Air flow in Convergent divergent nozzle

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  • #1
gkraju
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i understood the air flow properties variation in convergent divergent / condi nozzle or laval nozzle for subsonic flow based on the formula, for a incompressible flow :

density * Area * velocity = constant
as the total pressure in the flow is constant dynamic pressure is minimum at the throat and static pressure is much more.

for subsonic flow, as the area decreases at the throat of the nozzle, velocity increases and after the throat as the area increases, velocity decreases, that is how velocity is maximum at the throat of the condi/laval nozzle.

once speed of the flow is slowly increased as the velocity of the flow reaches the velocity of sound i.e. M=1, the compressibility effects come into picture, how the flow properties i.e. velocity and dynamic pressure variations change.

1. it is said that for supersonic flow the velocity at the throat decreases where as the velocity increases as the throat diverges, why is it so, why the velocity variations are opposite to that of subsonic flow ?

2. And when does the shock wave forms in the condi nozzle ? how the shock wave forms if at all it forms ?

3 why the velocity in condi nozzle never exceeds M =1 ?
can any one please explain me in simple terms rather than math formulae.
 

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  • #2
cjl
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So obviously this is very old and you probably don't need an answer any more, but on the off chance that this is helpful for someone:

The goal of a nozzle is to accelerate air (or some fluid). However, if you take a slow moving fluid and end up with a fast moving fluid, you have added kinetic energy to it, and that energy comes from somewhere. Since a nozzle has no way to add energy to the fluid, this energy must have already been present in the fluid just in a different form. The most convenient form (and what a nozzle takes advantage of) is pressure. It turns out there's actually quite a lot of energy stored in a high pressure gas, and a nozzle trades some of this off to accelerate the fluid.

If you aren't accelerating the fluid very much, the pressure change will be small, and thus you can kind of pretend that no compression happens and the volume of the fluid does not change. This leads to the "traditional" result we're all familiar with where convergence accelerates the flow, because if the volume of each bit of fluid doesn't change, and we aren't magically adding additional fluid or removing it anywhere, the only way to make it flow faster is to reduce the cross sectional area.

However, as you want to accelerate the flow faster and faster, you need more and more energy, especially since kinetic energy scales with the square of the velocity so accelerating a flow to 200mph is 4x as much energy as accelerating it to 100mph. It also turns out that for math-related reasons that I'll avoid here (since you requested minimal math), the energy contained in a pressurized gas is related to the sound speed, so we can use the speed we want to accelerate the gas to relative to the speed of sound in that gas as a good metric of how much of the pressure energy we need to use. Once that number starts to get above 30% of the speed of sound or so, we need to use so much of the pressure energy that the pressure drop becomes large enough that we can no longer ignore the change in volume of the gas. However, aside from some slight corrections, the above explanation still mostly works, and you still accelerate the gas with a convergent nozzle (though the exit velocity is a bit higher than you'd expect if you assumed incompressibility, since the gas volume coming out is a bit larger than what went in because of the pressure increase.

Once you accelerate it even further and hit mach 1, something interesting happens. Now, you're pulling so much energy out of the gas that the pressure has to drop dramatically, and furthermore, because the kinetic energy being added scales with velocity squared, each little bit of additional speed takes more and more of that energy. This means the volume of gas starts to increase dramatically. If you keep trying to neck it down further, the only way for it to expand would be to further accelerate down the tube, but that's actually not possible because we're already having to expand it to generate the energy for the prior bit of acceleration, so we end up in a situation where there's physically not enough energy available to keep accelerating it. This is the answer to your question 3, and is also what's referred to as a "choked" condition for the nozzle. Because there's no energy to keep accelerating the flow under these conditions, if you try to cram more and more flow through it, it'll just stay at mach 1 at the throat.

However, the way around this and to keep accelerating the flow is to give the flow a way to expand without increasing in velocity. This is why you want a diverging section. As it expands, you can continue to accelerate it, but it has to expand faster than it accelerates so you need to keep adding more cross sectional area to make this possible (sorry if this is a bit handwavey and non-specific, I'm trying to do the best I can without math here).

As for a shock? Ideally you don't want one anywhere. The flow entering a shock wave is supersonic and the flow slows down across the shock, so the only place one would be possible in a converging-diverging nozzle would be in the diverging section (since it's subsonic everywhere before the throat), but the presence of a shock would indicate that the nozzle is poorly designed or operating well outside of its design condition (since as I said, shocks slow the flow down and usually a nozzle is trying to increase the speed, plus shocks also create irreversible losses that end up just heating up the flow).
 
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  • #3
boneh3ad
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To expand on what @cjl said (with math, for anyone who stumbles across this who isn't the OP and since that is more or less necessary for topics like these)...

1.
For a converging-diverging nozzle, the flow is quasi-1D, so you can generally treat it as a function of only one spatial variable, say, ##x##. We can then consider the conservation of mass
[tex]
d(\rho u A) = 0 \quad\to\quad \frac{du}{u} + \frac{d\rho}{\rho} + \frac{dA}{A} = 0,
[/tex]
and conservation of momentum
[tex]
d(\rho u^2A + pA) = p\;dA, \quad\to\quad \rho u\; du = -dp
[/tex]
where ##\rho## is density, ##u## is velocity, ##A## is cross-sectional area in the duct, ##p## is static pressure, and ##d## are differential operators (i.e. small changes in a variable).

You can combine these, along with the definition of the speed of sound, to come up with a more general area-velocity relationship that holds for both incompressible and compressible flows:
[tex]
\frac{du}{u} = -\frac{1}{1-M^2}\left(\frac{dA}{A}\right).
[/tex]
This tells us that if ##M>1##, then ##dA > 0## (diverging duct) leads to ##du<0## (decelerated flow). The opposite is true for ##M<1##.

2.
Whether a shock forms in the nozzle is dependent on the pressure ratio across the nozzle. That's a much longer topic.

3.
The flow in a condi nozzle absolutely can exceed ##M=1##.
 
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