Simulations of dolphin kick swimming using smoothed particle hydrodynamics.

Abstract

In competitive human swimming the submerged dolphin kick stroke (underwater undulatory swimming) is utilized after dives and turns. The optimal dolphin kick has a balance between minimizing drag and maximizing thrust while also minimizing the physical exertion required of the swimmer. In this study laser scans of athletes are used to provide realistic swimmer geometries in a single anatomical pose. These are rigged and animated to closely match side-on video footage. Smoothed Particle Hydrodynamics (SPH) fluid simulations are performed to evaluate variants of this swimming stroke technique. This computational approach provides full temporal and spatial information about the flow moving around the deforming swimmer model. The effects of changes in ankle flexibility and stroke frequency are investigated through a parametric study. The results suggest that the net streamwise force on the swimmer is relatively insensitive to ankle flexibility but is strongly dependent on kick frequency.

Full text

Page 1

Simulations of dolphin kick swimming using smoothed
particle hydrodynamics
Raymond C.Z. Cohena,⇑, Paul W. Clearya, Bruce R. Masonb
aMathematics, Informatics & Statistics, CSIRO, Gate 5 Normanby Rd., Clayton, VIC 3168, Australia
bAquatics Testing and Training Unit, Australian Institute of Sport, Leverrier St., Bruce, ACT 2617, Australia
a r t i c l ei n f o
Article history:
Available online xxxx
PsycINFO classification:
3720
Keywords:
Human swimming
Dolphin kick
Underwater undulatory swimming
Smoothed particle hydrodynamics
Computational fluid dynamics
Stroke optimisation
a b s t r a c t
In competitive human swimming the submerged dolphin kick
stroke (underwater undulatory swimming) is utilized after dives
and turns. The optimal dolphin kick has a balance between mini-
mizing drag and maximizing thrust while also minimizing the
physical exertion required of the swimmer. In this study laser
scans of athletes are used to provide realistic swimmer geometries
in a single anatomical pose. These are rigged and animated to clo-
sely match side-on video footage. Smoothed Particle Hydrodynam-
ics (SPH) fluid simulations are performed to evaluate variants of
this swimming stroke technique. This computational approach
provides full temporal and spatial information about the flow mov-
ing around the deforming swimmer model. The effects of changes
in ankle flexibility and stroke frequency are investigated through a
parametric study. The results suggest that the net streamwise force
on the swimmer is relatively insensitive to ankle flexibility but is
strongly dependent on kick frequency.
Crown Copyright ? 2011 Published by Elsevier B.V. All rights
reserved.
1. Introduction
In competitive swimming, the underwater phase of dives and turns are characterized by motion at
much higher speeds than those maintained during free swimming. An optimal swimming technique at
these times seeks to minimize the momentum loss experienced by the swimmer (Lyttle & Keys, 2004).
This involves the trade off between minimizing the drag forces, maximizing the propulsive forces and
0167-9457/$ - see front matter Crown Copyright ? 2011 Published by Elsevier B.V. All rights reserved.
doi:10.1016/j.humov.2011.06.008
⇑Corresponding author. Tel.: +61 3 9545 8064; fax: +61 3 9545 8080.
E-mail address: Raymond.Cohen@csiro.au (R.C.Z. Cohen).
Human Movement Science xxx (2011) xxx–xxx
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Human Movement Science
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minimizing the physical exertion required of the swimmer. Compared to surface swimming, sub-
merged gliding and swim techniques have decreased overall drag forces because the contribution
from wave drag is negligible at depth (Vennell, Pease, & Wilson, 2006). For these reasons, submerged
dolphin kick swimming (underwater undulatory swimming) is typically used after starts and turns.
The body assumes a streamlined pose with the arms outstretched beyond the top of the head and
the hands placed on top of each other. Transverse (sagittal plane) undulations pass along the length
of the body towards the feet while growing in amplitude. This is a type of subcarangiform motion sim-
ular to that utilized by cetaceans including dolphins. Dolphins, however, have a much higher swim-
ming efficiency than humans, which can be understood from the disparity in morphologies caused
by different evolutionary adaptive priorities (Connaboy, Coleman, & Sanders, 2009). A number of fac-
tors contribute towards the effectiveness of submerged dolphin kick swimming including the kick fre-
quency, kick amplitude and human swimmer morphology.
Alves, Lopes, Veloso, and Martins-Silva (2006) investigated the effects of body orientation on the
kinematics of dolphin kick by analyzing footage of swimmers performing dolphin kick at maximal
speed after diving. They found that a lateral body positioning caused an increase in kicking motion
amplitudes and a dorsal body positioning led to increased ankle extension. Collard, Auvray, and Bel-
launay (2008) postulated that the lateral orientation should be more hydrodynamically efficient than
dorsal orientation and studied this by conducting trials using fourteen swimmers. They found signif-
icant speed increases for lateral orientation over 15 m but a slight disadvantage over 25 m. Sugimoto,
Nakashima, Ichikawa, Miwa, and Takeda (2008) investigated the effects of maximum ankle plantar
flexion angle using a simplified human swimming model with a-priori known hydrodynamic force
coefficients. They found that increasing the plantar flexion angle by around 5? can have a significant
effect on the thrust generated. Using the same approach Nakashima (2009) studied the effect of trunk
undulation and found that for maximum speed using in-phase undulation is optimal and for maxi-
mum efficiency that out-of-phase (see-saw) undulation is best. From this it was concluded that the
torso motion is critical for determining efficiency. Von Loebbecke, Mittal, Fish, and Mark (2009a) con-
ducted a comprehensive kinematic analysis of dolphin kick video footage taken of nine female and
thirteen male Olympic level swimmers. They found that humans and cetaceans have comparable
non-dimensional kick amplitudes but humans have much higher kick frequencies. In non-dimensional
terms, humans have much higher kicks per body length travelled as well as much higher propulsive
Strouhal numbers (St = fD/u1) which are indicative of less efficient swimming. Arellano, Pardillo,
and Gavilan (2003) studied the dolphin kick performance of thirty-two swimmers ranging from na-
tional to international levels. Lower Strouhal numbers were shown to correlate to higher swimming
speeds.
Recently, three-dimensional computational fluid dynamics (CFD) studies have been undertaken
into dolphin kick swimming. Lyttle and Keys (2004, 2006) used a commercial grid-based flow solver
to perform quasi-steady simulations of dolphin kick in order to study the difference between high-fre-
quency low-amplitude kicking versus low-frequency high-amplitude kicking. They concluded that
their large amplitude kick was more effective at speeds less than 2.4 m/s. They also noted that much
of the propulsion came from the legs rather than from the feet. Another research group has performed
dolphin kick investigations using an immersed boundary unsteady fluid solver (Mittal, Dong, Loeb-
beck, Bozkurttas, & Naijar, 2006; von Loebbecke, Mittal, Fish, & Mark, 2009b; von Loebbecke, Mittal,
Mark, & Hahn, 2009). These studies have shown the creation of a vortex ring during the extension kick
which is associated with most of the thrust generation. They also quantified the efficiency (defined in
terms of useful work divided by total work) of human dolphin kick as being between 11% and 29%
across five elite level athletes in comparison to 56% for a cetacean.
Connaboy et al. (2009) recently conducted a review of dolphin kick swimming or as they term it
undulatory underwater swimming (UUS). From this synthesis of a number of studies, several key
shortcomings of the literature were identified. First, there is a need for parametric studies of different
kicking frequencies. Second, studies should be performed to determine in what way large hand oscil-
lations affect the drag forces. There is debate as to whether this is detrimental due to the increases in
frontal area, or whether this is actually beneficial due to vortex control that delays flow separation and
reduces drag. Finally, this review also suggested that the effects of different phase relationships be-
tween the heaving and pitching motions of the feet needs further investigation.
2
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In the present study, dolphin kick swimming video footage and laser body scans of a single national
level male athlete were used. They provide the anthropometric data necessary for generating a kine-
matic model of the swimmer to be used in simulations of the fluid flow around the body. To simulate
the fluid flow around the swimmer model, the Lagrangian mesh-free method known as Smoothed Par-
ticle Hydrodynamics (SPH) is used. The major strengths of SPH are its ability to handle both complex
deforming boundaries and very complex splashing fluid free surfaces. This is demonstrated by its pre-
vious application to dam break problems (Colagrossi & Landrini, 2003; Cummins, Silvester, & Cleary,
2011; Monaghan, 1994), geophysical modeling (Cleary & Prakash, 2004), ship dynamics (Cartwright,
Xia, Cannon, McGuckin, & Groenenboom, 2006), liquid sloshing problems (Rudman, Cleary, & Prakash,
2009; Souto-Iglesias, Delorme, Pérez-Rojas, & Abril-Pérez, 2006) and to casting processes (Cleary, Ha,
& Ahuja, 2000; Cleary, Ha, Alguine, & Nguyen, 2002). There have also been review articles on the SPH
method (Li & Liu, 2002) and on its industrial applications (Cleary, Prakash, Ha, & Stokes, 2007). More
recently SPH has been applied to the problem of marine animal swimming (Cohen & Cleary, 2010).
The aim of this study is to determine the relative importance of the extension kick (often called
downbeat) compared to the flexion kick (often called upbeat) in dolphin kick swimming. This study
also explores the effects of ankle flexibility and stroke frequency on both the propulsion and corre-
sponding flow structures. It is hypothesized that the varying ankle flexibility will only have a marginal
effect and that increases in stroke frequency will have diminishing returns.
2. Numerical method
2.1. Fluid dynamics solver
The present study uses a numerical approach to solve the Navier-Stokes equations which govern
the motion of the fluid. This is done using the Lagrangian mesh-free method known as Smoothed Par-
ticle Hydrodynamics (SPH). While originally developed for astrophysics, it was adapted for use in com-
putational fluid dynamics (Monaghan, 1994). In the SPH approach a scalar quantity / can be evaluated
at point r using the expression,
X
where the summation is over all particles within radius 2h of r. The kernel W has been chosen to use a
cubic spline function with compact support radius of 2h (Monaghan & Lattanzio, 1985). Similarly, gra-
dients of scalar quantities may be evaluated using the expression,
X
where rW is the gradient of the kernel. The SPH continuity equation is expressed as
dq
dt¼
b
/ðrÞ ¼
b
mb
qb
/bWðr ? rb;hÞ;
ð1Þ
r/ðrÞ ¼
b
mb
qb
/brWðr ? rb;hÞ;
ð2Þ
X
mbvab? rWab;
ð3Þ
and the conservation of momentum is expressed as (Cleary, 1998),
?
If b is a boundary particle then the pressure term in the momentum equation is replaced by the
boundary force fak(Monaghan, 1995). Here rab= ra? rb, g is the gravitational acceleration, n is a con-
stant calibrated for the particular kernel (Cleary, 1998) and h is a small parameter to avoid singular-
ities. SPH uses a quasi-compressible approach to avoid having to solve a pressure-Poisson equation at
each time-step. Typically the speed of sound is chosen to be ten times the maximum characteristic
fluid speed to ensure density fluctuations are less than 1% (Monaghan, 1994). The equation of state
used in the solver is
dva
dt
¼ ?
X
b
mb
Pb
q2
a
þPa
q2
a
?
?
n
qaqb
4lalb
ðlaþlbÞ
vab? rab
r2
abþg2
??
þ g:
ð4Þ
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P ¼ P0
q
q0
??c
? 1
??
;
ð5Þ
and the dynamic pressure scale is given by
gP0
r0
¼ 100V2¼ c2:
ð6Þ
The normal component of the interaction of solid surfaces with the fluid are modeled using a pen-
alty force based on the distance normal to the surface at each boundary node. Deforming boundaries
are handled by applying deformation velocities to the nodes on the boundary surface. The surface of
the swimmer in this study is modeled as a deforming solid boundary with prescribed kinematics.
2.2. Human swimmer model
Highly realistic human swimmer models are required to enable all pertinent fluid dynamical effects
to be captured by simulation. To obtain the geometry of an elite level male swimmer, a three-dimen-
sional body scan of an athlete was performed using a VITUS Smart XXL machine (Human Solutions
GmbH, Kaiserslautern, Germany). The swimmer assumed a dolphin kick pose with arms outstretched
beyond his head. Fig. 1 shows the three-dimensional triangular surface mesh which was created. Er-
rors and holes in this mesh were corrected prior to the mesh being ‘‘rigged’’ with internal joints in the
commercial animation package Maya (Autodesk Inc., San Rafael, CA, USA). Subsequent manipulation
of the joint positions and orientations enables smooth deformation of the surface mesh into other de-
sired swimmer poses.
Side-on video footage of an elite level backstroke swimmer performing submerged dolphin kick
swimming was captured using an underwater video camera mounted on the boom of a trolley being
pushed along the side of the pool at approximately the same speed. In this footage the swimmer main-
tains a fixed location within the video frame. From this a representative full period of the stroke was
chosen. The raw footage was in a 25 Hz interlaced standard definition format which upon de-interlac-
ing increased the temporal resolution to 50 Hz but with a halved vertical resolution. This increase in
temporal resolution is particularly important in dolphin kick which in this case has a period of 0.46 s,
corresponding to only 11.5 interlaced frames or 23 de-interlaced frames per swimming cycle. Fine de-
tails of the swimming kinematics would be missed if the footage was not de-interlaced. This present
stroke period of 0.46 s is the same as the averaged value from twenty athletes as measured by Gavilán,
Arellano, and Sanders (2006).
The body surface of the swimmer was carefully animated to visually match each frame of footage
to high precision. This is done by manipulating the joints of the swimmer model alongside the video
footage within the animation software as shown in Fig. 2. The resulting stroke cycle (denoted variant
1) of period T s is shown in Fig. 3, beginning with the flexion (downwards) kick and finishing with the
extension (upwards) kick. The legs undergo asymmetrical up and down strokes because of the joint
Fig. 1. Side-on and top-down views of the swimmer geometry obtained from a laser scan of the athlete. The joints which
constitute the rigging are also shown.
4
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Fig. 2. A single pose of the animated swimmer mesh alongside the corresponding frame of video footage (variant 1).
Fig. 3. Some of the 46 frames which form the dolphin kick animation (variant 1). The significant times denoted are the top
stroke reversal (tEF), the middle of the flexion down-stroke (tF), the bottom stroke reversal (tFE) and the middle of the extension
up-stroke (tE).
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asymmetry, especially in the knees and ankles. The corresponding projected frontal area history
shown in Fig. 4a has a higher peak at the end of the flexion kick (tFE) than at the end of the extension
kick (tEF), which is consistent with the reporting of Nicolas, Bideau, Colobert, and Berton (2007) for
monofin swimming. This is significant because the pressure drag is proportional to the frontal area.
The minimums in projected frontal areas occur when the legs are at the mid-points of the flexion
(tF) and extension kicks (tE). Fig. 4b shows corresponding head-on visualizations of the swimmer at
all these critical times when the projected frontal area reaches minimums and maximums. This single
cycle of motion is repeated for fluid simulations over multiple strokes. The joint angle kinematics of
the hip and knee are shown in Fig. 5a, while the ankle joint angle kinematics are shown in Fig. 5b.
The hips and knees undergo almost sinusoidal rotations with the knees being phase lagged behind
the hips. The ankle rotations are further phase lagged after the bending of each knee. These phase rela-
tionships are consistent with a transverse wave passing along the length of the body.
To investigate the effects of varying ankle flexibility, a number of variants of the stroke were also
generated, as shown in Fig. 5b. Variant 1 matches the video footage. Variant 2 has reduced ankle flex-
ibility where the maximum plantar flexion angle was reduced by approximately 15?. In variant 3,
there is a delayed ankle flick at the end of the extension kick as compared to variant 1. Finally, variant
4 has further reduced ankle flexibility where the maximum ankle plantar flexion angle was reduced by
approximately 25? from that of variant 1.
These stroke variants are identical except for the changes in the ankle kinematics described here.
This ability to isolate and vary single kinematic parameters is one of the major strengths of this mod-
eling approach. The baseline stroke rate matches the video footage and is f = 2.17 Hz. Four other stroke
rates with two slower (f = 1.74, 1.96 Hz) and two faster (f = 2.39, 2.61 Hz) are investigated. These fre-
quency ranges are consistent with average values given by Arellano et al. (2003) for their international
swimmers (f = 2.139 Hz) and their national level swimmers (f = 1.755 Hz). The remaining anthropo-
metric parameters used in this study are given in Table 1.
Simulations are conducted either using a tethered swimmer (for prescribed velocity simulations) or
a dynamic swimmer where the speed of the swimmer is determined by the dynamical forces. Tethered
simulations enable non-equilibrium constant speeds to be considered. This was the approach taken by
von Loebbecke et al. (2009b), von Loebbecke, Mittal, Mark et al. (2009) as well as Lyttle and Keys
(2006). Tethered simulations are likely to result in larger variations of the thrust and drag but are more
representative of commonly conducted experiments where a swimmer is towed by a cable at constant
speed. Additionally, tethered simulations can provide insights into the method of deriving active drag
Fig. 4. (a) The frontal area of the swimmer throughout the dolphin kick cycle (variant 1); and (b) the frontal views of the
swimmer at times tEF, tF, tFEand tE.
6
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from towing cable data (Alcock & Mason, 2007). In dynamic simulations, the swimmer accelerates or
decelerates in response to the balance of forces applied to their body by the water and by their body
on the water. The model swimmer quickly attains a mean equilibrium speed for which there is a bal-
ance between the propulsion and drag over a swimming cycle. Table 2 shows the list of cases inves-
tigated including the four different ankle flexibility variants, four different kicking frequencies and
four different speeds.
2.3. Computational domain and simulation parameters
The computational domain shown in Fig. 6 consists of a rectangular tank full of water with a free
surface at the top. The translational motion of the swimmer is constrained to be in the x-direction
along the tank and the swimmer maintains a depth of 1 m below the water free surface which approx-
imately matches the video footage. In separate trials of starts and turns for a male and female inter-
national level swimmer, the maximum depths reached ranged from 0.8 m to 1.2 m, justifying the
simulation depth chosen here. At this depth the wave drag will have a negligible contribution to
the overall drag. All simulations were conducted with an SPH particle spacing of 4p= 20 mm which
corresponds to approximately 5?106fluid particles within the tank. The boundary particle spacing
is also 4b= 20 mm and the swimmer model contains approximately 1.5?104boundary nodes. The
simulations took approximately 30 cpu days each for the swimmer to complete six cycles of the swim-
ming stroke.
3. Results
3.1. Flow fields for the base case tethered swimmer simulation
The first case considered uses kinematics which closely match the video footage (variant 1 with
f = 2.17 Hz) for the tethered swimmer moving at u1= 1.5 m/s. Visualization of the flow on the sagittal
plane of this virtual swimmer throughout one stroke cycle is shown in Fig. 7. The legs initially flex
to maximum knee bend (t ? 2.0–2.1 s) before kicking upwards (extension kick) to full knee exte-
nsion (t ? 2.1–2.3 s), finally kicking back downwards (flexion kick) and starting to knee flex
Fig. 5. Kinematics of the leg joints for the dolphin kick swimmer. The zero-angles correspond to the pose of the laser scan
shown in Fig. 1. (a) Joint angles of the hips and knee. (b) Ankle joint angles for the different stroke variants (video footage,
reduced flexibility, delayed flick and further reduced flexibility respectively).
Table 1
Parameters of the dolphin kick animated swimmer mesh.
Parameter Value
Toe peak-to-peak kicking amplitude, D [m]
Fingers to toes swimmer height, L [m]
Kicking period from video footage, T [s]
Swimmer internal volume [m3]
0.462
2.35
0.46
0.0774
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(t ? 2.3–2.5 s). These kicks alternately generate fluid structures which travel diagonally up or down
away from the swimmer, as seen by the high speed fluid (shown as red) in Fig. 7a. Also shown are
the instantaneous in-plane streamline patterns in the reference frame of the moving swimmer.
Fig. 7b shows the sagittal plane colored by spanwise vorticity, indicating the amount of rotational mo-
tion of the fluid. This highlights the shear layers formed along the body and the significant vortex
shedding that not only occurs from the legs, but also from the upper body. The extension and flexion
kicks create pairs of alternately signed positive (red) and negative (blue) vortices that interact with the
natural vortices shed by the body. The vortices generated by the extension kick advect diagonally up-
wards and the vortices generated by the flexion kick advect diagonally downwards.
Table 2
Dolphin kicking simulation cases.
Stroke variantSpeed, u1[m/s]
1.5
2.0
2.5
Dynamic
1.5
2.0
2.5
Dynamic
1.5
2.0
2.5
Dynamic
1.5
2.0
2.5
Dynamic
1.5
1.5
1.5
1.5
2.0
2.0
2.0
2.0
2.5
2.5
2.5
2.5
Dynamic
Dynamic
Dynamic
Dynamic
Stroke period, T [s]Stroke frequency, f [Hz] Strouhal number, (St)
1
1
1
1
2
2
2
2
3
3
3
3
4
4
4
4
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0.46
0.46
0.46
0.46
0.46
0.46
0.46
0.46
0.46
0.46
0.46
0.46
0.46
0.46
0.46
0.46
0.575
0.511
0.418
0.383
0.575
0.511
0.418
0.383
0.575
0.511
0.418
0.383
0.575
0.511
0.418
0.383
2.17
2.17
2.17
2.17
2.17
2.17
2.17
2.17
2.17
2.17
2.17
2.17
2.17
2.17
2.17
2.17
1.74
1.96
2.39
2.61
1.74
1.96
2.39
2.61
1.74
1.96
2.39
2.61
1.74
1.96
2.39
2.60
0.67
0.50
0.40
–
0.67
0.50
0.40
–
0.67
0.50
0.40
–
0.67
0.50
0.40
–
0.54
0.60
0.74
0.80
0.40
0.45
0.55
0.60
0.32
0.36
0.44
0.48
–
–
–
–
Fig. 6. The computational domain consisting of a tank of water with air above used for the dolphin kick simulations.
8
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Three-dimensional iso-surfaces of wake structures identified using the Jeong and Hussain (1995) k2
method are shown in Fig. 8. Both the extension and flexion kicks alternately generate vortical
Fig. 7. Flow visualizations on the sagittal plane of the dolphin kick swimmer (variant 1, u1= 1.5, f = 2.17). (a) Colored by
velocity magnitude with in-plane flow streamlines overlayed. These streamlines are integrated over the velocity field u = (ux–
u1, uy) which is in the reference frame of the moving swimmer. (b) Colored by spanwise vorticity.
Fig. 8. Isosurfaces of wake structures identified using Jeong and Hussain (1995) k2method, colored by vorticity magnitude (|x|)
for the dolphin kick swimmer at t = 2.0 s (variant 1, u1= 1.5, f = 2.17).
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structures which emanate from the feet and lower legs. These structures move away from the body in
the same direction as the kick which created them. The structures associated with the extension (up-
wards) kick appear to be coherent toroidal shaped vortex rings which slowly expand as they move
away from the body.
3.2. Forces on a tethered swimmer performing dolphin kick
Fig. 9 shows time histories of the net streamwise forces on the swimmer for each of the ankle mo-
tion variants at each of the tethered velocities for f = 2.17 Hz. The corresponding phases of the stroke
cycle at times t = 2.0–2.5 s are found in Fig. 7. In Fig. 9 the positive force corresponds to a net thrust
and a negative force corresponds to a net drag. Each of these plots shows a similar characteristic pro-
file throughout each cycle. The large decrease in streamwise force prior to time tFEoccurs during the
stroke reversal when the knee attains maximum flexion. This corresponds to a maximum in frontal
area (see Fig. 4) for the swimmer which is the likely cause of this increase in drag. The short duration
high peak in thrust after this (tFE? tE) corresponds to the start of the extension kick. At the end of this
kick cycle (prior to tEF), when the feet are at their highest point there is another local minimum in the
streamwise force. Finally, during the flexion kick (tEF? tF) there is a long duration lower peak in thrust.
As the tethered speed is increased from 1.5 m/s up to 2.5 m/s in Fig. 9, the net streamwise force
decreases due to the increase in drag. The overall amplitude of the force variations however, is un-
changed as the speed is varied, indicating that the thrust generation of the stroke is insensitive to
the speed.
For the ankle variants there are noticeable differences in the heights of the major thrust peaks of
the force profiles (Fig. 9), especially during and after the extension kick (tE) when the corresponding
ankle kinematics actually differ. The minimum values of the streamwise forces attained in the troughs
of the force profiles are similar across the ankle variants. From these profiles it is difficult to discern
which ankle variants produce the largest mean thrust over an entire cycle because the curves fluctuate
from being above and below one another. The time averaged values of these streamwise forces
Fig. 9. Net streamwise forces on the tethered dolphin kick swimmer for different stroke variants (video footage, reduced
flexibility, delayed flick and further reduced flexibility respectively) with f = 2.17 Hz; (a) u1= 1.5 m/s. (b) u1= 2.0 m/s. (c)
u1= 2.5 m/s.
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calculated from the final three kicking periods of each simulation are given in Fig. 10. The amount of
spread in these mean streamwise force values was largest for the intermediate speed case of
u1= 2.0 m/s rather than the faster or slowest case. As the tethered speed increases the mean force
is observed to decrease due to the increase in drag which increasingly overwhelms the thrust gener-
ation that is independent of speed. Since the mean streamwise forces are all negative, there is a net
drag force on the swimmer in all the tethered cases considered.
Fig. 11 shows the evolution of the net streamwise force on the swimmer using variant 1 kinematics
for a range of kicking frequencies and speeds. The different frequency cases have similar force profiles
with no phase differences when the time is non-dimensionalized by the kicking period, T. One notice-
able difference is that higher frequency cases have higher peaks and also lower troughs in the forces.
Increasing the tethered speeds again results in a negative translation in the streamwise force profiles
due to an increase in drag. The amplitude of the force variations do not change with speed meaning
the thrust mechanisms are not changing with the relative velocity of the fluid. Fig. 12 shows that
the mean streamwise force on the tethered swimmer increases linearly with the kick frequency for
all the speeds considered.
Fig. 10. Mean streamwise force versus speed for different variants (video footage, reduced flexibility, delayed flick and further
reduced flexibility respectively) of tethered dolphin kick swimming at f = 2.17 Hz.
Fig. 11. Net streamwise forces on the tethered dolphin kick swimmer for different kicking frequencies with variant 1
kinematics; (a) u1= 1.5 m/s. (b) u1= 2.0 m/s. (c) 01= 2.5 m/s.
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3.3. Forces on a dynamic swimmer performing dolphin kick
Dynamic simulations where the speed of the swimmer is controlled by the forces generated by the
stroke were also conducted for variant 1 kinematics. After a short transient phase, the swimmer at-
tains a speed which has a balance between the propulsion and drag forces over each stroke cycle.
The time histories of the streamwise force and the speed for these dynamic cases are shown in
Fig. 13 for two stroke cycles. The force and speed equilibrate so the mean force over each cycle is
approximately zero. The force profiles are broadly similar in shape to those of the earlier tethered
cases. The beginning phase of the extension kick (around tFE) is observed to generate a peak in thrust
and prior to the stroke reversals (prior to tFEand tEF) generate the peaks in drag. The first part of the
flexion kick (tEF? tF) results in streamwise forces that are constant and close to neutral, meaning this
is a recovery phase of the stroke.
The dynamic swimmer experiences peak-to-peak speed fluctuations that are 30–34% of their mean
speed in response to the variation in the force balances over the swimming cycle. A more common
statistical measure for this is the coefficient of variation for the speed (COV ¼ urms=u1). For the dy-
namic simulations, the COV decreases slightly from 9% down to 7% as the frequency is increased.
The knee flexion period (just prior to tFE) and end of the extension kick (tE? tEF) are responsible for
most of the deceleration of the swimmer. The minimum speed occurs at the end of large drag force
knee flexion period just prior to the extension kick (prior to tFE). The large thrust extension kick
(tFE? tE) creates a large acceleration which results in a peak speed when the net streamwise force
drops back to zero. The flexion kick only has a minor effect on the speed of the swimmer.
Fig. 12. Mean streamwise force as a function of stroke frequency for different speeds of tethered dolphin kick swimming using
variant 1 kinematics.
Fig. 13. Dynamic dolphin kick swimming for different kicking frequencies with variant 1 kinematics; (a) Net streamwise force
history. (b) Speed history.
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The magnitude of the forces (both peaks and troughs) increases modestly with increasing fre-
quency. This means that the higher frequency of kicking increases the thrust generation leading to
higher swimming speeds which in turn leads to increased drag on the swimmer. The speed histories
have similar shapes for all frequencies but the average speed increases and the speed fluctuations
throughout the cycle also increase as the frequency increases. The average speeds achieved by the dy-
namic swimmer are around 0.9 m/s for f = 1.74 Hz and increase linearly up to 1.4 m/s for f = 2.61 Hz.
4. Discussion
This numerical study into dolphin kick swimming has shown that the leg kicks generate vortex
rings; that the extension kick generates more thrust than the flexion kick; that the net streamwise
thrust is relatively insensitive to the changes in ankle flexibility variants considered; and increases
in stroke frequency result in a proportional increase in speed. In this section the implications of these
results are assessed and compared with previously reported studies.
4.1. Flowfields for the base case tethered swimmer simulation
The asymmetry in the kicking technique creates asymmetry in the resulting flow structures. In
Fig. 7a, the relative magnitudes of the flow structures indicate that the extension kick generates higher
speed (and momentum) fluid than the flexion kick. The instantaneous streamline patterns indicate
significant flow separation along the length of the torso during different stages of the stroke. Flow sep-
aration results in large pressure drag which is countered by the motion of the lower half of the swim-
mer which generates propulsion. Fig. 7b indicates that the forcing from the body generates a wake
consisting of diverging lines of alternately signed vortices. Considering Fig. 8, these are revealed to
be three-dimensional toroidal shaped vortex rings. Such structures induce a strongly directed jet
through their core and dissipate slowly so are highly suited to thrust generation (von Loebbecke, Mit-
tal, Mark et al., 2009). The flexion (downwards) kick produces a less coherent vortical structure which
quickly dissipates. From these qualitative observations it appears that the extension kick would pro-
duce most of the thrust for the swimmer. For an athlete to improve this part of their stroke they may
optimize the timing and amplitude of their leg motion along with working to strengthen the required
musculature.
4.2. Forces on a tethered swimmer performing dolphin kick
Tethered swimming simulations enable specified swimming speeds to be investigated. In Fig. 9
there are minimums in streamwise forces just before the stroke reversals (tEFand tFE). This is expected
because of both the increased frontal area which increases drag and the slowing of the stroke which
reduces thrust. This minimum in streamwise force is smaller for the flexion stroke reversal (tFE) which
has a larger projected frontal area (Fig. 4). The extension kick (tFE? tE) generates short duration large
streamwise forces whilst the flexion kick (tEF? tF) produces longer duration smaller streamwise
forces. This is consistent with the flow visualizations which indicate that the extension kick produces
higher momentum flow structures than the flexion kick. This is also consistent with the CFD studies of
von Loebbecke, Mittal, Mark et al. (2009) who showed that peaks in drag occur prior to the stroke
reversal and peaks in thrust occur just afterwards. The forces for the second cycle shown are similar
but not identical to the first cycle, this is due to different timescales of the natural vortex shedding of
the body and the forced vortex shedding caused by the legs. The intracyclic variation in the kinematics
for real atheletes would be another cause of force aperiodicity.
The effect of increasing the tethered speed is to decrease the streamwise force while maintaining
its overall amplitude. As the speed is increased, the peak thrust from the flexion kick becomes closer to
the peak thrust of the extension kick. The implications of this are that when moving at low speeds the
extension kick is far more important to thrust generation but at higher speeds the thrust contribution
from the flexion kick is important also.
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The forces for the ankle flexibility variants have only small differences during the stroke reversal
after the flexion kick (tF? tFE). Since the ankle kinematics across the variants are identical during this
phase of the stroke, this shows that the impact of the ankle flexibility is short lived and does not affect
the large force trough at this time. The differences in the force profiles become less pronounced as the
speed of the cases are increased. The mean values of the streamwise forces for the tethered simula-
tions are given in Fig. 10. The degree of ankle flexibility does not consistently show any particular var-
iant as being more beneficial than the others. For the current dolphin kick leg kinematics the net force
is insensitive to the variations in ankle flexibility. It could be anticipated that the larger displacements
of the lower legs may have more bearing on the thrust generation than the details of the feet motion.
This was also the finding of Lyttle and Keys (2004). Although not shown here, there will be some min-
imum range of motion required by the ankle to provide a propulsive surface. Additionally, flexibility
does play a major role in preventing injury.
For the tethered swimming simulations (Fig. 11), the higher frequency cases have higher peaks and
lower troughs in the streamwise forces. Again increasing the tethered speed decreases the mean force
while maintaining the force amplitudes which indicates that the thrust mechanisms are insensitive to
the relative velocity of the fluid. The mean streamwise forces for these frequencies and speeds given in
Fig. 12 show that for a given increase in frequency there is a consistent increase in force, regardless of
the speed of the swimmer.
4.3. Forces on a dynamic swimmer performing dolphin kick
In comparison to the tethered simulations at the same frequencies, the amplitudes of the force fluc-
tuations in the dynamic cases is smaller (Fig. 13a). This is because the body is able to accelerate and
decelerate in response to the forces which acts to dampen the force peaks. From Fig. 13b there is a
linear increase in mean speed with frequency which is consistent with this swimming motion satis-
fying a constant propulsive Strouhal number relationship (Connaboy et al., 2009). Since the Strouhal
number is indicative of the propulsive efficiency, the efficiency is therefore constant for the different
stroke rates considered for these swimming kinematics. As the frequency is increased, the amplitude
of the speed fluctuations increases but the coefficient of variation for the speed remains small (7–9%).
This indicates that there are only subtle speed surges occurring throughout the stroke. Colman, Persyn,
and Ungerechts (1999) performed dolphin kick experiments on a single athlete which display a similar
coefficient of variation of 8%. By including added mass for the water in a center of mass calculation,
they found a very constant velocity for the center of mass during this stroke. They speculated that this
may explain the faster swimming of this stroke compared to surface butterfly swimming. Barbosa
et al. (2005) found that high intracyclic variation of speed during butterfly stroke is correlated to less
efficient swimming. For butterfly stroke, their intracyclic variations were between 28–59% which are
much larger than those for the present dolphin kick simulations.
While the present SPH method is capable of simulating this application, the significance of second-
ary modeling effects will be addressed in future studies. The fluids modeling may be improved by add-
ing turbulence models designed to handle the transitional flow. Additionally, careful application of
variable resolution particle spacing to improve resolution in regions with smaller scale structures
should result in increased accuracy of skin friction drag. The human swimmer model may be improved
by including two-way coupling of the skin deformation with the fluid. This requires the biomechanical
model to include skeletal structures, fat layers and muscles underneath the surface mesh.
5. Conclusions
A computational study of human underwater dolphin kick swimming was conducted using the
Smoothed Particle Hydrodynamics (SPH) method which enables detailed quantitative analysis of
swimming stroke variations and kineanthropometrical variations.
Flow visualizations revealed that the extension kick generates stronger vortex rings than the flex-
ion kick. The extension kick was shown to be more important to the thrust generation than the flexion
kick. The troughs in forces occurred just prior to the stroke reversals, with the lowest trough
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corresponding to the end of the flexion kick which has the largest projected frontal area. Increased an-
kle flexibility did have an effect on the streamwise force time histories, primarily during and after the
extension kick. However the mean streamwise force values did not consistently show any flexibility
variant as being superior across all the simulated tethered speeds. For tethered simulations at pre-
scribed speeds, the higher frequency cases yielded higher peaks in both thrust and drag. The mean
forces increased linearly with frequency and the rate of increase was the same for the different speeds
considered. Dynamic swimming simulations showed that the peak speed occurs at the end of the
extension kick while the minimum speed occurs just prior to this kick.
The practical implications of this study are that swimmers should focus on maximizing their exten-
sion kick as this generates most of the thrust. The periods prior to stroke reversals correspond to max-
imums in streamwise drag forces so swimmers should pay attention to quick direction reversals. After
starts and turns, increasing the stroke frequency imposes a linear increase in speed. Adopting these
recommendations will help to optimize the underwater stroke technique of individual athletes.
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Please cite this article in press as: Cohen, R. C. Z., et al. Simulations of dolphin kick swimming using smoothed
particle hydrodynamics. Human Movement Science (2011), doi:10.1016/j.humov.2011.06.008