Airship Stream Function | Physics Library

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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

ψ = 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

∑N ( ) ψp = − Qn- rp − rp + 1V ∞y2 n=1 4π n−1 n 2 p

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

-yp-- sin(αn− 1) = rp n−1

but

( y ) αn −1 = tan−1 ----p----- xp − xn−1

Therefore

rp = ---------yp(-------)-- n−1 sin(tan−1 ---yp--- ) xp−xn−1

Similarily,

y rpn = ---------p(------)-- sin (tan −1 --yp-- ) xp−xn

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.

∘ ----x2- yp = ±b 1 − --n a2

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

⌊ ⌋ ∘ -----x2-|| || ( ∘ -------)2 ∑N Qnb 1 − na2| 1 1 | 1 x2n ψp = − -------------|| ---(-------(--∘----2)-)--− ---(-------(--∘----2)-)- ||+ -V ∞ b 1 − -2- n=1 4π |⌈ − 1 b 1− xan2 −1 b 1− xan2 |⌉ 2 a sin(tan ( xp−xn−1) ) sin(tan ( -xp−xn-) )

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

1- A11Q1 + A12Q2 + ...+ A1nQn = 2 V∞y1

A Q + A Q + ...+ A Q = 1-V y 21 1 22 2 2n n 2 ∞ 2

...

1 An1Q1 + An2Q2 + ...+ AnnQn = -V∞yn 2

The matrix equation is then

AQ = Y

−1 Q = A Y

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

⌊ Q1 ⌋ ⌊ A11 A12 A13 A14 ⌋⌊ 1V∞y1 ⌋ | | | || 21 | | Q2 | = inv | A21 A22 A23 A24 || 21V∞y2 | ⌈ Q3 ⌉ ⌈ A31 A32 A33 A34 ⌉⌈ 2V∞y3 ⌉ Q4 A31 A32 A33 A44 12V∞y4

0.1 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.

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