Question regarding incompressible airflow

[DatLemonDoe]

Hi

I am trying to teach myself some of the basics of aerodynamics and one situation was problematic for me. I was unable to make sense of it. I thought that you guys could most likely help me. Just please keep in mind that my knowledge of these things is very limited since I've never actually studied it at school. Hence my question may be somewhat strange, but a very simplified answer will help me a lot.

So I'm looking at a flow going through the following structure. I'm assuming that we're working with an incompressible flow.

https://imgur.com/pRwg1Zf

From what I've learned, the pressure at the left of the structure will be lower, and the velocity at the left end of the structure will be higher

However, when I think about it in more detail, I can't quite figure out why that's right. I'll take you through my thoughts and hopefully, you can help me find my error. I will make assumptions, however, keep in mind that I know very well that one or more of these assumptions has a flaw. That's what I'm trying to find. Here it goes.

Since we're working with an incompressible flow, any parcel of the air will have a constant density as it moves with the flow. Therefore I would expect the density on the left end to be the same as that of the right end. While the flow is moving faster on the left end, there is also less space which I would assume compresses the particles. In my mind, we would end up with the same density across the structure.

I believe that the equation for Pressure P=pRT can be used here. Consequently, my conclusion is that the pressure on the left end would be identical to that of the right end, since R and T remain constant, leaving the pressure and density with a direct proportionality.

Can you help me find my mistake(s), so that I can keep studying what I don't understand? Thanks a lot, I truly appreciate it

Edit: After some more research I think that I may have found an issue. I'm thinking that using the ideal gas law was not the right thing to do in that situation. Instead, I'd have to use Bernoulli's principle to work out the relationship between the pressure, density, and velocity. Doing that would see an increase in pressure as the velocity goes down, always remaining at constant density. Is that the way to approach the situation?

Thanks again

 

[rb1957]

I'm not sure why we're assuming incompressible flow, but ok (maybe it's water, not air).

First, mass flow rate is constant. So that means velocity is higher at the left end, why? (I'm not giving you all the answers !)

Second, Bernoulli says total pressure is constant so that means static pressure is lower at the left, again why?

there are many good online course materials for this. I like MIT's opencourseware.

another day in paradise, or is paradise one day closer ?

 

[DatLemonDoe]

Hi, Thanks for your help!

I believe that we can assume that the airflow is incompressible if its velocity is low enough compared to the speed of sound. That's where my assumption came from.

-First, I believe that if the velocity were the same on the two sides, the mass flow rate (and hence density?) would be higher on the left side, since the airflow is compressed to a narrower space. For the mass flow rate to remain constant, the flow must be faster on the left side.

-Second, I believe that the total pressure is the sum of static pressure and dynamic pressure. If the velocity is higher on the left side, then the dynamic pressure is also higher on the left side. Hence the static pressure on the left side is lower.

I think that the source of my problems was a misunderstanding of the relationship between static pressure and density. In a way, I was looking at it as though we were studying a "steady" gas instead of a flow. Now I understand that dynamic pressure plays a role in the system, hence static pressure and density are not directly proportional.

Does that seem right? Thank you very much!

I will give the MIT OpenCourseWare a go, thank you for referring me to it.

 

[rb1957]

"the flow must be faster on the left side" correct, but you're only restating what I said. Mathematically, why?

"I believe that the total pressure is the sum of static pressure and dynamic pressure" ... don't need to "believe" that, it is (for your example; advanced question ... what could you change in your example to make a change in total pressure?)! Again, I refer you to the Bernoulli equation.

"hence static pressure and density are not directly proportional" ... correct (AFAIK).

another day in paradise, or is paradise one day closer ?

 

[DatLemonDoe]

-well mathematically, I would think that we should look at the following equation

Total Pressure (constant) = P + ρv²/2

Assuming that the density remains constant, we can see that a lower static pressure results in an increase in velocity. Is that what you had in mind or am I missing something?

-This is indeed an advanced question for me, but I believe that an incompressible air flow will always have a constant total pressure, at any point in the system. Consequently, I would think that we would need to accelerate the airflow to a point where it would act as a compressible air flow.

Perhaps we could have an increase in total pressure if the dynamic pressure was by itself greater than the initial total pressure. The static pressure could not be negative so we would see an increase in total pressure. To me, it makes sense since it would require a big velocity.

I have not studied this type of flow whatsoever so that's really as far as I'm comfortable speculating

Thank you

 

[rb1957]

that's most of Bernoulli (but not all !; google it, there are three terms), but doesn't talk to mass flow. if mass flow is constant, why should the velocity decrease ? (what assumptions are you making)

the point to the advanced question was most hinting at what could you do so that pressure (static + dynamic) is not constant ? Compressiblity is one thing.

um, you're out of your comfort zone ... so don't speculate (and certainly don't feel comfortable doing it !!). I think you can have a negative static pressure (I think that's suction as opposed to +ve pressure), but I could be wrong !?.

another day in paradise, or is paradise one day closer ?

 

[DatLemonDoe]

Isn't the third term irrelevant in this scenario? There is no elevation change so I'm not sure that I understand the necessity for the third term (gz)

Maybe the way to explain the decrease in velocity is with this equation?
mass flow rate = ρ.v.A

Making the assumption that the mass flow rate and density remain constant at all points of the system, then we see that an increase in Area results in a decrease in velocity.
I would then go back to the Bernoulli equation to explain the difference in static pressure, now that we have found the change in velocity.

Is that the correct approach?

As for a change in total pressure, I can't find another element than compressibility to explain it. I really don't know anything regarding a change in total pressure, but I'm happy to learn

Thank you

 

[rb1957]

that's pretty much where I was going (assume constant density, then mass flow rate equals volume flow rate, which sounds consistent with incompressible flow)
Compressible flow has variable density (as you may surmise from "compressible").

yes, the 3rd term isn't applicable to this problem, but if you get into the habit of not listing all the terms you can easily get lost later. List all the terms, then simplify as your problem limits things.

for changing total pressure, I was thinking more along the lines at an external force (yes, not part of your example) would invalidate bernoulli.

good luck with your studies.

another day in paradise, or is paradise one day closer ?

 

[DatLemonDoe]

Ok great!

Thank you once again for being so patient