Flow characteristics in a transition duct

[abrewmaster]

So I was trying to understand how fluid flow is affected through a transition duct when I found http://nopr.niscair.res.in/bitstream/123456789/152/1/IJEMS%2014%281%29%202007%2012-18.pdf that was done. In the study it was noticed that there was a dramatic drop in velocity near the end of the transition near the edges and was also dependent on the length in the transition (Which makes sense). I was curious if anyone knew anything about the math behind this relationship to basically calculate the pressure drop and or velocity change based on the initial cross section, final cross section,and length of the transition (possibly even angle of transition and other things like that).

 

[SteamKing]

The simplest analysis would be to apply the Bernoulli equation. If the fluid in the duct is air, at low speeds (below about 0.3 Mach), air can be treated as incompressible with little loss in accuracy of results.

 

[abrewmaster]

Yea I know that, the thing is that there are additional losses associated with a change in cross section shape that I do not know how to account for. I would use the Bernoulli equation but would need to calculate the head losses a different way.

 

[boneh3ad]

First, you would expect the velocity to decrease as you move along the duct simply due to continuity. Mass flow has to be constant, so when they increase the cross-sectional area of the duct, the velocity should fall accordingly and the static pressure should rise. You would also completely expect the velocity to tend toward zero as you get close to the wall, a phenomenon known as a boundary layer. That said, there are a number of things wrong with this experiment that would make the results unreliable.

First, they put their blower upstream of the test section without any means to mitigate the incredible amount of turbulence being put out by that blower, meaning who knows what the flow quality in their tunnel is. There are very likely large vortical fluctuations in their flow that will affect the results. Given, in this situation that would probably be a fairly minor effect, but it is possible that their entire flow is swirling around instead of flowing straight through the pipe, and they don't account for that.

Second, they cite the result that you need 25 to 40 diameters to establish fully-developed flow and they only give it 27 diameters, so odds are the flow is not fully developed when it enters their test section. You would expect fully-developed flow to change velocity inversely proportionally to the change in area. You don't that here until the last third of their duct, and that very likely could be because they still have boundary layer growth over the first portion of their test section rather than a fully-developed duct flow. In fact, you can pretty much see that they don't have fully developed flow because if they did, the velocity wouldn't be flat toward the center of the duct; it would have curvature the entire way and have a maximum at the centerline.

 

[abrewmaster]

Yea, according to what they were studying they were trying to look at turbulent flow so it would make sense that they would try to induce that however you are right about how it seemed as though they had no way of controlling the overall flow.

Would you say then that their test looked at mostly the entrance region of transition flow? Then again wouldn't the flow never actually fully develop since the cross section is continuously changing? I'm trying to understand more about this concept and the math behind it.

 

[boneh3ad]

It depends on how fast the boundary layer is growing versus how fast the duct is diverging. It sure looks like in that study the flow never reaches a fully-developed state given the flat region in their velocity profile. If, on the other hand, the boundary layer was growing faster than the duct was diverging, then, given a long enough duct, you would eventually see fully-developed flow. There is also really no way of saying if the flow they were looking at was laminar or turbulent or somewhere in between (transitional). They were using a hot-wire anemometer, which would give you that information by looking at the power spectrum, but they didn't report that result.

All that said, there is not going to be an analytical solution to the problem you are trying to solve. The best you could hope for is some kind of empirical relationship developed from one or more experiments. Otherwise, you would need to simulate the flow using a CFD package. Also, in the duct you are describing, you would actually have a pressure rise, not a pressure drop. You may have some luck searching for flows through a diffuser, as this type of duct with an increasing cross-section functions as a diffuser.

 

[abrewmaster]

ah ok, I guess that makes sense. I saw that they calculated Reynolds number and that the values suggested that the flow was turbulent, are you saying that they calculated them incorrectly?

Sorry for the confusion, the application I would be using this for would be for a decreasing area, nozzle instead of their situation so that's why I said pressure drop. I was hoping to possibly get a calculation that would be more accurate than a nozzle/diffuser calculation but I guess I'll just have to manage.

 

[boneh3ad]

ah ok, I guess that makes sense. I saw that they calculated Reynolds number and that the values suggested that the flow was turbulent, are you saying that they calculated them incorrectly?

They are using the Reynolds number based on the hydrualic diameter, which will define when a flow is going to eventually become turbulent in a pipe or a duct. It only applies to fully-developed pipe/duct flow and then not really to flows with changing areas. Even if this was a straight, constant-area pipe that was fully-developed, there is no closed-form answer as far as I know that says how far downstream the transition to turbulence is complete. A sufficiently high Reynolds number in a pipe only means that the flow will at some point transition, not that it is turbulent from the outset.

Sorry for the confusion, the application I would be using this for would be for a decreasing area, nozzle instead of their situation so that's why I said pressure drop. I was hoping to possibly get a calculation that would be more accurate than a nozzle/diffuser calculation but I guess I'll just have to manage.

You would need to likely use one of the various empirical relations for head loss. You will often find such things discussed using a loss coefficient that relates head loss to the velocity of the flow. If you felt particularly motivated, doing an experiment to determine your loss coefficient would be rather straightforward.

If you have access to it, maybe check out Chapter 8 in "Fundamentals of Fluid Mechanics" by Munson, Young and Okiishi. That would give you a good start. There are doubtless other sources you could use, too, but I happened to have this one at arm's reach. You may even be able to get a decent estimate of your losses without an experiment using known estimated values and the Darcy-Weisbach equation.

At any rate, the typical pressure loss is incredibly tiny in a nozzle. Note that when talking about pressure loss this way you have to be careful in discussing static pressure versus total pressure. For example, a nozzle will have two sources of "pressure loss": viscous dissipation and flow acceleration. If you completely ignore viscosity and just accelerate the flow through a nozzle you get a pressure drop due to the acceleration of the flow. If you were just just have flow through a pipe and take into account viscous dissipation, you would have total pressure loss that would manifest as a loss in static pressure since dynamic pressure (velocity for an incompressible flow) is constant. In other words, you just have to keep in mind that you have two sources of pressure loss, acceleration and viscosity, and for a nozzle you will have both present. The loss due to acceleration is recoverable, for example through a diffuser, but the loss due to dissipation can only be recovered using a pump or elevation change.