What About Tip Stall?

With the sweep and small chord near the wingtip of a crescent-shaped elliptical planform, it’s natural to wonder if the wingtips will stall before the rest of the wing. This would be undesirable since it could contribute to a tendency to roll off to one side or the other during a stall. Worse yet, the stalled region of the wing is right where the ailerons are located, meaning they could lose effectiveness as the stall is approached.

Understanding Stall Progression

It all starts with lift distribution, which is influenced by the planform. The following diagram shows how lift distribution changes with taper ratio (). The span efficiency () is a measure of how closely the lift distribution matches an elliptical distribution. This also illustrates how the bending moment at the wing root () is affected by the planform.

Because the chord varies with respect to the lift distribution in the three wings above, the lift coefficient also varies along the span. The lift coefficient at any point along the span can be expressed as:

$$ \scriptsize c_l=\frac{l \text{ (lift per unit of span, lb/ft)}}{q \text{ (dynamic pressure, lb/ft}^2\text{)} \cdot c \text{ (chord, ft)}}$$

Therefore the lift distribution curves above can be divided by the chord to give us the lift coefficient distribution:

For the Hershey bar wing on the left, we see that the root of the wing is operating at the highest of any point along the wing and therefore that is the region that will stall first. For the highly-tapered wing on the right, the highest ratio of lift to chord (that is, ) exists closer to the tip, and therefore is vulnerable to tip stall.

For an elliptical planform, the lift distribution is elliptical (although twist and interaction with the fuselage and root fillets affect this); therefore the chord distribution is proportional to the lift distribution and (as a ratio of lift to chord) is theoretically equal to unity across the entire span. This means that the whole wing should “let go” at once. Needless to say, this isn’t a desirable characteristic.

The location of the highest along the span is why you’ll see diagrams in textbooks that look like this:

Scale Effect

Another problem with small chord near the tips is scale effect. The varying chord along the span means that the Reynolds Number () varies also. And since decreases with diminishing , this means the tips are even more likely to stall first. John Thorp, designer of the Piper Cherokee and Thorp T-18 (among many other designs) wrote an excellent little article in the April 1960 issue of Sport Aviation magazine about this, in defense of his preference for Hershey bar planforms. It’s recommended reading.

Thorp points out that there are several methods for dealing with the tip-stall problem:

• Wing twist (decreased incidence at the wing tip)
• Increased section camber at the tip
• Blunt radius on the leading edge of the the tip airfoil
• Wing slots

Thorp rejects twist since it introduces complexity into construction, and rejects the other three strategies since they generally increase drag. However, for the ellipsair, simple construction is not the defining goal. We’ll be using a small amount of washout (twist) and selecting root and tip airfoils to introduce aerodynamic twist.

Sweep

Because the sweep of our planform increases (and rapidly so near the tip), it would be natural to assume this will have a detrimental effect on the lift produced near the tip since lift varies with the cosine of the sweep.

The sweep () of the wing axis (quarter-chord line) can be defined as the arctangent of slope of the wing axis:

$$ \scriptsize \text{sweep [deg]:}\:\: \beta=\tan^{-1}{\left(\frac{d}{dy}x_{c/4}\right)}$$

Now we can graph the lift penalty due to sweep by taking the cosine of :

$$ \scriptsize 1-cos{\beta}$$

…and the lift penalty (as a fraction of total lift available) is represented by the area of the purple region below (as a fraction of the dashed purple box):

$$ \scriptsize \frac{2}{b} \int_{0}^{\frac{b}{2}} 1-cos{\beta} \ dy \ =0.027=2.7\text{%}$$

We’ll take this 2.7% penalty into account later when we determine what the whole-wing lift coefficient really is.

Vortex Flow: A Mitigating Factor

As mentioned briefly in previous posts, swept elliptical wings create a vortex flow near the tips at high angles of attack (not unlike a highly-swept delta wing) which attaches to the upper surface and tends to delay the stall as compared to elliptical wings without sweep. Look for the papers by van Dam and Mineck in the resources page for more on this.

In the next post, we’ll examine how our choices of washout and aerodynamic twist affect stalling characteristics and drag.