# Why is the velocity of fluid on top a hydrofoil higher than on the bottom?

I am trying to learn how hydrofoil works. A big part of it is Bernoulli's principles.

I found this http://web.mit.edu/2.972/www/reports/hydrofoil/hydrofoil.html on MIT webisite.

There are two explanation for how it works:

1. The conservation of linear momentum or Newton's third law.If water is pushed down, the water pushes back on the hydrofoil.

http://web.mit.edu/2.972/www/reports/hydrofoil/hydrofoil-1.gif

For the change for momentum of water, their is a equal but opposite change in momentum of the hydrofoil, directed upward.

2. Bernoulli'principles: $$P_o=P+1/2dv^2+dgy$$

$P_o$ is called the stagnation of pressure and is a constant. P is pressure. d is density. v is velocity of fluid. and g is gravitational acceleration. So if v increases on top, the top pressure will decreases, and as v of fluid decreases on the bottom, the pressure with increase. The hydrofoil then starts to rise when the bottom pressure overcomes the top pressure. It rises till there is no water on top.

QUESTION

1. Why the velocity of fluid on top is greater than that of the fluid on bottom? The link says "this is due in part to viscous effects which lead to formation of vertices at the end of foil." But I am not sure what those "effect" and "vertices" are.

2. What keeps the top velocity increasing, and the bottom velocity decreasing?
I guess it's this enlarging difference in velocities that produces larger and larger pressure difference, and thus larger net force.

rcgldr
Homework Helper
Why the velocity of fluid on top is greater than that of the fluid on bottom?
For a somewhat simpler explanation, the higher velocity on top coexists with lower pressure. The article mentions curvature of streamlines, which requires a pressure gradient where pressure decreases in the direction of curvature. As the fluid flows over the convex upper surface of a foil, it fills in what would otherwise be a void, by mostly following the convex upper surface (a boundary layer is formed), or if separation and/or stall occurs, vortices are generated. in order to follow the upper surface, the flow is curved, which coexists with lower pressure and higher velocity near the upper surface (just outside the boundary layer).

What do people use to measure fluid velocity? Do they place a piece of paper and let it get blown by the air and measure its velocity?

What does mean as in the definition of streamline that "streamlines in a flow field have no mass flow crossing them"?

rcgldr
Homework Helper
What do people use to measure fluid velocity? Do they place a piece of paper and let it get blown by the air and measure its velocity?
Using pulsed smoke in a wind tunnel, along with a high speed camera could be used for this purpose. A model wing could be instrumented with a bunch of static port sensors to measure pressure, but those could interfere with the flow. A mathematical model is often used to estimate how air will flow across a wing section. One popular mathematical model is XFOIL (do a web search if interested).

What does mean as in the definition of streamline that "streamlines in a flow field have no mass flow crossing them"?
It's just that, a definition. Mass only flows in the direction of the streamline, without any component perpendicular to the direction of the streamline. A streamline may be curved, so there could be a component of acceleration and pressure gradient perpendicular to the direction of a streamline.

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People say Beenoulli equation is just a statement of conservation of energy. But there must be a external force. Otherwise, the fluid won't move. Is it like you push the fluid and let its momentum carry it?

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Using pulsed smoke in a wind tunnel, along with a high speed camera could be used for this purpose.

Pulsed smoke is generally not used for this purpose very often, though there is no reason it couldn't be. The act of introducing the smoke may well interfere with the flow, and the spatial resolution is likely to be poor.

There is a sort of variation on this method, however, called molecular tagging velocimetry. Instead of smoke, you seed your gas flow with a small amount of another gas. You then "write" a line into the flow with a laser of a specific wavelength that causes a thin line of that seed gas to change to a different species. You can then use a different laser to make that line fluoresce and take a pair of images of that line to see how far it moved over a known time to get velocity.

A model wing could be instrumented with a bunch of static port sensors to measure pressure, but those could interfere with the flow.

A properly-designed static port will not interfere with the flow in a meaningful way (for the purposes of measuring a mean velocity). These may be used to determine the velocity provided that other information about the flow is known a priori, such as its total pressure and the free-stream static pressure. The one caveat is that static ports absolutely may (and often will) introduce small disturbances that will lead to boundary-layer transition, so no such features may be present upstream of a region in which you plan to study laminar-turbulent transition.

What do people use to measure fluid velocity? Do they place a piece of paper and let it get blown by the air and measure its velocity?

There are several common methods, but perhaps the most common is the Pitot tube (or a Pitot-static probe). Essentially, you place a tube in the flow facing upstream and measure the pressure at the tip, which is stagnation pressure, and then either add a side port for static pressure or measure it elsewhere, then use Bernoulli's equation to back out the velocity at the tip of the probe.

There is also a method known as hot-wire anemometry, where a tiny wire is heated and placed in the flow. When air flows over the wire, it cools down. The voltage required to maintain a constant wire temperature can be calibrated to the velocity at the point measured by the probe.

Another popular technique is particle-image velocimetry. The flow is seeded with a multitude of tiny particles that are small enough that they can be assumed to follow the path of the flow. If you use lenses to form a laser beam into a sheet and shine that sheet through the flow at the same time that you take two images in quick succession with a known time gap, you can determine how far the particles have been displaced and convert that into velocity. From this technique you can get velocity at "every" point within your field of view and within the plane illuminated by the laser.

There are other techniques as well. It just depends on your goals and experimental requirements.

People say Beenoulli equation is just a statement of conservation of energy. But there must be a external force. Otherwise, the fluid won't move. Is it like you push the fluid and let its momentum carry it?

Bernoulli's equation is a simplification of the equations governing fluid flow. Under certain conditions, it describes a moving fluid quite well. Under others, it fails. For the conditions under which it applies, it amounts to an energy balance. If you want to think in terms of forces, you can think about the pressure. If you were to pick two point at which to evaluate Bernoulli's equation, the pressure difference between them corresponds to the the net force acting on the chunk of fluid between those two points.

If you prefer to think more in terms of ##F=ma##, then you might try one of the fuller-featured equations, such as the Euler equations for inviscid flow, or the Navier-Stokes equations, which are the full equations that govern viscous fluid flows in a continuum. Each amounts essentially to a force (per unit volume) balance. Both are quite a bit more complicated and more difficult to work with than Bernoulli's equation. Additionally, you can derive Euler's equations from the Navier-Stokes equations by assuming zero viscosity, and you can derive Bernoulli's equation from the Euler equations under several more assumptions. That's sort of a different topic than the main one in this thread, however.

So the energy input from the net force is taken care of by the "PV" part. The conservation of energy is more about "1/2mv^2+mgh"??

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There's no energy input. If there was energy input then Bernoulli's equation would not be valid. You might visualize static pressure as a sort of stored spring energy per volume. Bernoulli's equation has essentially three terms: static pressure (a sort of potential energy per volume), dynamic pressure (the one with velocity, representing kinetic energy per volume), and hydrostatic pressure (the one with gravity, which is gravitational potential energy per volume). For Bernoulli's equation to work, no additional energy can be added or removed from the system. It can merely be transferred from one form to another.

rcgldr
Homework Helper
Why don't more articles related to Bernoulli include the equations related to acceleration perpendicular to the flow (curved streamline), such as section 3.3 (equation 3.12) in:

http://www.me.ncu.edu.tw/energy/ME381/Ch3.pdf

I don't get why the article in the FIRST link says Bernoulli is more practical than Newton.

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I don't get why the article in the FIRST link says Bernoulli is more practical than Newton.

Pressure and velocity are very easy to measure. The total effect that an object in an air flow has on all of that air is not, and this is required for the direct application of Newton's laws. Calculating the two would be roughly similar in difficulty for a computer code, but at that point, knowing pressure distributions and velocities is a lot more useful anyway.

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Why don't more articles related to Bernoulli include the equations related to acceleration perpendicular to the flow (curved streamline), such as section 3.3 (equation 3.12) in:

http://www.me.ncu.edu.tw/energy/ME381/Ch3.pdf

Mostly because it is not really germane to the discussion of Bernoulli's equation (given that that equation is not actually Bernoulli's equation). Further, conservation of energy would imply that you don't really need to know that curvature in order to calculate the same values so long as you know the velocity already. Given that, in that example, both ##\vec{v}## and ##\mathscr{R}## must be known a priori in order to calculate that normal pressure gradient, it means you have to already know the velocity field in order to actually use that equation quantitatively (since both the velocity itself and the radius of curvature of the streamlines appear explicitly). This works fine for something simple like a vortex (their tornado example) where the streamlines are known to be circular. It doesn't works so well when you don't already know those two quantities.

rcgldr
Homework Helper
Why don't more articles related to Bernoulli include the equations related to acceleration perpendicular to the flow (curved streamline)

Mostly because it is not really germane to the discussion of Bernoulli's equation (given that that equation is not actually Bernoulli's equation).

From wiki: "This also is a way to intuitively explain why airfoils generate lift forces."

http://en.wikipedia.org/wiki/Euler_equations_(fluid_dynamics)#Streamline_curvature_theorem

Based on some other articles I've read, streamline curvature is one of the factors used to calculate the velocity field in the case of some mathematical models. Seems like this would be an iterative process that converges towards some final velocity field, since the pressure gradient and velocity field are coupled / codependent.

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First: Wikipedia articles, especially on issues like lift, are susceptible to the same kind of nonsense that I discussed at length in my Insight and than NASA discusses at length in the links I posted and referenced in said article.

Second: The article you link is not, in fact, about Bernoulli's equation, but about Euler's equations. In fact, the effect of streamline curvature is inherently contained in the Euler equations, and a suitable change in coordinates to a streamline coordinate system results in the expression for stream-normal pressure gradient and its effects. So, in effect, streamline curvature is often discussed in the context of the Euler equations, just not Bernoulli's equation. As Euler's equations are considerably more complicated, they simply don't generally come up in basic-level discussions of fluid dynamics such as those relying on Bernoulli's equation.

Third: This still doesn't solve the issue of needing to know the streamlines a priori in order to actually calculate anything using this approach. You still need to know the velocity field so you can come up with the streamlines and find their radius of curvature to solve that stream-normal equation. I suppose this may be used as an iterative step in some codes (I am certainly not familiar with every code, or even all that many of them), but it is still not germane to the discussion of Bernoulli's equation or beginner-level fluid dynamics.

Fourth: I see no reason why this couldn't be used as a method of illustrating lift. However, it would need to be a fairly qualitative discussion (much like the rest of these) since it will naturally lead to the questions about knowing "how curved" the streamlines are, which is functionally equivalent to asking about how fast the air moves over an airfoil and why. The answer to these are more complicated than just "streamline curvature" or "Newton's laws" or "Bernoulli's equation."

rcgldr
Homework Helper
The article you link is not, in fact, about Bernoulli's equation, but about Euler's equations. In fact, the effect of streamline curvature is inherently contained in the Euler equations ...
I didn't mean to imply it was Bernoulli, since the wiki article is specifically about Euler equations. I provided the link to what I thought was an alternate (non-Bernoulli) explanation for the OP's question about why the velocity on the "upper" surface of a foil is faster. If streamline curvature is a bad example for the OP's questions, I can just strike through or delete my prior posts about it.