AerodynamicsB, AE2115 I, Chapter 5

1
Chapter 5 Incompressible Flow over Finite Wings

• 5.1 Introduction finite wings, downwash,
  induced drag
• 5.2 Vortex Theory principle the vortex filament
• Biot-Savart law
• Helmholtzs vortex theorems
• 5.3 The Classical Lifting-Line Theory
• elliptical and general lift distribution
• the effect of Aspect Ratio
• 5.4-5 Extensions numerical implementation
• lifting-surface/vortex-lattice
• ADDITIONAL MATERIAL (see www.hsa.lr.tudelft.nl/b
  vo/aerob)
• 5.A Numerical Example of the Wing Equation

2
The flow over finite wings

• In what respect is the flow around a true wing
  different from an airfoil (an infinite wing)?

Airfoil 2D flow (cl , cd) Real Wing 3D flow
(CL , CD) (1) finite extent (2) variation of
sections along the wing span

• spanwise flow component due to leakage flow
  around the tips

3
Trailing vortices and downwash

• Results
• trailing vortices (tip vortices)
• and downwash
• (vertical flow component)

downwash
tip vortex
4
Downwash and the effective flow direction

• 1. The downwash modifies the effective flow
  direction and reduces ?
• effective angle of attack
• ? - geometric angle of attack
• ?i - induced angle of attack

2. The lift vector is inclined backwards
induced drag Note total drag induced
drag profile drag
flight direction
5
Distribution of lift (1)

• L total lift of the wing
• L sectional lift, local lift per unit span

• Along the wing span variation of
• chord c
• airfoil properties (aerodynamic twist)
• geometric ? (geometric twist)
• induced ?

• Hence, also variation of
• lift coefficient
• sectional lift L
• circulation ?

Note
6
Distribution of lift (2)

• Note Lift is zero at the tips (pressure
  equalization)

Central subject of wing theory Relation between
wing shape and lift distribution 1. Analysis
determine the lift distribution for given wing
shape 2. Design determine wing shape for
desired lift distribution
Lifting line theory the wing is replaced by a
vortex filament with variable circulation ?(y) at
the quarter-chord line free vortices
7
3-D Vortex Theory the vortex filament
flow around a real wing ? uniform flow
vortices
V?
2D Straight vortex line 3D
general curved vortex filament
r
P
induced velocity
8
3-D Vortex Theory Helmholtzs vortex theorems

• (compare the velocity induced by the vortex
  filament
• to the magnetic field induced by an electrical
  current)
• The circulation strength ? remains constant along
  the filament
• a vortex filament cannot end in the flow, but
• extends to infinity
• ends at a boundary
• forms a closed loop
• consequence

9
3-D Vortex Theory The Biot-Savart Law
The contribution dV of a filament section dl to
the induced velocity in P
The Biot-Savart Law
Direction is perpendicular to
and Magnitude
Note ? is the angle
10
Properties of a straight vortex filament segment
(1)
B

• Finite segment AB, constant ?

A
B
r
l
P
h
A
11
Properties of a straight vortex filament segment
(2)
B

• Note ?A and ?B are the internal angles of ? ABP

h
P

• Special cases
• infinite vortex filament
• ?A ?B 0
• semi-infinite filament
• ?A 90º ?B 0

A
(same as 2D vortex)
A
P
12
5.3 The Lifting-Line Theory
?
?
?

• The Horseshoe vortex as a simple model of a
  finite wing
• the wing itself ? a bound vortex at the
  1/4-chord line
• is fixed, hence, experiences lift (L ?V? ?)
• the tip vortices ? free-trailing vortices
• free to adjust to the local flow direction, no
  lift

• All vortices have the same circulation strength
  ?
• the free trailing vortices extend to infinity
  downstream

13
The horseshoe vortex
Downwash induced along the wing by the two
trailing (wing tip) vortices
?
?
right tip vortex
left tip vortex

• Remarks
• w lt 0 when ? gt 0 the induced flow is indeed
  downwards for positive lift
• Problems with the simple horseshoe-vortex model
  of a wing
• ?(y) constant lift distribution
• w ? ? at the tips

not realistic!
14
Extension of the horseshoe vortex model towards
the lifting-line model

• Instead of a single horseshoe vortex
  superposition of many vortex systems
• Each vortex has a different span but the bound
  vortex segments coincide on the same line and
  form the lifting line ( the wing)
• The circulation ? along the lifting line is no
  longer constant, but it varies along the span in
  a stepwise fashion
• Extrapolate to infinite number of horseshoe
  vortices to obtain continuous ?(y)

15
Principle of the lifting line
? d?
d?
?

• The wing is replaced by a bound vortex with
  (continuously) varying circulation ?(y)
• The trailing vortices create a vortex wake in
  the form of a continuous vortex sheet
• local strength of the trailing vortex at position
  y is given by
• the change in ?(y) d? (d?/dy) dy
• the vortex sheet is assumed to remain flat (no
  deformation)
• Validity good approximation for straight,
  slender wings at moderate lift

16
Determining the downwash of the lifting line
w
?(y)

• Strength of the trailing vortex at position y
• along the wing span
• Take small segment of the lifting line, dy,
• at position y
• Over this segment the change in circulation of
• the lifting line is d? (d?/dy) dy
• This is equal to the strength of the trailing
  vortex
• The contribution dw to the induced velocity at
  position y0

y0
y0 - y
dy
d? (d?/dy) dy
Total velocity at position y0 induced by the
entire vortex wake
induced angle of attack
17
The relation between circulation and wing shape

• Use 2D airfoil theory, but modified by the
  effective flow direction
• From the relation between lift and circulation
• combination

The fundamental equation of Prandtls
lifting-line theory
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Prandtls lifting-line equation (the wing
equation)

• Some remarks
• 1. This equation describes the relation between
  circulation and wing properties
• 2. It is linear in ?
• 3. The circulation ? is proportional to V?
  (Lift ?V? ? ?V?2 )
• 4. For a wing without twist (? and ?L0 are
  constant)
• circulation ? is proportional to ? ?L0
• for every value of ? the lift distribution has
  the same form
• (which depends on a0(y), c(y) and b, therefore,
  on the wing shape)
• the total lift is zero when ? ?L0 and then ?
  ? 0
• 5. For a wing with twist (? and ?L0 are not
  constant) THIS IS NOT SO
• in particular total zero lift is in general not
  accompanied by ? ? 0

19
Wing properties for given circulation ?(y)

• 1. Lift distribution
• 2. Total lift
• 3. Induced angle of attack
• 4. Induced drag

20
The elliptical lift distribution (1)
Consider the following elliptical lift
distribution
?(y)
?0 max.circulation
Compute the downwash velocity from
y
b/2
-b/2
coordinate transformation
?
Downwash and induced angle of attack are constant
over the span of the wing!
21
The elliptical lift distribution (2)

• Calculation of the total lift

• Relation between ?0 and CL

• The induced angle of attack

A b2/S is called the aspect ratio (AR) of
the wing (slankheid) typical values 6-8 for
subsonic aircraft 10-22 for glider aircraft
22
The elliptical lift distribution (3)
Calculation of the induced drag
Note that is constant here

• Conclusions
• The inducd drag is the drag due to lift
• Remember total drag
• quadratic dependence
• large AR decreases induced
  drag

23
The elliptical lift distribution - wing shape

• What wing shape can generate an elliptical lift
  distribution?
• assume no twist so ? and ?L-0 are constant
  
• assume lift slope a0 dcl /d? ( ? 2?) is
  constant
• consequence (with also ?i constant)
• required variation of the chord

Remark Proof
The wing must have an elliptical planform
24
The elliptical wing shape
An elliptical wing planform (note straight
1/4-chord line)
1/4-chord line
An elliptic lift distribution, an elliptic wing
planform and a constant downwash
25
The Supermarine Spitfire
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Aerodynamic properties of the elliptic wing

• ( constant)
• ( constant)
• where

• We found that

for an elliptic wing
for a general wing
Combining
solve for CL note CL 0 when ?
?L0 and
27
Effect of Aspect Ratio on the lift-curve CL(?)
for an elliptic wing
The lift slope is reduced. physical explanation
the downwash reduces the effective angle of
attack
28
The elliptical lift distribution - summary

• Constant downwash along the span
• Induced drag
• Lift slope
• effect of increasing the wing aspect ratio -
  induced drag smaller
• - lift-slope larger (a ? a0)
• Practical significance of the elliptical wing
• optimum wing shape minimal induced drag for
  given lift
• reference wing reasonable approximation for real
  wings

29
General lift distribution
For the elliptical wing
with and
a constant depending linearly on CL, hence, on ?
Describe the circulation of a general wing with a
Fourier sine series
constants that depend on ? Elliptical wing N1
A1CL/?A

• Note
• The number of terms N should be taken
  sufficiently large
• ? 0 at the tips
• Questions to be answered
• what are the aerodynamic properties (lift,
  induced drag)?
• what is the relation between the coefficients and
  the wing geometry?

30
General lift distribution total lift
Calculation of the lift coefficient
Standard integrals 0 when n ? 1 ?/2
when n 1
A
(Depends only on the first coefficient)
31
General lift distribution downwash
Calculation of the induced angle of attack
Standard integrals
32
General lift distribution induced drag
Calculation of the induced-drag coefficient
0 when n ? m ?/2 when n m
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General lift distribution summary and conclusions

• Conclusion
• the elliptic wing (? 0, e 1) gives the
  lowest possible induced drag (for given lift and
  aspect ratio)

34
The effect of twist

• For a wing without twist
• The shape of the lift distribution is
• the same for every ??
• At zero lift the circulation is
• identical to zero
• The induced drag is zero when the lift is zero

• For a wing with twist
• The shape of the lift distribution is
• not the same for every ??
• At zero lift the circulation is not identical
  zero
• The induced drag is not zero when the lift is zero

35
Illustration Wing with twist at zero lift
Example
(total lift zero)

• Lift distribution

L ?


-
b/2
-b/2

• Induced angle of attack

?i
-

• Contribution to the induced drag

(total induced drag gt 0)
36
The Lift-curve CL(?) of a general wing

• Conceptual comparison with an elliptic wing
• Assume an average/effective constant downwash
• Lift
• lift slope
• Compare induced drag

• Actual value of ? depends on the wing shape!
• Lift follows from
• To find A1(?) requires solution of the wing
  equation
• In general

37
The dependence of the lift distribution on ?
? the AOA of the wing

• Prandtls wing equation, for a general wing
• Effect on the circulation distribution ?(y) of
• changing the angle of attack of the wing ?
  ??(y)/?? ??(y)
• differentiate the wing equation w.r.t. ?
• as a consequence

geometric aerodynamic twist
??(y) ??(y)/?? is independent of ? and of the
twist
dCL/d? is independent of ? (and of the twist)
38
The dependence of the lift distribution on CL

• Change of circulation with the wing lift
  coefficient, CLCL(?)
• General form of the lift (circulation)
  distribution
• In terms of the coefficients An

??(y)/?CL is independent of ? and, hence, of CL
Basic lift distribution lift distribution
at zero lift
Additional lift distribution
independent of ?
An bn an CL
39
The relation between the An and the wing geometry

• Solve Prandtls wing equation
• substitute
• Numerical solution method
• Take a truncated series with N unknown
  coefficients A1, A2,AN
• Take N different spanwise locations on the wing
  where the equation is to be satisfied ?1, ?2, ..
  ?N (but not at the tips, so 0 lt ?1 lt ?)
• System of N equations with N unknowns (Solve N ?
  N matix)
• Note it is not possible to solve for only one
  coefficient, as in Chapter 4!

40
Numerical example of the wing equation (5.51)

• Consider rectangular wing c constant
  span b b/c A without twist ?
  constant ?L0 0
• evaluate the wing equation at the N control
  points at ?i
• The wing is symmetrical ? A2, A4, are zero
• take only A1, A3, as unknowns
• take only control points on half of the wing 0 lt
  ?i ? ?/2
• Example for N3
• take A1, A3, A5 as unknowns
• take control points (equidistant in ?) ?1
  ?/6, ?2 ?/3, ?3 ?/2
• take lift-slope of the airfoils a0 2?, and wing
  aspect ratio A 2?

41
Numerical example the rectangular wing (N3)

• The set of equations becomes
  with solution
• Evaluation of the properties of the rectangular
  wing (with A a0 2?)
• Note with ? ? 0.05 only 5 more induced drag
  than elliptical wing!

N3 N20
42
Effect of wing planform and aspect ratio

• Values of ? and ? depend on planform and aspect
  ratio of the wing

• Effect of wing planform on ? for a tapered wing

example
A tapered wing with taper ratio ct/cr 0.3 is
almost as good as an elliptical wing!
43
Final conclusions the effect of wing planform on
the induced drag

• In order to reduce the induced drag it is more
  important to increase the aspect ratio A than
  trying to approach the elliptic lift distribution
  accurately
• A tapered wing with taper ratio ct/cr 0.3 is
  almost as good as an elliptical wing and is much
  easier to manufacture
• Note that the parameter ? is a constant (i.e.,
  independent of ?) only for a wing without twist!
• Remember
• total drag induced drag profile drag (
  viscosity)

44
Wing theory - a summary

• Lifting-line theory
• The wing is replaced by a bound vortex at the
  1/4-chord line of the wing with varying
  circulation ?(y) the lifting line
• The trailing vortices form a flat sheet of
  distributed vorticity the vortex wake
• Limitations of the classical theory
• slender wings (large aspect ratio, or
  spangtgtchord)
• straight wings (no wing sweep)
• moderate aerodynamic loading (no deformation of
  the vortex wake)
• linear relation
• Extensions
• (5.4) non-linear lifting-line theory
• (5.5) methods where the wing is represented by a
  vortex-sheet (instead of a line)
• lifting-surface / vortex-lattice methods

45
5.4 A numerical nonlinear lifting-line method

• Given the wing shape and the angle of attack ?
• 1. Divide the wing in spanwise positions yn
• 2. Assume an initial circulation distribution
• ?n?(yn), e.g. elliptical
• 3. Calculate the induced angle of attack
• 4. Calculate
• 5. Calculate lift coefficient
• 6. Update circulation

(evaluate the integral numerically)
iterate until convergence (under relaxation
46
5.5 Lifting-surface theory (principle)
wing
wake (streamwise vorticity)
Lifting line wing represented by a vortex
filament (only spanwise vorticity) valid only for
slender wings
Lifting surface wing represented by a vortex
sheet with distributed spanwise and chordwise
vorticity
47
Lifting-surface theory - numerical implementation

• 3D vortex-panel methods
• the wing is represented by panels with
  distributed vorticity
• (three-dimensional extension of the vortex-panel
  method in section 4.9)
• Vortex-Lattice methods
• distributed vorticity is concentrated into a
  lattice of horseshoe vortices

A single horseshoe vortex
The vortex-lattice system on a finite wing
48
Chapter 5 Final remarks

• BASIC MATERIAL (2nd ed.)
• Study thoroughly
• Sections 1 to 3 (and the summary)
• Read very carefully (be familiar with the
  contents)
• Section 4 Numerical Nonlinear Lifting-Line
  Method
• Section 5 Lifting-Surface Theory Vortex
  Lattice Numerical Method
• Optional
• Section 6 Delta Wings
• Section 7-8 Historical Notes
• ADDITIONAL MATERIAL (see www.hsa.lr.tudelft.nl/b
  vo/aerob)
• 5.A Numerical Example of the Wing Equation
• Make the Related Problems from the set of
  Exercises!