2. 1
Table of Contents
Introduction………………………………………………2
Objective…………………………………………………2
Description of the System………………………………..2
Rajiform Locomotion……………………………...2
Near Substrate Conditions…………………………7
How System Relates to Fluid Mechanics………………...7
Submerged Curved Surfaces………………………7
Fluid Friction………………………………………9
Design Enhancement……………………………………10
Buoyancy & Angle of Attack…………………….10
Conclusion………………………………………………11
References………………………………………………12
3. 2
Introduction:
The locomotive methods of ray fish are unique and efficient. The population of rays is
composed of over 500 different species and each displays one of two main forms of
locomotion: Mobuliform (oscillatory) and Rajiform (undulatory) [4]. This report will be
focusing on the latter; the Rajiform motion and its undulatory propulsion method. This
unique propulsion technique along with Mobuliform has allowed ray fish to be known as
some of the most efficient swimmers in the sea.
Objective:
Nature is full of systems that have been designed and refined over the past 150 million
years. As nature strives for efficiency, product designs based off of natural systems can
prove to provide an advantage over many traditional man-made designs. Nature is a
valuable resource that can serve as inspiration for countless engineering design based
projects and products. Someday I hope to be able to create products or technologies that
have been inspired by nature, as I believe many effective designs lie hidden in plain sight.
I believe that by looking at nature and at some of the mechanisms nature has created
many products and man-made mechanisms can be redesigned to work more affectively
and operate more efficiently.
Description of the system:
Rajiform Locomotion:
Most fish utilize multiple control surfaces that interact with surrounding fluids in order to
induce locomotion. Undulatory rays, however, use only one control surface, the pectoral
disc, in order to swim steadily, accelerate and perform other maneuvers [1]. Rajiform
deforms the pectoral disc into a traveling sinusoidal waveform, which is utilized in order
to achieve locomotion. Wave generation is induced by pectoral muscles located on left
and right sides of the ray within the pectoral fin. The stingray anatomy can be seen in
Figure 1 on the following page.
4. 3
Figure 1
These muscles are responsible for the movement of the leading edge of the pectoral fin.
As the leading edge is moved up and down, the wave takes form and travels the length of
the ray’s pectoral fin from anterior to posterior with multiple wavelengths present at one
time [3]. Modulation of this wave allows for various swimming and maneuvering
capabilities.
Thrust is generated from forces applied to the surrounding fluid. These forces are
resultant and are created from the curved surfaces of the pectoral fin. About 25 percent of
the pectoral disc undulates with significant amplitude greater than 0.5cm [1]. Amplitude
of these waves increases from anterior to posterior until approximately mid-pectoral disc.
After, the amplitude remains constant. Muscles within the length of the pectoral fin
contract and relax in order to maintain the waveform along the length of the fin. Semi-
circular fluid structures form along the fin and are utilized in order to propel the ray in the
desired direction [4].
Figure 2
5. 4
As the sinusoidal-type wave travels along the ray’s fin from front to back, the fin applies
pressure to the surrounding fluid in the same direction the wave is traveling. Forces are
applied to the surrounding water from the top-side of the pectoral fin immediately
following a peak, and from the bottom-side of the pectoral fin immediately following a
trough. Because every action has an equal and opposite reaction, the force applied to the
surrounding fluid results in an opposing force in the opposite direction. This force propels
the ray.
The traveling waves of Rajiform locomotion are what make this specific locomotion type
so interesting and efficient. These waves have the ability to suppress turbulence as the ray
travels through the water and allows the ray to swim in a vertical direction. Efficiency of
this motion is strongly correlated to the Strouhal Number (St) [3].
€
€
St =
fU
d
f = flapping frequency
d = wake width
U = external flow velocity
This is the non-dimensional parameter of the ray’s flapping frequency. According to MIT
research, optimal thrust conditions occur when the Strouhal Number is 0.30 ± 0.05. This
is because within this range, the generated wake has a pattern in which the velocity
profile provides levels of drag that are significantly reduced when compared to higher or
lower St values [3].
Another work-related, non-dimensional parameter that applies to this undulatory, aquatic
propulsion system is the c/U ratio. Where c is the wave speed. This ratio is related to the
turbulence generation of the undulatory motion. When this value is larger than 1, the
wave speed is faster than the external fluid flow speed (in the same direction). This
reduces the possibility of stagnation points to occur in the troughs of the wave because
the fluid is able to flow into and out of the troughs of the wave. Within the troughs,
sustained vortices act as “fluid roller bearings” and result in the reduction of drag forces
[1]. When the external fluid flow velocity is faster than the undulatory wave speed, the
fluid flows over the peaks (because of its faster relative speed) and cannot flow properly
into the troughs. This generates stagnation points and thus creates drag and turbulence
[3].
6. 5
Figure 3
The c/U ratio also affects the power consumption of the animal and efficiency of the
locomotion method. Two energy consumption factors to consider when analyzing the
efficiency of this locomotion method are 1) energy used in order to perform the
swimming motion and 2) energy needed to overcome drag forces. The energy required to
overcome drag forces decreases as the frequency of the wave increases. This being said,
top speeds for rays occur when the frequency of the wave is increased, not the amplitude.
Increasing the amplitude causes larger stagnation points within the troughs of the
undulatory wave and directly causes an increase in drag due to an increase in projected
area [3].
In Figure 4, it can be seen how an increased swimming speed changes the wave timing
along the pectoral fin of a P. orbignyi. On the left hand side (A) the specimen is
swimming at 1.5 disc lengths per second (DL/s). On the right hand side (B) the specimen
is swimming at 2.5 DL/s. From 0 to 100 percent, the ray can be viewed at intervals of
25% of one finbeat (time it takes for one wave to travel the length of the pectoral fin) [1].
It is clear to see that wave frequency increases with swimming speed while average
amplitude remains relatively constant. Also, it can be seen that the central portion of the
ray does not undulate. This is due to the vertebrae column that is composed of a stiffer
cartilage.
7. 6
Figure 4
As stingrays increase swimming speed, the angle between the oncoming flow and the
dorsal surface increases. This increase in angle is relatively small, but significant.
Changes in angle from approximately 5.18° to 7.75° have been seen [1]. However, this
small angle is not used in order to gain swimming altitude. This increased angle is used to
counterbalance inherent negative buoyancy as well as uneven torques generated during
travel. As the thickness of a stingray decreases from anterior to posterior, the cross
section of a stingray resembles an airfoil. However, this shape does increases lift forces if
swimming speed is increased and the angle of attack relative to the oncoming flow is
great.
8. 7
Near Substrate Conditions:
Unlike most other fishes, stingray’s flattened body form allows them to swim in close
proximity to the sea floor or other substrate. This close proximity provides some altered
fluid characteristics due to the ground effect on the fluid layer near substrate. Research
has shown that traveling near substrate can offer benefits to propulsive motions.
Undulating fins have been seen to allow the stingray to travel faster when near a surface.
This allows the ray to reduce the rate of energy exhaustion by approximately 25% as flow
velocity decreases to zero at the boundary [1].
This boundary layer allows stingray and other species to seek shelter from stronger
currents that are present within more open and unbounded fluid regions [1]. Swimming
performance is not just due to swimming in close proximity to substrate, it is also
dependent on changes in pressure and direction related to fin movement. Swimming
technique within stingrays becomes altered when near substrate due to these ground
effects within the boundary layer.
How System Relates to Fluid Mechanics:
Fluid mechanics is the science of fluids and the forces on them. As stingrays natural
habitat is salt and/or freshwater, their mode of transportation relies upon forces being
applied to the surrounding water in order generate thrust. Fluid factors such as viscosity,
density, drag, lift, buoyancy, velocity profile, boundary layer, forces acting on curved
surfaces, stagnation points, turbulence and flow velocity are relevant terms when
analyzing this biological system.
Submerged Curved Surfaces:
Stingrays utilize the concept of forces on submerged curved surfaces in order to travel.
These forces are similar to those related to static curved surfaces, however the curved
surfaces of the stingray’s pectoral fin are not stationary. Instead of generating a force
opposed with an equal and opposite force in order to maintain a static equilibrium state,
the equal forces accelerate the stingray due to the density of water relative to the average
density of a stingray.
Figure 5
9. 8
Calculating Resultant Force:
€
FR = FV
2
+ FH
2
FR = Resultant Force (black arrow in Figure 5)
FV = Vertical Force (blue arrow in Figure 5)
FH = Horizontal Force (red arrow in Figure 5)
Composed of an infinite amount of forces with an infinite amount of directions, resultant
forces act in the opposite direction of travel and are generated during the undulation
action. As resultant forces are generated on the top and bottom sides of the pectoral fin,
the forces generated on the top of the fin accelerate the ray in a forward and downward
direction and the forces generated on the bottom of the fin accelerate the ray in the
forward and upward direction. The vertical components (FV) are used to aid in swimming
altitude and determine the angle of the resultant force (FR). The angle of the resultant
force can be determined using the equation below.
€
tanα =
FV
FH
€
α = angle of resultant force relative to horizontal axis
These forces are transferred from the anterior of the stingray’s pectoral fin to the
posterior. Because of the uniqueness of undulatory locomotion, forces applied to the
surrounding fluids are continuous and contribute to the overall efficiency of this
propulsion method. When a resultant force is applied to the surrounding water, it is
transferred down the length of the stingray. By the time this force is dissipated towards
the rear of the stingray, multiple forces have been generated from the leading edge of the
pectoral fin and are transferred in the same manner. This allows the stingray to achieve
continuous propulsion unlike many other fish species.
The sum of multiple forces generated at once determines the stingray’s rate of travel. As
swimming speed increases, wave frequency increases as well. In theory, an increase in
wave amplitude would be an alternative method in order to increase swimming speed. An
increase in amplitude would generate a larger resultant force acting on the surrounding
fluid, thus providing the stingray with acceleration up to a higher swimming speed.
However, increasing the amplitude would cause the wave speed to decrease. Fluid flow
velocity would be faster than the wave speed and would create stagnation points in the
troughs of the undulatory waves, which would increase drag. Stingrays are known for
being efficient travelers, and increasing frequency rather than amplitude is one of their
efficiency strategies. Increasing the frequency of the waves would allow a greater number
of waves to be present along the pectoral fin at one time. As a larger number of waves
10. 9
occupy the pectoral fin, the total amount of thrust generated increases due to each
additional resultant force. Also, wave speed increase with frequency, which prevents the
occurrence of stagnation points within the wave troughs.
Fluid Friction:
Stingrays secrete an epidermal mucus that helps to heal wounds rapidly due to its
antimicrobial properties. This mucus also reduces fluid friction between the water and the
skin of the ray [2]. Again this biological adaptation has aided stingrays in the pursuit of
energy efficiency.
This defense against water resistance is effective due to the fact that oil (the secreted
mucus) and water do not mix. The interaction is hydrophobic due to the differences in
composition of water and oils. Water molecules are polar so they are attracted to each
other. Oil molecules are non-polar so they do not attract or repel other oil molecules [5].
Figure 6 below shows the behavior of water droplets in close proximity as they attract
and form larger droplets.
Figure 6
Oils do not have this attractiveness, so the forces that water molecules feel do not apply
between oil molecules or between a water molecule and an oil molecule. The lack of
attractive forces keeps the oil separate from the water and reduces the friction as the
water flows over the oil layer. As this mucus is secreted through the skin, a layer is
created that sticks the surface of the stingray due to its high viscosity. This layer acts as a
barrier that keeps the water from coming into direct contact with the skin. Due to the
interaction between oils and water as described above, water can more easily flow over
the body of the stingray with less friction than if there was not a mucus layer.
11. 10
Design Enhancement:
The design of a stingray is one that has been perfected over millions of years of
evolution. This being said, enhancing a design that has been designed by nature is a
difficult task. As I stated in the beginning of this report, many designs found in nature
have been refined to a point of near maximum efficiency. This is precisely why I find this
topic so fascinating. When an engineer designs a product or mechanism, he or she uses
their brain think and to problem-solve. Nature’s designs are not “thought up” and tend to
follow the idea of the path of least resistance among an infinite amount of alternate
routes. This is an entirely different kind of problem solving that eliminates the human
factor. Humans are amazing creatures, but because we are human we tend to
overcomplicate things. Of the many different paths to choose when designing or
re-designing a product or system, chances are the chosen one is not the best or most
efficient for the desired function. Even if the chosen concept is in fact the best choice in
order to achieve a particular function, we do not posses the technology to reproduce
designs as efficient as many in nature. This leaves many opportunities to enhance man-
made designs. Nature, however, does not leave such obvious gaps in design enhancement
possibilities.
Buoyancy & Angle of Attack:
If a stingray stops swimming, it sinks. This is because it is not a buoyant creature as they
do not have large livers containing buoyancy-enhancing oils, nor do they have swim
bladders. Many fish have a swim bladder, which is a gas filled organ that allows them to
control their buoyancy. This organ relieves these fish of the duty of having to keep
swimming in order to maintain a specific depth [6]. Stingrays compensate for this by
adjusting their angle of attack relative to the oncoming water flow. This angle increases
or reduces lift, allowing the ray to adjust its swimming altitude. Although this is an
effective strategy, increasing the angle of attack also increases the overall projected
profile of the ray. By increasing the projected profile, the amount of drag imposed on the
ray increases. The amount of energy needed to overcome this increased-drag state also
increases. This attack angle can also be seen when the ray is simply trying to swim at a
maintained depth. Due to their inherent negative buoyancy, a slight angle is required in
order to oppose the force of gravity as seen earlier in Figure 4. Stingrays’ inability to
control their buoyancy also means they must always be swimming in order to maintain a
specific depth. Because this is true, they are required to exhaust energy while seeking a
vantage point above prey.
A design enhancement that would help stingrays to be more energy efficient would be the
addition of a swimming bladder. This addition would eliminate the need for an increased
and angle of attack while swimming at a constant depth as well as give the stingray the
ability to save energy while idle at a maintained depth. This “hovering” ability would
only be beneficial for hunting purposes if the eyes of the ray were relocated to the front
edge of the snout as their wide flat body would prevent them from seeing any potential
prey on the seafloor. Mainly, this modification would save energy while traveling long
12. 11
distances upwards of 600 miles during migration [6]. By reducing the incurred drag by a
seemingly insignificant amount, energy savings during long journeys such as migration
routes would prove to be substantial. Swimming with zero degree angle of attack, would
effectively reduce the overall amount of drag.
Conclusion:
The Rajiform locomotion method of undulatory stingrays is a unique and efficient means
of underwater travel. Fish are able to swim through water due to propulsive forces their
bodies impose on surrounding fluids; this same concept also applies to an undulatory type
stingray. Stingrays utilize a traveling wave along the edge of their pectoral fin in order to
generate thrust. As the horizontal forces (FH) and vertical forces (FV) acting on the curved
surfaces of the pectoral fin consolidate into a single resultant force (FR) at angle α, the
sum of each resultant force contributes to the overall thrust and locomotion of the ray.
The swimming speed of the ray is dependent upon the frequency of the undulations along
the pectoral fin rather than increasing the amplitude of each wave. Increasing amplitude
slows down the wave speed relative to the oncoming flow and results in stagnation point
generation within the troughs of the waves. Stagnation points incur increased levels of
drag and ultimately result in increased energy consumption. By increasing frequency
rather than amplitude, wave speed is faster than the oncoming flow and mitigates the
generation of stagnation points and drag. In addition to increasing frequency over
amplitude, drag is reduced via epidermal mucus secretions. Epidermal mucus secretions
also contribute to energy efficiency of the stingray. Therefore, the hydrophobic mucus
layer that sticks to the skin of stingrays allows water to slip over the relatively large
surface area of ray fish. Because the oily mucus is a non-polar fluid, the polar molecules
of water are not attracted to the non-polar molecules within the oil. The lack of attractive
forces reduces friction that would otherwise occur between the animal and the water.
Although stingrays are known to be efficient swimmers, the addition of a swim bladder
would net an overall increase in locomotive efficiency. Stingrays have an inherently
negative buoyancy, as they are not equipped with large livers filled with buoyant oils or a
swim bladder. Because of this, the angle of attack is required to maintain a specific depth.
This angle increases the overall projected area of the ray, thus increasing drag. In order to
mitigate this drag while traveling, the addition of a swim bladder would increase the
buoyancy and eliminate the need for this attack angle. During longer journeys, such as
600-mile migration routes, the effect would result in significant energy savings using the
Rajiform locomotion method.
13. 12
References:
1. Blevins, Erin. Undulatory Locomotion in Freshwater Stingray Potamotrygon
Orbignyi: Kinematics, Pectoral Fin Morphology, and Ground Effects on Rajiform
Swimming. Dissertation, Harvard. Cambridge, Massachusetts: DASH. Harvard,
March 18, 2014.
2. Luer, Carl. Novel Compounds From Shark and Stingray Epidermal Mucus With
Antimicrobial Activity Against Wound Infection Pathogens. U.S. Army Medical
Research and Materiel Command Fort Detrick, Maryland: April 12, 2014. 21702-
5012
3. Beem, Heather and Triantafyllou, Michael. Biomimetic Design of an Undulatory
Stingray AUV Fin. Massachusetts Institute of Technology. Cambridge,
Massachusetts: March 3, 2014.
4. Rosenberger, Lisa and Westneat, Mark. FUNCTIONAL MORPHOLOGY OF
UNDULATORY PECTORAL FIN LOCOMOTION IN THE STINGRAY
TAENIURA LYMMA (CHONDRICHTHYES: DASYATIDAE). University of
Chicago. Chicago, Illinois: March 3, 2014
5. Hydrophobic interactions (Hydrophobic interactions)
http://staff.jccc.net/pdecell/chemistry/hydrophob.html
6. Southern Stingray (Dasyatis Americana) (Stingray)
7. http://bioweb.uwlax.edu/bio203/s2009/long_nico/Adaptation.htm
