PROFOIL-WWW

Four-Segment Airfoil (notes page)

As a reminder, the "design page" should be opened in a separate browser window and used interactively while going through these notes. For now the design page can be made by clicking on "next page" once the design page is shown. The values in the template can then be set to those in the discussion that follows.

Several example airfoils will be presented in these notes to illustrate the design method. All airfoils in this series will have four segments - the minimum number of segment. Nevertheless, a surprisingly wide variety of airfoils can be designed with just four segments.

The values presented in the tables are suggested values for the design pages. The notes correspond to these values; but, of course, any values can be used.

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Example 1

The default values of the design page are shown below in the order in which they appear in the form. In the design page, submit the form to design the first airfoil. 
Segment   PHI        ALPHA
1        15.5           10
2        32.2           10
3        45.5            4
4        60.0            4
Cm = -0.15
Ks =  0.35
After the airfoil is designed, the velocity distributions, airfoil shape, converged input file (with PHI2 adjusted, see details below) and airfoil coordinates are returned for display.

As seen above, the design angle of attack for the second segment is 10 deg. For this angle of attack the corresponding velocity distribution is constant for the second segment. Likewise, for the third segment, the velocity is constant for 4 deg.

(As a reminder, the upper curves in the velocity distribution correspond to the upper surface, while the lower curves are for the lower surface. As the angle of attack increases, the difference between the upper and lower surface velocity increases - the greater the difference between the curves, the higher the lift coefficient. Thus, the top curve and the lower curve correspond to the highest angle of attack, which is 14 deg.)

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Example 2

To move the recovery point S2 (see definitions page) further back on the upper surface, the arc limit PHI1 is moved from 15.5 toward the trailing edge to 10.5. 
Segment   PHI        ALPHA
1        15.5 -> 10.5   10
2        32.2           10
3        45.5            4
4        60.0            4
Cm = -0.15
Ks =  0.35
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Example 3

Starting with airfoil 2, the pitching moment is increased from -.15 to -.25 to provide more aft loading and aft camber. Note that still the velocity is constant along the desired segments at the corresponding design angles of attack. 
Segment   PHI        ALPHA
1        10.5           10
2        32.2           10
3        45.5            4
4        60.0            4
Cm = -0.15 -> -0.25
Ks =  0.35
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Example 4

As mentioned in the definitions section, KS is the trailing edge thickness parameter. A small value of makes the trailing-edge thickness KS essentailly zero, while a relatively large value of 2 produces a thick trailing edge. 
Segment   PHI        ALPHA
1        15.5           10
2        32.2           10
3        45.5            4
4        60.0            4
Cm = -0.15
Ks =  0.35 -> 2
(Note: A value of 0 for KS for this example produces an error and no figures or output are produced.)

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Example 5

Starting from example 1, this next airfoil illustrates the effects of changing the design angles of attack. For a design angle of attack of 10 deg for the second segment, the velocity distribution is constant over that segment at that angle of attack. When the angle of attack increases (see results for 12 and 14 deg), the velocity distribution becomes adverse; that is, a high velocity is reached near the nose of the airfoil followed by a rapid deceleration. This feature of the velocity distribution eventually is so extreme that the flow will separate and cause the airfoil to stall.

In a direct design method, the approach usually taken to prevent the airfoil from stalling too early is to increase the airfoil camber or thickness at the nose. One advantage of the current inverse method is that the velocity distribution (in particular, the gradient of the velocity distribution) can be specified to avoid an "adverse pressure gradient" (a rapid deceleration in velocity) until the desired angle of attack is reached.

For the baseline airfoil (Example 1), the design angle of attack for the second segment is 10 deg, which corresponds to a lift coefficient near 1. Stall can be expected to occur shortly after this angle of attack (or lift coefficient) is reached. In the current example, the angle of attack on the upper-surface second segment AFLA2 is changed to 12 deg. Now the velocity distributions are favorable up to 12 deg and stall will occur shortly thereafter. In this case, the lift coefficient is near 1.2. 

Segment   PHI        ALPHA
1        15.5           10
2        32.2           10 -> 12
3        45.5            4
4        60.0            4
Cm = -0.15
Ks =  0.35
As seen the airfoil is somewhat thicker. This has happened because the lower surface was not changed; it must still operate with a constant velocity on the lower surface down to an angle of attack of 4 deg. Below this angle of attack the velocity distributions become unfavorable, and the airfoil can be expected to separate off the nose of the lower surface.

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Example 6

Airfoil 5 is the starting point for this last example. The design angle of attack of the lower surface is changed from 4 to 8 deg. Airfoil 5 behaved well down to 4 deg - the lower surface design angle of attack. The new airfoil only operates well down to 8 deg - the lower surface design angle of attack. The airfoil, therefore, has a narrower operating range (8 to 12 deg) as compared with airfoil 5 (4 to 12 deg). Likewise, the airfoil is also thinner, which could have been anticipated. 
Segment   PHI        ALPHA
1        15.5           10
2        32.2           12
3        45.5            4 -> 8
4        60.0            4
Cm = -0.15
Ks =  0.35
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Some Details

The PHIs and ALPHAs (and a few others not listed) are the only parameters needed to solved all the equations to generate an airfoil. These parameters alone do not guarantee that the resulting airfoil will be realistic. The pitching moment coefficient CM and trailing-edge thickness parameter KS are needed to help ensure that the airfoil is somewhat reasonable. Still, however, the airfoil can be cross and other pecular things can happen. But the airfoil at least will appear on the plot.

To achieve the desired CM and KS values, two of the design parameters are needed for iteration. These two parameters include the leading-edge arc limit PHI2 and the velocity level (not discussed). Thus, PHI2 on the design page will be adjusted by the method in which case making changes to PHI2 on the design page will have no effect.

In the iteration process, the maximum number of iterations is set to 40. If the specified design parameters do not converge within the given number of iterations, the airfoil will not have the desired CM and KS.

For those with some experience with the Eppler code, the pitching moment coefficient CM takes the place of the OMEGAs. The K values have been set equal to 1 in PROFOIL-WWW, but this is not a requirement. The MUs are now determined by the method and cannot be explicitly prescribed unless further iteration performed; that is, in addition to prescribing CM and KS, the MUs can also be specified, provided that additional design variable are allowed to be free for iteration.

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