# Student guest blog post: the Wagner effect

The starting vortex at the trailing edge, as visualized by Prandtl in 1934 using a water channel and aluminum particles.

The fourth and final student guest post explains the Wagner effect, and its role in animal flight. We hope you enjoy the series of posts from GW students of animal locomotion!

The previous guest posts are:

This post completes the series, which was inspired by the entertaining cross-comment thread to a Google+ post of Michael Habib (Sept.22, 2013).

# The Wagner effect

by Olivier Mesnard, Prithviraj Pawar, Liangwei (Starson) Li

Both experiments and numerical simulations have shown that the aerodynamic forces acting on a wing accelerating from rest are lower than the values predicted by quasi-steady models (which assume steady-state forces are produced at each instant in time). A transition period is needed before the forces reach the steady-state value. This latency in the establishment of lift was first proposed by Wagner (1925) and studied experimentally by Walker (1931).

Wagner effect: growth of the circulation to a steady-state due to the shedding of the starting vortex (from Sane, 2003).

The Wagner effect can be described in terms of circulation around an airfoil: as a wing starts impulsively from rest, the circulation around it does not immediately attain its steady-state value. Thus, when any wing first starts moving through the air, it must travel several chord-lengths before reaching the steady-state circulation around it (Dickinson & Götz, 1993 & Sane, 2003; see figure, left).

The Wagner effect is the consequence of two phenomena. First, when the wing is accelerating from rest, vorticity is generated around it (called the bound vortex) and shed at the trailing edge, where it rolls up forming the starting vortex (see feature image, above). This trailing-edge vortex can be observed on insect wings, but also on the wings of a commercial jetliner. The starting vortex induces a velocity field that counteracts that of the circulation bound to the wing. The influence of the starting vortex on the airfoil decreases as it is convected away from the trailing edge by the free-stream velocity.

Second, the fluid viscosity slows the development of the bound vortex, so we could conclude that the delay in reaching steady state will be amplified at low Reynolds numbers. However, the experiments of Dickinson & Götz (1993) with model insect wings showed that the delay in build-up of lift is less noticeable at lower Reynolds numbers, in the range $10. (The experiments of Walker used $Re=140,000$.) It appears that the dependence of the Wagner effect on Reynolds number is not firmly established.

### Wagner's function

When designing either an aircraft or a bio-inspired robot, engineers have to explore many design alternatives to optimize performances of the device. This can be very time-consuming. That is why usually only quasi-steady aerodynamic models are employed (which are fast and robust). But those models do not take into account the possible unsteady effects that can occur during rapid manoeuvres or takeoff, such as the Wagner effect.

Using linearized thin-airfoil theory for the impulsive motion of an airfoil in an incompressible flow, a relation can be derived for the unsteady lift generated, as a function of time (Fung, 2002, p. 207). The unsteady lift for an airfoil of chord $2b$, impulsively started to velocity $U$, has the following form:

$L= 2 \pi b \rho U w \Phi(\tau)$

Here, $\Phi(\tau)$ is called Wagner’s function (see a plot, below). It corrects the quasi-steady models when obtaining the aerodynamic forces during the transitional period. The angle of attack, $\alpha$, is assumed small, and $w$ is the downwash velocity: $w=U \sin \alpha \approx U \alpha$.

Wagner's function for an impulsively started airfoil in an incompressible fluid. The value of lift starts at 50% of the steady-state value. (Fung, 2002; p. 207)

Although Wagner's function can be derived analytically, good approximate expressions have been calculated (some of them can be found in Fung’s book).

### The clap-and-fling mechanism

Some insects, as well as some birds, use mechanisms to quickly build the circulation around their wings during take-off, and therefore attenuate the Wagner effect. The clap-and-fling mechanism was first described by Weis-Fogh (1973) to explain how certain insects and birds are able to increase the amount of generated lift.

The mechanism starts before the pronation (rotational movement on the wing between the upstroke and the downstroke) with the apposition of the two wings. The wings are then rotated about the trailing edge. The void created by the separation of the leading edges is rapidly filled with air generating a circulation. It also produces counter-rotating vortices above the animal that give extra lift. Once the pronation is done, the downstroke begins.

The clap-and-fling mechanism is mostly used by insects; however, some birds have developed this function as well. For example, the pigeon uses this technique during lift-off (see photograph) producing a recognizable sound. But mostly, it is observed that the birds do not often use the clap-and-fling mechanism to overcome the Wagner effect (Sane, 2003). This suggests that the Wagner effect might not have much effect on bird flight. Some birds (e.g., pigeon) may be using a clap-and-fling motion just to obtain maximum amplitude by the wings, rather than to overcome the Wagner effect.

Photograph of a pigeon clapping its wings at the top of the upstroke, for take-off. Credit: Joe Hancuff, @joehancuff on Twitter.

### Does the Wagner effect apply to pterosaur flight?

As previously stated, the latency to build circulation applies to any accelerating wings. According to Habib, the Wagner effect is an important factor for any animal flyer during launch and is often underestimated. Even if this delay is more significant at low Reynolds numbers, one could wonder if the Wagner effect affects a pterosaur to get off the ground?

There are two schools of thought about the take-off of the pterosaur: bipedal launch and quadrupedal launch. Witton and Habib (2010), some of the avid proponents of a quadrupedal launch, maintain that the bipedal launch position would have placed the wings in a stall configuration, a real problem to quickly build circulation. Habib (2008) mentioned that the pterosaur could reach steady-state lift faster by having high launch accelerations (more plausible with a quadrupedal launch, using the forelimbs and the hindlimbs) and opening the wing rapidly by flinging it outwards and forward during the ballistic phase, thereby overcoming the Wagner effect.

Alexander and Vogel (2004) suggest that the Wagner effect is magnified at low Reynolds numbers (where viscous effects dominate over inertial effects). However, some results contradict this statement. The experiments of Dickinson and Gotz (1993) at Re=10–103 show little evidence of Wagner’s effect. For the first one-chord-length of travel, the lift increases quickly, under the effect of a finite acceleration. Upon reaching uniform velocity, the lift dips sharply, then slightly increases for about one extra chord length of travel. The authors identify this last stage of lift increase as evidence of Wagner’s. But by $2c$ of travel, steady-state lift is established.

We were not able to find Walker’s paper (1931), but other sources say his experiments at Re=140,000 agree with Wagner’s prediction. Beckwith and Babinsky (2009) conducted some experiments on a flat plate at a Reynolds number about 60,000 that show the Wagner effect is negligible after about one chord length of travel. Considering that the Reynolds number is estimated to range from 104 to 106 for pterosaurs, it is reasonable to think that the latency would be negligible for pterosaurs. Therefore, using quasi-steady models could be adequate to evaluate aerodynamic forces acting on the pterosaur.

Moreover, it’s entirely plausible that the angle of attack during the first downstroke is in the post-stall region. In that case, other unsteady effects, such as delayed stall, will be more important than the Wagner effect.

## References:

• Alexander D.E. and Vogel S. (2004). Nature’s flyers: birds, insects and the biomechanics of flight. JHU Press.
• Beckwith, R. M. H. and Babinsky, H. (2009). Impulsively started flat plate flow, Journal of Aircraft, Vol. 46(6):2186–2188.
• Dickinson M.H. and Gotz K.G. (1993). Unsteady aerodynamic performance of model wings at low Reynolds numbers. J. Exp. Biology, 174:45-64.
• Dickinson M.H., Lehmann F.O. and Sane S.P. (1999). Wing rotation and the aerodynamic basis of insect flight. Science, 284:1954-1960.
• Fung, Y. C. (2002). An introduction to the theory of aeroelasticity. Courier Dover Publications. // Read in Google Books
• Habib M.B. (2008). Comparative evidence for quadrupedal launch in pterosaurs. Wellnhofer Pterosaur Meeting: Zitteliana B28. Pp 161-168.
• Sane S.P. (2003). The aerodynamics of insect flight. J. Exp. Biology, 206:4191-4208.
• Walker, J. A. (2002). Rotational lift: something different or more of the same? Journal of Experimental Biology, 205(24), 3783-3792.
• Witton M.P. and Habib M.B. (2010). On the size and flight of the giant pterosaurs, the use of birds as pterosaur analogues and comments on pterosaur flightlessness. PLoS ONE 5(11).
• Walker, P. B.(1931). Growth of circulation about a wing and an apparatus for measuring fluid motion. Reports and Mem., Aeronaut. Res. Com. no. 1402.
• Michael Habib

Sorry to be jumping in late on this one; I had some travel that kept me away from the blogosphere for a bit. I really enjoyed this article. In particular, this post in the series probably had the most impact on my own thinking. Based on your summary and the additional literature you supplied, I have started to take a second look at the potential importance of circulation delays during launch in pterosaurs, and it may not have been as important as I originally thought. That's always the best part of research - when things turn out differently than you expected!

A few notes that might be of interest:

- For pigeons, it may be that they use clap and fling in order to more rapidly climb out after launch. Pigeons are able to launch and climb out at very steep angles and high accelerations (called burst launch), but they don't have as many hind limb specializations for this behavior as other burst launchers, such as pheasants and quail. It might be that pigeons compensate by reaching maximum CL faster after the initial launch.

- One of the reasons that I have previously suggested that the Wagner Effect needs to be considered in animal launch is the ubiquity of high acceleration launches, even in suspensory animals. Flying animals as diverse as gliding snakes, gliding lizards, birds, and bats use leaping to initiate launch - even in species that can hover. Bats launching from a suspensory position cycle the wings 2-3 times before they let go of the perch, effectively "revving" up the wings. These features could be explained (in part) as adaptations to beat circulation delays and therefore improve initial climb out performance. However, other explanations are possible (for leaping, the primary factor is probably just starting acceleration: pushing against a substrate performs much better than lift starting at zero speed. It gets you "out of the gate" faster. That's why fish use drag to fast-start).

- Pterosaurs probably used relatively low amplitude flapping cycles, which could mean that even a 1-2 chord delay in reaching full circulation might be slightly detrimental. If the Beckwith and Babinsky results applied to large pterosaurs, then it is likely that they would not *require* a more rapid circulation start than what a quad launch already allowed, but it might still slightly improve performance. Rough sums to check this possibility would be an interesting exercise (one which I've been playing with over the last day or so). The wing was probably opened rapidly after launch regardless - a slight improvement in starting CL might have been more of a useful side effect than a primary adaptation, based on what you all have presented here.

- I mentioned the stall issue with bipedal launch. Another issue, if a running bipedal launch is modeled, is trailing edge flutter (since the wing was attached to the hind limbs), which might be something fun to look at. In fact, the whole subject of aeroelastic limits in pterosaurs needs work, and might something for you all to look at in the future. Somewhat paradoxically, quad launch was probably better for tensioning the membrane than biped launch, because a quad situation allows the hind limbs (and thus the trailing edge of the wing) to be set well before the wing opens.

Cheers everyone! Some great work. I am duly impressed.

• Olivier Mesnard

Prof. Habib, thank you very much for your very kind comment! We really enjoyed working on the Wagner effect that is quite unknown and controversial.

It seems that researchers have been investigating it for birds and insects, but not for larger animals. They noticed the presence of the Wagner effect, but mentioned that this effect is dominated by some other unsteady mechanisms at post-stall angle of attack (Weis-Fogh mechanism, delayed stall, wake capture). Those studies have been done for reynolds numbers found in insects/birds. Since Walker in 1931, I cannot find other studies for higher Reynolds numbers.

I got the idea of the bat flapping its wings before releasing from the perch. A nice trick to build circulation before getting airborne. I guess this is not relevant to the pterosaur and its quadrupedal launch.

Based on the kinematics you proposed for the quadrupedal launch, the pterosaur is pushing skyward and moving forward when taking-off. Once in the air, the animal is opening its wings rapidly. If the pterosaur was doing it at high acceleration, that would have been a smart way to get away faster from the influence of the starting vortex.

About the bipedal launch: is it completely inadequate to think that those flyers might have been able to enhance lift, flapping their wings at post-stall angle of attack, taking benefit of a possible leading edge vortex, while pushing upward and forward with their hindlimbs?

I agree with the fact that large pterosaur had to use their forelimbs to launch. But what about smaller species? Do you think a bird-like take-off is a serious hypothesis for those ones?

Another controversy is about the shape of the membrane. The shape is different among models that have been reconstructed during the last decades. You mentioned that the wing was attached on the hindlimbs. What is your opinion about the location where the wing was attached? On its ankle, on its femur?

Again, thank you for your help in this work, we really enjoyed having you on Google hangout last time.

Olivier

• Michael Habib

Great thoughts, Oliver. In terms of bipedal launch, trying to engage post-stall vorticity upon launch is very difficult to do. Delayed stall works well for many flying animals if they already have a bound vortex on the wing. The stall is delayed, in part, because it takes time for the vorticity on the wing to be reduced by the sudden change in wake structure. However, starting at that sort of angle would presumably require an additional mechanism, such as clap and fling (which pterosaurs likely could not use because their shoulder anatomy lacked the required range of motion). This may be why, in practice, only the smallest birds (hummingbirds) get much assistance from their wings during launch. Hummingbirds get about 50% of their launch force from the hind limbs and the other 50% from the wings, according to work by Tobalske and colleagues. All other birds measured to date achieve 80%-95% of their launch power from the hind limbs. There are some water launch birds that push against the water with the wings and feet to take off (a quad launch scenario). These birds achieve much more rapid takeoffs than those that run/leap across the surface of the water with the hind limbs. Otherwise, any bird larger than about 5 grams seems to rely almost entirely on the hind limbs for takeoff.

That said, just because bipedal launch is less powerful does not mean that small pterosaurs never used it. The question largely comes down to whether any pterosaurs were bipedal. All of the trackways we have show a quadrupedal gait. However, we lack good trackways from early pterosaurs. Kevin Padian, in particular, has suggested that pterosaur ancestors were bipedal (see Padian 2008 in Zitteliana). Chris Bennett has suggested a semi-bipedal arboreal precursor for pterosaurs, as well. If pterosaurs had a bipedal ancestry, then early pterosaurs themselves may still have been bipedal, with quadrupedality appearing further up the pterosaur phylogeny. Since the early pterosaurs were also not particularly large in most cases, it may be that the smaller, early forms were bipedal launchers, with quadrupedal launch appearing later (and then allowing for the evolution of giant size, among other traits). If early pterosaurs were bipedal launchers, then I would still expect it to have occurred via leaping acceleration, rather than running.

In terms of the wing attachment, there are a few specimens showing an ankle attachment, and one species that seems to show a thigh attachment. Beyond that, we have very little direct evidence of attachment location. Because the known specimens have hind limb attachments, and these are usually near the ankle, that is the configuration I typically use in analysis. However, it could be that there was more variation in membrane attachment location within pterosaurs than we see in the fossil record. Some species might have had hip attachments, for example, and thigh attachments might have been more common than it appears. For now, the most conservative approach is to assume that ankle attachment dominated. [Interestingly enough, I used to be a proponent of hip attachments for most pterosaur wings - I subsequently changed my mind on this after being shown some critical fossils in Munich by my colleague, David Hone].

• Philip Parsons

Mike Habib,
If you get to read this I challenge you to provide the maths behind your outrageous claims that Quetzalcoatlus Northropi could fly at 80mph; could fly for 12,000 miles non stop (literally); could reach 15,00 ft altitude and yet had a body about the same size as a Labrador dog: a shoulder to hip measurement of less than 2 feet to hold the muscle mass to flap a 34ft wing.
Habib, unless you can provide some maths for the above, I place you as a charlatan and a snake-oil salesman. You are one of those so called 'scientists' who invent facts to fit the answers they have already decided they want. Shame on you. Give me the maths behind these outrageous claims above or withdraw them. Put up - or shut up!
Phil Parsons

• Dear Mr Parsons — Thank you for commenting and noticing our blog posts. But I feel compelled to remind you that this is my home. Some of your language is not appropriate for the respectful dialogue that I hope to host here, an informal but still professional and academic space.
Thus, I would insist that you behave courteously to my guests, in my home. You are also more likely to get a genuine and meaningful reply.

• Phil Parsons MRAeS

Dear Dr Barba,
I have been trying for over a year to get a sensible reply from Mike Habib. My language is strong and disrespectful because my respect for Mike and his theatrical postulations is non-existant. If you do not like the arguments in your 'home' may I suggest you get out into the public arena and debate them there. Just look in Wikipedia and notice the nonsense quoted. You might also ask Mike Habib why his postulations are qualified in parenthesis as (Habib - unpublished). Why hasn't he published and how can he quote from un-published material without causing scepticism in his reader?
Where on the web can we go to continue this discussion without offending your domestic sensibilities?
Incidentally, what are your own views on the aerodynamic capabilities of the late Cretaceous azhdarchidae?

Phil Parsons

• Michael Habib

One other quick note on the Wagner Effect: it can also apply during flight, rather than just at its initiation. From the 2013 thesis of Kristy Schlueter: “…the Wagner effect postulates that in cases where the steady state bound circulation around a wing changes instantaneously, there will be a delayed growth in circulation and lift. Examples of such instances are an impulsive translational motion or an impulsive change in angle of attack”

• JoeMarfice

I was just wondering about that. I come from an entirely different discipline (optical engineering), so my understanding of aeronautic principles is sketchy at best... but it seems to me from the explanations that the Wagner Effect might also limit the lift of a hovering flier (such as a hummingbird), since there's no laminar flow possible (the columns of air ahead and behind the wings are static, so there has to be some turbulence... I think?).

• Michael Habib

Indeed, Wagner Effect constraints have been historically examined mostly in animals with low advance ratios (like hummingbirds and insects). The expectation was that circulation delays would affect hovering and rapid-frequency flapping flight more than it would in larger animals. In practice, the expected delays don't seem to be observed in most cases, and this is probably an indication that small flyers are using some tricks to reduce the delay in reaching maximum circulation. Clap-and-fling is a well known example, but the most prevalent mechanism is probably rapid wing rotation at the end of each half stroke.