Why do curved flows accelerate?

So I’m a bit of a fluid dynamics nerd, but for a while now this has been bothering me. We know that wings produce lift in part because of the flow accelerating over the top of the curved wing, and as a result of conservation of mass and energy Bernoulli showed that this results in a drop in static pressure. But why does the flow accelerate around a curved shape? I have some vague feeling it is related to circulation around the wing (Γ) but I really can’t work it out; anyone know?

• One thing's certain: Bernoulli's Principle does NOT explain lift provided by an airfoil, grc.nasa.gov/www/k-12/airplane/wrong1.html, despite masses of people believing so.
– Gert
Jul 23, 2021 at 19:58
• That’s the equal distance fallacy. Go read what the slides say further on. You could also read what Ludwig Prandtl wrote in a NACA report in 1921. He derives an equation of lift rather beautifully. Half from integrating the pressure difference and half from impulse. Jul 23, 2021 at 20:31
• The higher speed over the curved path comes from the mass flow rates being the same. Just assume incompressible at first to get the intuition. Once the air stream splits, you have to get as much mass flow around the top as the bottom. So it has to go faster if that path is longer (longer if curved) because has to go further in same time. Think of incompressible case first. Still higher velocity even in that case as just described, and in reality, like w water. Now if also lower pressure, then lower density so even higher volumetric flow rate for that mass flow rate. Jul 27, 2021 at 2:20
• Al Brown the above web link May help there. The equal time fallacy is still the equal time fallacy mass flow rate or otherwise Jul 28, 2021 at 7:28

Maybe it would help to understand the flow around a flat plate, where the velocity of the flow is directed normal to the plate.

In the potential flow model, the fluid wraps around the plate. Why does it want to accelerate around the corners?

In reality, there is a wake behind the plate because the flow loses some energy to friction.

Why does the flow want to wrap around and occupy that space? Because there is nothing there and the fluid has pressure.

Imagine the black blob as a vacuum. Taking out a fluid blob near the vacuum, we see that there are no neighboring fluid particles towards the vacuum to supply a pressure. Thus, the pressure imbalance drives the fluid particles towards the vacuum.

To understand your wing example, now try tilting the plate and then adding some curvature.

• So it’s just the vacuum and Coanda effect? Jul 23, 2021 at 20:36
• I'm not sure. When you say, "why does flow accelerate around a curved shape," then are you referring to speed changes or directional changes in the velocity of the flow? I've tried to help explain to you the directional changes.
– Evan
Jul 23, 2021 at 20:43
• I think both. So the speed of the flow and the net velocity perpendicular to the airfoil chord increase over the top of the wing, I think if I understand what you’re saying is that if the flow simply separated at the top of the curve a low pressure area (call it a vacuum) would occur and flow would be pulled into it (although of course fluid pressure is always positive). This pressure differential causes the increase in velocity/speed Jul 24, 2021 at 21:05
• Exactly right! The first tendency is for there to be empty space behind the airfoil with no fluid material in it. But then the pressure gradients drive the flow at the top down and toward the left (if the flow is coming from the right) and into the vacuum. The flow on the bottom doesn't have such a tendency as it's harder for it to wrap around the sharp corner.
– Evan
Jul 26, 2021 at 18:08

Lets start with knowing that the fluid close to the airfoil surface will follow the contour of the airfoil, otherwise fluid would either penetrate the surface or produce a vacuum.

Based on that premise alone it is possible to infer what kinds of pressure gradients we might expect to see, and then make a guess at the nature of the velocity field.

Because the fluid follows curved paths around the airfoil, it is experiencing accelerations in various directions as it passes the airfoil. These accelerations are aligned with pressure gradients in the fluid.

In this example image the upper streamline is strongly curved near regions 2 and 3, the red arrows show the direction fluid elements must accelerate in order to follow those curves.

Knowing that fluids accelerate away from regions of higher pressure towards regions of lower pressure, we can guess that the area just below the leading edge of the airfoil is a region of higher pressure, and the area just above and behind the leading edge is a region of lower pressure.

Now that we have established which areas are expected to be above and below ambient pressure based on streamline curvature, we can infer how the fluid velocity will change along the surface of the airfoil.

Between regions $$1 \rightarrow 2$$ the pressure is increasing, and the velocity is decreasing as the fluid increases in pressure.

Between regions $$2 \rightarrow 3$$ the pressure is decreasing, the velocity is therefore increasing as the fluid accelerates from a high pressure region into a low pressure region.

The fluid then gradually slows down through regions $$3 \rightarrow 4 \rightarrow 5$$ as it approaches the trailing edge where we expect to find a weak high pressure region.

The fluid weakly accelerates between regions $$5 \rightarrow 6$$, as it recovers to ambient pressure after leaving the trailing edge.

Now let's look at the pressure field calculated with potential flow. Ambient pressure is marked green, high pressure yellow, and low pressure blue and violet.

Not a bad guess, let's take a look at the velocity field below just to be sure. Here low velocities are violet, medium velocities blue, and high velocities green and yellow.

Now we see just what we expected to see, an airfoil where fluid moves faster over the upper surface than under the lower surface.

This doesn't just apply to airfoils, any region of curved flow is accompanied by a pressure gradient, and any pressure gradient that is not perfectly parallel with the flow will be accompanied by flow curvature.