- #1
WhiteWolf98
- 86
- 5
- Homework Statement
- A flow is defined by the complex potential:
##F=U_{\infty}(z+ \frac {R^2} {z})##
Show that the surface pressure coefficient distribution is given by:
##{C_p}=1-4\sin^{2}\theta##
- Relevant Equations
- ##z=x+iy=re^{i\theta}##
##F=\phi+i\psi##
##\frac {dF} {dz}=u-iv##
So, it's a long way to the solution, but I'm finding it difficult to find a starting point. I'm going to say that as a first step, I should find what the value of the stream function ##\psi## is, at the surface. In order to do this, I need to use the following equation:
##F=\phi+i\psi##
If I can decompose ##F## into its real and imaginary parts, then I can find what ##\psi## is (##\psi=Im(F)##). I would just like to add that for a solid body, ##\psi=0##. In the case of a doublet in a uniform flow, you end up getting a flow around a circle. As no fluid passes through into this boundary, nor out of it (and it's a closed streamline), it can be see as a solid body and thus ##\psi## must be ##0##. I want to prove it anyway as this might be the case here, but perhaps not in a different case. My problem with decomposing ##F(z)## is the ##z## in the denominator. So you'd end up with:
##F(z)=U_{\infty}(z)+\frac {U_{\infty}R^2} {z}=U_{\infty}(x+iy)+\frac {U_{\infty}R^2} {x+iy}##
It's quite easy to know what to do with the first part ##(U_{\infty}x+iU_{\infty}y##), we quite nicely have a real and imaginary part there. But I've no clue what to do with the second part, where we have ##x+iy## in the denominator. Ultimately if I want pressure, I need to know what ##u## and ##v## are at the boundary (I'm assuming anywhere along the circle as no specific point has been given), and to know those I need to know what ##\psi## is.
##F=\phi+i\psi##
If I can decompose ##F## into its real and imaginary parts, then I can find what ##\psi## is (##\psi=Im(F)##). I would just like to add that for a solid body, ##\psi=0##. In the case of a doublet in a uniform flow, you end up getting a flow around a circle. As no fluid passes through into this boundary, nor out of it (and it's a closed streamline), it can be see as a solid body and thus ##\psi## must be ##0##. I want to prove it anyway as this might be the case here, but perhaps not in a different case. My problem with decomposing ##F(z)## is the ##z## in the denominator. So you'd end up with:
##F(z)=U_{\infty}(z)+\frac {U_{\infty}R^2} {z}=U_{\infty}(x+iy)+\frac {U_{\infty}R^2} {x+iy}##
It's quite easy to know what to do with the first part ##(U_{\infty}x+iU_{\infty}y##), we quite nicely have a real and imaginary part there. But I've no clue what to do with the second part, where we have ##x+iy## in the denominator. Ultimately if I want pressure, I need to know what ##u## and ##v## are at the boundary (I'm assuming anywhere along the circle as no specific point has been given), and to know those I need to know what ##\psi## is.