Thin Airfoil Theory

The analysis methods of low-speed aerodynamics start with the idea that the effect of a lifting body on a flowfield can be simulated by solving the Laplace equation for the right boundary conditions. Thus we will see that the effects of an airfoil section can be simulated if the airfoil is replaced by a "vortex sheet" combined with the freestream. We will adjust the strength of the vortex sheet as needed to ensure that the flow remains tangential along the contours of our desired airfoil (i.e., the solid surface is satisfactority simulated by specifying that the flow cannot cross the contour of the surface.  In its simplest form, the airfoil can be represented by a vortex sheet along the line equidistant from the upper and lower surfaces. The unknown then will be the strength of the vortex sheet required at each point along this mean "camber" line (defined below). In other words,

"What distribution of vortex sheet strength will ensure that the desired boundary conditions are satisfied? "

In most of aerodynamics, we will find ourselves writing these sorts of "boundary condition" equations, and solving for the unknown "strengths" of the "singularities". So where do we actually solve the Laplace equation? Well, the Laplace equation is already solved for us: the flowfield of a source, or a sink, or a doublet, or a vortex, or a vortex sheet, is each a solution of the Laplace equation. Since the Laplace equation is a "linear" equation, its solutions can be "linearly superposed". In other words, you take the solutions for sources, sinks etc., multiply each by some constant, and add them all together, and the result is the solution of the Laplace equation, too.  Each such elementary solution, e.g., source, sink, doublet, vortex, is called a "singluarity".  So, when we find all the unknown "singularity strengths" to represent the airfoil in a freestream, this is a solution of the Laplace equation.

In the following, we use the derivation written out by Professor Sankar.
 

Terminology and Definitions

 

 



An airfoil is defined by first drawing a “mean” camber line. The straight line that joins the leading and trailing ends of the mean camber line is called the chord line. The length of the chord line is called chord, and given the symbol ‘c’.To the mean camber line, a thickness distribution is added in a direction normal to the camber line to produce the final airfoil shape. Equal amounts of thickness are added above the camber line, and below the camber line.

                An airfoil with no camber (i.e. a flat straight line for camber) is a symmetric airfoil.

                The angle that a freestream makes with the chord line is called the angle of attack.

Forces along the x- and y- axes: The forces and moments acting on an airfoil may be computed as follows.

Let Y be the pressure force per unit span (i.e. per unit distance normal to the plane of the paper) along the y- axis. This force may be computed as:

If we subtract off a constant value p¥ from the upper and lower side pressure values, then the force will not change. Thus,

We can non-dimensionalize the above dimensional form by dividing the pressure by the dynamic pressure, and the distances by the chord c. Then,
 


 

We can likewise find the component of force acting along the x- axis. This force and its non-dimensional form are given below:

Lift and Drag: The X and Y- forces act along the x- and y- axes, respectively. Lift is defined as the component of pressure force that is normal to the freestream direction, and drag is defined as the component of pressure force along the freestream direction. If the airfoil was initially located so that the chord line is along the x- axis, then the angle between the freestream direction and the x- axis is the angle of attack a.

Lift and drag are related to the X- and Y- forces as follows:

The quantities Cl and Cd are called the lift, and drag coefficients, respectively. By convention, the lower case subscripts are used in 2-D flows, while upper case subscripts are used to denote lift and drag coefficients of three-dimensional configurations such as wings.

Pitching Moment: We can also define the pitching moment about any point on the chord line. Nose up moment is considered positive. About a general point on the x- axis whose co-ordinates are given by (a,0), the pitching moment per unit span is given in dimensional form by:

The non-dimensional form is given by:

While the pitching moment can be defined about any point in space, it is customary to compute the pitching moment M and the pitching moment coefficient Cm about the quarter chord, i.e. a location 25%c downstream of the leading edge.

Center of Pressure: the center of pressure is defined as the point about which the pitching moment is zero. As the flow conditions change (example, angle of attack a changes), the center of pressure will change.

Aerodynamic Center: The aerodynamic pressure is defined as the point where the pitching moment (or the pitching moment coefficient) is independent of a. That is, if we computed the pitching moment about the aerodynamic center,

The thin airfoil theory to be covered will yield the following results:

The quantity a0 is called the angle of zero lift, since lift is zero at a=a0. In real flows, Cl, Cd and Cm will differ from our theory (hopefully, only slightly) due to viscous effects.

As may be expected, symmetric airfoils will have zero lift at zero angle of attack. Thus,a0 is zero for symmetric airfoils. For Cambered airfoils a0 can have positive or negative, depending on whether they have a positive camber (camber line is convex, i.e. curved up) or negative camber (camber line is concave, i.e. curved down).

Airfoil Geometry Naming Conventions: The following is an excerpt

from: the web site: http://www.desktopaero.com/appliedaero/appliedaero.html
To respect the copyright , and to acknowledge the considerable effort the author of the web site hasmade in preparing the material, this excerpt is printed in italics.

Airfoil geometry can be characterized by the coordinates of the upper and lower surface. It is often summarized by a few parameters such as: maximum thickness, maximum camber, position of max thickness, position of max camber, and nose radius. One can generate a reasonable airfoil section given these parameters. This was done by Eastman Jacobs in the early 1930's to create a family of airfoils known as the NACA Sections.

Example of a NACA 4-Digit Series: Consider the airfoil NACA 4412. The first digit gives maximum camber in % of chord, the second digit gives in tenth of a chord where the maximum camber occurs, and the last two digits give the maximum thickness in %chord.

4                      4                                  12

            max camber                position                       max thickness

        in % chord              of max camber           in % of chord

            in 1/10 of c

After the 4-digit sections came the 5-digit sections such as the famous NACA 23012. These sections had the same thickness distribution, but used a camber line with more curvature near the nose. A cubic was faired into a straight line for the 5-digit sections. 

NACA 5-Digit Series:

2                      30                                12

approx max                 position                                   max thickness

camber                        of max camber                        in % of chord

                        divided by 1/50 of c

The 6-series of NACA airfoils departed from this simply-defined family. These sections were generated from a more or less prescribed pressure distribution and were meant to achieve some laminar flow. 

NACA 6-Digit Series:

63,2-           2       12

Six-locationhalf width  ideal Clmax thickness

Seriesof min Cpof low drag      in tenthsin % of chord

in 1/10 chordbucket in 1/10 of Cl

After the six-series sections, airfoil design became much more specialized for the particular application. Airfoils with good transonic performance, good maximum lift capability, very thick sections, very low drag sections are now designed for each use. Often a wing design begins with the definition of several airfoil sections and then the entire geometry is modified based on its 3-dimensional characteristics.


How does the airfoil develop lift?

                In the previous handouts, we demonstrated that a vortex solution is essential for generating circulation and lift. For some bodies (cylinder, sphere, baseball, golf ball, tennis ball) circulation may be generated by spinning the body. How about airfoils? How is the vortex generated? Where is it stored? Is the vortex strength unique?


To answer these questions, we need to look at the flow over an airfoil when it has impulsively started from rest. At time t=0+ (i.e. shortly after the flow starts), the streamlines look as follows:
 

                In the above figure, there are two stagnation points, one on the lower surface near the nose, and the second on the upper surface near the trailing edge. Some of the flow will have to go around the trailing edge from the bottom surface to the top, before eventually leaving the airfoil.

Because of viscosity, rotation develops at the solid surface and vorticity develops. On the upper surface, the vorticity will be generally clockwise. On the lower surface, the vorticity will be generally counterclockwise. In the small strip of surface downstream of the upper surface stagnation point, the vorticity is counterclockwise.

If we look at the streamline at the trailing edge, it needs to turn around a sharp corner, and move from a low- pressure region at the trailing edge towards the high-pressure region at the rear stagnation point. Viscous effects prevent the flow from doing any of these things. Instead, this streamline separates off the trailing edge. In the process, it convects away the clockwise vorticity on the upper surface between the rear stagnation point and the trailing edge. This clockwise vorticty is thus “shed” into the wake. The resulting flow may be visualized as follows:

 

 

 



                This vortex that has been shed into the wake is called a starting vortex. What is left over on the airfoil is “bound vortex”, i.e. the vorticity that is tied to the airfoil and the upper and lower surface boundary layers. If the net amount of vorticity  shed starting vortex is counterclockwise (+ve), then the net amount of bound vorticty  will be negative (counterclockwise) and vice versa. If we add the bound vorticty around the airfoil, and the vorticity in the starting vortex, the sum will be zero. This is known as Conservation of Total Vorticity.
 
 

                How much vorticity will be shed into the wake? Just the right amount, until the flow near the trailing edge leaves the trailing edge smoothly, without any pressure difference between the upper side trailing edge and the lower side trailing edge.

This requirement that the flow should leave the trailing edge smoothly, and that there be no pressure difference between the upper and lower sides of the trailing edge is called the “Kutta Condition.”

 

 


            The starting vortex has been visualized in wind tunnel based visualization of the flow over an airfoil. Here is a typical image for an airfoil at a negative angle of attack.
 

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