Lifting line theory [Aerodynamics #16]

Summary

This video delves into Prandtl's lifting line theory, a foundational concept in aerodynamics used to predict the performance and losses of finite wings. It explains how wingtip vortices impact lift and induce drag, and introduces the horseshoe vortex model to estimate these effects. The script walks through the derivation of the theory, explores the elliptical circulation distribution for efficiency, and discusses the iterative process for solving arbitrary gamma distributions. It concludes by highlighting the theory's practical applications and limitations in modern aircraft design.

Takeaways

• 🌀 Prandtl's lifting line theory is an analytical approach to finite wings, developed around World War One, and still used today for modern design analysis.
• 📊 The theory provides three key insights: lift distribution along the wing span, total lift produced by the wing, and induced drag, which indicates wing efficiency.
• 🔍 The concept of horseshoe vortices is introduced to model the tip vortices and the vortex sheet shed at the trailing edge, improving the representation of real-world aerodynamics.
• 📚 The Biot-Savart law is used to describe the vertical velocity induced by the vortices, which is crucial for understanding the downwash effect on the wing.
• 🔧 The theory involves an iterative process to solve for the vortex strength distribution, gamma, which is essential for determining lift and drag characteristics.
• 📉 The induced drag is found to increase dramatically with lift and decrease with higher aspect ratios, emphasizing the importance of wing design for efficiency.
• 📈 An elliptical circulation distribution is identified as the most efficient, producing the lowest induced drag, and is a common goal in wing design.
• 🔄 The process of solving for arbitrary gamma distributions involves an iterative guess and check method, converging to the actual gamma value for a given wing.
• 🚫 The lifting line theory has limitations and may not be suitable for low aspect ratio or highly swept wings, where more advanced numerical techniques are required.
• 🛠 Despite its limitations, the theory is a valuable tool for initial design estimates and influences the pursuit of high aspect ratio and specific wing shapes for efficiency.
• 🎓 The video concludes by highlighting the educational value of the theory in understanding aerodynamics and its practical applications in aircraft design.

Q & A

What is the main focus of the video script on aerodynamics?

The main focus of the video script is on Prandtl's lifting line theory, which is used to predict finite wing losses, including lift distribution, total lift, and induced drag.

What are the three major outcomes from Prandtl's lifting line theory?

The three major outcomes from Prandtl's lifting line theory are the lift distribution along the wing span, the total lift produced by the wing, and the induced drag which indicates the wing's efficiency in moving forward.

Why was the concept of a horseshoe vortex introduced in the lifting line theory?

The concept of a horseshoe vortex was introduced to model the tip vortices and the vortex sheet shed at the trailing edge of the wing, which helps in better representing the reality of the flow field around a finite wing.

How does the strength of the bound vortex in the horseshoe vortex model change along the span?

In the horseshoe vortex model, the strength of the bound vortex changes based on how many horseshoes are overlapping at a given location along the span. It varies stepwise, with a maximum at the center and lesser at the edges.

What is the significance of the Biot-Savart law in the context of the lifting line theory?

The Biot-Savart law is significant in the lifting line theory as it is used to describe the vertical velocity induced by a vortex, which is crucial for calculating the downwash and the induced angle of attack in the flow field around the wing.

How is the fundamental equation of Prandtl's lifting line theory derived?

The fundamental equation of Prandtl's lifting line theory is derived by expressing the known set angle of attack as a function of the unknown vortex strength distribution gamma, using the relationships between the effective angle of attack, induced angle of attack, and the circulation around the wing.

What is an elliptical circulation distribution and why is it important in aerodynamics?

An elliptical circulation distribution is a specific distribution of vortex strength along the wing span that peaks in the center and goes to zero at the edges. It is important because it represents the most efficient distribution, producing the lowest induced drag of any other gamma distribution.

How does the induced drag in a wing change with the aspect ratio?

The induced drag decreases with an increase in the aspect ratio. This is why high aspect ratio wings are pursued for efficient flight, as they result in lower induced drag.

What is the process for solving for arbitrary gamma distributions in lifting line theory?

The process involves an iterative method of guess and check. One starts with an initial guess for the gamma distribution, calculates the induced angle of attack, effective angle of attack, lift coefficient distribution, and then compares the resulting gamma distribution with the initial guess. The process is repeated until convergence is achieved.

What are the limitations of using Prandtl's lifting line theory for wing analysis?

Prandtl's lifting line theory has limitations for wings with low aspect ratios, highly swept wings, and delta wings, as it may not accurately represent the complex flow fields around these shapes. In such cases, more modern numerical techniques like the lifting surface theory or vortex lattice method may be employed.

Outlines

00:00

😲 Introduction to Prandtl's Lifting Line Theory

This paragraph introduces the concept of aerodynamics, specifically focusing on the finite wings and the challenges they present compared to two-dimensional foils. It highlights the tip vortex and its effects on the wing's performance, such as inducing a downwash and additional drag. The paragraph sets the stage for discussing Prandtl's lifting line theory, which aims to predict finite wing losses by modeling the wing with trailing edge vortices as a horseshoe vortex system. The historical context of the theory's development by Prandtl is provided, along with an overview of what the theory offers: lift distribution, total lift, and induced drag estimations.

05:02

🔍 Exploring the Horseshoe Vortex and Its Mathematical Representation

The second paragraph delves into the specifics of the horseshoe vortex, a concept used in Prandtl's lifting line theory to model the wing tip vortices and their effects on the wing's performance. It discusses the mathematical representation of the vertical velocity induced by the vortices using the Biot-Savart law and the challenges faced when the vertical velocity becomes infinite at the wing tips. The paragraph then explores the solution involving multiple horseshoe vortices to create a more realistic model and the use of calculus to derive a continuous distribution of vortex strength along the wing span.

10:03

📚 Derivation of the Fundamental Equation for Lifting Line Theory

This paragraph focuses on the derivation of the fundamental equation for Prandtl's lifting line theory. It discusses the relationship between the effective angle of attack, the induced angle of attack, and the circulation (gamma) distribution along the wing span. The paragraph explains how the known angle of attack can be expressed as a function of the unknown gamma distribution, leading to the formulation of the fundamental equation. It also touches on the importance of identifying spanwise functions such as the angle of attack with zero lift and the chord variation, which are essential for solving the differential equation for gamma.

15:05

📉 Analyzing Lift Distribution and Induced Drag with Elliptical Circulation

The fourth paragraph examines the application of the lifting line theory using an elliptical circulation distribution as an example. It explains how to calculate the lift distribution, total lift, and induced drag for this specific case. The paragraph introduces the use of a circular coordinate system (theta space) to simplify the integrals involved in the calculations. It also highlights the properties of the elliptical gamma distribution, such as producing the lowest induced drag and being a practical goal for wing design due to its efficiency and structural benefits.

20:06

🚀 Conclusion and Application of Lifting Line Theory in Aerodynamics

The final paragraph summarizes the lifting line theory and its practical applications in aerodynamics. It emphasizes the iterative process of solving for arbitrary gamma distributions through guess and check methods until convergence is achieved. The paragraph also discusses the limitations of the theory, such as its applicability to straight wings with moderate to high angles of attack and its limitations with low aspect ratio and highly swept wings. Finally, it acknowledges the importance of the theory in driving wing design towards higher aspect ratios and efficient shapes, while also noting the need for more modern numerical techniques for more aggressive aircraft designs.

Keywords

💡Aerodynamics

Aerodynamics is the study of the motion of air and other gases in relation to solid objects, such as wings of an aircraft. In the video, aerodynamics is the central theme, as it discusses the performance and losses of finite wings, including the effects of wing tip vortices and induced drag, which are critical for understanding flight dynamics.

💡Finite Wings

Finite wings refer to real-world wings with a limited span, as opposed to the idealized infinite or two-dimensional wings often used in basic aerodynamic studies. The script discusses how finite wings produce additional features and losses, such as tip vortices, which affect the overall performance and induce new forms of drag.

💡Tip Vortex

A tip vortex is a swirling flow of air that forms at the wingtip due to the pressure difference between the upper and lower surfaces of the wing. In the script, the tip vortex is highlighted as a significant factor in the downwash effect, which changes the effective angle of attack on the wing and contributes to induced drag.

💡Downwash

Downwash is the downward deflection of air caused by the wing's lift, which affects the airflow behind the wing. The script explains how the tip vortex induces downwash, altering the effective angle of attack and adding to the drag experienced by the wing.

💡Prandtl's Lifting Line Theory

Prandtl's lifting line theory is an analytical approach developed by Ludwig Prandtl to predict the performance and losses of finite wings. The script delves into this theory, which models the wing and its tip vortices as a system of horseshoe vortices to estimate lift distribution, total lift, and induced drag.

💡Horseshoe Vortex

A horseshoe vortex is a vortex system in fluid dynamics that resembles a U-shape, formed by connecting the trailing vortices at the wingtips with a bound vortex along the wing's span. The script describes how this concept is used in lifting line theory to model the complex flow around finite wings.

💡Biot-Savart Law

The Biot-Savart law is used to calculate the velocity induced by a vortex at a given point. In the script, it is mentioned in the context of estimating the downwash near an idealized line vortex, which is integral to understanding how the vortices contribute to the overall aerodynamic forces on a wing.

💡Induced Drag

Induced drag is the drag caused by the wing's generation of lift, primarily due to the downwash and the pressure difference created by the wing. The script discusses how lifting line theory helps in estimating this type of drag, which is a critical factor in the efficiency of an aircraft's flight.

💡Aspect Ratio

Aspect ratio is the ratio of the square of the wing's span to its planform area and is a key parameter in aerodynamics, affecting lift and drag. The script notes that induced drag decreases with increasing aspect ratio, which is why high aspect ratio wings are pursued for efficient flight.

💡Elliptical Circulation Distribution

An elliptical circulation distribution refers to a theoretical distribution of lift along the span of a wing that is most efficient, producing the lowest induced drag. The script uses this as an example to illustrate how lifting line theory can be applied to calculate lift distribution, total lift, and induced drag.

💡Iterative Solution

An iterative solution in the context of the script refers to the mathematical process of guessing and checking gamma distributions to converge on the correct distribution that satisfies the lifting line theory's fundamental equation. This process is necessary when dealing with non-elliptical circulation distributions and is crucial for accurately predicting wing performance.

Highlights

Introduction to aerodynamics, focusing on the effects of finite wings and the reality of wing tip losses.

Explanation of the tip vortex and its impact on the effective angle of attack and induced drag.

Introduction of Prandtl's lifting line theory for predicting finite wing losses.

Derivation of the method to estimate a foil with trailing edge vortices as a horseshoe vortex.

Discussion on the three major outcomes from lifting line theory: lift distribution, total lift, and induced drag.

Use of Biot-Savart law to estimate downwash near an idealized line vortex.

Concept of the horseshoe vortex in fluid dynamics and its relation to wing tip vortices.

Problem of vertical velocity blowing up at the wing tips and the adaptation using multiple horseshoe vortices.

Infinitesimally small horseshoe vortices and their contribution to the vortex strength distribution.

Derivation of the formula for induced velocity at the bound vortex along the span.

Expression for the induced angle of attack and its relation to the downwash distribution.

Integration of thin airfoil theory to find the lift slope and its relation to the lift coefficient.

Fundamental equation of Prandtl's lifting line theory and its role in solving for the vortex strength distribution.

Calculation of lift distribution, total lift, and induced drag using the elliptical circulation distribution.

Significance of the elliptical gamma distribution as the most efficient in aerodynamics.

Iterative process for solving arbitrary gamma distributions through mathematical guess and check.

Limitations of lifting line theory and its applicability to straight wings with moderate to high angles of attack.

Practical applications of lifting line theory in driving wing design for efficiency and structural superiority.

Transcripts

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 biots of

our 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 one

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

and 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 beauts of our

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 kodachikowski 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 wing span is s and the span goes

from minus

s over two to s over two

it produces lift and drag and at the

wing tips you have vortices

prandtl's idea here was to remove the

surface entirely

we can model the tip vertices as a

semi-infinite lined 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 vertices 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-savara law we introduced

in a previous video

the vertical velocity induced by a for

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

vertices

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 two

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 little or 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 vertices

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

what 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 the 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 free stream 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 the 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 dcld-alpha

equaling 2 pi regardless of the camber

plotting it you would see something like

this

the slope before separation occurs is at

or near 2 pi

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 cut at jakowski 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 cl

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 cutter jakowski 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 define 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 down wash

function v

here the downwash function is defined by

the integral of this

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 free stream

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 0 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 plan-form 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 0

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 plant form 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 katachokowski to get a new

gamma distribution

check that gamma against your guess if

it's different

you can repeat steps one through five

with the new gamma distribution

and you will 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 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 infamous 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

in 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'll find that the lifting line theory

is of 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 platform 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 can 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 lift

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