246 lines
7.9 KiB
TeX
246 lines
7.9 KiB
TeX
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\subsection{Skin friction drag}
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Skin friction is one of the most notable sources of model rocket
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drag. It is caused by the friction of the viscous flow of air
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around the rocket. In his thesis Barrowman presented formulae for
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estimating the skin friction coefficient for both laminar and
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turbulent boundary layers as well as the transition between the
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two~\cite[pp.~43--47]{barrowman-thesis}. As discussed above, a fully
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turbulent boundary layer will be assumed in this thesis.
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%Two calculation methods will
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%be presented, one for ``typical'' rockets and one for those with very
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%fine precision finish.
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The skin friction coefficient $C_f$ is defined as the drag coefficient
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due to friction with the reference area being the total wetted area
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of the rocket, that is, the body and fin area in contact with the
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airflow:
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%
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\begin{equation}
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C_f = \frac{D_{\rm friction}}{\frac{1}{2} \rho v_0^2\;A_{\rm wet}}
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\end{equation}
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%
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The coefficient is a function of the rocket's Reynolds number $R$ and
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the surface roughness.
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%, defined
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%as
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%
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%\begin{equation}
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%R = \frac{v_0\;L_r}{\nu}
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%\end{equation}
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%
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%where $v_0$ is the free-stream velocity of the rocket, $L_r$ is the
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%length of the rocket and $\nu$ is the local kinematic viscosity of
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%air.
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The aim is to first calculate the skin friction coefficient,
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then apply corrections due to compressibility and geometry effects,
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and finally to convert the coefficient to the proper reference area.
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\subsubsection{Skin friction coefficients}
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\label{sec-skin-friction-coefficient}
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The values for $C_f$ are given by different formulae depending on the
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Reynolds number. If $R<5\cdot10^5$ the flow is assumed to be
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completely laminar, and the corresponding skin friction coefficient is
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%
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\begin{equation}
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C_f = \frac{1.328}{\sqrt{R}}.
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\label{eq-laminar-friction}
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\end{equation}
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%
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Correspondingly, for completely turbulent flow (also for low Reynolds
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numbers when forced by some protrusion from the surface) the
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coefficient is given by
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%
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\begin{equation}
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C_f = \frac{1}{(1.50\; \ln R - 5.6)^2}.
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\label{eq-turbulent-friction}
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\end{equation}
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%
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Above $R=5\cdot10^5$ some of the flow around the rocket is turbulent
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and some laminar. Measured data of the transition results in an
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empirical formula for the transition from
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equation~(\ref{eq-laminar-friction}) to
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equation~(\ref{eq-turbulent-friction}) as
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%
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\begin{equation}
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C_f = \frac{1}{(1.50\;\ln R - 5.6)^2} - \frac{1700}{R}.
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\label{eq-transition-friction}
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\end{equation}
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%
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This equation gives a continuation from the laminar equation. The
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exact point of switch to the transitional equation is the point where
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equations (\ref{eq-laminar-friction}) and
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(\ref{eq-transition-friction}) are equal, $R=5.39\cdot10^5$.
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% Exact transition point from laminar to transitional
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% R = 539154
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The above formulae assume that the surface is ``smooth'' and the
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surface roughness is completely submerged in a laminar sublayer. At
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sufficient speeds even slight roughness may have an effect on the skin
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friction. The critical Reynolds number corresponding to the roughness
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is given by
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%
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\begin{equation}
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R_{\rm crit} = 51\left(\frac{R_s}{L_r}\right)^{-1.039},
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\end{equation}
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%
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where $R_s$ is an approximate roughess height of the surface. A few
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typical roughness heights are presented in Table~\ref{tab-roughnesses}.
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For Reynolds numbers above the critical value, the skin friction
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coefficient can be considered independent of Reynolds number, and has
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a value of
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%
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\begin{equation}
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C_f = 0.032\left(\frac{R_s}{L_r}\right)^{0.2}.
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\label{eq-critical-friction}
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\end{equation}
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%
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% TODO: ep<65>jatkuvuus karkeudessa
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% mit<69> jos karkeus suurta?
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% Katso Hoernerista!!
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\begin{table}
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\caption{Approximate roughness heights of different
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surfaces~\cite[p.~XXX]{hoerner}}
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\label{tab-roughnesses}
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\begin{center}
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\begin{tabular}{lc}
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Type of surface & Height / \um \\
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\hline
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Average glass & 0.1 \\
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Finished and polished surface & 0.5 \\
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Optimum paint-sprayed surface & 5 \\
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Planed wooden boards & 15 \\
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% planed = h<>yl<79>tty ???
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Paint in aircraft mass production & 20 \\
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Smooth cement surface & 50 \\
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Dip-galvanized metal surface & 150 \\
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Incorrectly sprayed aircraft paint & 200 \\
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Raw wooden boards & 500 \\
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Average concrete surface & 1000 \\
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\hline
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\end{tabular}
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\end{center}
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\end{table}
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Finally, a correction must be made for very low Reynolds numbers. The
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experimental formulae are applicable above approximately
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$R\approx10^4$. This corresponds to velocities typically below 1~m/s,
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and have therefore negligible effect on simulations. Below this
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Reynolds number, the skin friction coeffifient is assumed to be equal
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as for $R=10^4$.
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As a summary, when assuming the rocket to be finished with enough
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precision to have a significant portion of laminar flow, the value of
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$C_f$ is computed by
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%
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\begin{equation}
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C_f = \left\{
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\begin{array}{ll}
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1.33\cdot10^{-2}, & \mbox{if $R<10^4$} \\
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\mbox{Eq.~(\ref{eq-laminar-friction})}, & \mbox{if $10^4<R<5.39\cdot10^5$} \\
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\mbox{Eq.~(\ref{eq-transition-friction})}, & \mbox{if
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$5.39\cdot10^5 < R < R_{\rm crit}$} \\
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\mbox{Eq.~(\ref{eq-critical-friction})}, & \mbox{if $R>R_{\rm crit}$}
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\end{array}
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\right. .
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\end{equation}
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%
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When assuming a fully turbulent flow, $C_f$ is computed by
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%
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\begin{equation}
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C_f = \left\{
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\begin{array}{ll}
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1.48\cdot10^{-2}, & \mbox{if $R<10^4$} \\
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\mbox{Eq.~(\ref{eq-turbulent-friction})}, & \mbox{if $10^4<R<R_{\rm crit}$} \\
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\mbox{Eq.~(\ref{eq-critical-friction})}, & \mbox{if $R>R_{\rm crit}$}
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\end{array}
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\right. .
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\end{equation}
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%
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These formulae are plotted in Figure~\ref{fig-skinfriction-plot}.
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\begin{figure}
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\centering
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\epsfig{file=figures/drag/skinfriction.eps,width=11cm}
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\caption{Skin friction coefficient of fully laminar
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(Eq.~(\ref{eq-laminar-friction})), fully turbulent
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(Eq.~(\ref{eq-turbulent-friction})), transitional
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(Eq.~(\ref{eq-transition-friction})) and roughness-limited
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(Eq.~(\ref{eq-critical-friction})) boundary layers.}
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\label{fig-skinfriction-plot}
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% TODO: kuva matlabilla, kapeampi pystysuunnassa
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\end{figure}
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\subsubsection{Compressibility corrections}
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The laminar skin friction coefficient is constant at both subsonic and
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low supersonic speeds, ${C_f}_c = C_f$. However, at subsonic speeds
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the smooth turbulent value of equation~(\ref{eq-turbulent-friction})
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and the roughness-limited value of
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equation~(\ref{eq-critical-friction}) must be corrected with
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%
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\begin{equation}
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{C_f}_c = C_f\; (1-0.1\, M^2).
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\end{equation}
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%
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In supersonic flow, the smooth turbulent skin friction coefficient
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must be corrected with
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%
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\begin{equation}
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{C_f}_c = \frac{C_f}{(1+0.15\, M^2)^{0.58}}
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\end{equation}
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%
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and the roughness-limited value with
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%
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\begin{equation}
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{C_f}_c = \frac{C_f}{1 + 0.18\, M^2}.
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\end{equation}
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%
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However, the corrected roughness-limited value should not be used if
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it would yield a value smaller than the corresponding smooth turbulent
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value. In order not to cause discontinuities, the transition point
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from of laminar to turbulent corrections and from subsonic to
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supersonic is done gradually over a suitable Reynolds number or Mach
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number range.
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\subsubsection{Skin friction drag coefficient}
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\label{sec-skin-friction-drag}
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After correcting the skin friction coefficient for compressibility
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effects, the coefficient can be converted into the actual drag
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coefficient. This is performed by scaling it to the correct reference
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area. The body wetted area is corrected for its cylindrical geometry,
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and the effect of finite fin thickness which Barrowman handled
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separately is also included~\cite[p.~55]{barrowman-thesis}. The total
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friction drag coefficient is then
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%
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\begin{equation}
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(C_D)_{\rm friction} = {C_f}_c \; \frac{
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\del{1 + \frac{1}{2f_B}} \cdot A_{\rm wet,body} +
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\del{1 + \frac{2t}{\bar c}} \cdot A_{\rm wet,fins}}
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{\Aref}
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\label{eq-friction-drag-scale}
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\end{equation}
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%
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where $f_B$ is the fineness ratio of the rocket, and $t$ the thickness
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and $\bar c$ the mean aerodynamic chord length of the fins. The
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wetted area of the fins $A_{\rm wet,fins}$ includes both sides of the
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fins.
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