We all are used to pressure being a dimensional quantity having units of $ \frac{N}{m^2} $, but in the aerospace world, since it is used to often, there is much use for a pressure coefficient, or $ C_p $, which is dimensionless.
The pressure coefficient can be define as
[ 1 ]
$$ C_p = \frac{P - P_{\infty}}{q_{\infty}} $$
where
[ 2 ]
$$ q_{\infty} = \frac{1}{2} \rho_{\infty} v_{\infty}^2 $$
Pressure coefficient is used very commonly throughout aerospace and its use can range from incompressible flows to hypersonic flows (over 5 times the speed of sound)!
Now, for incompressible flows, $ C_p $ can only be expressed in terms of velocity. We can derive this using the Bernoulli's eq where we start out with a freestream pressure and velocity and choose an arbitrary flow point where the pressure and velocity are $ P $ and $ v $, respectively.
[ 3 ]
$$ P_{\infty} + \frac{1}{2} \rho v_{\infty}^2 = P + \frac{1}{2} \rho v^2 $$ $$ \Rightarrow P - P_{\infty} = \frac{1}{2} \rho (v_{\infty}^2 - v^2) $$
Hence, we can make some substitutions in Eq. (1):
[ 4 ]
$$ C_p = \frac{P - P_{\infty}}{q_{\infty}} $$ $$ = \frac{\frac{1}{2} \rho (v_{\infty}^2 - v^2)}{\frac{1}{2} \rho v_{\infty}^2} $$
We can them simplify this down to get that
[ 5 ]
$$ C_p = 1 - (\frac{v}{v_{\infty}})^2 $$
I should say again, Eq. (5) is only good for incompressible flows.
Remembering back to a stagnation point, where velocity is zero, the pressure coefficient would actually be equal to 1.0 in an incompressible flow. This is because $ C_p $ can be seen as how many multiples of $ q_{\infty} $ is equal to the difference in stagnation and static pressure, or $ P - P_{\infty} $. We can see this if we rewrite Eq. (1) above as
[ 6 ]
$$ P - P_{\infty} = q_{\infty} C_p $$
This makes sense because we know from the tutorial on Pitot Tubes that
[ 7 ]
$$ P_{\infty} + q_{\infty} = P $$
where the static pressure ($ P_{\infty} $) plus the dynamic pressure ($ q_{\infty} $) is equal to the stagnation, or total, pressure ($ P $). If we subtract our pressures, then we have a multiple of 1 $ C_p $. Now let's say the pressure coefficient was equal to -3, then $ P = P_{\infty} - 3 q_{\infty} $.