Finite wing effects [Aerodynamics #15]

Summary

This 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

• 🛫 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.
• 📚 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.
• 📉 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.
• 🔁 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.
• 🌀 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.
• 📊 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.
• 📉 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.
• 📐 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.
• 🔧 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.
• 📚 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.
• 🌐 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

00:00

🛫 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.

05:02

🌀 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.

10:02

📚 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.

15:06

🔧 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.

Keywords

💡Aerodynamics

Aerodynamics is the study of the motion of air and other gases in relation to solid bodies moving through them. In the context of the video, aerodynamics is crucial for understanding how forces such as lift and drag act on an aerodynamic body, such as a wing. The video script delves into the complexities of aerodynamics, particularly focusing on the effects of finite span wings and the induced drag caused by tip vortices.

💡Drag

Drag is a force that opposes the motion of an object moving through a fluid, like air. In the script, drag is described as having two main sources: pressure drag, which is due to the pressure difference between the front and rear of the body, and viscous drag, which is due to the friction between the fluid and the body's surface. The video emphasizes how the presence of a tip vortex on a finite span wing can significantly increase the induced drag.

💡Finite Span Wing

A finite span wing is a wing with a limited length, as opposed to an idealized infinite span wing. The video script explains that real-world wings are finite and introduces the concept of how this affects aerodynamic performance, particularly through the creation of tip vortices and induced drag, which are not present in the idealized two-dimensional case.

💡Tip Vortex

A tip vortex is a swirling flow of air that forms at the wingtips of an aircraft due to the pressure difference between the upper and lower surfaces of the wing. The script discusses how the tip vortex induces a downwash, affecting the effective angle of attack and leading to induced drag, which is a critical aspect of understanding real-world aerodynamics.

💡Downwash

Downwash is the downward vertical velocity induced by the presence of a tip vortex. The script describes how downwash changes the effective angle of attack on the wing and tilts the lift vector, resulting in a decrease in lift and an increase in induced drag, which are detrimental to the performance of the wing.

💡Induced Drag

Induced drag is a component of drag caused by the wingtip vortices that tilt the lift vector and turn some of the lift into drag. The script explains that induced drag is a significant factor in the performance of finite span wings and is a direct result of the downwash created by the tip vortices.

💡Lift Coefficient

The lift coefficient is a dimensionless number that is used to quantify the amount of lift generated by a wing for a given velocity and wing area. In the script, the lift coefficient is discussed in relation to the performance of a NACA 4412 airfoil section and how it changes when transitioning from a two-dimensional to a three-dimensional wing.

💡Angle of Attack

The angle of attack is the angle between the oncoming airflow (relative wind) and the wing's chord line. The script mentions that the presence of a tip vortex induces a downwash, which in turn changes the effective angle of attack on the wing, affecting the lift and drag characteristics.

💡Biot-Savart Law

The Biot-Savart Law is used to calculate the velocity induced by a vortex at a given point in the fluid. In the script, this law is applied to understand the behavior of the tip vortex and its effect on the downwash experienced by a finite span wing, providing a mathematical framework for analyzing the aerodynamic effects of vortices.

💡Helmholtz Vortex Theorems

The Helmholtz Vortex Theorems are a set of principles that describe the behavior of vortex filaments in fluid dynamics. The script briefly mentions these theorems to emphasize the rules governing vortices, such as their strength being constant along the filament and the requirement for a vortex to either terminate at a solid boundary or form a closed loop.

💡Aspect Ratio

The aspect ratio of a wing is the ratio of its span to its average chord. The script discusses the importance of aspect ratio in aerodynamics, noting that higher aspect ratio wings are more efficient because they are less affected by tip vortices and induced drag, which is a key consideration in aircraft design.

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