Bernoulli's Eq/Static Pressure/Stagnation Pressure question

[iuaero]

I have a few questions regarding Bernoulli's equation:

1) Nomenclature: can "stagnation pressure" and "total pressure" be used interchangeably? I realize that at a stagnation point, p_total = p_stag = p_static, but can you always say p_total = p_stag (along a given streamline)?

2) Does static pressure or stagnation pressure (or total pressure) depend on frame of reference?

3) For an airfoil moving through the atmosphere, will p_static at a large distance away from the airfoil essentially be p_atm?

The reason why I ask is that I had a lot of confusion with an incompressible fluid mechanics homework problem last week. Basically, a submarine at a given depth was moving at a given constant velocity, and using Bernoulli's equation we were to calculate the static pressure at a point on the submarine where the water was flowing past it with twice the freestream velocity.

I viewed the problem from the frame of the submarine. I used hydrostatics to calculate water pressure (p_w = (ro)*g*z), then I set that to the static pressure of the fluid a large distance in front of the sub. Then using Bernoulli's equation along a streamline close to the surface of the submarine, I said:

p_w + 1/2*(ro)*V^2 = p_static + 1/2*(ro)*(2V)^2

where p_w = hydrostatic water pressure, ro is density, V is freestream velocity, and p_static is the pressure I'm solving for.

So the left hand side of the equation is the streamline in front of the sub, and the right hand side is the same streamline but where the velocity across the sub is 2V.

A fellow student argued that the entire left hand side of the equation was supposed to be p_stag=p_w. He has taken a few more fluids courses than I have (and I actually missed the lecture where Bernoulli's eq was derived... whoops...) so I really wasn't able to back myself up except with examples (like the pitot-static system on an airplane and how the static port essentially measures ambient pressure) which were dismissed in the argument for some resaon.

So basically, who's right? If he's right, that would mean that total pressure does not depend on the reference frame (since he set p_total = p_w), which would also mean that p_static depends on the reference frame (in this case, p_static in the moving frame would be less than p_w). Or, am I right, which would mean that static pressure does not depend on reference frame, but total pressure does?

I've been having a hard time wrapping my head around this and it's really been bugging me, so thanks for any answers!

 

[Navin A S]

1) Generally, the terms "stagnation pressure" and "total pressure" can be used interchangeably. However, in certain cases the two terms may not be equal. For example, when a flow is accelerating, the stagnation pressure will be greater than the total pressure. This is due to an increase in kinetic energy, which causes an increase in total pressure. 2) Both static pressure and stagnation pressure do depend on the frame of reference. For example, if a fluid is flowing with velocity V in a certain frame of reference, then the stagnation pressure in that frame of reference will be p_stag = p_static + 1/2*(ro)*V^2. If you were to change the frame of reference so that the fluid was flowing with velocity 2V, then the stagnation pressure in the new frame of reference would be p_stag = p_static + 1/2*(ro)*(2V)^2.3) In general, yes, p_static at a large distance away from the airfoil will essentially be p_atm. However, this may not always be the case, especially when the airfoil is moving quickly and the air is being accelerated around it. In these cases, the static pressure may be higher or lower than the ambient atmospheric pressure.