Hello and welcome to the next video in aerodynamics. Last time we discussed the sources of drag on aerodynamic bodies. Force transmits from the fluid to the body through two mechanisms.
The pressure that acts normal to the surface and shear which is parallel to the surface. Typically the drag of an aerodynamic body is mostly due to viscous forcing, although if flow separates it quickly becomes dominated by pressure drag. Today Tonight we will learn about a different source of pressure drag in our exploration of finite span airfoils and the effects that come with the wingtip.
We will explore induced drag, effective angle of attack, and downwash as well as build tools with the Biot-Savart law to explain the impact of the vortex near the foil. Let's jump in. All real wings are finite.
Eventually, nothing can have an infinite span. But most of what we've learned so far applies only to an ideal two-dimensional wing. Here, we introduce the physical effects that come with a finite span wing, and we begin to thinking about the impacts of three-dimensional reality in addition to the two-dimensional ideal case. Let's start by considering the performance of a NACA 4412 foil section. If we looked up the performance of this foil, We would see the lift coefficient and drag coefficients as functions of angle of attack.
At, say, an angle of attack of 8 degrees, we would get a lift coefficient of 1.2 and a drag coefficient of 0.0068. However, these parameters are for 2D ideal sections. What happens if we go from 2D to 3D and add in the span?
From our graphs above, we've noted the lift and drag coefficients for this foil at a given angle of attack. Notice, these coefficients have lower case letters in the subscript, lower case L and lower case D. This indicates that these are used to calculate per unit span quantities. Recall our lift equations from a much earlier video, arranged to solve for the coefficients.
The lift per unit span coefficient is calculated using the lift per unit span, and the lift coefficient is calculated using the total lift force with the addition of span in the denominator. At first it might seem like by definition these two things are the same, because lift per unit span is the same as lift divided by the span. However, there's an important distinction.
The lift and drag coefficients per unit span are not necessarily equal to the total lift and total drag coefficients. Per unit span quantities assume an ideal wing with infinite span. In a sense, no finite wing effects. The lift and drag only come from 2D sources.
However, in real cases the span of the wing is not infinity, and there are some interesting end effects that occur. It's these end effects that cause the lift coefficients to be different to the lift per unit span coefficients. Today, our focus will be on the tip vortex and the downwash it produces, which leads to a change in the effective angle of attack and adds induced drag.
Again, let's consider our foil from before. The foil produces both lift and drag. Specifically, in order to create lift there is a low pressure region on top of the foil and a relatively higher pressure region at the bottom. When the wing is finite, it has outer tips or edges. When there is a pressure difference across the surface, near the edges the flow will want to leak from the higher pressure to the lower pressure.
So, The flow rolls around the tips and escapes to the top. This rolling motion creates something called a tip vortex. The flow continuously rolls from underneath to the top during flight and creates a trailing vortex in the wake of the wing.
Interestingly, this vortex now has influence over the foil itself. Consider a section on the foil near the tip. In the ideal sense, the foil is at an angle of attack at some forward velocity.
However, we now have a pesky tip vortex in our vicinity, and this vortex induces a downwash, or a downward vertical velocity on our foil. Just look at the orientation of the vortex, and the flow rolling up and over. It's clear that near this vortex we get a vertical velocity component, also called induced velocity.
and that induced velocity is downwards. This induced velocity is called a downwash. Now, we add this vertical velocity V with the freestream velocity component, and the resultant total velocity vector has an angle of alpha I with respect to the travel direction. This induced angle alpha I from the downwash actually works to decrease the effective angle of attack of the foil. Additionally, the lift force generated is now perpendicular to this new total velocity vector, and no longer perpendicular to our direction of motion.
If we stay in the reference frame of the direction of motion, this acts to decrease the vertical lift slightly and adds a new drag in the direction of travel. This drag is called the induced drag. It's because the downwash tilts our lift vector and turns some of it into drag. Ultimately, downwash does two critical things.
First, it changes the effective angle of attack locally, meaning it changes the expected lift performance we should be getting. Second, it tilts a portion of the lift in a way that induces the new drag on the foil and we lose lift. Both of these things, decreased lift and added drag, work to hurt our foil performance. So, it's safe to say that downwash is typically bad. Now we can consider this among our other force producing components.
A finite span foil has drag from three sources. Skin friction comes from viscous forcing. Separation, which ultimately happens because of the boundary layer and viscous things, is actually a pressure drag.
And now we have induced drag, a second form of pressure drag that comes from the lift tilting. All this downwash also leads to something called lift distribution. Across our span, our lift per unit span varies where it decreases near the tips and maxes out as far from the tips as we can get. This is because near the tips the flow is allowed to travel from the bottom to the top and it balances out the pressure so there's no longer a large pressure difference from the bottom and top of the foil.
And what we know from Kuda Joukowsky is that the lift per unit span varying across the span means the circulation also varies across the span, which will become important in the next video. Downwash isn't the only thing that causes lift distribution. A lot of the time, lift distribution is purposefully designed into the wing.
Most commonly, we see this in the variation of the cord length with the span. Anything other than a rectangular foil has a varying cord, which covers most aircraft, meaning a non-uniform lift distribution. Second is geometric twist.
Literally twisting the foil in a way that the angle of attack changes with the span. This change in angle leads to variation in lift. Lastly, there is something called aerodynamic twist.
This is where the aerodynamic characteristics of the foil change with the span, meaning the starting foil shape might be different than the leading edge foil shape. This is common among more modern and complex aircraft. Everything in this video so far has been due to one physical phenomena, the tip vortex. The tip vortex represents a straight, semi-infinite vortex, and it will help us to build some tools for aerodynamic analysis of vortices moving forward. For that, we call on the Biot-Savart law, which will hopefully tell us something about the semi-infinite vortex and its downwash.
Let's consider an arbitrary vortex filament. Label a point P some distance off of the filament. This point feels an induced velocity due to the neighboring vortex. Specifically, we get a delta V due to a segment of the filament, delta S, when point P is distance r from the vortex.
To describe this velocity induced, we can use the Biot-Savart law, which covers this. It's defined as follows. The induced velocity increases with the vortex circulation, or the strength of the vortex, and decreases with increased distance from the vortex.
This law is actually common even outside of aerodynamics and works in general. Specifically, you might see it in fields like electromagnetism. Now let's apply this equation to something like a real vortex. Let's say we restrict our vortex to be straight only, and extend it from positive to negative infinity.
To get the total induced velocity, we need to take Biot-Zawart and add up all the vortex segments acting on our point of interest, point P. This is done by integrating the equation from negative to positive infinity. But, before moving forward, we need to make some geometric changes to the equation.
Let's draw a diagram of what's happening. We have our straight vortex and point P off to the side. Technically, point P is distance r away from the segment at angle theta. Let's also mark a different distance, h, which goes from point P and connects to the filament at a 90 degree angle.
Back to the equation. We will make two simplifications. The definition of a cross product between two vectors is applied. And we note that we only care about the magnitude of the velocity induced. This simplifies our equation slightly.
Let's bring up our angle diagram. Our goal is to change the r and s into h and theta. For that, we apply trigonometry. This lets us define the r, s, and ds as a function of theta and h.
Also, we want to change our bounds to be from minus infinity to infinity into the angle, which means that the angle goes from pi to zero respectively. Add this and the trigonometry equations into our main equation, and we get a relatively simple integral to solve. This leaves us with the equation for the induced velocity for an infinite vortex, gamma over 2 pi h.
This means that the induced velocity increases linearly with vortex strength, and decreases linearly with the distance away. This is the vertical velocity you would feel if you were standing distance h away from an infinite vortex. Maybe you're standing in the runway of an airport, after an aircraft goes by and is some distance away. But, But we don't want to stop here.
The tip vortex has a starting point that's finite and does not extend between infinity and negative infinity. This vortex goes from some finite location to infinity and is called a semi-infinite vortex. Our analysis is the same, we just change our integration bounds to go from pi to pi over 2, effectively cutting our window in half. And it's all done.
we have the equation for the induced velocity of a semi-infinite vortex, which is half that of the infinite vortex. This is the velocity that a point along the foil span, which is distance h from the tip, feels due to the vortex presence. This equation will be integral in our calculation of the induced velocity and downwash effect. And, while we're thinking about vortices, this is a good spot to point out that vortex filaments have rules.
First, the vortex filament strength is constant along the filament. This is also interpreted as the vortex filament strength is not changing in time without external forcing. Second, a vortex cannot just end at any point in a fluid. A vortex can either terminate at some sort of solid boundary, like our wingtip, or it can connect to itself and form a closed loop.
This is called a vortex ring when it closes. Lastly, if there is nothing to externally cause rotation, an irrotational flow stays irrotational. This is something we've talked about in the past when we discussed rotational flows at length. These three rules are theorems, and they're called the Helmholtz vortex theorems.
Although they're technically not laws, just theorems, they have been largely shown to be true through measurement and observation. Moving forward. we're going to take our knowledge of vortex filaments and design ways to predict and describe the effect of downwash. This includes induced drag due to the tilting of the lift vector, the new total lift of the wing so we know if we'll stay in the air, and the lift distribution along the span.
We'll work to be able to predict these in the future for finite span wings. With regards to these finite wing effects, In practice, they are super important because all physical aircraft have a finite span. The tip-vortex pair that comes off of an aircraft has a tremendous influence on the flow after the plane goes by, and dictates the frequency that the aircraft can take off and land at airports, because they want to be outside of the wake of the aircraft before it. Secondly, it has led to technology innovation to avoid this tip-vortex. Because generally end effects are bad, so we want to avoid them.
We now have wingtips and end plates to stop this rolling velocity leakage. Last, it leads us to strive for high aspect ratio wings when we can, because higher aspect ratio wings have a lesser impact from the tip and lead to more efficient flight. And that's it! Let's review.
We started by introducing why ideal 2D flows are different from real 3D wings in terms of performance. The FiniteSpan foil has flow spill over the edges due to the pressure difference, creating the tip vortex. The tip vortex induces downwash on the foil, leading to changes in the angle of attack and added drag. Using the Biot-Zavar law, we derive the induced velocity equation for the semi-infinite vortex that represents the tip vortex, so that we can now calculate the downwash from relatively known quantities.
And we finished with a discussion of all of the practical impacts of the tip vortex. I hope you enjoyed the video, and thanks for watching.