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What is the boundary condition if the pipe is not long enough to fully develop? 

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October 26, 2016, 09:49 
What is the boundary condition if the pipe is not long enough to fully develop?

#1 
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Aaron
Join Date: Apr 2016
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Let me first explain what I mean. Suppose a Poiseuille flow just happens to be well developed at 1m outlet. I want the data at 0.5m, so in the fully developed condition, I can get the correct result I want.
But I would like to simulate a pipe only 0.5m, I want the same or very close results compare to 0.5m result in 1m pipe then what kind of the the boundary conditions should I use? actually, now I encountered a problem which pipe length may be not long enough to reach fully develop, so I confused and cannot figure out which boundary condition should I use. can you give me some suggestions? Thanks for any advice provided. 

October 26, 2016, 11:49 

#2 
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Kevin van As
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Euh?
Boundary conditions are a tricky thing for any mathematical problem. We, physicists, typically only know the boundary conditions in infinity. For example, take a wind turbine. The wind turbine will have a very complicated influence of the surrounding air, making it impossible to predict what the velocity will be like 1m behind the wind turbine. Actually, that is why we are running the simulation in the first place!, right? However, we do know one thing: sufficiently far away from the wind turbine, the flow must be undisturbed, as the wind turbine should not be able to affect a location infinitely far away, right? Therefore, to simulate a wind turbine you'll need a domain that is sufficiently big to simulate the problem accurately. It is simply impossible to define a boundary condition at some plane closer to the wind turbine, as we don't know what to expect there. Your pipe poses a similar problem: you plain simply do not know the velocity profile, hence you cannot demand one and must really simulate a longer pipe. After all, if you set the velocity profile at 0.5m and then measure at 0.5m, I can tell you in advance what you obtain: your input information. However, if your pipe flow is laminar, then it is possible to find (or estimate I should say) the exact solution of the developing profile. But even then: then your simulation outcome will simply yield what you already put in, as you are trying to probe the boundary. Now I assumed implicitly that you want to set the velocity itself, but of course, you may also set different BCs, like setting a gradient. > That is what we do for infinity. You could try to come up with such BCs from the equations, e.g. the continuity equation says that the normal gradient (du/dx) must be equal to minus the tangential gradient (dv/dy). However, I cannot imagine that you will be able to close the system this way (especially because pressure is annoying) for if you could, you can simulate anything by looking at only a small part of the domain. You will probably end up demanding "that your equations must hold true on the boundary", but that is not a BC. Conclusion: Simulate the entire pipe, or find the exact solution (or more realistically: an approximate solution using e.g. an integral method). 

October 27, 2016, 22:28 

#3 
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Aaron
Join Date: Apr 2016
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Dear Kevin,
Thanks for your detailed answers, I understand your meaning, but my pipe is a given length, it is a mixed convection problem, so under buoyancy effect, every position in pipe is different, so I doubt it cannot reach fully developed at outlet position. I doubt if there exist a boundary condition which like this: velocity ,pressure and temperature gradient equal to the gradient near the outlet region? is this boundary condition suit for this uncertain boundary condition problem? or should I come up with other simplified model? best wishes, Aaron 

October 28, 2016, 03:09 

#4  
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Kevin van As
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If your physical pipe is a fixed length which is shorter than the fullydeveloped length of a pipe, then you might want to add a containercube behind your pipe with "infinity" conditions "sufficiently far away".
Quote:
So the thing at your outlet is: it does not only depend on what happens before the outlet. It does also very much depend on what happens after the outlet. To understand this, think of simple pipe flow: the pressure gradient (which required the pressure at both the inlet and outlet) determines the magnitude of the velocity. Therefore, to set a boundary condition at your outlet, you must somehow take into account what happens after your outlet  which you probably do not know in general, and may be impossibly difficult to do, accurately at least. As far as I know, you must either demand wrong boundary conditions at 0.5m to simplify your problem (like demanding fullydeveloped conditions), or you must simulate a larger domain to obtain a greater accuracy (like the containedcube I mentioned above). 

October 28, 2016, 07:28 

#5  
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Aaron
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Quote:
extend the pipe length(or = add a dummy region)? Quote:
I will consider extend my pipe length 

October 28, 2016, 08:48 

#6 
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Kevin van As
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Extending the pipe length is one way to solve the problem.
However, if the problem you study really consists of a pipe of 0.5m, enlarging the pipe will yield different results than you might want. An alternative is therefore using a "containercube", meaning adding a dummy region behind your pipe. So you'd have (from left to right): inflow patch > pipe > box glued to the pipe > "infinity" patch. This figure might clarify what I mean, if not, please do ask: Code:
    inflow "infinity conditions"  > zeroGradient    

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boundary condition, cfd, fully developed flow 
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