PARAGLIDER DESIGN HANDBOOK
CHAPTER 2. AEROFOILS
2.3 Joukowsky transformation

2.1 Introduction

It is a fact of common experience that a body in motion throught a fluid experiences a resultant force. In most cases, is mainly a resistence to the motion. In the case of an aerofoil the resultant force normal to the direction of motion is many times grater than the component resisting the motion. The possibility of the flight of the paraglider depends on the use of an aerofoil for the wing structure.

2.2. Cylinder in an airflow

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2.3. Joukowsky transformation

The Joukowsky transformation converts a circle in an airfoil section. Explained here.

2.4 Airfoil geometry and parameters

Airfoil geometry can be characterized by the coordinates (x,y) of the upper and lower surface, and the following:

- Trailing edge
- Chord: line joining the centres of curvature of the leading anf trailing edges.
- Mean or camber line

Geometric parameters that summarize the airfoil:

- maximum thickness (% of chord)
- position of max thickness (% of chord)
- maximum camber (% of chord)
- position of max camber (% of chord)
- trailing edge angle

Fig. 2.1. Airfoil geometry

2.5 Aerodynamic forces

Angle of incidence: alpha defined as the angle between the chord and the direction of motion relative to the fluid

Resultant aerodynamic force: resultant of the pressure distribution

The line of action of the resultant aerodynamic force intersects the chord in a point CP center of pressure

The resultant force is resolved into two components:

The Lift L at right angle to the direction of motion
The Drag D parallel to direction of motion but opposing

Resultant moment

Its value depends of the reference point for moments. The sense is such that a positive moment tends to increase the angle of incidence. Using the leading edge A as a point of reference for moments the magnitude is:

M(A) = -CP · (L cos (alpha) + D sin (alpha))

where CP is the distance of the centre of pressure behind the leading edge of the chord

It can be shown that there is a point on the chord, for which the aerodynamic moment does not vary with the angle of incidence. dM/dalpha=0 This point is called the aerodynamic center AC and has been found experimentally and theoretically that is within a quarter of the chord from the edge.

Then assuming CP=AC+(CP-AC) = AC + e  ( e =
distance between the aerodynamic center and center of pressure )

For small alpha values ​ ​and assuming L>> D can be simplified as follows:

M(A) = -CP · L  or

M(A) = -(AC + e) · L = -AC · L - e · L

By unitary chord AC=1/4 then

M(A) = - e · L - L/4

M(A) = M(AC) - L/4

M(AC) =
moment about the aerodynamic center (does not vary with the angle of incidence)

For L = 0 , M(A)=M(AC)=M(0)        (M(AC) = constant with alpha)

Then

M(A) = M(0) - L/4

The forces and moments can also be expressed by their respective lift, drag, and moment coefficients:

L = Cl · (1/2) · rho · V^2 · S          (rho air density, V= air speed, S = wing projected area)
D = Cd · (1/2) · rho · V^2 · S

Cma = -CP · (Cl cos (alpha) + Cd sin (alpha))

Cma = Cm0 - Cl/4     (Cmo = Cm at zero lift)

The value and sign of Cm0 has an important role in the behavior and stability of the wing:

If Cmo<0 Cma will be more negative when alpha (Cl) increases, and CP moves backward
If Cmo>0
Cma will be positive for small alpha and CP moves forward
If Cm0=0 Cma follows Cl, and CP does not move located at
a quarter of the chord from the leading edge

Cmo < 0 aerofoils will be called "unstable"
Cmo > 0 aerofoils will be called "stable"
Cmo < 0 aerofoils will be called "indiferent"

You may use the three types of Cm0 aerofoils in paragliding. In general, the gliders with Cm0 <0 behave in a more lively and the easy to turn. While whith Cm> 0 we have a more stable wing, but difficult to turn (reflex airfoils). Depending on the use of the wing and its performances can be used either type. The trend is to use profiles Cm0 zero or slightly positive on the wings of competition to temper his natural aggressive behavior. But there is no fixed rule on this issue.

The overall stability of the wing is controlled by the position of center of gravity. Should be discussed in another section.

Fig 2.2. Airfoil resultant

2.6 NACA profiles

The NACA aerofoils can be generated by the use of a set of simple polynomial equations.

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2.7 Paraglider aerofoils

The paraglider airfoils, despite being pressurized fabric, behave like the rest of the gliders.

But besides the geometric and aerodynamic parameters described above, the following are also interesting:

- Position of air inlets
- Position of anchors
- Internal rib holes

Position of air inlets: Must ensure the air entry inside the wing to keep the pressure inside slightly higher than outside pressure. Can be located by theoretical approaches, or even more frequent, by past experiences. Air intakes: 1% to 6% . Values in % of chord.

Position of anchors: It has to do with the transmission of loads to the inflated structure. Usual place from 5 to 2 anchorage points along the chord. As normal in many years has been to use 4 anchors "A", "B", "C", "D". In more hight AR competition wings, may be used only 2 locations on the chord, in order to reduce drag. The position of these anchors must be determined experimentally and from previous experiences. The reduction of anchors may require the use of internal reinforcements to secure the transmission of loads over a wide area.

Internal holes:
The general rule is, to distribute the holes to allow free air circulation between cells, and at the same time maintaining structural strength of the rib.

 Example Max thickness Thickness location Max camber Camber location Inlet begin Inlet end Lo A Loc B Loc C Loc D 1992 Comp 17.09 23.3 3.09 29.8 2.16 6.6 9.6 30 51.1 72.4-85.3 1995 Bi 19.19 24.3 2.06 21.9 1.38 5.83 9.4 32.02 55.9 79.9 gnuEASY 15.52 24.2 4.8 31.6 0.25 6 7.8 33.3 57.7 79.4 gnuLAB1 14 21.4 1.83 39.6 0.12 3 8.5 31 53.5 75.8 Ascender 18 21.9 2.03 15.7 1.2 5 8.5 27.5 53 78.5 gnuLAB2 18 21.9 2.03 15.7 1.2 5.5 8.5 27.5 53 78.5
Table 2.1 Paragliders airfoils. All values expresed in % of airfoil chord

Fig 2.3 Paragliders airfoils

Fig 2.4: 1992 DHV-3 competition airfoil

Fig 2.5: 1995 Bi DHV-2 paraglider

2.8 XFOIL program

XFOIL is an interactive program for the design and analysis of subsonic isolated airfoils. It consists of a collection of menu driven rutines which perform various useful functions such as:

- Analysis of an existing airfoil
- Airfoil design and redesign by interactive modification of surface speed distributions
- Airfoil redesign by interactive modification of geometric parameters
- Plotting of geometry, pressure distributions, and multiple polars

XFOIL is released under GNU General Public License. The detailed manual and the program can be obtained from the website: http://web.mit.edu/drela/Public/web/xfoil/

2.9 XFLR5 program

http://xflr5.sourceforge.net/xflr5.htm

2.10 AIrfoil fitting

LE procedure to obtain the paraglider aerofoil of any wing. Method based on the grid ripstop fabric.
Aerofoil fitting

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