Velocity potential for wind over a semi cylinder

[MaxManus]

Homework Statement

A wind flows along the horizontal ground with strength U and over a long semi cylinder. The flow can be described using two dimensional potential theory
Find the velocity potential

Homework Equations

The Attempt at a Solution

I'm not sure what to do.

The equations for velocity potential are
v_r = $$\frac{d\phi}{dr}$$ and v_$$\theta$$ = $$\frac{d \phi}{rd \theta}$$

Can I say that there is no wind flowing trough the walls so v_r = 0
and that v_$$\theta$$ = U because air is incompressible?

I'm not sure at all.

The Attempt at a Solution

 

[Valkarie]

The velocity potential for this situation can be found using the Laplace equation for a two dimensional potential flow. The Laplace equation is given by \frac \partial \partial x^2 \phi + \frac \partial \partial y^2 \phi = 0 The solution to this equation is \phi (x,y) = f(x) + g(y) where f and g are arbitrary functions. In this case, we can assume that the wind is flowing in the x-direction, so we can set f(x) = Ux + C, where U is the wind speed and C is an arbitrary constant. Then, we can set g(y) = 0 since there is no flow in the y-direction. Thus, the velocity potential is \phi (x,y) = Ux + C