Stagnation pressure and Stagnation Points

[mikeyboy22]

Homework Statement

Hi there,
Im currently doing my A2 physics project on Supersonic flight. I am really struggling to understand the concept of Stagnation pressure, it just really hasn't clicked yet. I understand that the fluid 'responds' to pressure waves by building up stagnation pressure and stagnation points for the air to flow over or under the aircraft. Could somebody maybe give me what explanation or scenario worked for them to grasp the concept?

Homework Equations

Bernoulli's equation: P(total)= 0.5*rho*velocity^2 + P(static)

The Attempt at a Solution

I've tried to understand this but I just need some guidance, another way to describe it would be great, it will click eventually!
Thanks alot.

 

[minger]

Stagnation (or total) pressure is kind of a measure of a fluids energy, as I think of it. According to Bernoulli's, a fluid has three forms of energy, which we are already familiar with. Internal, kinetic and potential. Kinetic and potential we know from statics or kinematics, and internal is something we picked up thermodynamics.

Now, the idea of total pressure is that if we can isentropically convert all of the potential and kinetic energy into internal energy, then we have this value. So, a shock is essentially a huge energy killer. If we assume a perfect gas, where cp is a constant, then we can write:
$$s_2 - s_1 = c_p \ln\frac{T_2}{T_1} - R \ln\frac{p_2}{p_1}$$
If we consider thsi equation across a shock, then:
$$s_{2a} - s_{1a} = c_p \ln\frac{T_{2a}}{T_{1a}} - R\ln\frac{p_{2a}}{p_{1a}}$$
However, $$T_{2a} = T_0 = T_{1a}$$, so:
$$s_2 - s_1 = -R \ln\frac{p_{0_1}}{p_{0_2}}$$
or:
$$\frac{p_{0_1}}{p_{0_2}} = e^{-(s_2-s_1)/R}$$
Because $$s_2>s_1$$ the total pressure always decreases across a shock wave.

Anyways, hope this helped a little with your understanding, if you have any other questions, feel free to ask.

 

[mikeyboy22]

Hmm that helped.
In the case of an object moving through the air at sub-sonic speeds, why does the fluid set up a stagnation point?? Is it so that there is no kinetic energy, i.e so the fluid can flow over the airfoil or below the airfoil?? I'm struggling with this :(

 

[minger]

The stagnation point where the fluid "stops". At we said, the stagnation pressure is the pressure that a fluid would have if it was isentropically slowed to zero.

If you look at an airfoil, or even easier yet, a sphere moving through a fluid. You expect half of the fluid to move around the sphere above, and half below. As you "zoom" in on the leading edge you approach that separation point. Theoretically, if we zoom in enough, there exists a point there where the fluid goes neither above nor below and simply dead-ends right into the sphere; that's the stagnation point. It exists because of the geometry of the object.

Because the fluid has "stopped" at this point, it has no kinetic energy, and the pressure is the stagnation pressure.

 

[mikeyboy22]

Ah that makes sense, thanks so much!
At supersonic speeds - no stagnation points are able to be established because of the aircraft moving quicker - thus resulting in a shock wave?

 

[minger]

Well, there will be a stagnation point, but the pressure at that point will be significantly lower than the total pressure of the freestream flow. If you imagine a sphere moving supersonically; there will be a detached shock wave in front of it. The flow will "shock down" but will still flow subsonically afterwards. After the shock, it will hit a stagnation point 'right' at the front of the sphere. However, the stagnation properties dropped going through the shock.

 

[mikeyboy22]

This clears some stuff up - thanks so much for your help its really appreciated btw.
so the shock wave is the result of the spontaneous change in pressure, density, temperature, speed of the airflow, energy from each is converted into a shockwave?

 

[minger]

Kind of. The best analogy that I've heard is based on information. "Information" essentially propagates at the speed of sound. You can think of each molecule of the fluid as a little guy. At subsonic speeds the guys in front "see" the plane (or whatever) coming and they yell backwards to the rest of the guys. They can then nicely get out of the way because they "knew" it was coming.

At supersonic speeds, they try and yell back to each other, but the object is moving faster than they can communicate. So one minute I'm just a lonely fluid molecule minding my own business, and a moment later *BOOM* I'm hit by a frickin' airplane. So, it can kind of be thought as changing the the fluid to accommodate the moving object.

...in a way.

 

[mikeyboy22]

hahah lovely! thanks a lot man, you've really helped :P