Tuesday, July 10, 2012

Aquaflyers Again: Skates and Rays

Previously I wrote a bit about the wonders of aquaflying in penguins.  This time, I thought it would be fun to write briefly on some of the interesting details of aquaflying in skates and rays.

Not all rays are aquaflyers in the sense I am using here.  Many rays propel themselves by moving a series of waves down either pectoral complex like this.  I'd like to talk more about that in the future, but for now, I am talking about those rays that propel themselves by flapping underwater flight - that is, reciprocating the entire pectoral complex on either side of the body as wings, like this.

Aquaflying rays, such as cownose rays, often move in large groups (see photograph above by Chris Hobaugh.  She retains all copyright; do not use without permission).  There are some quite interesting potential dynamics there in terms of motion in neighbor wakes and combined tip vortex effects, but so far as I am aware there are no data on those aspects for rays, so there isn't much that can be said on it for now (though there might be something on it down the road, hint hint).

One thing that is known, however, is that aquaflying rays move with almost absurdly large advance ratios.  Ridiculous, even.  To understand what this means, we need to examine the idea of advance ratios.

For an airplane with a propeller, advance ratio is simple: it is the ratio of the forward speed over the product of the revolution rate of the propeller and the diameter of the circle made by the propeller blades.  So, we have:

Advance Ratio = v/(f * d)

Where v is forward speed, f is the rate of propeller spin, and d is the diameter of the swept disc.

For a flapping animal, we have to take into account the reciprocating wings/fins, and this can be done using amplitude as an added variable (see Ellington, 1984; Vogel, 2003).  So, this gives us:

Advance Ratio = v/(2*r*f*l)

Where v is forward speed, r is the amplitude of the stroke (in radians), f is the flapping frequency, and l is the wing length.  To get a number you can compare to an airplane or other machine using a propeller, multiple by π.

Now, flapping swimmers often do quite well.  Penguins, for example, manage advance ratios around 0.5, which is quite good for motion in water (Hui, 1988).  However, cownose rays exceed an advance ratio of 2 (Heine, 1992).  This is an extraordinary amount of forward motion for each wing cycle.  The trick is that they use their entire bodies as aquafoils, and therefore get lift (mostly as thrust) not just from motion of the "wings", but also from motion of their bodies.

Now, one thing that's interesting about this in rays is that, theoretically, they should be able to get a highly mirrored stroke.  I mentioned the issue of mirrored strokes in the penguin post, and if you want a more technical discussion check out Habib (2010).  The upshot is that if both the upstroke and downstroke produce similar amounts of thrust, then the animal will proceed at a relatively constant speed, rather than lunging forward on each downstroke.  That "lunging" is called a surge acceleration.  The orthogonal motion (up and down for an aquaflyer) is called a heave acceleration.

Aquaflying animals can waste a lot of energy in surge accelerations if they don't have equal phases to their swimming strokes (example: puffins).  Rays probably have very small surge accelerations, because their stroke cycle is close to a true sine wave and their bodies (which are the aquafoil) are relatively symmetrical in the dorsal and ventral aspects. However, to my knowledge this has not been examined in detail despite the fact that accelerometer data do exist for rays.  If this prediction is accurate, however, rays are getting the best of two worlds of aquatic efficiency: high advance ratios from using the entire body for thrust, and low surge accelerations through stroke mirroring.  Presumably this comes at the cost of some additional heave acceleration, but it's still an awfully good bargin, and some of the rays can get a good head of speed going, too.  So much so, that they can do things like leap multiple body lengths out of the water.  See photo at left (taken from here).  More photos of leaping mobula rays by Barcroft here.


Ellington CP. 1984. The aerodynamics of hovering insect flight. Philosophical Transacations of the Royal Society of London, Series B. 305: 1-181

Habib M. 2010. The structural mechanics and evolution of aquaflying birds. Biological Journal of the Linnean Society. 99(4): 687-698

Heine C. 1992. Mechanics of flapping fin locomotion in the cownose ray, Rhinoptera bonasus (Elasmobranchii: Myliobatidae). Ph.D. dissertation, Duke University, Durham NC

Hui  CA. 1988. Penguin swimming. I. Hydrodynamics. Physiological and Zoology. 61: 333-343

Vogel S. 2003. Comparative Biomechanics. Princeton University Press. 580 pp


  1. Mike, as a naval architect, I think of advance ratios in terms of marine propellers and one thing that the design charts show is that as advance ratio increases, so does efficiency. It is also accompanied by a reduction in disc loading - so for example in boats you want a large diameter, slow turning propeller for maximum efficiency, but are generally limited in what is possible by practicalities of draft and the size of the reduction gearbox. By using their whole body and the full span of their wings, batoids are achieving low disc loading and can consequently operate at high advance ratios.

    The other thing that comes into this is the Strouhal number since high efficiency in flapping propulsion is limited to a narrow band of St values (Nudds et al 2004 for example). It would be interesting to look at this for rays.


    1. Greetings Colin. Excellent commentary regarding the disc loading angle. One thing I didn't mention, but would be interesting to study, is that the relatively amplitude of the body section vs the wing tips appears to shift quite a bit in batoids at faster speeds. As such, there may be some interesting effects of gait on advance ratio in those animals beyond what is seen in other swimmers/flyers.

      I'm not sure if St Number has been examined in rays, but I think it has. The tricky bit there is making sure that the value is cross-comparable: while Strouhal number is technically applicable to any propulsion by reciprocating fins or wings, it does have a "characteristic length" term that needs to be applied consistently. Since the relevant length value can sometimes be slightly subjective, it is not always clear that St values calculated for different animal body plans are as cross-comparable as we would like. Using the stroke amplitude works well, and this is what Taylor et al. 2003 and Nudds et al. 2004 have done to good effect.

      For those interested, the original paper on St Number in animals, by Taylor et al., is here: http://www.nature.com/nature/journal/v425/n6959/abs/nature02000.html

      It will require a subscription to read. If you don't have one, a good summary (complete with some of the original figures and data) is here: http://style.org/strouhalflight/

  2. Mike,

    I was watching a nature program on the box tonight - with pictures of a turtle swimming. They appear to be more or less neutrally buoyant yet they have a very asymmetric swimming stroke, and the propelling flippers are clearly have a cambered section - to be more effective on the down-stroke. Any thoughts on why they do not exploit the advantages of a mirrored stroke?

    These Youtube vids show what I saw.



    1. Hey Colin,

      I do have an idea of what is going on there, but it's important to first note the difference between a mirrored stroke in terms of fluid force production, and a kinematically symmetric stroke. The most efficient system, without any constraints, is to have a stroke pattern that is both kinematically symmetric and hydrodynamically mirrored. However, physical constraints sometimes prevent this.

      In birds, for example, constraints of ancestry and competing methods of motion prevent auks from obtaining either form of symmetry. Penguins have fewer constraints (no aerial locomotion), and this allows them a nearly mirrored stroke cycle hydrodynamically speaking. However, penguins still have the ancestral baggage, and so they can't use a kinematic on each halfstroke that's identical. They work at a greater angle of attack on the upstroke than the downstroke, to compensate for the slower upstroke speed. This means that they do produce a bit more drag than the ideal case.

      Marine turtles seem to have the same issue as penguins. Aquaflying turtles are derived, most likely, from drag-propelled ancestors. I've actually been collecting some comparative data from a range of turtles, including marine and freshwater aquaflyers, and most of the morphological change happens in the distal limb - they still run the apparatus off of relatively conserved pectoral and pelvic girdles.

      All that said, there is also another factor likely involved for turtles, which is that they are a bit negatively buoyant (this is apparent because they can lay on the bottom to sleep). I'm not sure how deep they have to go before becoming negatively buoyant, but I suspect that issue is driving some of the cambering of the flipper section.