PlanetPhysics/Airship Stream Function

This is a [[../Work/|work]] in progress... \\

Here we will calculate the stream [[../Bijective/|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 [[../CoIntersections/|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

$\psi =0$

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.

{\mathbf Stream Function}

${\psi }_p=-\sum _{n=1}^N\frac{Q_n}{4\pi }(r_{n-1}^p-r_n^p)+\frac{1}{2}{V}_{\mathrm{\infty }}{y}_p^2$

In the below figure we need expressions for the [[../Vectors/|vector]] [[../AbsoluteMagnitude/|magnitude]] from the source to the point on the surface

$\sin ({\alpha }_{n-1})=\frac{y_p}{r_{n-1}^p}$

but

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

Therefore

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

Similarily,

$r_n^p=\frac{y_p}{\sin ({\tan }^{-1}\left(\frac{y_p}{x_p-x_n}\right))}$

\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

${\psi }_p=-\sum _{n=1}^N\frac{Q_{n}b\sqrt{1-\frac{x_n^2}{a^2}}}{4\pi }[\frac{1}{\sin \left({\tan }^{-1}\left(\frac{b\sqrt{1-\frac{x_n^2}{a^2}}}{x_p-x_{n-1}}\right)\right)}-\frac{1}{\sin \left({\tan }^{-1}\left(\frac{b\sqrt{1-\frac{x_n^2}{a^2}}}{x_p-x_n}\right)\right)}]+\frac{1}{2}{V}_{\mathrm{\infty }}{\left(b\sqrt{1-\frac{x_n^2}{a^2}}\right)}^2$

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

$A_{11}{Q}_1+A_{12}{Q}_2+...+A_{1n}{Q}_n=\frac{1}{2}{V}_{\mathrm{\infty }}{y}_1$ $A_{21}{Q}_1+A_{22}{Q}_2+...+A_{2n}{Q}_n=\frac{1}{2}{V}_{\mathrm{\infty }}{y}_2$ $...$ $A_{n1}{Q}_1+A_{n2}{Q}_2+...+A_{nn}{Q}_n=\frac{1}{2}{V}_{\mathrm{\infty }}{y}_n$

The matrix equation is then

$𝐀𝐐=𝐘$ $𝐐={𝐀}^{-1}𝐘$

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

$\left[\begin{array}{c}{Q}_1\\ Q_2\\ Q_3\\ Q_4\end{array}\right]=inv\left[\begin{array}{cccc}{A}_{11}& A_{12}& A_{13}& A_{14}\\ A_{21}& A_{22}& A_{23}& A_{24}\\ A_{31}& A_{32}& A_{33}& A_{34}\\ A_{31}& A_{32}& A_{33}& A_{44}\end{array}\right]\left[\begin{array}{c}\frac{1}{2}{V}_{\mathrm{\infty }}{y}_1\\ \frac{1}{2}{V}_{\mathrm{\infty }}{y}_2\\ \frac{1}{2}{V}_{\mathrm{\infty }}{y}_3\\ \frac{1}{2}{V}_{\mathrm{\infty }}{y}_4\end{array}\right]$

References

[1] Kundu, P.E., Cohen, I.M. "Fluid [[../Mechanics/|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.