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Airship Stream Function (Topic)

This is a work in progress...

Here we will calculate the stream function 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 algebraic equations to solve for the strengths

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

$\displaystyle \psi = 0$

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.

Stream Function

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

In the below figure we need expressions for the vector magnitude from the source to the point on the surface

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

but

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

Therefore

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

Similarily,

$\displaystyle r^p_{n} = \frac{y_p}{\sin(\tan^{-1}\left( \frac{y_p}{x_p - x_{n}} \right ))} $
Figure: Airship Stream Function Setup
\includegraphics[scale=.85]{AirshipStreamFunction.eps}

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.

$\displaystyle 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

$\displaystyle \psi_p = -\sum_{n=1}^N \frac{Q_n b \sqrt{1 - \frac{x_n^2}{a^2}}}{... ...ght ] + \frac{1}{2} V_{\infty} \left (b \sqrt{1 - \frac{x_n^2}{a^2}} \right )^2$

Yikes! Following the example in [2] the matrix form of the above equation when put together for N equations for the top surface becomes

$\displaystyle A_{11} Q_1 + A_{12} Q_2 + ... + A_{1n} Q_n = \frac{1}{2} V_{\infty} y_1$

$\displaystyle A_{21} Q_1 + A_{22} Q_2 + ... + A_{2n} Q_n = \frac{1}{2} V_{\infty} y_2$

$\displaystyle ...$

$\displaystyle A_{n1} Q_1 + A_{n2} Q_2 + ... + A_{nn} Q_n = \frac{1}{2} V_{\infty} y_n$

The matrix equation is then

$\displaystyle {\bf A} {\bf Q} = {\bf Y}$

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

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

$\displaystyle \left[ \begin{array}{c} Q_{1} \ Q_{2} \ Q_{3} \ Q_{4} \end{... ... \frac{1}{2} V_{\infty} y_3 \ \frac{1}{2} V_{\infty} y_4 \end{array} \right]$

References

[1] Kundu, P.E., Cohen, I.M. "Fluid mechanics" 2nd Edition. Academic Press, San Diego, 2002.

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



"Airship Stream Function" is owned by bloftin.

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Cross-references: mechanics, matrix, magnitude, vector, algebraic, function, work

This is version 10 of Airship Stream Function, born on 2006-10-26, modified 2006-10-29.
Object id is 233, canonical name is AirshipStreamFunction.
Accessed 1029 times total.

Classification:
Physics Classification: 47.85.Gj (Aerodynamics)
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