AerodynamicsB, AE2115 I, Chapter 4

1
Aerodynamics B - contents overview
Chap. 1 Introduction Chap. 2 Fundamental
Principles and Equations (basic concepts and
definitions) Chap.36 Inviscid, Incompressible
Flow (Potential flows in 2D and 3D)
FUNDAMENTALS
Chap. 4 Incompressible Flow over Airfoils Chap.
5 Incompressible Flow over Finite Wings
APPLICATIONS
2
Review of the results of Potential flow theory

• Assumptions
• irotational
• inviscid
• incompressible
• steady

• Properties
• velocity field is governed by a linear equation
  (Laplace)
• superposition of solutions
• pressure follows from Bernoulli

• Results for a closed body placed in a uniform
  flow
• Drag 0 (paradox of dAlembert)
• Lift only when there is circulation L ? V? ?
  (Kutta-Joukowski)
• Value of circulation ? is not unique (Kutta
  condition)
• solution for ? 0 with source distribution on
  the contour
• solution for ?? 0 with vortex distribution on the
  contour

3
Chapter 4 Incompressible Flow over Airfoils

• 4.1-3 Introduction the Airfoil concept
• 4.4-6 Airfoil Theory principle the vortex sheet
• the Kutta condition
• Kelvins circulation theorem
• 4.7-8 Classical Thin Airfoil Theory for
  symmetrical and cambered airfoils
• 4.9 Lifting Flow over Arbitrary Bodies the
  vortex panel method
• 4.11 Flow over an Airfoil - The Real Case the
  effect of viscosity
• ADDITIONAL MATERIAL (see www.hsa.lr.tudelft.nl/b
  vo/aerob)
• 4.A The Design Condition of an Airfoil
• 4.B Discrete Vortex Representation

4
The concept of the airfoil (wing section)

• Prandtls approach to the analysis of airplane
  wings
• (1) the study of the section of the wing (the
  airfoil)
• (2) the modification of airfoil properties to
  account for the complete wing

z

• What is an airfoil?
• an infinite wing in 2D flow
• the local section of a true wing

x

y
Airfoil section

• Motivation for looking at airfoils
• the wing properties follow from the local airfoil
  properties
• a good model for slender wings (i.e. with large
  aspect ratio)

V?
5
Airfoil Nomenclature
Mean camber line
Trailing edge
thickness
Leading edge
Chord line
Chord c

• NACA method to generate standard airfoil
  series
• airfoil contour mean camber line thickness
  distribution

6
Airfoil Characteristics
Attached flow cl a (inviscid) airfoil theory
7
Limitations of the (inviscid) airfoil theory

• Assumptions - inviscid, irrotational flow
• - incompressible
• What is correctly predicted the pressure
  distribution over the airfoil
• lift and pitching moment
• What is absent viscous effects - boundary
  layer development - friction forces -
  flow separation
• no prediction of drag (D 0!) or maximum lift

Conclusion airfoil theory can reasonably predict
lift and pitching moment as long as the flow
does not separate
8
Example Results of the (thin) airfoil theory
for the NACA 2412 airfoil

• Lift

• Pitching moment

9
Theory the vortex sheet

• Basic idea to reconstruct the lifting flow
  around a body (airfoil) by placing
• many elementary vortices at convenient locations
  in the flow
• (airfoil on the contour, the camber line or the
  chord line)

point vortex vortex sheet distributed
vorticity along a line with variable strength
?(s) ? A segment of length ds acts as a point
vortex with strength ?(s).ds
10
Properties of the vortex sheet (1)
A segment of length ds acts as a point vortex
with strength ?(s).ds

• Induced Velocity (vectorial addition)

Velocity Potential (skalar addition)
11
Properties of the vortex sheet (2)
1. Total circulation around the vortex sheet
total vortex strength
2. Across the vortex sheet there is a jump in
the tangential velocity that is equal to the
local vortex strength
Proof circulation total vortex strength
let now dn ? 0
12
Properties of the vortex sheet (3)
3. There is a pressure difference across the
vortex sheet proportional to the local vortex
strength
(Bernoulli)
4. This pressure difference generates lift on the
vortex sheet
(Kutta-Joukowski)
Total lift
13
Application of the vortex sheet to airfoil
analysis

• 1. Arbitrary shape (thick airfoil)
• vortex sheet on airfoil surface

- Task determine vortex strength ?(s) such that
airfoil surface becomes a streamline of the flow
(numerical solution) - The vorticity sheet can be
seen to represent the (vorticity in the) thin
boundary layer - The lift follows from
2. Approximation for thin airfoil vortex sheet
on the camber line
14
The Kutta condition

• Potential flow with lift is not unique!
• (Circulation ? may have any value)

Potential flow around a cylinder
The same happens for the flow around an airfoil
Which flow occurs in reality? The flow that
leaves smoothly at the trailing edge The Kutta
condition
15
The Kutta condition

• Be aware that the Kutta condition is an
  artificial, additional condition introduced to
  describe an effect that is the result of
  viscosity
• This does not mean that the entire effect of
  viscosity is included correctly, for example,
  there is still no drag!

16
Implementation of the Kutta condition

• Consequences of the Kutta condition

No pressure loading at the trailing edge
Velocity at the trailing edge
Strength of the vortex sheet at the trailing edge
17
The basic concept of the thin airfoil theory

• The airfoil is replaced by a vortex sheet along
  the camber line
• The (variable) strength of the vortex sheet is to
  be determined, such that the camber line is a
  streamline of the flow
• (the flow tangency condition)
• The Kutta condition is imposed to fix the value
  of the circulation of the airfoil ??TE 0

18
The flow-tangency condition (1)

• The (variable) strength of the vortex sheet is
  to be determined, such that the camber line is a
  streamline of the flow

For the total velocity component normal to the
camber line
normal component of the freestream
camber
induced velocity of the vortex sheet
Simplification For thin airfoil the effect of
the vortex can be calculated as if the vortex
sheet is along the chord
19
The flow-tangency condition (2)
normal component of the freestream
slope of the camber line
velocity induced by the vortex sheet
(x is fixed ? is running variable)
20
Resume the basic equations of the thin airfoil
theory
1. The fundamental equation of the thin airfoil
theory the flow-tangency condition (making the
camber line z(x) a streamline)
2. The relation that determines the circulation
of the airfoil the Kutta condition (making the
flow smooth at the trailing edge)
21
The symmetrical airfoil

• Symmetrical airfoil

Coordinate transformation
Solution is given by
22
verification

• is the solution?

Standard integrals (n0,1,2)
23
The symmetrical airfoil (continued)

• Vorticity distribution for the symmetrical airfoil

?
Notethe vorticity distribution is proportional
to the lift distribution on the airfoil
Is the Kutta condition at the trailing edge
satisfied? i.e. ? 0 for ? ?
x
TE
LE
LHopitals rule
YES!
24
The symmetrical airfoil lift
Calculation of the lift
Lift coefficient
Lift slope
25
The symmetrical airfoil pitching moment
Calculation of the pitching moment about the
leading edge
Moment coefficient about leading edge
26
The symmetrical airfoil the center of pressure
and the aerodynamic center
Moment coefficient about leading edge
Lift coefficient
L
Center of pressure
LE
xCP
x
Moment coefficient about quarter-chord point
quarter-chord point is also the aerodynamic
center is independent of ?!
27
The symmetrical airfoil summary

• Vorticity distribution (lift distribution)

Lift coefficient
Lift slope
Moment coefficient about quarter-chord point
quarter-chord point is both the center of
pressure and
the aerodynamic center is
independent of ?
28
4.8 The cambered airfoil
Condition to make the camber line z(x) a
streamline of the flow
The solution for this more general problem can be
written as a Fourier series

• the coefficients An (n0,1,2,...) depend on the
  shape of the camber line z(x)
• the coefficients A0 depends also on ?

Additional terms
Basic solution for the symmetrical airfoil A0
?
Note ?(?) 0, so the Kutta condition is
satisfied
Substitution of the proposed solution in the
upper equation gives (use again standard
integrals)
29
The cambered airfoil finding the coefficients An
The solution can be interpreted as a Fourier
expansion of the function dz/dx
This Fourier series can be inverted to find the
explicit relations for the individual
coefficients An
We can use these expressions in two ways 1.
Analysis determine the coefficients An for a
given camber line z(x) 2. Design determine
camber line z(x) for given coefficients An
30
The cambered airfoil the aerodynamic
coefficients (1)
The lift coefficient
Note for the lift coefficient only A0 and A1
required!
Independent of ?
Lift slope
for every (thin) airfoil!
Zero-lift angle
31
The cambered airfoil the aerodynamic
coefficients (2)
The moment coefficient about the LE
Note for the moment coefficient only A0, A1 and
A2 required!
moment about the quarter-chord point
Independent of ?!
For every (thin) airfoil the aerodynamic center
is located at the quarter-chord point The
quarter-chord point is (in general) not the
center of pressure
32
The cambered airfoil summary

• Vorticity distribution
• (lift distribution)

Relation with the camber line shape z(x)
Aerodynamic coefficients
33
Chapter 4 Incompressible Flow over
AirfoilsAdditional Topics A
4.A The Design of an Airfoil 4.A.1 The design
of a camber line 4.A.2 The design condition of
an airfoil
34
4.A.1 The Design of a Camber Line
Objective to determine the camber line
shape z(x) for given vorticity distribrution
(pressure or lift distribution)

• For given choice of parameters A0, A1,
• this equation is a 1st order dif.eq. for z(x)
• There are two boundary conditions z(0)z(c)0
  (both LE and TE on the x-axis)

Conclusions 1. We cannot prescribe ?, but its
value follows from the solution 2. For a
different value of A0 the same z(x) and (?-
A0) 3. The value of A0 is not important for the
shape of z(x) The camber line is determined by
the coefficients An, with n?1
35
4.A.2 The Design Condition of an Airfoil
What is the reason for applying cambered airfoils?
cd
airfoil without camber
airfoil with camber
? 0 Cl 0 Cm,AC 0
? 0.5 Cl 0.51 Cm,AC -0.106
design condition
cl

• Application of a positive camber gives
• lower drag the minimum drag occurs at positive
  lift
• increase in the maximum lift
• negative Cm,AC

36
The ideal angle of attack
LE
TE

• For the (theoretical inviscid) flow around the
  camber line
• Trailing edge smooth flow at every ? (Kutta
  condition)
• Leading edge smooth flow only for a specific
  value of ? ?opt
• which is the optimal or ideal angle of
  attack

• For any other angle of attack
• the potential flow around the camber line gives
  infinitely large velocities
• the real flow around the true airfoil
  displays large velocity gradients near the nose

37
The design condition (1)

• The design condition in terms of the thin-airfoil
  theory
• compare
• smooth flow at trailing edge (Kutta condition)
• similarly
• smooth flow at leading edge

Implications for the vortex distribution of an
arbitrary airfoil
Near the leading edge
Condition for
38
The design condition (2)

• Conclusion The design condition (smooth flow at
  leading edge) occurs for
• implying

With the earlier results
Note for the symmetrical airfoil
39
The design condition of the parabolic camber line

• For the parabolic camber line we found (Problem
  4.1a)

kzmax /c is the maximum relative camber
So for the design condition we have

• For general camber lines that are symmetrical
  w.r.t. the half-chord c/2

• the lift distribution at the design condition is
  symmetrical
• CP is then at half-chord

Lift
Note longitudinal stability requires
40
Design of a camber line with cl,optgt0 and
cm,c/40
Assume
Computation of the camber line z(x)
z
Integration with B.C. z(0)z(c)0
x
Note negative camber at the tail of the airfoil
41
Chapter 4 Incompressible Flow over
AirfoilsNumerical implementations of the
vortex-sheet method
4.9 The Vortex Panel Method (4.10 in 3rd
ed.) 4.B Discrete Vortex Representation of
the camber line (Additional Topic)
42
The Vortex Panel Method (principle)
Arbitrary shape (thick airfoil) vortex sheet on
airfoil surface

• Numerical implementation
• Approximate the true contour by n straight
  panels
• i1,2,n
• Describe the vortex strength on each panel, e.g.,
  by a constant value of ?i.
• Take on each panel a control point where the
  flow-tangency condition is to be satisfied, e.g.,
  the center of each panel
• Evaluate this condition, for each control point
• here, Ai,j is the contribution of panel j on
  the velocity in control point i
• This system of n equations for n unknowns (?i) is
  singular (the circulation is undetermined), and
  one of the equations is to be replaced by (a form
  of) the Kutta condition, ?(TE) 0.

43
4.B Discrete Vortex Representation of the
camber line (Additional Topic)
Continuous vortex representation of the camber
line

• Simplification
• Discrete vortex representation of the camber
  line
• each panel has one vortex ?1 and one
  control-point
• How must the locations of the vortex and the
  control point be chosen?

?2
?3
?1
44
Discrete vortex representation of the symmetrical
airfoil

• Approximation with a single vortex

• Exact thin-airfoil theory

This result can be reproduced by a single
vortex of strength
placed at x c/4

• Considerations
• Because of the simplification with one vortex,
  the flow-tangency is no longer satisfied at every
  point the camber line is no longer a streamline!
• At what point is the flow-tangency condition
  satisfied, i.e., what point would have served as
  control point?

45
One-vortex representation of the symmetrical
airfoil
w
The velocity induced by this vortex at point x on
the camber line is
?
x
x c/4
V?
The total velocity normal to the camber line is

• Conclusion
• the flow-tangency condition is satisfied only at
  the 3/4-chord point
• we obtain the correct lift and pitching moment
  point by placing the vortex at the 1/4-chord and
  taking the control point at 3/4-chord.

46
Two-vortex representation of the symmetrical
airfoil

• Approximation with two vortices
• Take two vortices, ?1 and ?2 , placed in x1 and
  x2 , respectively.
• Divide the camber line in panels and place a
  vortex in the 1/4-chord point of each panel

w
?
x
x1
V?
x2

• Require lift and moment to be in agreement with
  the thin-airfoil theory

• Determine normal velocity component

• Flow-tangency at
  the 3/4-chord points of each panel!

47
Discrete vortex representation Recipe for the
General Case

• A generalization of the previous results for the
  symmetrical airfoil leads to
• the following recipe to treat an arbitrary
  camber line
• Divide the chord line in n panels j1,2,n
• Place a vortex ?j on the 1/4-chord point of each
  panel j
• Choose the 3/4-chord point of each panel as a
  control point, x j
• Evaluate the condition that in each control point
  the flow must be tangent to the camber line
• This gives n equations for the n unknowns, ?j
• From the values of ?j the lift and pitching
  moment can be calculated
• (Kutta-Joukowski)

Velocity induced in control point xi , by all
the vortices
slope of the camber line at control point xi
48
Chapter 4 Final remarks

• BASIC MATERIAL (2nd ed.)
• Study thoroughly
• Sections 1 to 5 and 7 to 8 14 (summary)
• Read very carefully (be familiar with the
  contents)
• Section 6 Kelvins circulation theorem
• Section 9 The Vortex Panel Method
• Section 11 The Flow over an Airfoil - The Real
  Case
• ADDITIONAL MATERIAL (see www.hsa.lr.tudelft.nl/b
  vo/aerob)
• 4.A The Design Condition of an Airfoil
• 4.B Discrete Vortex Representation
• Make the Related Problems from the set of
  Exercises!