choose the pitch angle to be in the same plane as the angle of attack, \ie the plane defined by the velocity vector and the centerline of the rocket. Thus, the normal force generates the pitching moment and no other moments. \subsection{Aerodynamic force coefficients} When studying rocket configurations, the absolute force values are often difficult to interpret, since many factors affect them. In order to get a value better suited for comparison, the forces are normalized by the current dynamic pressure $q=\frac{1}{2}\rho v_0^2$ and some characteristic area \Aref\ to get a non-dimensional force coefficient. Similarly, the moments are normalized by the dynamic pressure, characteristic area and characteristic length $d$. Thus, the normal force coefficient corresponding to the normal force $N$ is defined as % \begin{equation} C_N = \frac{N}{\frac{1}{2}\rho v_0^2 \, \Aref} \label{eq-CN-def} \end{equation} % and the pitch moment coefficient for a pitch moment $m$ as % \begin{equation} C_m = \frac{m}{\frac{1}{2}\rho v_0^2 \, \Aref\, d}. \label{eq-Cm-def} \end{equation} % A typical choice of reference area is the base of the rocket's nose cone and the reference length is its diameter. The pitch moment is always calculated around some reference point, while the normal force stays constant regardless of the point of origin. If the moment coefficient $C_m$ is known for some reference point, the moment coefficient at another point $C_m'$ can be calculated from % \begin{equation} C_m'd = C_md - C_N\Delta x \label{eq-moment-reference} \end{equation} % where $\Delta x$ is the distance along the rocket centerline. Therefore it is sufficient to calculate the moment coefficient only at some constant point along the rocket body. In this thesis the reference point is chosen to be the tip of the nose cone. The {\it center of pressure} (CP) is defined as the position from which the total normal force alone produces the current pitching moment. Therefore the total normal force produces no moment around the CP itself, and an equation for the location of the CP can be obtained from (\ref{eq-moment-reference}) by selecting setting $C_m'=0$: % \begin{equation} X = \frac{C_m}{C_N}\,d \end{equation} % Here $X$ is the position of the CP along the rocket centerline from the nose cone tip. This equation is valid when $\alpha>0$. As $\alpha$ approaches zero, both $C_m$ and $C_N$ approach zero. The CP is then obtained as a continuous extension using l'Hôpital's rule % \begin{equation} X = \left.\frac{\;\frac{\partial C_m}{\partial\alpha}\;} {\;\frac{\partial C_N}{\partial\alpha}\;}\,d\right|_{\alpha=0} = \frac{\Cma}{\CNa}\,d \label{eq-CP-position} \end{equation} % where the normal force coefficient and pitch moment coefficient derivatives have been defined as % \begin{equation} \CNa = \left.\frac{\partial C_N}{\partial\alpha}\right|_{\alpha=0} \hspace{5mm}\mbox{and}\hspace{5mm} \Cma = \left.\frac{\partial C_m}{\partial\alpha}\right|_{\alpha=0}. \label{eq-CNa-derivative} \end{equation} At very small angles of attack we may approximate $C_N$ and $C_m$ to be linear with $\alpha$, so to a first approximation % \begin{equation} C_N \approx \CNa\,\alpha \hspace{5mm}\mbox{and}\hspace{5mm} C_m \approx \Cma\,\alpha. \label{eq-CNa-approx} \end{equation} % The Barrowman method uses the coefficient derivatives to determine the CP position using equation~(\ref{eq-CP-position}). However, there are some significant nonlinearities in the variation of $C_N$ as a function of $\alpha$. These will be accounted for by holding the approximation of equation~(\ref{eq-CNa-approx}) exact and letting \CNa\ and \Cma\ be a function of $\alpha$. Therefore, for the purposes of this thesis we define % \begin{equation} \CNa = \frac{C_N}{\alpha} \hspace{5mm}\mbox{and}\hspace{5mm} \Cma = \frac{C_m}{\alpha} \label{eq-CNa-definition} \end{equation} % for $\alpha>0$ and by equation~(\ref{eq-CNa-derivative}) for $\alpha=0$. These definitions are compatible, since equation~(\ref{eq-CNa-definition}) simplifies to the partial derivative~(\ref{eq-CNa-derivative}) at the limit $\alpha\rightarrow0$. This definition also allows us to stay true to Barrowman's original method which is familiar to many rocketeers. Similar to the normal force coefficient, the drag coefficient is defined as % \begin{equation} C_D = \frac{D}{\frac{1}{2}\rho v_0^2 \, \Aref}. \label{eq-CD-def} \end{equation} % Since the size of the rocket has been factored out, the drag coefficient at zero angle of attack $C_{D0}$ allows a straightforward method of comparing the effect of different rocket shapes on drag. However, this coefficient is not constant and will vary with \eg the speed of the rocket and its angle of attack. If each of the fins of a rocket are canted at some angle $\delta>0$ with respect to the rocket centerline, the fins will produce a roll moment on the rocket. Contrary to the normal force and pitching moment, canting the fins will produce a non-zero rolling moment but no corresponding net force. Therefore the only quantity computed is the roll moment coefficient, defined by % \begin{equation} C_l = \frac{l}{\frac{1}{2}\rho v_0^2 \, \Aref\, d} \label{eq-Cl-def} \end{equation} % where $l$ is the roll moment. It shall be shown later that rockets with axially-symmetrical fin configurations experience no forces that would produce net yawing moments. However, a single fin may produce all six types of forces and moments. The equations for the forces and moments of a single fin will not be explicitly written out, and they can be computed from the geometry in question. \subsection{Velocity regions} Most of the aerodynamic properties of rockets vary with the velocity of the rocket. The important parameter is the {\it Mach number}, which is the free-stream velocity of the rocket divided by the local speed of sound % \begin{equation} M = \frac{v_0}{c}. \end{equation} % The velocity range encountered by rockets is divided into regions with different impacts on the aerodynamical properties, listed in Table~\ref{tab-sonics}. In {\it subsonic flight} all of the airflow around the rocket occurs below the speed of sound. This is the case for approximately $M<0.8$. At very low Mach numbers air can be effectively treated as an incompressible fluid, but already above $M\approx 0.3$ some compressibility issues may have to be considered. In {\it transonic flight} some of the air flowing around the rocket accelerates above the speed of sound, while at other places it remains subsonic. Some local shock waves are generated and hard-to-predict interference effects may occur. The drag of a rocket has a sharp increase in the transonic region, making it hard to pass into the supersonic region. Transonic flight occurs at Mach numbers of approximately 0.8--1.2. In {\it supersonic flight} all of the airflow is faster than the speed of sound (with the exception of \eg the nose cone tip). A shock wave is generated by the nose cone and fins. In supersonic flight the drag reduces from that of transonic flight, but is generally greater than that of subsonic flight. Above approximately Mach 5 new phenomena begin to emerge that are not encountered at lower supersonic speeds. This region is called {\it hypersonic flight}. \begin{table} \caption{Velocity regions of rocket flight} \label{tab-sonics} \begin{center} \begin{tabular}{cr@{ -- }l} Region & \multicolumn{2}{c}{Mach number ($M$)} \\ \hline Subsonic & \hspace{10mm} 0 & 0.8 \\ Transonic & 0.8 & 1.2 \\ Supersonic & 1.2 & $\sim5$ \\ Hypersonic & $\sim5$ & \\ \hline \end{tabular} \end{center} \end{table} Methods for predicting the aerodynamic properties of subsonic flight and some extensions to supersonic flight will be presented. Since the analytical prediction of aerodynamic properties in the transonic region is quite difficult, this region will be accounted for by using some suitable interpolation function that corresponds reasonably to actual measurements. Hypersonic flight will not be considered, since practically no model or high power rockets ever achieve such speeds. \subsection{Flow and geometry parameters} There exist many different parameters that characterize aspects of flow or a rocket's geometry. One of the most important flow parameters is the {\it Reynolds number} $R$. It is a dimensionless quantity that characterizes the ratio of inertial forces and viscous forces of flow. Many aerodynamic properties depend on the Reynolds number, defined as % \begin{equation} R = \frac{v_0\; L}{\nu}. \end{equation} % Here $v_0$ is the free-stream velocity of the rocket, $L$ is a characteristic length and $\nu$ is the kinematic viscosity of air. It is notable that the Reynolds number is dependent on a characteristic length of the object in question. In most cases, the length used is the length of the rocket. A typical 30~cm sport model flying at 50~m/s has a corresponding Reynolds number of approximately 1\s000\s000. Another term that is frequently encountered in aerodynamical equations has been defined its own parameter $\beta$, which characterizes the flow speed both in subsonic and supersonic flow: % \begin{equation} \beta = \sqrt{\envert{M^2-1}} = \left\{ \begin{array}{ll} \sqrt{1-M^2}, & {\rm if\ } M<1 \\ \sqrt{M^2-1}, & {\rm if\ } M>1 \end{array} \right. \end{equation} % As the flow speed approaches the transonic region $\beta$ approaches zero. This term appears for example in the {\it Prandtl factor} $P$ which corrects subsonic force coefficients for compressible flow: % \begin{equation} P = \frac{1}{\beta} = \frac{1}{\sqrt{1-M^2}} \label{eq-prandtl-factor} \end{equation} It is also often useful to define parameters characterizing general properties of a rocket. One such parameter is the {\it caliber}, defined as the maximum body diameter. The caliber is often used to indicate relative distances on the body of a rocket, such as the stability margin. Another common parameter characterizes the ``slenderness'' of a rocket. It is the {\it fineness ratio} of a rocket $f_B$, defined as the length of the rocket body divided by the maximum body diameter. Typical model rockets have a fineness ratio in the range of 10--20, but extreme models may have a fineness ratio as low as 5 or as large as 50. \subsection{Coordinate systems} During calculation of the aerodynamic properties a coordinate system fixed to the rocket will be used. The origin of the coordinates is at the nose cone tip with the positive $x$-axis directed along the rocket centerline. This convention is also followed internally in the produced software. In the following sections the position of the $y$- and $z$-axes are arbitrary; the parameter $y$ is used as a general spanwise coordinate when discussing the fins. During simulation, however, the $y$- and $z$-axes are fixed in relation to the rocket, and do not necessarily align with the plane of the pitching moments. \clearpage \section{Normal forces and pitching moments} Barrowman's method~\cite{barrowman-thesis} for determining the total normal force coefficient derivative \CNa, the pitch moment coefficient derivative \Cma\ and the CP location at subsonic speeds first splits the rocket into simple separate components, then calculates the CP location and \CNa\ for each component separately and then combines these to get the desired coefficients and CP location. The general assumptions made by the derivation are: % \begin{enumerate} \item The angle of attack is very close to zero. \item The flow around the body is steady and non-rotational. \item The rocket is a rigid body. \item The nose tip is a sharp point. \item The fins are flat plates. \item The rocket body is axially symmetric. \end{enumerate} The components that will be discussed are nose cones, cylindrical body tube sections, shoulders, boattails and fins, in an arbitrary order. The interference effect between the body and fins will be taken into account by a separate correction term. Extensions to account for body lift and arbitrary fin shapes will also be derived. \subsection{Axially symmetric body components} The body of the rocket is assumed to be an axially symmetric body of rotation. The entire body could be considered to be a single component, but in practice it is divided into nose cones, shoulders, boattails and cylindrical body tube sections. The geometry of typical nose cones, shoulders and boattails are described in Appendix~\ref{app-nosecone-geometry}. The method presented by Barrowman for calculating the normal force and pitch moment coefficients at supersonic speeds is based on a second-order shock expansion method. However, this assumes that the body of the rocket is very streamlined, and it cannot handle areas with a slope larger than than $\sim30^\circ$. Since the software allows basically any body shape, applying this method would be difficult. Since the emphasis is on subsonic flow, for the purposes of this thesis the normal force and pitching moments produced by the body are assumed to be equal at subsonic and supersonic speeds. The assumption is that the CP location is primarily affected by the fins. The effect of supersonic flight on the drag of the body will be accounted for in Section~\ref{sec-drag}. \subsubsection{\CNa\ of body components at subsonic speeds} The normal force for an axially symmetric body at position $x$ in subsonic flow is given by % \begin{equation} N(x) = \rho v_0 \; \frac{\partial}{\partial x}[A(x)w(x)] \label{eq-normal-force} \end{equation} % where $A(x)$ is the cross-sectional area of the body, and the $w(x)$ is the local downwash, given as a function of the angle of attack as % \begin{equation} w(x) = v_0 \sin\alpha. \end{equation} % 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 coefficient~(\ref{eq-CN-def}) and equation~(\ref{eq-normal-force}) we obtain % \begin{equation} C_N(x) = \frac{N(x)}{\frac{1}{2}\rho v_0^2\;\Aref} = \frac{2\; \sin\alpha}{\Aref}\; \frac{\dif A(x)}{\dif x}. \label{eq-CNx} \end{equation} % Assuming that the derivative $\frac{\dif A(x)}{\dif x}$ is 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 = \frac{2\; \sin\alpha}{\Aref}\; [A(l)-A(0)]. \end{equation} % We then have % \begin{equation} \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 = höylätty ??? 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 $10^4R_{\rm crit}$} \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}