How Does Jet Injection Work in Fluid Dynamics?

[Twigg]

Hi all,

Sorry to post an open ended question, but I'm struggling because I don't know the right terms I need to do a literature search on my own. Any kind of feedback here is helpful!

I'm interested in the dyanmics of a class of device I've heard described as a "jet injector". I know of two examples. The first is naturally-aspirated propane burners, like hand torches, where a nozzle creates a stream of high velocity, low pressure propane gas which sucks in atmospheric air via negative pressure. The air then mixes with the fuel to make a flammable mixture flowing out the torch. The second is a steam injector on a steam locomotive, where a jet of steam passes through liquid water and "pulls" a small mass of water into the jet. The steam and water droplets mix, and the steam flow carries the water to the boiler to generate more steam. To summarize, I'm interested in the process of using a high momentum jet to "suck in" some amount of a second fluid at rest. What does the thermodynamic / mechanical analysis of this process look like?

Per the rules, here's my attempts at understanding: There's some momentum exchange between the inducing fluid (propane or steam) and the induced fluid (air or water). In my head I imagine something like a Langevin equation. For the case where the secondary fluid is a gas, I imagine the decay time is inversely proportional to the inter-species collisional cross section (I think that goes as the product of the two molecules' kinetic diameters, right?). Where I get tripped up is how to turn the single-particle Langevin description into a control volume description.

Again, sorry for such an open-ended post. If you want more info, or more clarification, anything, just let me know in the replies.

 

[anorlunda]

Good question. This Wiki article can help you get started. Usually, the links to references at the bottom of the article help you to dig deeper.

https://en.wikipedia.org/wiki/Injector

 

[bigfooted]

Also related to:
https://en.wikipedia.org/wiki/Entrainment_(engineering)

There are several mechanisms for entrainment, but usually there is a strong concentration gradient or pressure gradient. All these processes can probably be described with Fokker-Planck or Langevin equations, but this is usually done only for dispersed phases. There are entrainment equations that you can look into that might help in building your understanding about the physics.

 

[Twigg]

I'm more interested in the gas-to-gas case, so dispersed phases works for me.

 

[Twigg]

I've done a little reading here and there as I have time, and what I've encountered seems pretty disheartening (but not surprising in hindsight). It seems that in jet injection the dominant mixing mechanism is turbulent eddies generated by the shear at the boundary of the jet.

I was reading a chapter out of what appears to be a http://www.dartmouth.edu/~cushman/books/EFM/chap9.pdf (can't tell the author because it's just one chapter in the pdf), and they claim that for a jet intruding into a stationary fluid of equal density, there's a universal angle of $11.8^{\circ}$ for the diverging jet envelope. This is hard to explain in words, but check out Fig 9.2 of the link.

I know we're talking about turbulent flow, and I shouldn't expect neat derive-able results, but here's to hoping. Is this universal angle something that can be obtained from first principles or is it just from numerics/observation? I'm asking because I'm interested in the non-equal density case. Thanks!

 

[bigfooted]

You can view the entire (unfinished) book of Cushman here:
https://www.dartmouth.edu/~cushman/books/EFM.html

You can get this jet angle from an analysis of the momentum equation but it still leads to a nonlinear ODE with no known analytical solution, at least not given in for instance the book of Wilcox 'Turbulence Modeling for CFD'.

Interestingly, different (simplified) turbulence models like the k-epsilon model predict slightly different jet angles and they fail to simultaneously predict the correct angle for a planar jet and an axisymmetric jet. This is the round jet anomaly, and is discussed a lot in the literature as well as the book of Wilcox (there are some corrections proposed).

If you want to know how such problems can be dealt with using a Lagrangian approach (with Langevin type models), you can have a look at Stephen Pope's book 'turbulent flow'. Stephen was also one of the first persons to solve the round jet anomaly.

If you want to play with some simplified models to get a feeling for it, you could have a look at the momentum analysis in Wilcox' book, although this is treated in most books on turbulent fluid dynamics, like White: 'viscous fluid flow'.

 

[Twigg]

That anomaly is pretty bonkers. Thanks for the info! I'll be trying to get my hands on a copy of Wilcox and Pope in some format.