Finite wing effects [Aerodynamics #15]
Summary
TLDRThis aerodynamics video delves into the effects of finite span wings and wing tips on an aerodynamic body. It explains the concept of induced drag, effective angle of attack, and downwash caused by the tip vortex. The script introduces Biot-Savart's law to calculate the induced velocity of a semi-infinite vortex, which is crucial for understanding lift distribution and performance impacts on real-world aircraft. Practical implications include the design of wing tips to mitigate vortex effects and the preference for high aspect ratio wings for efficiency.
Takeaways
- đ« The script introduces the concept of drag on aerodynamic bodies, explaining that it's primarily due to viscous forces but can be dominated by pressure drag when flow separates quickly.
- đ It explains that the ideal two-dimensional (2D) wing principles don't fully apply to real-world three-dimensional (3D) wings, leading to the exploration of finite span wings and their effects.
- đ The script discusses the performance of a NACA 4412 airfoil section, highlighting the difference between 2D ideal sections and 3D real-world applications, particularly the change in lift and drag coefficients.
- đ The concept of end effects is introduced, particularly focusing on the tip vortex and downwash, which are caused by the pressure difference near the wing edges and roll around the tips, affecting the wing's performance.
- đ The tip vortex is explained as a result of the flow rolling from underneath to the top of the wing, creating a trailing vortex in the wake and influencing the wing's angle of attack and lift.
- đ The script differentiates between total lift and drag coefficients and those per unit span, emphasizing that they are not necessarily equal due to finite span effects.
- đ Induced drag is introduced as a new form of drag that occurs when the lift vector is tilted due to downwash, reducing lift efficiency and adding to the overall drag.
- đ The script discusses lift distribution across the span of a wing, noting that it varies and is influenced by factors such as the tip vortex and the design of the wing.
- đ§ Practical applications of understanding tip vortices are highlighted, including their impact on aircraft performance, frequency of takeoffs and landings, and the development of wingtip devices to mitigate their effects.
- đ Biot-Savart Law is introduced as a tool for analyzing the induced velocity of a vortex, with a detailed explanation of how it applies to the semi-infinite vortex representing the tip vortex.
- đ The script concludes with a discussion of the Helmholtz vortex theorems, which describe the behavior of vortex filaments in fluid dynamics, and their relevance to understanding aerodynamic effects.
Q & A
What are the two mechanisms through which force is transmitted from a fluid to an aerodynamic body?
-The two mechanisms are pressure, which acts normal to the surface, and shear, which is parallel to the surface.
Why is the drag of an aerodynamic body typically due to viscous forcing?
-The drag is mostly due to viscous forcing because of the friction between the fluid and the body's surface. However, if flow separates quickly, it becomes dominated by pressure drag.
What is the primary focus of the video regarding the effects that come with the wing tip?
-The primary focus is on the tip vortex and the downwash it produces, which leads to a change in the effective angle of attack and adds induced drag.
How does the lift coefficient and drag coefficient change when considering a 3D wing compared to a 2D ideal section?
-In a 3D wing, the lift and drag coefficients are different from those of a 2D ideal section due to finite span effects, such as the tip vortex and downwash, which alter the performance.
What is the significance of the tip vortex in the context of aerodynamics?
-The tip vortex is significant because it induces a downwash that changes the effective angle of attack on the wing, leading to a decrease in lift and an increase in induced drag.
What is the relationship between the lift per unit span and the total lift force?
-The lift per unit span is calculated using the lift divided by the span, whereas the total lift force uses the total lift with the addition of span in the denominator.
Why does the lift distribution across the span of a wing vary?
-The lift distribution varies because near the tips, the flow can travel from the bottom to the top, balancing out the pressure and reducing the pressure difference from the bottom and top of the foil.
What is the Biot-Savart Law and how is it applied in this context?
-The Biot-Savart Law describes the induced velocity on a point due to a segment of a vortex filament. It is applied here to calculate the downwash effect caused by the tip vortex on the wing.
How does the tip vortex influence the performance of a finite span wing?
-The tip vortex induces a downwash that changes the effective angle of attack, decreases lift, and adds induced drag, which negatively impacts the wing's performance.
What are the practical impacts of the tip vortex on aircraft operations?
-The tip vortex affects the frequency that aircraft can take off and land at airports, influences technology innovation to avoid the negative effects of the vortex, and encourages the design of high aspect ratio wings for more efficient flight.
What are the Helmholtz vortex theorems and why are they important in understanding vortex behavior?
-The Helmholtz vortex theorems are rules stating that vortex filament strength is constant along the filament, a vortex cannot end arbitrarily in a fluid but must terminate at a solid boundary or form a closed loop, and irrotational flow remains irrotational without external forcing. These theorems are important for predicting and describing the effects of vortices like the tip vortex.
Outlines
đ« Introduction to Finite Span Aerodynamics
This paragraph introduces the concept of finite span aerodynamics, contrasting it with ideal two-dimensional (2D) wings. It explains that while 2D aerodynamics principles apply to theoretical, infinite-span wings, real-world wings have a finite span and are subject to additional effects. The focus is on the tip vortex and its effects, such as induced drag, effective angle of attack, and downwash. The paragraph sets the stage for exploring the impact of the vortex near the wing and the three-dimensional (3D) reality of aerodynamics.
đ Understanding Tip Vortex and Induced Drag
This paragraph delves into the phenomenon of the tip vortex, which forms at the wingtips of finite span wings due to pressure differences. The vortex induces a downwash, a downward vertical velocity that affects the wing's angle of attack and introduces induced drag. The explanation covers how the lift vector is tilted by the downwash, converting some lift into drag and impacting the wing's performance. It also touches on the concept of lift distribution across the wing span and how it's influenced by the tip vortex, leading to a non-uniform distribution with less lift near the tips.
đ Applying Biot-Savart Law to Vortex Analysis
The paragraph introduces the Biot-Savart Law as a tool for analyzing the effects of vortices in aerodynamics. It discusses how the induced velocity from a vortex is related to the vortex's circulation and the distance from it. The law is applied to an infinite vortex and then adapted for a semi-infinite vortex, which is more representative of a wing's tip vortex. The paragraph explains the process of integrating the Biot-Savart Law to derive the induced velocity experienced by a point near the vortex, considering the geometric and trigonometric relationships involved.
đ§ Practical Implications of Tip Vortex Effects
This final paragraph summarizes the practical implications of the tip vortex and its effects on aircraft performance. It highlights the importance of understanding finite span effects for real-world aircraft design, including the impact on takeoff and landing frequencies at airports and the development of technologies to mitigate the negative effects of tip vortices. The paragraph also emphasizes the benefits of high aspect ratio wings in reducing tip vortex effects and improving flight efficiency. It concludes with a brief review of the key points covered in the video.
Mindmap
Keywords
đĄAerodynamics
đĄDrag
đĄFinite Span Wing
đĄTip Vortex
đĄDownwash
đĄInduced Drag
đĄLift Coefficient
đĄAngle of Attack
đĄBiot-Savart Law
đĄHelmholtz Vortex Theorems
đĄAspect Ratio
Highlights
Drag on aerodynamic bodies is transmitted through pressure and shear forces.
Aerodynamic body drag is predominantly due to viscous forcing, but pressure drag dominates with flow separation.
The concept of finite span airfoils introduces a different source of pressure drag related to the wing tip.
Induced drag, effective angle of attack, and downwash are explored in the context of finite span wings.
Real wings are finite, and the physical effects of a finite span wing are distinct from ideal two-dimensional wings.
Lift and drag coefficients for a NACA 4412 foil section are presented as functions of the angle of attack.
The distinction between per unit span and total lift and drag coefficients is clarified.
The tip vortex and its downwash effect on the effective angle of attack and induced drag are discussed.
The tip vortex is created by the pressure difference near the wing edges, causing flow to roll around the tips.
Downwash, induced by the tip vortex, decreases the effective angle of attack and adds induced drag.
Induced drag is a result of the lift vector tilting due to downwash, turning some lift into drag.
Lift distribution across the span is affected by the tip vortex, with lift per unit span decreasing near the tips.
The Biot-Savart law is introduced to explain the impact of the vortex and calculate induced velocity.
The Biot-Savart law is applied to derive the induced velocity equation for a semi-infinite vortex.
Helmholtz vortex theorems are highlighted, explaining the behavior of vortex filaments in fluid dynamics.
Practical implications of the tip vortex include its influence on aircraft takeoff and landing frequencies, and the development of wing tip technologies to mitigate its effects.
The pursuit of high aspect ratio wings is encouraged for more efficient flight due to reduced tip effects.
Transcripts
hello
and welcome to the next video in
aerodynamics
last time we discussed the sources of
drag on aerodynamic bodies
force transmits from the fluid to the
body through two mechanisms
the pressure that acts normal to the
surface and shear which is parallel to
the surface
typically the drag of an aerodynamic
body is mostly due to viscous forcing
although if flow separates it quickly
becomes dominated by pressure drag
today we will learn about a different
source of pressure drag in our
exploration of finite span airfoils
and the effects that come with the wing
tip we will explore induced drag
effective angle of attack and downwash
as well as build tools with the bios of
art law
to explain the impact of the vortex near
the foil
let's jump in all real wings are finite
eventually nothing can have an infinite
span
but most of what we've learned so far
applies only to an
ideal two-dimensional wing here
we introduce the physical effects that
come with a finite span wing
and we begin to thinking about the
impacts of three-dimensional reality
in addition to the two-dimensional ideal
case
let's start by considering the
performance of a naca 4412 foil section
if we looked up the performance of this
foil
we would see the lift coefficient and
drag coefficients as functions of angle
of attack
at say an angle of attack of 8 degrees
we would get a lift coefficient of 1.2
and a drag coefficient of 0.0068
however these parameters are for 2d
ideal sections
what happens if we go from 2d to 3d and
add in the span
from our graphs above we've noted the
lift and drag coefficients for this foil
at a given angle of attack
notice these coefficients have lower
case letters in the subscript
lowercase l and lowercase d this
indicates that these are used to
calculate
per unit span quantities recall our lift
equations from a much earlier video
arranged to solve for the coefficients
the lift per unit span coefficient is
calculated using the lift per unit span
and the lift coefficient is calculated
using the total lift force with the
addition of span
in the denominator at first it might
seem like by definition these two things
are the same
because lift per unit span is the same
as lift divided by the span
however there's an important distinction
the lift and drag coefficients per unit
span are not necessarily equal to the
total lift and total drag coefficients
per unit span quantities assume an ideal
wing with infinite span
in a sense no finite wing effects
the lift and drag only come from 2d
sources
however in real cases the span of the
wing is not infinity
and there are some interesting end
effects that occur
it's these end effects that cause the
lift coefficients to be different to the
lift per unit span coefficients
today our focus will be on the tip
vortex and the downwash it produces
which leads to a change in the effective
angle of attack and adds induced drag
again let's consider our foil from
before
the foil produces both lift and drag
specifically in order to create lift
there is a low pressure region on top of
the foil
and a relatively higher pressure region
at the bottom
when the wing is finite it has outer
tips or edges
when there is a pressure difference
across the surface near the edges the
flow will want to leak from the higher
pressure to the lower pressure
so the flow rolls around the tips and
escapes to the top
this rolling motion creates something
called a tip vortex
the flow continuously rolls from
underneath to the top during flight
and creates a trailing vortex in the
wake of the wing
interestingly this vortex now has
influence over the foil itself
consider a section on the foil near the
tip
in the ideal sense the foil is at an
angle of attack at some forward velocity
however we now have a pesky tip vortex
in our vicinity
and this vortex induces a downwash or a
downward vertical velocity on our foil
just look at the orientation of the
vortex and the flow rolling up and over
it's clear that near this vortex we get
a vertical velocity component
also called induced velocity and that
induced velocity is downwards
induced velocity is called a downwash
now we add this vertical velocity v with
the free stream velocity component
and the resultant total velocity vector
has an angle of alpha i
with respect to the travel direction
this induced angle alpha i from the
downwash actually works to decrease the
effective angle of attack of the foil
additionally the lift force generated is
now perpendicular to this new total
velocity vector
and no longer perpendicular to our
direction of motion
if we stay in the reference frame of the
direction of motion
this acts to decrease the vertical lift
slightly and adds a new drag in the
direction of travel
this drag is called the induced drag
it's because the downwash tilts our lift
vector
and turns some of it into drag
ultimately
downwash does two critical things
first it changes the effective angle of
attack locally
meaning it changes the expected lift
performance we should be getting
second it tilts a portion of the lift in
a way that induces the new drag on the
foil
and we lose lift both of these things
decrease lift and added drag
work to hurt or foil performance so it's
safe to say that downwash is typically
bad
now we can consider this among our other
force producing components
a finite span foil has dragged from
three sources
skin friction comes from viscous forcing
separation which ultimately happens
because of the boundary layer and
viscous things
is actually a pressure track and now we
have induced drag
a second form of pressure drag that
comes from the lift tilting
all this downwash also leads to
something called lift distribution
across our span our lift per unit span
varies where it decreases near the tips
and maxes out as far from the tips as we
can get
this is because near the tips the flow
is allowed to travel
from the bottom to the top and it
balances out the pressure so there's no
longer a large pressure difference from
the bottom and top of the foil
and what we know from kuda jakowski is
that
the lift per unit span varying across
the span
means the circulation also varies across
the span
which will become important in the next
video
downwash isn't the only thing that
causes lift distribution
a lot of the time lift distribution is
purposefully designed
into the wing most commonly we see this
in the variation of the cord length with
the span
anything other than a rectangular foil
has a varying cord
which covers most aircraft meaning a
non-uniform lift distribution
second is geometric twist
literally twisting the foil in a way
that the angle of attack changes with
the span
this change in angle leads to variation
and lift
lastly there is something called
aerodynamic twist
this is where the aerodynamic
characteristics of the foil change with
the span
meaning the starting foil shape might be
different than the leading edge foil
shape
this is common among more modern and
complex aircraft
everything in this video so far has been
due to one physical phenomena
the tip vortex the tip vortex represents
a straight
semi-infant infinite vortex and it will
help us to build some tools for
aerodynamic
analysis of vortices moving forward
for that we call on the biot-savara law
which will hopefully tell us something
about the semi-infinite vortex
and its downwash let's consider an
arbitrary vortex filament
label a point p some distance off of the
filament
this point feels an induced velocity due
to the neighboring vortex
specifically we get a delta v due to a
segment of the filament
delta s when point p is distance r
from the vortex to describe this
velocity induced
we can use the b savar law which covers
this
it's defined as follows
the induced velocity increases with the
vortex circulation
or the strength of the vortex and
decreases with increased distance from
the vortex
this law is actually common even outside
of aerodynamics and works in general
specifically you might see it in fields
like electromagnetism
now let's apply this equation to
something like a real vortex
let's say we restrict our vortex to be
straight only
and extend it from positive to negative
infinity
to get the total induced velocity we
need to take biots of art
and add up all the vortex segments
acting on our point of interest
point p this is done by integrating the
equation from negative to positive
infinity
but before moving forward we need to
make some geometric
changes to the equation let's draw a
diagram of what's happening
we have our straight vortex and point p
off to the side
technically point p is distance r away
from the segment
at angle theta let's also mark a
different distance
h which goes from point p and connects
to the filament
at a 90 degree angle
back to the equation we will make two
simplifications
the definition of a cross product
between two vectors is applied and we
note that we only care about the
magnitude of the velocity induced
this simplifies our equation slightly
let's bring up our angle diagram our
goal is to change the r
and s into h and theta for that we apply
trigonometry
this lets us define the r s and d s
as a function of theta and h
also we want to change our bounds to b
from minus
infinity to infinity into the angle
which means that the angle goes from pi
to zero respectively
add this and the trigonometry equations
into our
main equation and we get a relatively
simple integral to solve
this leaves us with the equation for the
induced velocity for an infinite vortex
gamma over 2 pi h this means that the
induced velocity increases linearly with
vortex strength
and decreases linearly with the distance
away
this is the vertical velocity you would
feel if you were standing distance
h away from an infinite vortex
maybe you're standing in the runway of
an airport after an aircraft goes by
and is some distance away but
we don't want to stop here the tip
vortex has a starting point that's
finite
and does not extend between infinity and
negative infinity
this vortex goes from some finite
location to infinity
and is called a semi-infinite vortex
our analysis is the same we just change
our integration bounds to go from pi to
pi over 2
effectively cutting our window in half
once it's all done we have the equation
for the induced velocity of a
semi-infinite vortex
which is half that of the infinite
vortex
this is the velocity that a point along
the foil span
which is distance h from the tip feels
due to the vortex presence
this equation will be integral in our
calculation of the induced
velocity and downwash effect and
while we're thinking about vortices this
is a good spot to point out that vortex
filaments have rules
first the vortex filament strength is
constant along the filament
this is also interpreted as the vortex
filament strength is not changing in
time without
external forcing
second a vortex cannot just end at any
point in a fluid
a vortex can either terminate at some
sort of solid boundary
like our wing tip or it can connect to
itself and form a closed loop
this is called a vortex ring when it
closes
lastly if there is nothing to externally
cause rotation
an irrotational flow stays irrotational
this is something we've talked about in
the past when we discussed rotational
flows at length
these three rules are theorems and
they're called the helmholtz vortex
theorems
although they're technically not laws
just theorems they have been largely
shown to be true
through measurement and observation
moving forward we're going to take our
knowledge of vortex
filaments and design ways to predict and
describe the effect of downwash
this includes induced drag due to the
tilting of the lift vector
the new total lift of the wing so we
know if we'll stay in the air
and the lift distribution along the span
we'll work to be able to predict these
in the future for finite span wings
with regards to these finite wing
effects in practice they are super
important because all physical aircraft
have a finite span
the tip vortex pair that comes off of an
aircraft has a tremendous influence on
the flow
after the plane goes by and dictates the
frequency that the aircraft can take off
and land at airports
because they want to be outside of the
wake of the aircraft before it
secondly it has led to technology
innovation to avoid this tip vortex
because generally end effects are bad so
we want to avoid them
we now have wing tips and end plates to
stop this rolling velocity leakage
last it leads us to strive for high
aspect ratio wings when we can
because higher aspect ratio wings have a
lesser impact from the tip
and lead to more efficient flight
and that's it let's review
we started by introducing why ideal 2d
flows are different from
real 3d wings in terms of performance
the finite span foil has flow spill over
the edges due to the pressure difference
creating the tip vortex the tip vortex
induces downwash on the foil
leading to changes in the angle of
attack and added drag
using the b outs of our law we derive
the induced velocity equation for the
semi-infinite vortex that represents the
tip vortex
so that we can now calculate the
downwash from relatively known
quantities
and we finished with a discussion of all
of the practical
impacts of the tip vortex i hope you
enjoyed the video
and thanks for watching
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