How to calculate pressure field in potential flow

Question

I haven't ever studied fluid dynamics before and may mix something here, so please, be patient :).

Given flow potential of the form (homogeneus flow over a dipole):

$$ \phi = u_\infty x -\frac{M}{2\pi} \frac{x}{x^2 + y^2} $$

how do I calculate pressure field? I know the answer is like below, and it's derived using Bernoulli's principle. I just don't know how to get there analytically.

In my materials the answer is:

$$ C_p = \frac{p-p_\infty}{\frac{1}{2} \rho u_\infty^2} = 1 - \left( \frac{u}{u_\infty}\right)^2$$

but I don't understand where it comes from.

Answer 1

The pressure coefficient at a certain point (at which the value of the pressure is $p$) is defined as

$C_p=\frac{p-p_\infty}{\frac12\rho_\infty u_\infty^2},$

where the $\infty$-Symbol denotes freestream quantities.

For an incompressible and steady fluid and assuming zero viscosity, Bernoulli's equation is given by

$p+\frac12\rho u^2=p_\infty+\frac12\rho u_\infty^2,$

which we can rearrange as

$\frac{p-p_\infty}{\frac12\rho u_\infty^2}=1-\frac{u^2}{u_\infty^2}.$

In order for this expression to be valid for given fluid, we have to show that it is incompressible and steady. Incompressibility means that the Laplacian of the flow potential $\phi$ vanishes, this can be shown to be true for the problem at hand. Furthermore, a fluid is steady if its flow does not depend explicitely on time, which is also the case.