Fluids, Streamlines and Pressure

[member 428835]

Hi PF!

Can anyone help me understand the relationship between streamlines and pressure in an inviscid/incompressible flow? I know a streamline is everywhere parallel to flow. Is it ever possible for streamlines to cross each other? If you have a recommended reading please let me know, or if you can answer the above and/or offer other insights about streamlines I would greatly appreciate it.

Thanks!

 

[haruspex]

Is it ever possible for streamlines to cross each other?

No, but of course, in a 2D projection of a 3D flow they may appear to cross.
You mentioned pressure. Can you be more specific about your question on that?

 

[member 428835]

No, but of course, in a 2D projection of a 3D flow they may appear to cross.

Why can't they? Does this have anything to do with incompressibility?

You mentioned pressure. Can you be more specific about your question on that?

Yea, can we deduce anything about pressure given, say, two regions of streamlines, one where they are very close and the other where they are more spread out?

 

[haruspex]

Does this have anything to do with incompressibility?

No, even compressible flows do not cross. It has more to do with lack of intent and memory. Two streamlines might merge, effectively, but having done so they are subject to the same forces, so have no reason to separate again according to their origins. They may retain different speeds but lie parallel.
We can imagine a sheet of falling water, with a hose directing a horizontal stream at it. This might punch a hole through the sheet, but still we would not say the streams flow through each other.

can we deduce anything about pressure given, say, two regions of streamlines, one where they are very close and the other where they are more spread out?

Yes. Where the streamlines come closer together the same volumetric flow occupies less width, so goes faster. That tells you something about pressure variation along the stream.

 

[boneh3ad]

Yes. Where the streamlines come closer together the same volumetric flow occupies less width, so goes faster. That tells you something about pressure variation along the stream.

Be careful with this. Picking where to draw streamlines is arbitrary. You could take the same flow field and simply add more streamlines to it and they will be closer together but the flow field will be exactly the same. However, if you pick two arbitrary streamlines and follow them through a flow, if they get closer together then the average velocity between them is greater. Comparing streamlines in two different flows is pretty useless, however.

Why can't they? Does this have anything to do with incompressibility?

The definition of a streamline is such that they are everwhere parallel to the velocity field. If two streamlines crossed, they would intersect at a point that would have to be multi-valued, which obviously cannot happen.

 

[haruspex]

if you pick two arbitrary streamlines and follow them through a flow, if they get closer together then the average velocity between them is greater

Yes, that is how I meant "streamlines coming closer".

 

[boneh3ad]

Yes, that is how I meant "streamlines coming closer".

Right but you have to be careful how you describe that. Otherwise it is very common for people to read it as "closer streamlines means faster flow", which is only true if we are talking about the same pair of streamlines in the same flow.

 

[russ_watters]

Is it ever possible for streamlines to cross each other?

Remember that streamlines map the movement of an arbitrarily small "packet" of a fluid. If streamlines crossed, it would mean two such "packets" (such as two drops of water) would have to be in the same place at the same time, which is of course impossible.

 

[Chestermiller]

If you properly establish stream function values for the stream lines, then the volumetric flow rate of fluid along the gap between any two of them is proportional to the difference in their stream function values. For a given flow, if you consider streamlines closer that are together, the difference in their stream function value will smaller.

 

[boneh3ad]

Remember that streamlines map the movement of an arbitrarily small "packet" of a fluid. If streamlines crossed, it would mean two such "packets" (such as two drops of water) would have to be in the same place at the same time, which is of course impossible.

That's actually not the definition of a streamline, and it is only true in a steady flow. The path a fluid particle takes through a flow is actually called a pathline. It's a subtle but important difference. Streamlines are defined as being lines that are everywhere tangent to the velocity field.

 

[russ_watters]

That's actually not the definition of a streamline, and it is only true in a steady flow.

I'm unclear on what you are saying here: are you saying that streamlines can cross in unsteady flow?

The path a fluid particle takes through a flow is actually called a pathline. It's a subtle but important difference. Streamlines are defined as being lines that are everywhere tangent to the velocity field.

Is one a subset of the other? You said they are different, but didn't say what the difference is...

I know wikipedia can be wrong, but it looks to me like the only difference between streamlines and pathlines is history and that my description is included in the definition of streamlines. Since I was referring to two particles traveling next to each other (in parallel streamlines or pathlines), it appears to me that your objection is moot. But either way, please provide a more detailed description of what you are trying to say.
https://en.wikipedia.org/wiki/Streamlines,_streaklines,_and_pathlines

• Streamlines are a family of curves that are instantaneously tangent to the velocity vector of the flow. These show the direction in which a massless fluid element will travel at any point in time.[3]
• Pathlines are the trajectories that individual fluid particles follow. These can be thought of as "recording" the path of a fluid element in the flow over a certain period. The direction the path takes will be determined by the streamlines of the fluid at each moment in time.

 

[boneh3ad]

I did say the difference. Pathlines trace the path of individual fluid particles. Streamlines are lines that follow the velocity field. For steady flow they are the same. For unsteady flow they are not.

In an unsteady flow, pathlines can cross. Streamlines still cannot.

 

[russ_watters]

I did say the difference. Pathlines trace the path of individual fluid particles. Streamlines are lines that follow the velocity field.

Given that for steady flow they are the same, I think an explanation of why they are different - as opposed to just being different ways of saying the same thing - was warranted.

In an unsteady flow, pathlines can cross.

But does that mean that two fluid elements can occupy the same space at the same time? I'm fine with being pedantic, but you painted with such a broad brush - imprecisely - that you implied something (the key thing) I said that was true wasn't true.

 

[haruspex]

Given that for steady flow they are the same, I think an explanation of why they are different - as opposed to just being different ways of saying the same thing - was warranted.

But does that mean that two fluid elements can occupy the same space at the same time? I'm fine with being pedantic, but you painted with such a broad brush - imprecisely - that you implied something (the key thing) I said that was true wasn't true.

I believe the objection is to this

Remember that streamlines map the movement of an arbitrarily small "packet" of a fluid.

If the flow keeps shifting, the streamline through a particular point at one time might be quite different from that at another time. Thus the path Iines taken by successive particles arriving there can indeed cross. Streamlines correspond to a snapshot of the flows at an instant.

 

[boneh3ad]

I believe the objection is to this

If the flow keeps shifting, the streamline through a particular point at one time might be quite different from that at another time. Thus the path Iines taken by successive particles arriving there can indeed cross. Streamlines correspond to a snapshot of the flows at an instant.

Correct. Thanks.

Given that for steady flow they are the same, I think an explanation of why they are different - as opposed to just being different ways of saying the same thing - was warranted.

But does that mean that two fluid elements can occupy the same space at the same time? I'm fine with being pedantic, but you painted with such a broad brush - imprecisely - that you implied something (the key thing) I said that was true wasn't true.

I am not being pedantic; I am being precise. I'll start at the beginning and hopefully that will be more clear. There are generally three types of flow path lines that can exist in a flow: pathlines, streaklines, and streamlines.

• Pathlines: Pathlines trace the paths of individual particles in the flow. Imagine a campfire where small tiny embers follow the flow field. If you took a long exposure photograph such that they now became paths drawn out by the light, that is a pathline. Alternatively, long-exposure photography of traffic produces pathlines for those cars (traffic often behaves much like a fluid).
  
• Streaklines: Streaklines form when you tag every particle passing through a single point in space with some sort of tracer and see the line traced out by every successive fluid particle. I am sure you've seen pictures of wind tunnels with a car in them and a guy stand there with a smoke wand holding that out in front of the car and watching the line pass over it. That is a streakline.
  
• Streamlines: Streamlines are lines that are everywhere tangent to the velocity field.

All of these can change and move in time. In general, they are entirely different lines that are traced out by each concept. For a steady flow, they all line up on top of each other. However, in no situation can two streamlines cross, whether steady or unsteady. Since your explanation above was using the definition of pathlines to try to make statements about streamlines, it does not apply to steady flow. That's why I gave the answer I did earlier in the thread: given the definition of a streamline, if two streamlines cross, then the velocity field at their intersection must be multivalued. This cannot happen and therefore, steady or unsteady, streamlines cannot cross.

Of course two fluid elements cannot occupy the same space at the same time, but that isn't what you were actually describing in general given the definition you provided. This isn't an example of pedantism. It's a matter of giving a correct answer on a topic that can sometimes be confusing. I'd suggest also viewing the following video. It's old but it's probably the best discussion, with visualizations, of why pathlines and streaklines and streamlines are not always the same and how pathlines can absolutely cross.

 

[member 428835]

Thank you all for the discussion! It's been very helpful for me.

 

[Nidum]

You might find it interesting to read about potential flow and equipotential lines .