Bernoulli's equation applicability question....

[fog37]

Hello,

I just want to make sure I am on the right track: the three terms in Bernoulli's equation add to the same exact constant for any two points along the same streamline if the fluid is:

• stationary
• incompressible
• inviscid

However, if the fluid is also irrotational, the the three terms add to the SAME constant for any pair of points in the fluid. Is that correct?
(exception: if the fluid is not irrotational but the streamlines derive from a region, upstream, of uniform flow, then the Bernoulli trinomial is equal to the same constant for every spatial point).

Am I correct?

 

[cjl]

The flow need not be stationary, just steady. Other than that, this looks correct.

 

[fog37]

Thanks cjl. But what is the difference between stationary and steady?

For "stationary" I mean that the local fluid properties (density, velocity, etc.) at any point in the fluid do not change over time, hence are time invariant. I thought steady and stationary were synonyms...

Glad you confirm that the Bernoulli trinomial is equal to the same constant even for pairs of points that are NOT on the same streamline as long, as the flow is also irrotational, so points on different streamlines can have the same constant. Almost everywhere (books, web, etc.), I always find that the constant is the same only for points on the same streamline...not sure why they don't include the irrotational condition to make the Bernoulli trinomial more general...

 

[russ_watters]

Thanks cjl. But what is the difference between stationary and steady?

For "stationary" I mean that the local fluid properties (density, velocity, etc.) at any point in the fluid do not change over time, hence are time invariant. I thought steady and stationary were synonyms...

Stationary is zero velocity. Steady is constant velocity (at a given point).

 

[cjl]

Thanks cjl. But what is the difference between stationary and steady?

As Russ said, stationary means the whole flowfield has zero velocity. Steady just means that the flowfield is not time-dependent (so the velocity is constant at any single point in space).

 

[fog37]

Thanks!

I still find very interesting that if the flow has zero curl (irrotational), then Bernoulli's equation can applied to any pair of points in the flow (not just pairs of points on a streamline). Why is that?

Most textbooks skip this important detail. Why? Is it really hard for a steady, inviscid flow to also be irrotational?

 

[cjl]

Fundamentally, it's because the flow then has the same total pressure everywhere. If the flow has rotation, the point in the middle of the rotation has a lower pressure than the points around it, even though it is not moving at a higher velocity, but if there's no rotation anywhere in the flow, the only way for the pressure to fluctuate is for there to exist velocity gradients associated with it, as described by Bernoulli. As for whether this is a difficult condition, it's actually pretty common, at least if you ignore the effects of the boundary layer.

 

[fog37]

ok. So velocity gradients imply the presence of curl (even if the streamlines are perfectly straight lines).

What do you mean by "total" pressure? I know we often distinguish between "dynamic" and "static" pressures. Static pressure is called static even if the flow is truly moving. It is just measured in a direction other than the direction of motion of the flow...Is that correct?

 

[cjl]

No, you can have velocity gradients without curl (and you frequently do). Curl implies pressure gradients without necessarily having associated velocity gradients. Total pressure is just static plus dynamic, and is constant for situations where bernoulli's relation applies.