Bernoulli's equation, different streamlines, and wings

[fog37]

Hello Everyone,

Bernoulli's equations expresses the conservation of mechanical energy for a particular fluid parcel moving inside a time-independent flow. The parcel is restricted to move and remain along a particular streamlines. The sum of the trinomial is equal to a constant on every different streamline but the various constants are different. In the case of a flying airfoil (wing), we assume that these constants are all the same for the incoming flow and for the curved flow that passes around the wing. Why? What allows us to apply this assumption and considered all the constants equal to each other?

When explaining how a wing flies, both Newton's 3rd law (action reaction) and Bernoulli's principle are applicable and correct but Bernoulli can only be applied outside of the boundary layer where the fluid is considered perfectly inviscid, correct? Newton's law gives a conceptual explanation (air is deflected downward providing a reaction upward force and and backward direction force (drag). Nonetheless, Bernoulli's equation still provides the correct pressure distribution to get the correct lift. Is it because the boundary layer is so thin that we can ignore its presence and therefore apply Bernoulli's equation to obtain the correct pressure distribution around the wing?

Thanks,
Fog37

 

[boneh3ad]

In the case of a flying airfoil (wing), we assume that these constants are all the same for the incoming flow and for the curved flow that passes around the wing. Why? What allows us to apply this assumption and considered all the constants equal to each other?

They all originate from a common reservoir (the far field upstream of the airfoil) where the Bernoulli constant is constant is uniform throughout. Consider that, by definition, $p$, $\rho$, and $V$ are all constant upstream and the hydrostatic term is generally considered negligible since the density of air is small and the height change is also small.

Bernoulli can only be applied outside of the boundary layer where the fluid is considered perfectly inviscid, correct?

This is correct.

Nonetheless, Bernoulli's equation still provides the correct pressure distribution to get the correct lift.

This is true provided you have some means of determining the velocity distribution.

Is it because the boundary layer is so thin that we can ignore its presence and therefore apply Bernoulli's equation to obtain the correct pressure distribution around the wing?

One of the key features of boundary layers is that the pressure gradient normal to the surface is zero throughout them. In other words, the pressure is the same at the wall as it is at the boundary-layer edge. If you can therefore calculate the pressure just outside the boundary layer at the edge, you have calculate the pressure at the surface as well.

 

[fog37]

Thanks boneh3ad,

a) You mention that Bernoulli's equation can provide the pressure distribution around the wing provided we have some means of determining the velocity distribution. When we see those colorful simulations of the velocity and pressure fields around airfoils, how was the velocity distribution numerically determined to later calculate from it the pressure distribution? From the quantitative standpoint, Bernoulli's equation is what is used in calculation. Newton's 3rd law cannot really used directly.

The incorrect lift explanation based on Bernoulli's equation explains that the air on top and on bottom must meet again a the trailing edge. Where did this erroneous fact come from? From the fear that a vacuum could happen if the air did not meet? We know that no vacuum shows up. Why? Is it simply because air can travel at any speed it wants?

b) It is always hard to pin point cause-effect relationships but in the case of a wing, could we say that the velocity field determines in this case the pressure field? Or does it never make sense to try to find a cause and an effect?

c) As far as pressure gradients: I know there is a positive pressure gradient ( pressure increases in the direction opposite to the direction of flight). This positive gradient is advantageous since it would seem to help the forward propulsion. For the boundary layer, there is also a negative pressure gradient which is responsible of separating the boundary layer from the wing surface, correct? What about the resistive drag that hinders the plane motion forward? Which pressure gradient is responsible for that?

There is also a transverse pressure gradient. On top of the wing, far from it, the atmospheric pressure is atmospheric $P_{atm}$. As we move from above the wing vertically and closer to the wing, the pressure decreases (pressure slightly less than atmospheric). The opposite happens below the wing. All this causes the overall lift. You mention that within the boundary layer the pressure is constant (zero gradient)...why? "If you can therefore calculate the pressure just outside the boundary layer at the edge, you have calculate the pressure at the surface as well." That is helpful since we can just calculate the pressure in the inviscid fluid right above the boundary layer and not worry about the viscous effects within the layer...

Thank you for the great help.

 

[boneh3ad]

When we see those colorful simulations of the velocity and pressure fields around airfoils, how was the velocity distribution numerically determined to later calculate from it the pressure distribution? From the quantitative standpoint, Bernoulli's equation is what is used in calculation. Newton's 3rd law cannot really used directly.

From a quantitative standpoint, Bernoulli's equation is used in simple calculations such as converting velocity fields into pressure fields. If you want to do actual detailed computations, you need something a bit more sophisticated. On the low fidelity end, you could calculate the velocity field using potential flow theory via vortex panel methods (though you'd want some degree of viscous correction in there). You can get a little more complete by solving the Euler equations, which are still inviscid and would need some kind of correction near the wall. Most modern CFD, though, is performed by solving some modeled form of the Navier-Stokes equations. They are complicated and costly to solve, but they take into account all known phenomena. Which model is chosen depends on the needs of the project.

The incorrect lift explanation based on Bernoulli's equation explains that the air on top and on bottom must meet again a the trailing edge. Where did this erroneous fact come from?

I suspect it comes from the fact that fluid dynamics is a very complex subject, but this is a very neat, tidy answer. People like easy answers. Sadly, it is in no way based on reality.

From the fear that a vacuum could happen if the air did not meet? We know that no vacuum shows up. Why? Is it simply because air can travel at any speed it wants?

Let's flip that around. Why would a vacuum appear? There's nothing in the physical rules that govern fluid flows that states that a vacuum would appear along a slip line where velocity is discontinuous.

It is always hard to pin point cause-effect relationships but in the case of a wing, could we say that the velocity field determines in this case the pressure field? Or does it never make sense to try to find a cause and an effect?

It still doesn't make any sense. You could argue that you need the velocity first to calculate the pressure. I could turn back around and argue that the velocity needs a pressure gradient to accelerate or decelerate. Cause and effect makes no real sense here (and it serves no real purpose, in my opinion).

I know there is a positive pressure gradient ( pressure increases in the direction opposite to the direction of flight).

To be honest, I am not sure what you mean by this, but the sign of the pressure gradient changes over different parts of an airfoil. It is negative near the front end where air accelerates and positive near the back end where air is decelerating.

This positive gradient is advantageous since it would seem to help the forward propulsion.

Propulsion comes from the engines of the plane, not the pressure distribution over the wings.

or the boundary layer, there is also a negative pressure gradient which is responsible of separating the boundary layer from the wing surface, correct?

No, a negative pressure gradient is what we call favorable. Negative pressure gradients have a pressure that decreases in the direction of flow, meaning the net pressure force tends to accelerate the flow and will help prevent separation. Positive pressure gradients are adverse and do the opposite.

What about the resistive drag that hinders the plane motion forward? Which pressure gradient is responsible for that?

No pressure gradient is responsible, per se. Drag due to the pressure field has to do with any region where there is a net pressure difference between the front and back of the airfoil. This is identically zero for inviscid flows, but viscous flows can have nonzero pressure (or form) drag. Consider that if you have a ball with a higher pressure on the front than on the back, it will experience a net backward force. This is no different on a wing. Of course, the pressure is not constant on the front and back or top and bottom, so this is an integrated effect.

Drag can also come from skin friction due to viscosity rather than pressure. Actually, this is a pretty major source of drag in most cases. For fast-moving airfoils (transonic or higher) there is also wave drag to consider, which is a pressure-related drag that forms due to the existence of shock waves.

You mention that within the boundary layer the pressure is constant (zero gradient)...why?

I am not sure I have a great answer for your why question here. Maybe you could call it one of those wonderful conveniences of nature. Perhaps the best explanation I can give is that viscosity retards the streamwise velocity but does nothing to wall-normal velocity, so it shouldn't affect the wall-normal pressure gradient. Further, the wall-normal velocity is identically zero at the wall, and since boundary layers are confined to being very close to the wall, the wall-normal velocity stays very close to zero, so the pressure gradient in that direction should be very close to zero. I should mention that it is not necessarily identically zero, but is, in practice, so close to zero that it is generally ignored. In fact, it is one of the key assumptions used to derive the boundary layer equations that are used ubiquitously to study these problems.

 

[fog37]

Thank you! Very detailed and instructive questions.

• I just read that Euler equation is essentially the Navier-Stokes equation with zero viscosity (inviscid fluid). Bernoulli's equation derives from Euler's equation but assuming a time-invariant flow field. Does the nonzero viscosity term make NS much more difficult to solve than Euler's equation? Or does it only really pose a difficulty just for analytical problems? Maybe computers, with enough time, can solve any type of NS problem by just cranking numbers.

• One clarification on the boundary layer and its separation. To my limited knowledge, the boundary layer is that region of space close to the wing' top surface (I guess there is a boundary layer also at the bottom surface)?) where the viscous effects are present and there is a velocity variation from 0 up to free stream velocity). The streamlines inside the boundary layer are still lines but curved (laminar boundary layer). The phenomenon of "boundary layer separation" implies that a region of flow with velocity gradient and streamlines continues to exist but at one point on the wing's surface it "departs" upward and takes off in the surrounding air. From the separation point on, the air becomes highly vorticous and turbulent. In a sense, the boundary layer passes from being laminar boundary layer to becoming a turbulent boundary layer. The separation point is somewhere towards the trailing edge of the wing. Ideally, the separation never happens. Once it separates and turbulence forms in the back region of the wing, the wing starts experiencing extra drag. So boundary layer separation is never a good thing. Is this a correct basic physical interpretation of the boundary layer separation?

On the bottom surface of the wing, the situation is surely not same as above.