Lifting Line Theory

Hello and welcome to the next video on aerodynamics. Last time we discussed the reality of finite wings. The wing tip produces a lot of extra features and losses that we didn't see with two-dimensional foils, and generally the performance is hindered by reality. This included the tip vortex, which induced a downwash over the foil, changing the effective angle of attack, and adding a new drag. Additionally, we brought in the Beowtsevar law to learn how to estimate the downwash near an idealized line vortex. Today we'll be discussing Prandtl's Lifting Line Theory. Our goal is to predict these finite wing losses, which is successfully done by estimating a foil with trailing edge vortices as a horseshoe vortex with no surface. Today we'll derive the method, discuss its solutions, and practical importance. Let's jump in. Lifting line theory was another development by Prandtl near the time of World War I. It was first started in 1911 and is still used in modern design analysis today. This lifting line theory was an analytical approach to finite wings and is a predictive model for performance in losses. We get three major things from this theory. First, the lift distribution. or how the lift varies along the span of the wing. We get the total lift, or how much upward total force the wing produces, and we get the induced drag, or how efficient our wing is at moving forward. In essence, you will see similar flavors to how these types of models were done in the past. We will avoid surfaces at all costs and use a combination of the Biot-Zavar law to explain the behavior of semi-infinite line vortices. An essence of elementary flows, where we remove the concept of the surface and replace it with a vortex, and the Kutajkowski theorem, where we relate the circulation of a vortex to the lift. Say you have a finite wing in a flow field using a Cartesian coordinate system. The wingspan is s, and the span goes from minus s over 2 to s over 2. It produces lift and drag. and at the wingtips you have vortices. Prandtl's idea here was to remove the surface entirely. We can model the tip vortices as a semi-infinite line vortex. However, the Helmholtz vortex theorem tells us that these vortices cannot just start out of nowhere. But instead of using a surface like the wing, we're going to connect them with something called a bounded vortex. Here, the vortex is called a bound vortex because it is bound. or represents a boundary replacement. The tip vortices are called free trailing vortices because they exist and follow the flow. This connection of vortices which looks somewhat u-shaped is commonly referred to in fluid dynamics as the horseshoe vortex. Let's examine this horseshoe vortex more closely. Here we label the two trailing vortices as vortex 1 and vortex 2. These two vortices induce a vertical velocity field that varies in the spanwise direction, z. The induced velocity field in this case would look curved, where it would be smallest at the center, z equals zero, and would blow up near the edges. Recall the Biot-Zavara law we introduced in a previous video. The vertical velocity induced by a forfe... vortex was a function of the vortex strength, gamma, and how far you were from the vortex, h. Let's use this to describe the vertical velocity in this case where we're between two trailing vortices. In this case there are two terms, one from each vortex. This simplifies down to an expression that is a function of the strength of the vortices, the span, and the z location. However, At close inspection we might notice a bit of a problem. As the z-location approaches the tips, s over 2, the denominator goes to zero and the vertical velocity blows up. This isn't okay for this analysis, so we'll need to adapt. What if we tried more horseshoe vortices? You might say to yourself, hey, this no longer looks like what goes on behind a foil where there are just two main vortices at the tips. However, In this case you'd be a bit wrong. In reality, downstream of a foil is a vortex sheet that is shed at the trailing edge of the foil along the span. and those tend to roll up into littler vortices between the major ones. So, in essence, by adding in multiple horseshoe vortices throughout the wake, we're doing better at following reality. And it's also convenient because it will eventually fix our problem. So now we add up a few finite number of horseshoe vortices. In this case, three. Each horseshoe vortex has two trails with strengths d-gamma. However, The bound vortex now behaves a bit different. Instead of having one constant strength, it has a strength based upon how many horseshoes are overlapping. At the very edges, we have a strength of d gamma sub 1. One step inwards, we add the strength of the next vortex. Finally, in the middle, we sum up all three strengths. Our strength of the bound vortex is now a function of the span. Let's look at it from the front view. The strength varies in stepwise fashion, with a maximum at the center and lesser at the edges. If we had many steps, you might see how this could fix our problem. We can control it so that the strength of the bound vortex goes to zero at the edges and stops our vertical velocity induced from exploding. Great! In this case we had three horseshoe vortices. But what happens if we bring in a little bit of calculus and add in infinite infinitesimally small horseshoe vortices. Let's sketch it out, doing our best to draw infinity vortices. Each of these trailing vortices contributes to the distribution of the vortex strength, gamma, at the bound vortex, which is a continuous function of z. To derive the formula for the induced velocity at the bound vortex along z, let's start by considering a single vortex segment, d-gamma. This vortex segment is at the location z. and occupies space dz. Now, we consider the induced velocity by this segment at some other point along the span, which we'll note here is z sub zero since z is already taken. And, we're interested in the segment of the vertical velocity contributed by only this vortex segment. So, the equation for this single segment is written as follows. If we want the entire induced velocity distribution, v, we have to add up all of our vortex segments. This integral represents the downwash due to a strength distribution, gamma, which is still unknown. Next, we can use this to give us an expression for the induced angle of attack. Remember, this alpha sub i is the shifted angle of attack due to the new downwash. If we assume small angles, the induced angle of attack is just the negative of the downwash divided by the freestream velocity. Plugging our v expression into this gets a similar expression for the induced angle of attack due to the downwash distribution. And, like before, we still don't know gamma. Last, we also might want to consider the effective angle of attack. This is the actual angle of attack that the foil feels, with a mixture of the set original angle of attack alpha, and the unexpected downwash making alpha sub i. So the effective angle of attack is just the set angle of attack subtracting the induced. We can take this analysis a few steps further by considering some of the things we've learned in the past. In thin airfoil theory, we found that most relatively thin foils had a lift slope of dCl dAlpha equaling 2π, regardless of the camber. Plotting it, you would see something like this. The slope, before separation occurs, is at or near 2π in a perfect world. We can take this, separate variables, and get an expression for Cl as a function of alpha. There is also an offset that's possible here, specifically if our foil is cambered. So in the parentheses we have the effective angle of attack subtracted by the angle of attack for zero lift, which is generally known going in based on what FOIL profile you used. Let's use the lift equation to get lift per unit span as a function of this lift coefficient, and Kuda-Joukowsky tells us that the lift per unit span is a function of the circulation. Using all this information, we can rearrange to get the lift coefficient as a function of the circulation. We plug this back into the original equation for ZL, and we find the effective angle of attack as a function of the bound vortex strength distribution, gamma, which is still unknown. After all this, you might be seeing a trend. It'd be really helpful if we knew this gamma distribution. And if we're creative, we can't derive an expression for it. Right now, we know the effective angle of attack as a function of gamma, and we also know the induced angle of attack as a function of gamma. We do know that the original set angle of attack, alpha, is a function of the effective and induced angle. It's just the sum. What's nice is that, since we know alpha, it's our set flying condition by the pilot or controller, and since we know the other two terms as functions of gamma, we can write out a full expression for the known angle of attack as a function of gamma. Let's do that out now. This is called the fundamental equation for Prandtl's lifting line theory. It describes the angle of attack that's set, which is known, as a function of gamma and unknown. So effectively, it gives us a route to solving for gamma. In this expression, there are a number of spanwise functions that we should identify. Alpha is known, and it's possibly a function of z if there is geometric twist in our finite wing. The chord is also known, and also possibly a function of z if there's a chord variation, like wing tapering. The angle of attack with zero lift is also possibly a function of z for modern wings. If the profile shape is changing along the span using aerodynamic twist, that changes this value along the span. And this technically has one unknown gamma, which we can solve for, though it's a differential equation, so it won't be that easy. But once we've figured out the gamma distribution, we can solve for the lift distribution using the Kutta-Joukowsky theorem. We can integrate this lift distribution to get us the total lift on the wing as a function of gamma, and similarly we can find the induced drag on the wing as a function of gamma and the induced angle of attack. These three properties are super important during flight because they describe the forces the foil feels due to these finite wing effects. Let's try this out and see how it works. There's a bit of math in the next segment, so we'll rush through a little bit here. Don't get discouraged if you miss a step. Consider a wing that has an elliptical circulation distribution along the span. The equation would look something like this, where we're leaving it arbitrary so it has some peak gamma sub zero. The distribution, sketched out, would appear like this along the span, peaking in the center and zero at the edges. Straight away, we can calculate the lift distribution, L prime. To calculate the total lift, we will have to work a bit harder. Write out the equation with the integral. Here, we're going to deploy a tool we used in thin airfoil theory. It's more convenient to work in the theta space. where we define a circular coordinate system that projects onto the span. The edges go from 0 to pi instead of plus minus s over 2, and theta defines a point along the foil. With this transformation, z turns into s over 2 times cosine theta, and dz is s over 2 sine theta d theta. We'll use this coordinate transformation throughout our derivations in the next steps. and it usually produces convenient solvable integrals. When we apply this to the lift equation we can solve out the integral and get our final expression for lift. To get to the induced drag first we're going to need to solve for a number of things. We need the induced angle of attack and it will also be convenient for us to solve for the downwash function, v. Here the downwash function is defined by the integral of the slope of the gamma distribution which we can solve for directly. Plug this into the integral and we get the following expression. Let's use our coordinate transformation again so we get some simple solvable integrals. Now we have an integral with variable theta and dummy that variable theta sub zero. You might recognize this. We ran into this exact integral with our thin airfoil theory. Fortunately for us, math saves the day and we know the answer to this integral directly as a function of dummy variable theta sub zero. Plug this solution in and we get the expression for the downwash, which we save for later. Notice, this expression is not a function of the span. The downwash is constant, an interesting feature of an elliptical gamma distribution. Next, we consider the induced angle of attack, which is just the downwash that we know divided by the freestream velocity. Since we've already solved for the lift as a function of gamma sub zero above, we can work to get to the induced angle of attack as a function of the lift. First, find gamma sub zero as a function of lift. Then, use the lift equation to get it as a function of the lift coefficient. Plug this back into the induced angle of attack expression. You might recognize the ratio of chord over span as the inverse of the aspect ratio. Technically, aspect ratio is the span squared divided by the planform area of the foil. This accounts for wings of all shapes and sizes. However, here we can turn chord divided by span into the aspect ratio. This gives us our final form of the induced angle of attack. We need this for the induced drag. Write out the induced drag equation from above. This can simply be turned into the induced drag coefficient using the drag equation. Notice that the induced angle of attack is not a function of the span, so we can pull it out of the integral. Use our fancy coordinate transformation again, and we get easily solvable integrals. This gives us the induced drag coefficient that has gamma sub zero, and the induced angle of attack in it. Plugging in what we know about these two variables, we get a simplified expression for the induced drag coefficient. Let's pause here a moment and note some interesting things about this expression, because the induced drag is very important to aerodynamics. First, the induced drag increases dramatically with lift. It goes as lift squared. It's an interesting feature. The lighter you are, the more efficient you can be. Second, the induced drag goes down with aspect ratio. This drives wings to be high aspect ratio, chasing efficient flight. Now, you might be wondering why we started with this specific example. And it wasn't by accident. First and foremost, the elliptical gamma distribution is important to aerodynamics because it represents the most efficient distribution. meaning it produces the lowest induced drag of any other distribution of gamma. If you were to go back and solve everything out generally for arbitrary solutions of gamma, you would find that the induced drag equation looks a lot like the one for the elliptic distribution, but with a span efficiency factor e in the denominator. This e value can only go up to 1, so the fact that the elliptical distribution produces an e of 1 means it is as low as it gets. Second, elliptical distributions are not hard to achieve. An elliptical planform will get you exactly this, and specific tapered wing angles get you really quite close. But, let's say you didn't have an elliptical distribution, and we wanted to realistically solve for the fundamental equation for lifting line theory above. What you would do is the fancy mathematical strategy of guess and check. First, pick a gamma distribution. Probably an elliptic distribution is a good starting point. From the strength distribution, you can calculate the induced angle of attack. And then, you can get the effective angle of attack. From the effective angle of attack, you can estimate the lift coefficient distribution as a function of z. You can turn the lift coefficient into the lift per unit span, and then use Kuda-Joukowsky to get a new gamma distribution. Check that gamma against your guess. If it's different, you can repeat steps 1 through 5 with the new gamma distribution, and you'll repeat them until you get convergence. This will settle on the real gamma value for your wing, which you can then get your lift and drag characteristics from. It turns out this theory works fairly well for a wide range of applicable foils. It's good to use if you have a straight wing with moderate to high angles of attack. And this covers a surprising amount of aircraft today. However, if you have a more aggressive aircraft, it might not be so useful. Low aspect ratio foils are no good here. Also, highly swept wings are no good for this analysis, along with the infinites delta wing. For these types of foils, you might need to employ more modern numerical techniques. You could use something like the lifting surface theory and the vortex lattice method, but we can't get into that in this video. However, Lifting Line Theory is still a powerful tool despite these limitations. In practice, you will find that the Lifting Line Theory is a good first design estimate for the wings you will be working with. The result of this theory drives design. It pushes wings to have higher aspect ratios, chasing that efficient flight. It also drives planform shape, with elliptic and tapered wing designs. Conveniently. These wing shapes also happen to be structurally superior, which is a nice benefit. They can be both structurally best and most efficient. And that's it! Let's review. We started by introducing the lifting line theory. It is an idea that is centered around estimating a foil and the tip vortices as a horseshoe vortex. However, We use many small horseshoe vortices so that we have controllable vortex strength distribution at the bound vortex. This led us to the fundamental equation for lifting line theory, where we could technically solve for the gamma distribution. We tried with an elliptic strength distribution. where we solve for the ellipse distribution, the lift, and then the induced drag. Interestingly, the elliptic distribution is the most efficient in aerodynamics. To solve for arbitrary gamma distributions, we need to follow a set of iterative steps to converge to a solution, effectively mathematical guess and check. We finished by exploring the limitations of the theory and how it's applied today. I hope you enjoyed the video, and thanks for watching.

Today we'll be discussing Prandtl's Lifting Line Theory. Our goal is to predict these finite wing losses, which is successfully done by estimating a foil with trailing edge vortices as a horseshoe vortex with no surface. Today we'll derive the method, discuss its solutions, and practical importance.

Let's jump in. Lifting line theory was another development by Prandtl near the time of World War I. It was first started in 1911 and is still used in modern design analysis today. This lifting line theory was an analytical approach to finite wings and is a predictive model for performance in losses. We get three major things from this theory. First, the lift distribution.

or how the lift varies along the span of the wing. We get the total lift, or how much upward total force the wing produces, and we get the induced drag, or how efficient our wing is at moving forward. In essence, you will see similar flavors to how these types of models were done in the past. We will avoid surfaces at all costs and use a combination of the Biot-Zavar law to explain the behavior of semi-infinite line vortices. An essence of elementary flows, where we remove the concept of the surface and replace it with a vortex, and the Kutajkowski theorem, where we relate the circulation of a vortex to the lift.

Say you have a finite wing in a flow field using a Cartesian coordinate system. The wingspan is s, and the span goes from minus s over 2 to s over 2. It produces lift and drag. and at the wingtips you have vortices.

Prandtl's idea here was to remove the surface entirely. We can model the tip vortices as a semi-infinite line vortex. However, the Helmholtz vortex theorem tells us that these vortices cannot just start out of nowhere.

But instead of using a surface like the wing, we're going to connect them with something called a bounded vortex. Here, the vortex is called a bound vortex because it is bound. or represents a boundary replacement.

The tip vortices are called free trailing vortices because they exist and follow the flow. This connection of vortices which looks somewhat u-shaped is commonly referred to in fluid dynamics as the horseshoe vortex. Let's examine this horseshoe vortex more closely.

Here we label the two trailing vortices as vortex 1 and vortex 2. These two vortices induce a vertical velocity field that varies in the spanwise direction, z. The induced velocity field in this case would look curved, where it would be smallest at the center, z equals zero, and would blow up near the edges. Recall the Biot-Zavara law we introduced in a previous video.

The vertical velocity induced by a forfe... vortex was a function of the vortex strength, gamma, and how far you were from the vortex, h. Let's use this to describe the vertical velocity in this case where we're between two trailing vortices.

In this case there are two terms, one from each vortex. This simplifies down to an expression that is a function of the strength of the vortices, the span, and the z location. However, At close inspection we might notice a bit of a problem. As the z-location approaches the tips, s over 2, the denominator goes to zero and the vertical velocity blows up. This isn't okay for this analysis, so we'll need to adapt.

What if we tried more horseshoe vortices? You might say to yourself, hey, this no longer looks like what goes on behind a foil where there are just two main vortices at the tips. However, In this case you'd be a bit wrong.

In reality, downstream of a foil is a vortex sheet that is shed at the trailing edge of the foil along the span. and those tend to roll up into littler vortices between the major ones. So, in essence, by adding in multiple horseshoe vortices throughout the wake, we're doing better at following reality. And it's also convenient because it will eventually fix our problem. So now we add up a few finite number of horseshoe vortices.

In this case, three. Each horseshoe vortex has two trails with strengths d-gamma. However, The bound vortex now behaves a bit different.

Instead of having one constant strength, it has a strength based upon how many horseshoes are overlapping. At the very edges, we have a strength of d gamma sub 1. One step inwards, we add the strength of the next vortex. Finally, in the middle, we sum up all three strengths. Our strength of the bound vortex is now a function of the span. Let's look at it from the front view.

The strength varies in stepwise fashion, with a maximum at the center and lesser at the edges. If we had many steps, you might see how this could fix our problem. We can control it so that the strength of the bound vortex goes to zero at the edges and stops our vertical velocity induced from exploding. Great! In this case we had three horseshoe vortices.

But what happens if we bring in a little bit of calculus and add in infinite infinitesimally small horseshoe vortices. Let's sketch it out, doing our best to draw infinity vortices. Each of these trailing vortices contributes to the distribution of the vortex strength, gamma, at the bound vortex, which is a continuous function of z.

To derive the formula for the induced velocity at the bound vortex along z, let's start by considering a single vortex segment, d-gamma. This vortex segment is at the location z. and occupies space dz.

Now, we consider the induced velocity by this segment at some other point along the span, which we'll note here is z sub zero since z is already taken. And, we're interested in the segment of the vertical velocity contributed by only this vortex segment. So, the equation for this single segment is written as follows. If we want the entire induced velocity distribution, v, we have to add up all of our vortex segments.

This integral represents the downwash due to a strength distribution, gamma, which is still unknown. Next, we can use this to give us an expression for the induced angle of attack. Remember, this alpha sub i is the shifted angle of attack due to the new downwash. If we assume small angles, the induced angle of attack is just the negative of the downwash divided by the freestream velocity.

Plugging our v expression into this gets a similar expression for the induced angle of attack due to the downwash distribution. And, like before, we still don't know gamma. Last, we also might want to consider the effective angle of attack.

This is the actual angle of attack that the foil feels, with a mixture of the set original angle of attack alpha, and the unexpected downwash making alpha sub i. So the effective angle of attack is just the set angle of attack subtracting the induced. We can take this analysis a few steps further by considering some of the things we've learned in the past.

In thin airfoil theory, we found that most relatively thin foils had a lift slope of dCl dAlpha equaling 2π, regardless of the camber. Plotting it, you would see something like this. The slope, before separation occurs, is at or near 2π in a perfect world.

We can take this, separate variables, and get an expression for Cl as a function of alpha. There is also an offset that's possible here, specifically if our foil is cambered. So in the parentheses we have the effective angle of attack subtracted by the angle of attack for zero lift, which is generally known going in based on what FOIL profile you used.

Let's use the lift equation to get lift per unit span as a function of this lift coefficient, and Kuda-Joukowsky tells us that the lift per unit span is a function of the circulation. Using all this information, we can rearrange to get the lift coefficient as a function of the circulation. We plug this back into the original equation for ZL, and we find the effective angle of attack as a function of the bound vortex strength distribution, gamma, which is still unknown. After all this, you might be seeing a trend. It'd be really helpful if we knew this gamma distribution.

And if we're creative, we can't derive an expression for it. Right now, we know the effective angle of attack as a function of gamma, and we also know the induced angle of attack as a function of gamma. We do know that the original set angle of attack, alpha, is a function of the effective and induced angle.

It's just the sum. What's nice is that, since we know alpha, it's our set flying condition by the pilot or controller, and since we know the other two terms as functions of gamma, we can write out a full expression for the known angle of attack as a function of gamma. Let's do that out now. This is called the fundamental equation for Prandtl's lifting line theory. It describes the angle of attack that's set, which is known, as a function of gamma and unknown.

So effectively, it gives us a route to solving for gamma. In this expression, there are a number of spanwise functions that we should identify. Alpha is known, and it's possibly a function of z if there is geometric twist in our finite wing.

The chord is also known, and also possibly a function of z if there's a chord variation, like wing tapering. The angle of attack with zero lift is also possibly a function of z for modern wings. If the profile shape is changing along the span using aerodynamic twist, that changes this value along the span.

And this technically has one unknown gamma, which we can solve for, though it's a differential equation, so it won't be that easy. But once we've figured out the gamma distribution, we can solve for the lift distribution using the Kutta-Joukowsky theorem. We can integrate this lift distribution to get us the total lift on the wing as a function of gamma, and similarly we can find the induced drag on the wing as a function of gamma and the induced angle of attack.

These three properties are super important during flight because they describe the forces the foil feels due to these finite wing effects. Let's try this out and see how it works. There's a bit of math in the next segment, so we'll rush through a little bit here.

Don't get discouraged if you miss a step. Consider a wing that has an elliptical circulation distribution along the span. The equation would look something like this, where we're leaving it arbitrary so it has some peak gamma sub zero. The distribution, sketched out, would appear like this along the span, peaking in the center and zero at the edges.

Straight away, we can calculate the lift distribution, L prime. To calculate the total lift, we will have to work a bit harder. Write out the equation with the integral.

Here, we're going to deploy a tool we used in thin airfoil theory. It's more convenient to work in the theta space. where we define a circular coordinate system that projects onto the span.

The edges go from 0 to pi instead of plus minus s over 2, and theta defines a point along the foil. With this transformation, z turns into s over 2 times cosine theta, and dz is s over 2 sine theta d theta. We'll use this coordinate transformation throughout our derivations in the next steps.

and it usually produces convenient solvable integrals. When we apply this to the lift equation we can solve out the integral and get our final expression for lift. To get to the induced drag first we're going to need to solve for a number of things.

We need the induced angle of attack and it will also be convenient for us to solve for the downwash function, v. Here the downwash function is defined by the integral of the slope of the gamma distribution which we can solve for directly. Plug this into the integral and we get the following expression. Let's use our coordinate transformation again so we get some simple solvable integrals. Now we have an integral with variable theta and dummy that variable theta sub zero. You might recognize this.

We ran into this exact integral with our thin airfoil theory. Fortunately for us, math saves the day and we know the answer to this integral directly as a function of dummy variable theta sub zero. Plug this solution in and we get the expression for the downwash, which we save for later.

Notice, this expression is not a function of the span. The downwash is constant, an interesting feature of an elliptical gamma distribution. Next, we consider the induced angle of attack, which is just the downwash that we know divided by the freestream velocity.

Since we've already solved for the lift as a function of gamma sub zero above, we can work to get to the induced angle of attack as a function of the lift. First, find gamma sub zero as a function of lift. Then, use the lift equation to get it as a function of the lift coefficient. Plug this back into the induced angle of attack expression.

You might recognize the ratio of chord over span as the inverse of the aspect ratio. Technically, aspect ratio is the span squared divided by the planform area of the foil. This accounts for wings of all shapes and sizes. However, here we can turn chord divided by span into the aspect ratio.

This gives us our final form of the induced angle of attack. We need this for the induced drag. Write out the induced drag equation from above. This can simply be turned into the induced drag coefficient using the drag equation.

Notice that the induced angle of attack is not a function of the span, so we can pull it out of the integral. Use our fancy coordinate transformation again, and we get easily solvable integrals. This gives us the induced drag coefficient that has gamma sub zero, and the induced angle of attack in it.

Plugging in what we know about these two variables, we get a simplified expression for the induced drag coefficient. Let's pause here a moment and note some interesting things about this expression, because the induced drag is very important to aerodynamics. First, the induced drag increases dramatically with lift.

It goes as lift squared. It's an interesting feature. The lighter you are, the more efficient you can be. Second, the induced drag goes down with aspect ratio. This drives wings to be high aspect ratio, chasing efficient flight.

Now, you might be wondering why we started with this specific example. And it wasn't by accident. First and foremost, the elliptical gamma distribution is important to aerodynamics because it represents the most efficient distribution.

meaning it produces the lowest induced drag of any other distribution of gamma. If you were to go back and solve everything out generally for arbitrary solutions of gamma, you would find that the induced drag equation looks a lot like the one for the elliptic distribution, but with a span efficiency factor e in the denominator. This e value can only go up to 1, so the fact that the elliptical distribution produces an e of 1 means it is as low as it gets.

Second, elliptical distributions are not hard to achieve. An elliptical planform will get you exactly this, and specific tapered wing angles get you really quite close. But, let's say you didn't have an elliptical distribution, and we wanted to realistically solve for the fundamental equation for lifting line theory above.

What you would do is the fancy mathematical strategy of guess and check. First, pick a gamma distribution. Probably an elliptic distribution is a good starting point.

From the strength distribution, you can calculate the induced angle of attack. And then, you can get the effective angle of attack. From the effective angle of attack, you can estimate the lift coefficient distribution as a function of z. You can turn the lift coefficient into the lift per unit span, and then use Kuda-Joukowsky to get a new gamma distribution.

Check that gamma against your guess. If it's different, you can repeat steps 1 through 5 with the new gamma distribution, and you'll repeat them until you get convergence. This will settle on the real gamma value for your wing, which you can then get your lift and drag characteristics from. It turns out this theory works fairly well for a wide range of applicable foils. It's good to use if you have a straight wing with moderate to high angles of attack.

And this covers a surprising amount of aircraft today. However, if you have a more aggressive aircraft, it might not be so useful. Low aspect ratio foils are no good here. Also, highly swept wings are no good for this analysis, along with the infinites delta wing.

For these types of foils, you might need to employ more modern numerical techniques. You could use something like the lifting surface theory and the vortex lattice method, but we can't get into that in this video. However, Lifting Line Theory is still a powerful tool despite these limitations. In practice, you will find that the Lifting Line Theory is a good first design estimate for the wings you will be working with. The result of this theory drives design.

It pushes wings to have higher aspect ratios, chasing that efficient flight. It also drives planform shape, with elliptic and tapered wing designs. Conveniently.

These wing shapes also happen to be structurally superior, which is a nice benefit. They can be both structurally best and most efficient. And that's it! Let's review.

We started by introducing the lifting line theory. It is an idea that is centered around estimating a foil and the tip vortices as a horseshoe vortex. However, We use many small horseshoe vortices so that we have controllable vortex strength distribution at the bound vortex.

This led us to the fundamental equation for lifting line theory, where we could technically solve for the gamma distribution. We tried with an elliptic strength distribution. where we solve for the ellipse distribution, the lift, and then the induced drag. Interestingly, the elliptic distribution is the most efficient in aerodynamics. To solve for arbitrary gamma distributions, we need to follow a set of iterative steps to converge to a solution, effectively mathematical guess and check.

We finished by exploring the limitations of the theory and how it's applied today. I hope you enjoyed the video, and thanks for watching.

Transcript for:Lifting Line Theory

Transcript for:
Lifting Line Theory