Talk:PlanetPhysics/Airship Stream Function

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%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: Airship Stream Function %%% Primary Category Code: 47.85.Gj %%% Filename: AirshipStreamFunction.tex %%% Version: 10 %%% Owner: bloftin %%% Author(s): bloftin %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}

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This is a \htmladdnormallink{work}{http://planetphysics.us/encyclopedia/Work.html} in progress... \\

Here we will calculate the stream \htmladdnormallink{function}{http://planetphysics.us/encyclopedia/Bijective.html} for an arbitrary body of revolution. This will then let us calculate the lift on an airship for various hull geometries. Following the setup in [1], combine the uniform stream function with a line of sources and sinks along the axis of symmetry.

The steps of the calcuation are:

1) Get an expression for the stream function. Since, we have lots of sources/sinks along the axis and we don't know the strength of each one $Q_n$, we must must setup \htmladdnormallink{algebraic}{http://planetphysics.us/encyclopedia/CoIntersections.html} equations to solve for the strengths

2) N equations are created by using the property that the stream function is zero on the surface

$$ \psi = 0$$

So for N sources/sinks we have N points $P_n$ on the surface giving us N equations and N unknowns.

3) Numerically solve the equations for the N source/sinks strengths.

{\bf Stream Function}

$$\psi_p = -\sum_{n=1}^N \frac{Q_n}{4\pi} \left ( r^p_{n-1} - r^p_n \right ) + \frac{1}{2} V_{\infty} y^2_p$$

In the below figure we need expressions for the \htmladdnormallink{vector}{http://planetphysics.us/encyclopedia/Vectors.html} \htmladdnormallink{magnitude}{http://planetphysics.us/encyclopedia/AbsoluteMagnitude.html} from the source to the point on the surface

$$\sin(\alpha_{n-1}) = \frac{y_p}{r^p_{n-1}} $$

but

$$ \alpha_{n-1} = \tan^{-1}\left( \frac{y_p}{x_p - x_{n-1}} \right ) $$

Therefore

$$ r^p_{n-1} = \frac{y_p}{\sin(\tan^{-1}\left( \frac{y_p}{x_p - x_{n-1}} \right ))} $$

Similarily,

$$ r^p_{n} = \frac{y_p}{\sin(\tan^{-1}\left( \frac{y_p}{x_p - x_{n}} \right ))} $$

\begin{figure} \includegraphics[scale=.85]{AirshipStreamFunction.eps} \caption{Airship Stream Function Setup} \end{figure}

The last piece needed is to describe the airship geometry to give us $y_p$. Using the equation for an ellipse gives us a starting point.

$$ y_p = \pm b \sqrt{1 - \frac{x_n^2}{a^2}} $$

Combining all the equations gives us an expression for the stream function for the top surface points

$$\psi_p = -\sum_{n=1}^N \frac{Q_n b \sqrt{1 - \frac{x_n^2}{a^2}}}{4\pi} \left [ \frac{1 }{\sin \left (\tan^{-1}\left( \frac{b \sqrt{1 - \frac{x_n^2}{a^2}}}{x_p - x_{n-1}} \right ) \right )} - \frac{1}{\sin \left(\tan^{-1}\left( \frac{b \sqrt{1 - \frac{x_n^2}{a^2}} }{x_p - x_{n}} \right ) \right)} \right ] + \frac{1}{2} V_{\infty} \left (b \sqrt{1 - \frac{x_n^2}{a^2}} \right )^2$$

Yikes! Following the example in [2] the \htmladdnormallink{matrix}{http://planetphysics.us/encyclopedia/Matrix.html} form of the above equation when put together for N equations for the top surface becomes

$$A_{11} Q_1 + A_{12} Q_2 + ... + A_{1n} Q_n = \frac{1}{2} V_{\infty} y_1$$ $$A_{21} Q_1 + A_{22} Q_2 + ... + A_{2n} Q_n = \frac{1}{2} V_{\infty} y_2$$ $$...$$ $$A_{n1} Q_1 + A_{n2} Q_2 + ... + A_{nn} Q_n = \frac{1}{2} V_{\infty} y_n$$

The matrix equation is then

$${\bf A} {\bf Q} = {\bf Y}$$ $${\bf Q} = {\bf A}^{-1} {\bf Y}$$

therefore in matlab or octave we wil solve for the strengths, (i.e. 4 sources)

$$ \left[ \begin{array}{c} Q_{1} \\ Q_{2} \\ Q_{3} \\ Q_{4} \end{array} \right] = inv \left[ \begin{array}{cccc} A_{11} & A_{12} & A_{13} & A_{14} \\ A_{21} & A_{22} & A_{23} & A_{24} \\ A_{31} & A_{32} & A_{33} & A_{34} \\ A_{31} & A_{32} & A_{33} & A_{44} \end{array} \right] \left[ \begin{array}{c} \frac{1}{2} V_{\infty} y_1 \\ \frac{1}{2} V_{\infty} y_2 \\ \frac{1}{2} V_{\infty} y_3 \\ \frac{1}{2} V_{\infty} y_4 \end{array} \right]$$

\subsection{References}

[1] Kundu, P.E., Cohen, I.M. "Fluid \htmladdnormallink{mechanics}{http://planetphysics.us/encyclopedia/Mechanics.html}" 2nd Edition. Academic Press, San Diego, 2002.

[2[ Kuethe, A.M., Chow, C. "Foundations of Aerodynamics" 4th Edition. John Wiley \& Sons, New York 1986.

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