Is There a Pressure Discontinuity When Fluid Leaves a Pipe?

[nathan87]

Hi,
I'm just looking for some clarification on some flow concepts which I'm having a bit of trouble getting my head fully around :S any help on this would be greatly appreciated! thanks.

Basically, I understand the theory and derivations of Bernoulli's equation, the continuity principle and the steady flow momentum/energy equations. But when it then comes to applying them in real situations I am getting stuck or very confused over what seem to be pretty basic points. Here's one - I might post the others later

Say we had a straight streamline, down which an element A of fluid moves at constant velocity. This would mean that forces were exactly balanced on A. This could obviously be explained by the fact that, assuming inviscid, incompressible flow, fluid elements either side of A are exerting equal and opposite forces on it. And indeed, since particles are moving down this streamline at constant velocity, Bernoulli's equation verifies that they are all at the same pressure.

But say this was a situation with a straight, level pipe of constant cross section, with a constant pressure p on the left end and atmospheric pressure on the right, where p is greater than atmospheric pressure. Taking a cut at any two arbitrary locations along the pipe, continuity says that velocity must be equal at all points along the pipe. However, considering a fluid particle just moving out of the pipe to the right seems to lead to a contradiction. This particle must be subject to a force imbalance, and hence be accelerating to the right. In addition, if the fluid is incompressible, then surely this force imbalance would be transmitted through the fluid down the pipe, leading to an overall acceleration!

I am thinking that the solution to this might have something to do with compressibility, although I am not sure if have just misunderstood something here so thought it would be better to ask...also, am I correct in assuming that there is theoretically a pressure discontinuity as soon as fluid leaves the pipe?

many thanks.

 

[Langbein]

Lets say you have a 2 meter high tank with wather. In the bottom you have a plastic tube that you lock off with your thumb.

At the moment you relief your thumb from the tube the wather will start accellerating. As the water accelerate the "viscousious friction" (don't know the english word) will increase until there is a balance between the speed set up by the differential pressure and the "viscousious friction" in the tube.

At the steady state speed there will be a balance set up with the water tank and the pressure drop that is set up in the tube (Due to speed and friction.)

By the way there is basically to kind of flow pattern that can set up different "friction". Those two are "laminar flow" and "turbulent flow".

This picture show the two kind of flow over a air foil. The same prinsiple will also be valid in a tube transporting a fluid.

http://www.aviation-history.com/theory/lam-flow.htm

 

[ravenprp]

Hi there,
It sounds like you have a good understanding of the theoretical concepts behind fluid flow equations. I can definitely see why you are feeling confused when applying them to real situations. Let me try to address your specific example to help clarify things.

In the situation you described, with a straight, level pipe of constant cross section and a constant pressure on one end, it is important to consider the boundary conditions at each end of the pipe. At the left end, where the pressure is greater than atmospheric, there must be a force acting on the fluid to maintain that pressure. This could be due to a pump or some other external force. At the right end, where the pressure is atmospheric, there is no external force acting on the fluid.

Now, when we take a cut at two arbitrary locations along the pipe, continuity tells us that the velocity must be equal at those two points. However, this does not mean that the fluid particles are moving at the same velocity throughout the entire pipe. In fact, the fluid particles closer to the left end of the pipe will have a higher velocity than those closer to the right end, due to the pressure difference. This pressure difference is what causes the force imbalance on the fluid particle just leaving the pipe to the right. This particle will indeed accelerate to the right, but it is important to note that this acceleration is not transmitted through the entire pipe. It only affects the particles near the right end of the pipe, where the pressure is atmospheric.

In terms of compressibility, it is true that an incompressible fluid would not experience a pressure discontinuity as soon as it leaves the pipe. However, in real situations, most fluids are not completely incompressible and there will be some small changes in pressure as the fluid leaves the pipe. This is why it is important to consider the boundary conditions and external forces when applying fluid flow equations to real situations.

I hope this helps to clarify things for you. Keep in mind that fluid flow can be a complex topic and it is normal to have some confusion when first applying theoretical concepts to real situations. It's always a good idea to ask for clarification when needed. Best of luck!