[Aerodynamics] Bernoulli's equation and Pressure Coefficient

[Leo Liu]

We know that the definition of the pressure coefficient is $$C_p=\frac{p-p_\infty}{q_\infty}$$, where $p$ is the pressure at a point, $p_\infty$ is the ambient pressure (free-stream), and $q_\infty$ is the free-stream dynamic pressure.
We also know that the Bernoulli's equation is $$p_\infty+q_\infty=p+\frac{1}{2}\rho v^2$$
Let's assume the Mach number is below 0.3 so that the flow is incompressible. I can derive the following expression from the Bernoulli's equation:
$$C_p=\frac{p-p_\infty}{q_\infty}=1-\left(\frac{v}{v_\infty}\right)^2$$
This allows me to calculate the velocity at a point from the free-stream velocity and the pressure coefficient at that point, even if the pressure and air density is not explicitly provided.

However, when my friend tried to use the formula above to solve the problem below, he got 107 rather than 87.3.
Image:

My question is why my equation produced a different answer than the official solution. Thanks!

 

[cjl]

The problem here is that they've overdefined the problem. If the freestream velocity is 80m/s, and there's a point locally on the wing that has a local velocity of 110 m/s (and assuming the flow can reasonably be treated as incompressible), that point should have a cp of -0.89, not -1.5. You're also correct that for a cp of -0.8, the velocity should be 107ish. The only way for a point to have a local velocity of 110m/s and a cp of -1.5 is for the freestream to be 69.5m/s, not 80.

Amusingly though, they didn't even do the problem correctly for this alternate case. If we assume the 110m/s and -1.5 to correctly define the conditions and the 80m/s in in error, that would mean a velocity of a hair over 93 m/s for the point at -0.8, not 87.

It's worth noting that their final equation is correct, their problem is just using inconsistent input data. If you know freestream velocity and local pressure coefficient, you already have enough info to determine local velocity as well (assuming incompressibility holds). It's just that they're plugging in impossible values.

EDIT: Also, tagging @boneh3ad just in case I missed something dumb here (I've been tired all day and have been in a massive brain fog). I really can't see how the problem works as stated though.

 

[boneh3ad]

Assuming this is from a textbook, there's a good chance they published solutions for a new edition without updating it with new inputs. Publishers are terrible about that in their solutions manuals.

 

[Leo Liu]

Assuming this is from a textbook, there's a good chance they published solutions for a new edition without updating it with new inputs. Publishers are terrible about that in their solutions manuals.

Yes I took this question from a textbook (Intro. to Flight by Anderson), but I don't think its solution manual is relevant here because this is an example question.

 

[cjl]

Interesting. I just checked my copy of Anderson (6th ed, copyright 2008) and the error is present in mine too (I guess we never noticed it in class or when I was studying since I don't have any notes about it in the margins or anything). It's example 5.10 on page 284 in my edition vs example 5.15 in yours, but the inconsistent given numbers and incorrect resulting solution are all identical.