Lifting line theory [Aerodynamics #16]
Summary
TLDRThis 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
😲 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.
🔍 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.
📚 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.
📉 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.
🚀 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.
Mindmap
Keywords
💡Aerodynamics
💡Finite Wings
💡Tip Vortex
💡Downwash
💡Prandtl's Lifting Line Theory
💡Horseshoe Vortex
💡Biot-Savart Law
💡Induced Drag
💡Aspect Ratio
💡Elliptical Circulation Distribution
💡Iterative Solution
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
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