--- /dev/null
+
+
+\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.
+
+%Two calculation methods will
+%be presented, one for ``typical'' rockets and one for those with very
+%fine precision finish.
+
+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.
+%, defined
+%as
+%
+%\begin{equation}
+%R = \frac{v_0\;L_r}{\nu}
+%\end{equation}
+%
+%where $v_0$ is the free-stream velocity of the rocket, $L_r$ is the
+%length of the rocket and $\nu$ is the local kinematic viscosity of
+%air.
+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. If $R<5\cdot10^5$ the flow is assumed to be
+completely laminar, and the corresponding skin friction coefficient is
+%
+\begin{equation}
+C_f = \frac{1.328}{\sqrt{R}}.
+\label{eq-laminar-friction}
+\end{equation}
+%
+Correspondingly, for completely turbulent flow (also for low Reynolds
+numbers when forced by some protrusion from the surface) the
+coefficient is given by
+%
+\begin{equation}
+C_f = \frac{1}{(1.50\; \ln R - 5.6)^2}.
+\label{eq-turbulent-friction}
+\end{equation}
+%
+Above $R=5\cdot10^5$ some of the flow around the rocket is turbulent
+and some laminar. Measured data of the transition results in an
+empirical formula for the transition from
+equation~(\ref{eq-laminar-friction}) to
+equation~(\ref{eq-turbulent-friction}) as
+%
+\begin{equation}
+C_f = \frac{1}{(1.50\;\ln R - 5.6)^2} - \frac{1700}{R}.
+\label{eq-transition-friction}
+\end{equation}
+%
+This equation gives a continuation from the laminar equation. The
+exact point of switch to the transitional equation is the point where
+equations (\ref{eq-laminar-friction}) and
+(\ref{eq-transition-friction}) are equal, $R=5.39\cdot10^5$.
+
+% Exact transition point from laminar to transitional
+% R = 539154
+
+
+The above formulae assume that the surface is ``smooth'' and the
+surface roughness is completely submerged in a 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_r}\right)^{-1.039},
+\end{equation}
+%
+where $R_s$ is an approximate roughess 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_r}\right)^{0.2}.
+\label{eq-critical-friction}
+\end{equation}
+%
+
+% TODO: epäjatkuvuus karkeudessa
+% mitä jos karkeus suurta?
+% Katso Hoernerista!!
+
+
+\begin{table}
+\caption{Approximate roughness heights of different
+ surfaces~\cite[p.~XXX]{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,
+and have therefore negligible effect on simulations. Below this
+Reynolds number, the skin friction coeffifient is assumed to be equal
+as for $R=10^4$.
+
+
+As a summary, when assuming the rocket to be finished with enough
+precision to have a significant portion of laminar flow, the value of
+$C_f$ is computed by
+%
+\begin{equation}
+C_f = \left\{
+\begin{array}{ll}
+1.33\cdot10^{-2}, & \mbox{if $R<10^4$} \\
+\mbox{Eq.~(\ref{eq-laminar-friction})}, & \mbox{if $10^4<R<5.39\cdot10^5$} \\
+\mbox{Eq.~(\ref{eq-transition-friction})}, & \mbox{if
+ $5.39\cdot10^5 < R < R_{\rm crit}$} \\
+\mbox{Eq.~(\ref{eq-critical-friction})}, & \mbox{if $R>R_{\rm crit}$}
+\end{array}
+\right. .
+\end{equation}
+%
+When assuming a fully turbulent flow, $C_f$ is computed 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 in Figure~\ref{fig-skinfriction-plot}.
+
+\begin{figure}
+\centering
+\epsfig{file=figures/drag/skinfriction.eps,width=11cm}
+\caption{Skin friction coefficient of fully laminar
+ (Eq.~(\ref{eq-laminar-friction})), fully turbulent
+ (Eq.~(\ref{eq-turbulent-friction})), transitional
+ (Eq.~(\ref{eq-transition-friction})) and roughness-limited
+ (Eq.~(\ref{eq-critical-friction})) boundary layers.}
+\label{fig-skinfriction-plot}
+% TODO: kuva matlabilla, kapeampi pystysuunnassa
+\end{figure}
+
+
+\subsubsection{Compressibility corrections}
+
+The laminar skin friction coefficient is constant at both subsonic and
+low supersonic speeds, ${C_f}_c = C_f$. However, at subsonic speeds
+the smooth turbulent value of equation~(\ref{eq-turbulent-friction})
+and the roughness-limited value of
+equation~(\ref{eq-critical-friction}) must be corrected with
+%
+\begin{equation}
+{C_f}_c = C_f\; (1-0.1\, M^2).
+\end{equation}
+%
+In supersonic flow, the smooth 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 smooth turbulent
+value. In order not to cause discontinuities, the transition point
+from of laminar to turbulent corrections and from subsonic to
+supersonic is done gradually over a suitable Reynolds number or Mach
+number range.
+
+
+
+\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 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.
+
+
projects / debian / openrocket / blobdiff