A Practical Analysis of Unsteady Flow around a
Bicycle Wheel, Fork and partial Frame using CFD
Matthew N. Godo1
Intelligent Light, RutherFord, NJ, 07070
David Corson2
ACUSIM Software, Inc., Clifton Park, NY 12065
and
Steve M. Legensky3
Intelligent Light, RutherFord, NJ, 07070
CFD is used to study air flow around a rotating bicycle wheel in contact with the ground,
extending previous ‘wheel-only’ work on this problem by including the fork, head tube, top
tube, down tube, caliper and brake pads. Unsteady simulations, using a Delayed Detached
Eddy Simulation (DDES) turbulence model, were run for 9 different wheel and front fork
configurations, over 10 different operating conditions (5 yaw angles, repeated for two
different speeds, commonly encountered by cyclists), resulting in 90 transient design points.
A novel co-processing approach and a new automated concurrent postprocessing
methodology is introduced here to make the task of studying many transient cases practical.
These developments led to a greater than 30-fold reduction in disk storage requirements,
and up to a 20-fold reduction in the postprocessing time on a per time step basis.
Concurrent postprocessing, implemented with a job scheduling system, permitted as many
as 40 simultaneous postprocessing jobs to be run, further reducing the time to complete the
data analysis. The contribution of aerodynamic torque to the overall power requirements
for rotating wheels in contact with the ground is presented here for the first time.
Fundamental performance characteristics for the drag force on the wheel and fork, the
turning moment and overall power requirements are reported here. Differences noted for
the design configurations studied suggest that the selection of not only the wheel but wheel
and fork combination is a critical consideration for amateur and professional cyclists and
triathletes. The co-processing and postprocessing methodologies introduced here are
expected to have practical relevance to all transient CFD analyses.
Nomenclature
CD
CS
CV
D
FD
FS
FV
f
M
P
p
R
= drag coefficient (= FD/0.5 V2S)
= side force coefficient (= FS/0.5 V2S)
= vertical force coefficient (= FV/0.5 V2S)
= nominal bicycle wheel diameter
= axial drag force
= side (or lift force)
= vertical force
= frequency
= torque (aerodynamic)
= power
= pressure
= nominal bicycle wheel radius
1
FieldView Product Manager, Intelligent Light 301 Rt 17N, Rutherford NJ, Senior Member AIAA.
Senior Analyst, 634 Plank Road, Ste 205, Clifton Park, NY
3
General Manager, Intelligent Light 301 Rt 17N, Rutherford NJ, Associate Fellow AIAA.
2
1
American Institute of Aeronautics and Astronautics
S
St
Tx
t*
u
V




B
= reference area, D2/4
= Strouhal No. (= fD/V)
= x shear stress component, used in torque calculation
= dimensionless time (= tV/D)
= velocity vector
= bicycle speed (in direction of travel)
= yaw angle
= viscosity
= density
= rotational speed
I. Introduction
y closely examining the finishing times of professional cyclists and triathletes over the past few years in several
events such as the Tour de France, the Giro D’Italia and the Ironman World Championship, it has been
recognized that small performance gains, on the order of roughly 3%, have highly significant relevance1,2. For the
professional cyclists, such time gains can mean the difference between standing on the podium or not after many
days of competition in a grand Tour event. For professional and amateur triathletes who face strong competition to
secure very limited slots at qualifying events, these small differences determine whether they will be able to
compete at the World Championship event – a highly coveted goal in this athletic community.
In recent years, as manufacturers of racing bicycles and bicycle components have turned to wind tunnel testing
to optimize component design3, the athletes themselves are now able to purchase time in wind tunnels to refine and
perfect their riding positions. Comprehensive reviews by Burke4 and Lukes et.al.5 cite many efforts which validate
the conventional wisdom that the main contributors to overall drag are the rider, the frame including fork and
aerobars, and the wheels. Greenwell et.al.6 have concluded that the drag contribution from the wheels is on the
order of 10% to 15% of the total drag, and that with improvements in wheel design, an overall reduction in drag on
the order of 2% to 3% is possible. The wind tunnel results of Zdravkovich7 demonstrated that the addition of simple
splitter plates to extend the rim depth of standard wheels was seen to reduce drag by 5%. In the mid 1980’s,
commercial bicycle racing wheel manufacturers started to build wheels with increasingly deeper rims having
toroidal cross sections8,9. The experimental studies published by Tew and Sayers10 showed that the newer
aerodynamic wheels were able to reduce drag by up to 50% when compared to conventional wheels. Although
many wind tunnel tests have now been performed on bicycle wheels, it has been difficult to make direct
comparisons owing to variations in wind tunnel configurations and the testing apparatus used to support and rotate
the bicycle wheels studied. In the works published by Kyle and Burke11 and Kyle 12,13 it was observed that a very
significant reduction in drag was measured for rotating wheels when compared to stationary ones. Although the
experimental work of Fackrell and Harvey14,15,16 was applied to study the aerodynamic forces on much wider
automobile tires, it is worth noting that they also observed a reduction in drag and side forces when comparing
rotating and stationary wheels.
To date, far less work has been done to apply CFD to study the general problem of flow around a rolling wheel
in contact with the ground. Wray17 used RANS with standard k- and RNG k- turbulence models to explore the
effect of increasing yaw angle for flow around a rotating car tire. He observed that the yaw angle was seen to have a
strong association with the degree of flow separation occurring on the suction side of the wheel. A shortcoming of
this study was the lack of experimental data available for comparison and the authors note this as a direction for
future work. McManus et.al.18 used an unsteady Reynolds Averaged Navier Stokes (URANS) approach to validate
CFD against the experimental results of Fackrell and Harvey for an isolated rotating car tire in contact with the
ground. The one-equation Spalart-Allmaras19 (S-A) model and a two equation realizable k- turbulence model20
(RKE) were used. The time averaged results from this study showed good qualitative and quantitative agreement
with experimental data. The authors suggest that differences between their predictions on the rotating wheel and the
wind tunnel results for surface pressures at the line of contact was due to errors in experimental method and not the
2
American Institute of Aeronautics and Astronautics
result of shortcomings of
o the numericcal approach th
hat they emplooyed. The S-A
A turbulence model was nooted to
provide ressults which weere in better agrreement with th
he experimentaal data than thee RKE model.
With reespect to the prroblem of usin
ng CFD to stud
dy the performaance of bicycle wheels, we hhave previouslyy used
a novel methodology
m
to
o validate com
mputed drag results against w
wind tunnel ddata1 for an exxisting commercially
produced bicycle
b
wheel (Zipp 404). Steady and unsteady
u
compputations were performed ussing the comm
mercial
solver, Acu
uSolveTM,21 oveer a range of sp
peeds and yaw angles. We obbserved a transsition from the expected dow
wnward
acting forcce to the upwarrd direction at a yaw angle beetween 5 and 8 degrees. By breaking the rresolved forcess down
into individ
dual contributiions from the tiire and rim, hu
ub and spokes, something nott possible withh wind tunnel teesting,
we observeed this unexpeected transition
n to be limited to just the tiree and rim. Thiis work servedd to demonstraate that
the tire and
d rim were resp
ponsible for neearly all of thee aerodynamic performance eeffects observeed. We subseqquently
extended our
o work to cover
c
six diffeerent commerccially producedd wheels, nam
mely the Rolf Sestriere, HE
ED H3
TriSpoke, the Zipp 404,, 808 and 1080
0 deep rim wh
heels and the Zipp Sub9 dissc wheel2. Drrag coefficientts, CD,
calculated from the CF
FD simulationss compared very
v
well withh previously ppublished expperimental data (see
Greenwell et al.6, Kyle et
e al.4,11-13, Tew
w et al.10, Prassuhn22 and Kuuhnen23) and uunpublished daata provided byy Zipp
Speed Weaaponry, Indiana, USA. Unsteeady Delayed Detached
D
Eddyy Simulation (D
DDES) calculaations, run at a single
critical yaw
w angle of 10o, supported th
he general conclusions that pperiodic sheddding was obserrved, and, seenn to be
different fo
or each wheel. Further, the calculated
c
Strou
uhal number w
was seen to deccrease as the rim
m depth of the wheel
increased. The one exceeption to this ob
bservation wass for the HED H3 TriSpoke – in this case, the Strouhal nnumber
matched th
he passage of th
he characteristiic three wide, individual
i
spokkes.
In thiss work, we extend
e
our prrevious
efforts to model the flo
ow around a rotating
r
front wheeel of a bicyclee in contact with
w the
ground. Our
O objectives are two-fold. First,
we will make
m
the prob
blem geometry
y more
realistic by
y adding the front
f
fork, calliper &
brake pads, head tube, top tube and
d down
tube. Seco
ond, we recog
gnize that the physics
p
of a rotatiing wheel are best modeled
d using
transient as
a opposed to steady
s
state methods.
m
Notably, we recall from
f
our prrevious
modeling efforts that periodic trransient
shedding effects have a major imp
pact on
resolved forces
f
and con
ntrol. However, full
transient analyses
a
are prrohibitive in teerms of
time and disk
d space requ
uirements. Wee apply
a
newlly
develop
ped
co-proccessing
methodolo
ogy to dramaticcally reduce both the
time and reesource requireements needed
d to run
transient calculations. The co-proccessing
ogy makes it practical
p
to ex
xamine
methodolo
several different
d
wheeel and frontt fork
combinatio
ons over a widee range of yaw
w angles
and speedss.
Figure 11: Wheel and F
Fork geometrries studied
II. Methodology
M
A. Wheel,, Fork and Frame Geometrry
A profiile gauge was used
u
to measurre the cross secctional profile of the producttion Zipp 404 aand Zipp 1080 wheel
(Zipp Speeed Weaponry, Indiana,
I
USA) and the HED H3 TriSpoke w
wheel (HED C
Cycling Produccts). Each wheeel was
modeled with
w a Continen
ntal Tubular tire attached and inflated to 1200psi. A hub prrofile, also meaasured from the Zipp
404 wheel using the proffile gauge, wass used for the Zipp1080
Z
wheeel as well. The HED H3 TriS
Spoke wheel huub was
measured separately.
s
Spoke cross sectiions for the Zip
pp404 and Zippp1080 were baased on the com
mmercially avaailable
CX-Ray sp
poke (Sapim Race
R
Spokes, Belgium). The
T radius of ccurvature of thhe leading andd trailing edgees was
3
Americcan Institute off Aeronautics aand Astronautiics
estimated to
t be 0.5 mm. Spoke coun
nts and spacing
g matched the pproduction whheels. The proffile gauge wass again
used to obttain cross sectiional representations at several axial locatioons along two fforks studied, nnamely the Reyynolds
Full Carbo
on aero fork (R
Reynolds Cycliing, UT, USA, and the Blacckwell Time Bandit (Blackw
well Research, A
Austin
TX, USA). Axial cross sections were then combineed to create thee three dimenssional represenntation for eachh fork.
be geometry were
w based on Chris
C
King heaadset (NoThreaadSetTM, 2005). The top tubee had a
The headseet and head tub
constant diameter,
d
(30.5
5 mm) matchiing that of cu
ustom racing bbicycle (2005 Elite Razor, Elite Bicycless Inc.,
Pennsylvan
nia, USA). Th
he down tube cross section was constant aloong its length aand was measuured using the pprofile
gauge meth
hod. The 72o angle
a
between the top tube an
nd the front forrk (and head tuube), and the 49o angle betweeen the
top tube an
nd down tube matched
m
that of the previouslly mentioned ccustom racing bbicycle. A nottional caliper aand set
of brake pads were also included for the
t models con
ntaining forks. The final CF
FD mesh geom
metries used foor each
wheel are illustrated
i
in Figure 1.
The geeometry of tim
me trial and triiathlon
bicycles positions much of the weight of the
rider direcctly over the ffront wheel thhrough
the use of aerobars. To oobtain an estim
mate of
the contacct area betweeen the road annd the
front tire, a bicycle (20005 Elite Razorr, Elite
Bicycles Inc., Pennsyylvania, USA)) was
mounted oon a training ddevice, a 170lbb rider
was posittioned on thee bicycle, annd the
contact areea of the defleected tire was ttraced.
The contacct area was staandardized as ppart of
the modeel geometry uused in this study.
Under actuual riding condditions, variatiions in
the road surface woulld lead to vvertical
translationn of the wheeel, thus varyinng the
contact areea. It is felt thhat a ground ccontact
Figurre 2: Methodo
ology for mesh
hing the wheell geometry
considerattion is impoortant, and tthat a
standardizzed ground ccontact shape is a
reasonablee approximatioon.
D
B. Grid Development
Geometry and meshin
ng was fully au
utomated throu
ugh the develoopment of paraameterized jourrnals for GAM
MBIT24
25
and TGrid . The moddel domain foor the
problem w
was divided intto two sub-vollumes,
with one containing thhe spokes, huub and
inner edgee of the wheeel rim, and thee other
containingg the remaining toroidal wheel
surface, tthe fork and frame, the gground
contact annd the surrounnding volume. This
division iis needed to apply a rotaational
frame off reference too the inner wheel
section, peermitting realisstic movement of the
spokes inn subsequent transient moodeling
efforts. A non-conformaal interface waas used
between thhe inner wheell sub-volume aand the
surroundinng fluid volum
me. An illustrattion of
the methoddology is show
wn in Figure 2.
The bbase surface mesh size was
Figure 3: Volume mesh
m
with ‘near wheel’ refin
nement zones
consistentlly maintained for the tire, rim
ms and
forks. Onn the hub, spokkes, caliper andd brake
d tube, top tubee and down tu
ube, the surface mesh was iddentical for alll cases. Threee prismatic bouundary
pads, head
layers of constant
c
thickn
ness (0.025cm per
p layer) were generated foor all geometryy component suurfaces. Subseequent
y+ checks on convergeed RANS solu
utions were performed
p
to ensure that tthe prism layyer were within the
recommended range21 fo
or the turbulen
nce models beiing used. In adddition, two reectangular meesh refinement zones
4
Americcan Institute off Aeronautics aand Astronautiics
were creatted around eacch wheel to control the volum
me mesh near the wheel, fork and frame. The mesh uused to
study the Zipp
Z
1080 with
h the Blackwelll fork is shown
n in Figure 3.
A trunccated ellipsoid
d shape was used
u
to repressent the far fi eld boundary and is illustrated along wiith the
boundary conditions
c
described in the next
n
section in
n Figure 4. Suubsequent checcks on convergged RANS sollutions
were perfo
ormed to ensurre that the far field and dow
wnstream extennts were sufficcient to resolvve pressures att these
domain bo
oundaries. For this work, basseline computaational grids, fo
for each of the nine cases illuustrated in Figgure 1,
contained approximately
a
6 to 10 million
n tetrahedral ellements.
dary Conditions
C. Bound
The gro
ound plane waas modeled as a no-slip
surface, with
w a constant translational velocity
matching the forward speed
s
of the bicycle.
For this work
w
a speed of either 20
0mph or
30mph waas simulated. The fork, caaliper &
brake padss, head tube, to
op tube and down tube
were mod
deled as no-sliip surfaces wiith zero
relative veelocity. For the previous RANS
analyses1,2, a rotational frame
f
of reference was
applied to the outer wheeel, inner wheel, spokes
and hub ussing a rotation
nal velocity wh
hich was
consistent with the grou
und plane speeed being
studied. For
F the transien
nt computation
ns in this
work, a rotational
r
fram
me of referen
nce was
applied to the outer wheeel as was don
ne in the
RANS casse. On the inn
ner wheel sub-volume,
which contained the inneer wheel rim, hub
h and
spokes, a rotational
r
mesh motion was applied
to turn the
t
wheel at a rotationall speed
matching the
t outer wheell reference fram
me.
A unifo
form velocity profile
p
was ap
pplied to
the far fieeld boundary of the compu
utational
domain. The velocity vector directiion was
chosen to match one of
o five differeent yaw
angles span
nning the range from 0 to 20 degrees
(e.g. 0o, 5o, 10o, 15o or 20
2 o). A constaant eddy
viscosity far
fa field conditiion was specifiied to be
0.001 m2/ss. A pressuree outlet condition was
applied to the downstreeam boundary
y of the
model do
omain.
By using the trruncated
ellipsoid to represent th
he far field bo
oundary,
the same mesh
m
was re-ussed for all yaw
w angles
ure 4: Boundaary conditionss and model p
position within
n the
Figu
examined.
dom
main. Side vieew upper figurre; top view loower figure.
ology
D. Numerrical Methodo
In this work, the Nav
vier-Stokes equ
uations were solved using AccuSolveTM, a ccommercially aavailable flow solver
based on th
he Galerkin/Leeast-Squares (G
GLS) finite elem
ment method211,26,27. AcuSolvveTM is a generaal purpose CFD
D flow
solver thatt is used in a wide variety of application
ns and industrries. The flow
w solver is arcchitected for pparallel
execution on shared and distributed meemory computter systems andd provides fastt and efficient transient and steady
state soluttions for stand
dard unstructu
ured element topologies. A dditional detaails of the numerical methood are
summarizeed by Johnson et.al.28.
The GL
LS formulation
n provides seco
ond order accurracy for spatiall discretizationn of all variablees and utilizes ttightly
controlled numerical diffusion operattors to obtain
n stability andd maintain acccuracy. In adddition to satiisfying
conservatio
on laws globallly, the formullation implemeented in AcuSo
SolveTM ensuress local conservvation for indiividual
elements. Equal-order nodal
n
interpolation is used for all workinng variables, iincluding presssure and turbuulence
equations. The semi-disscrete generaliized-alpha metthod29 is used to integrate tthe equations iin time for traansient
5
Americcan Institute off Aeronautics aand Astronautiics
simulations. This approach has recently been verified as being second-order accurate in time30. The resultant
system of equations is solved as a fully coupled pressure/velocity matrix system using a preconditioned iterative
linear solver. The iterative solver yields robustness and rapid convergence on large unstructured meshes even when
high aspect ratio and badly distorted elements are present.
The following form of the Navier-Stokes equations were solved by AcuSolve TM to simulate the flow around the
bike wheel:

   u  0
t

(1)
u
 u  u  p     b
t
(2)
Where: ρ=density, u = velocity vector, p = pressure, τ = viscous stress tensor, b=momentum source vector.
Due to the low Mach Number (Ma ~ 0.04) involved in these simulations, the flow was assumed to be
incompressible, and the density time derivative in Eq. (1) was set to zero. For the preliminary steady RANS
simulations, the single equation Spalart-Allmaras (SA) turbulence model19 was used. The turbulence equation is
solved segregated from the flow equations using the GLS formulation. A stable linearization of the source terms is
constructed to provide a robust implementation of the model. The model equation is as follows:
2
~
1
~~
~ 
2
~
 u    cb1S   cw1 f w      (  ~ )~   cb 2 ~ 
t
d  

~
~
S  S  2 2 fv 2
 d
g  r  cw 2 ( r 6  r )
~


~
r ~ 2 2
S d
fv2  1 
f v1 

(3)

1  f v1
3
 3  cv31
1  u u 
Sij   i  j 
2  x j xi 
 1 c 
fw  g  6

g c 
6
w3
6
w3
1
6
S  (2 Sij Sij )1 / 2
Where: ~ =Spalart-Allmaras auxiliary variable, d=length scale, cb1 = 0.1355, σ=2/3, cb2 = 0.622, κ=0.41, cw1 =
cb1/κ2+ (1+cb2)/σ, cw2 = 0.3, cw3=2.0, cv1=7.1.
The eddy viscosity is then defined by:
 t  ~f v1
For the steady state solutions presented in this work, a first order time integration approach with infinite time
step size was used to iterate the solution to convergence. Steady state convergence was typically reached within 20
to 32 time steps for most simulations.
For the transient simulations, the Delayed Detached Eddy Simulation (DDES) model was used31. This model
differs from the (SA) RANS model only in the definition of the length scale. For the DDES model, the distance to
the wall, d, is replaced by d~ in Eq. (3). This modified length scale is obtained using the following relations:
rd 
 t 
ui , j ui , j  2 d 2
f d  1  tanh([8rd ]3 )
~
d  d  f d max(0, d  CDES )
6
American Institute of Aeronautics and Astronautics
Wheree: Δ=local elem
ment length scaale, and CDES = the des constaant.
This modification
m
of the
t length scale causes the model
m
to behavee as RANS witthin boundary llayers, and sim
milar to
the Smago
orinsky LES su
ubgrid model in separated flo
ow regions. N
Note that the abbove definitionn of the lengthh scale
deviates frrom the original formulation
n of DES, and makes the RA
ANS/LES trannsition criteria less sensitive to the
mesh desig
gn.
E. Scope of Work and Force
F
Resoluttion
Thee scope of thiss work encomp
passed the stu
udy of nine diffferent bicyclee wheel/fork coonfigurations. Each
configuratiion was run fo
or five differen
nt yaw angles (e.g.
(
0o, 5o, 10o , 15o or 20o) cchosen to matcch those that ccyclists
would norm
mally encounteer. Each yaw angle was run for two speedds: 20mph (com
mpetitive amateeur cyclist/triaathlete)
and 30mp
ph (professionaal cyclist/
triathlete). This resultted in 90
individual design points run for
two full wheel
w
revoluttions with
time stepss saved at every 2o of
wheel rotaation. Post-p
processing
was run fo
or all 90 design
n points on
the last 25
56 time steps (covering
512 degreees of wheel rotation)
only sincee it was noted
d that the
numerical solution requiired some
initial tim
me steps in order to
become staabilized. In tottal, 23,040
time step
ps were in
ndividually
analyzed. For each tiime step,
pressure and
a viscous fo
orces were
resolved to
t match: A) the axial
drag forcee that a cyclist would
experiencee in oppositio
on to the
direction of
o motion; B) th
he side (or
lift) force,, acting in a direction
perpendicu
ular to the dirrection of
Figuree 5: Force Res olution on thee rotating bicyycle wheel
motion, an
nd finally; C) th
he vertical
force actiing either to
owards or
away from
m the ground pllane, again actting perpendicu
ular to the direection of motioon. An illustraation of the resolved
forces with
h respect to th
he orientation of
o the bicycle wheel is show
wn in Figure 55. Also illustraated are the efffective
bicycle veelocity and th
he effective wind
w
velocity vectors
v
whichh would repreesent the expeerimental condditions
consistent with wind tun
nnel testing. The
T total (presssure+viscous) drag, side andd vertical forcees for each tim
me step
were furth
her resolved on
n a componentt-by-componen
nt basis for thee wheel, spokkes, hub, fork and caliper & brake
pads. In addition,
a
the tu
urning momentt was computed for each of tthe wheels bassed on the sidee force acting on the
wheel, spo
okes and hub.
It is off greatest intereest to know th
he power that a rider needs tto push a particular bicycle fforward at a sppecific
speed. Thiis total power requirement
r
is generally reco
ognized as the ssum of the folllowing contribuutions:
1.
2.
3.
4.
5.
Overall aero
odynamic resisttance
Wheel rotatiion
Rolling resisstance
Friction lossses
Gravity on in
nclines
The power needed to overcome the drag, FD, and the
t aerodynam
mic torque, M, aacting on the frront wheel is:
P
(4)
7
Americcan Institute off Aeronautics aand Astronautiics
The torque acting on one side of the wheel is given by:
(5)
∙
A simple validation case to compute the torque acting on a spinning disk in a fluid at rest was carried out. A flat
disc with a diameter and radius matching the nominal dimensions of the wheels being studied here was placed in a
large surrounding volume. The base surface mesh, the number and thickness of the prism layers at the disc surface
were all chosen to match the meshing characteristics for the wheel models. Disc rotational speeds matching the
wheel speeds of 20mph and 30mph were examined. Comparisons between the torque computed from the CFD
simulations and the experimental data, summarized from several sources by Schlichting et.al.32, were carried out.
Because of the repetitive and quantitative nature of the calculations required, all postprocessing of the simulation
results needed to resolve forces, moments, torques and power were automated through the use of the FVXTM
programming language feature available with FieldView33. Resolved forces, computed from FieldView were
verified directly with AcuSolveTM output, on a subset of the total data produced for this study.
F. Co-processing and Concurrent Postprocessing Methodology
To postprocess all of the transient results generated for this study, a new co-processing methodology, and, a new
concurrent postprocessing methodology were developed. With regard to the co-processing methodology, it was
recognized that saving the full solver output at each of the 256 time steps for all of the 90 design points would add a
significant amount of extra time and disk space to ultimately carry out the desired postprocessing calculations. To
address this problem, co-processing between AcuSolveTM and FieldView was implemented to:
1.
2.
3.
Reduce the disk space requirements needed to store the solver data during run-time
Keep the computational overhead low so as not to increase the solver run time, and
Remove the need for a (later) file conversion step requiring additional time and disk space
Since the primary objective of this study was to examine and compare the resolved forces, moments and power
requirements for each of the design points, only the data near and at the bicycle component surfaces (tire, rim, hub,
spokes, fork and so on) needed to be saved. Discarding the majority of the volume data, while keeping the solver
data at the surfaces of interest, was done as follows. First, a python script was developed to create a ‘subset mesh’
which contained just the surface elements and a user-definable number of volume element layers starting from the
relevant surfaces of interest. Provision for the unique specification of the surfaces of interest for the each of the nine
bicycle wheel configurations (see Figure 1) permitted flexible creation of the subset mesh. In this work, a single
volume mesh layer was used for each subset mesh.
Parallel AcuSolve
Processes
Socket Communication
User Requests (variables, elements, etc.)
Python Script
FieldView uns file
Figure 6: Co-processing methodology used during transient analyses
To generate the intermediate co-processing files during the solver run, AcuSolve was set up to listen on a userspecified port for a specific client connection call. The client python script, when called, gathered the information
8
American Institute of Aeronautics and Astronautics
from the subset mesh from memory, and wrote the results of the simulation for the current timestep onto disk using
the standard FieldView unstructured file format. This basic co-processing methodology is illustrated in Figure 6.
To perform all of the postprocessing calculations of interest, three FieldView FVX programs were developed to
automatically and sequentially run over all 256 timesteps, for the 10 operating conditions of speed and yaw angle,
for each of the 9 designs (see Figure 1), returning data in spreadsheet ready form. The first of the three programs
returned all of the resolved forces (drag, side and vertical), broken into pressure and viscous contributions for each
of the major components (tire and rim, hub, spokes, fork and caliper & brake pads).
The second program computed the turning moment by integrating the product of the side force and the axial
moment arm over a series of thresholded slices spanning the leading and trailing edge of the wheel. Side forces used
in the moment calculations included the contributions from the tire, rim, hub and spokes.
The third FieldView FVX program returned the aerodynamic torque, as specified in equation (5). Calculations
were repeated for both sides of the wheel by applying a threshold at the centerline in the direction normal to the
wheel. The pressure side, suction side and total torque were tabulated in the spreadsheet output. Shear forces used
in the torque calculations included contributions from the tire, rim, hub and spokes.
All postprocessing calculations were carried out remotely, in batch, on the co-processing output using a
concurrent approach. Shell scripts were used to queue the jobs to run any one of the three FieldView FVX programs
for all 90 design points at the same time. The queuing system (SLURM, Lawrence Livermore National Laboratory)
on the available cluster computing resource ultimately controlled how many postprocessing jobs were being run
simultaneously. However it was generally observed that more than 40 jobs were typically being run concurrently at
any given time.
III. Current Results
A. Drag Coefficients for DDES Calculations and Comparison to Wind Tunnel Data
Drag coefficients, CD, calculated from the DDES transient simulations are compared with previously published
experimental data (see Greenwell et al.6, Kyle et al.5,11-13, Tew et al.10, Prasuhn22 and Kuhnen23) and unpublished
data recently released for this work from Zipp Speed Weaponry, Indiana, USA in Figure 7. In each sub-figure, solid
lines represent the experimental data for CD as a function of yaw angle. The vertical bar, shown only at 0o yaw,
shows the range of experimental data reported from different sources for the wheel in question. The DDES data is
shown by the whisker plots (20mph in blue, 30mph in red). Each whisker plot summarizes all 256 values of CD for
each operating condition (yaw angle and speed) for each wheel configuration. The caps at the end of each box
indicate the extreme values (minimum and maximum), the box extents are defined by the lower and upper quartiles,
and the line in the center of the box is the median CD.
For the Zipp404, at 0o yaw, the average drag coefficient, CD, is calculated to be 0.0291 for 20mph, and is slightly
lower at 0.0272 for 30mph. These results tend to agree reasonably well with previously published ‘wheel-only’
experimental data (see Greenwell et al.6, Kyle et al.4,11-13, Tew et al.10, Prasuhn22 and Kuhnen23). The drag
coefficient trend with respect to yaw angle was also observed to be in good agreement with the published
experimental data. The drag coefficient was nearly the same on the Zipp 404 with or without either fork present,
and no significant differences between forks were observed. At higher yaw angles, the range between the minimum
and maximum values was seen to increase. The presence or absence of the front fork appears to have no damping
effect on this range.
On the deeper rimmed Zipp 1080 wheel, only limited drag coefficient data was available for comparison. At 0o
yaw, the average drag coefficient, CD, is calculated to be 0.0280 for 20mph, and is slightly lower at 0.0256 for
30mph. The drag coefficient was observed to be slightly lower at 30mph over the range of yaw angles studied. At
30mph only, drag forces exhibited a slight decrease at a 10o yaw angle. In our previous study2, RANS calculations
showed that the Zipp 1080 exhibited a significant drag coefficient reduction at the 10o yaw angle operating
condition. Further, the overall trend for the drag coefficient with respect to yaw angle was seen to be in better
agreement with the wind tunnel data for the RANS results compared to the DDES results. Similar to the Zipp 404,
there was no significant difference whether either of the forks was present or not. The DDES results did show a
dramatically increasing range of oscillation between the minimum and maximum calculated drag coefficient at both
speeds for the higher yaw angles. The presence of either fork did not provide any damping effect on this oscillating
behavior.
For the HED H3 TriSpoke, at 0o yaw, the average drag coefficient, CD, is calculated to be 0.0288 for 20mph, and
is slightly lower at 0.0271 for 30mph. Data for the HED H3 TriSpoke is seen to agree well with wind tunnel data
for several competitive trispoke wheels from other manufacturers at zero degrees yaw. The agreement with the drag
coefficient versus yaw, reported by Zipp, and to a lesser degree Greenwell et.al.6, is well within the range of the
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American Institute of Aeronautics and Astronautics
experimental data. How
wever, in con
ntrast to the experimental reesults, the draag coefficient is seen to inncrease
gradually with
w increasing
g yaw, and doees not exhibit th
he characteristiic maximum vaalue at around 5 to 10 degreees.
Figu
ure 7: Compa
arisons betweeen Wind Tunn
nel Data and D
DDES Results
In contrastt with the otheer two wheels, the HED H3
3 TriSpoke exhhibits a wider range betweeen the minimum
m and
maximum drag coefficieents at lower yaw
y
angles. This
T
range stayys relatively coonstant over thhe full range oof yaw
angles and
d does not increease at higher yaw angles in the same wayy that the Zipp 1080 is seen tto do. Finallyy, there
was no obsservable differeence on the draag coefficient whether
w
either fork was preseent.
B. Resolv
ved Drag Forces for DDES Calculations
C
on
o all Wheels
Time history
h
plots, and
a the corresp
ponding whisker plots, for thhe resolved draag force, on thhe tire, rim, huub and
spokes, forr all 9 design configurationss (see Figure 1), for all yaw
w angles, at a speed of 20m
mph are illustraated in
Figures 8a, 8b and 8c. In
n each of thesee three figures, the wheel onlyy configurationn is shown at ffar left; the Reyynolds
fork is sho
own in the mid
ddle, and the Blackwell
B
fork
k is shown at ffar right. The overall trendss and dependennce on
time with respect to yaw
w angle at 30m
mph are qualitaatively similarr to those obseerved at 20mpph, and conseqquently
these resullts are not shown here. In Figure
F
8a, we observed a faiirly uniform oscillation of drrag on the Zippp 404
around a mean
m
value at th
he higher yaw angles of 15o and
a 20o. As w
was noted with tthe drag coeffiicient, the abseence or
presence of
o the fork app
peared to have no significaant effect on thhe drag valuees. In Figure 8b, we see thhat the
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Americcan Institute off Aeronautics aand Astronautiics
oscillationss are less unifo
orm about a meean value, and at the higher yyaw angles, thee amplitude off oscillation inccreases
substantiallly. For the HED
H
H3 TriSpo
oke wheel, wee note that ovver 256 time steeps, a spoke w
will make a passsage
Figure 8a: Zipp 404 Dra
ag Force vs. Y
Yaw Angle at 220mph
through thee fork gap fourr times. In Fig
gure 8c, when either
e
the Reynnolds or Blackw
well fork is preesent, we see thhat for
Figure 8b: Zipp 1080 Drag Force vs. Y
Yaw Angle at 220mph
low yaw angles,
a
there arre four maxim
ma exhibited ov
ver the range oof solution tim
me steps shownn. As the yaw
w angle
increases, the behavior of
o the drag forcce with respectt to time becom
mes more com
mplex. At the hhighest yaw anngle of
20o, there appears to bee a dual peak on either sidee of the time aat which the sspokes pass thhrough the forkk gap.
Notably, th
he double peak
k occurs even in
n the absence of
o the fork, forr the wheel onlyy case shown aat left.
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Americcan Institute off Aeronautics aand Astronautiics
Figure 8c: HED
H
Trispokee Drag Force vvs. Yaw Anglee at 20mph
ved Drag Forces for DDES Calculations
C
on
o all Forks
C. Resolv
Time history
h
plots, and
a the corressponding whissker plots, for the resolved drag force foor both of the forks,
configured
d for each wheeel, for all yaw
w angles, at a speed of 20mpph are illustrat
ated in Figuress 9a, 9b and 9cc. No
experimental data for draag forces on iso
olated forks was available foor comparison. In each of theese three figurres, the
Reynolds fork
f
is shown at
a left, and the Blackwell fork
k is shown at rright. The overrall trends andd dependence oon time
with respecct to yaw angle at 30mph aree very similar to
t those observved at 20mph, and consequenntly these resuults are
not shown
n here. For all
three wheeels, we observ
ved
that there was a significant
drag
force
differen
nce
between eaach fork, with the
Blackwell fork consisten
ntly
exhibiting nearly dou
uble
the drag fo
orce computed for
the Reynollds fork.
With the Zipp 404
4
wheel, illu
ustrated in Figu
ure
9a, there is a differen
nce
with
resspect
the
to
dependencce of the fork
fo
drag forcee on yaw ang
gle.
For the Reeynolds fork, the
drag is seen reach a
minimum at 5o yaw, and
a
increase as the yaw an
ngle
increases.
With the
Blackwell fork, the drrag
force is hiighest at 0o yaaw,
and decrreases as yaw
y
Figure
F
9a: Forrk Drag Forcee vs. Yaw Anggle (Zipp 404) at 20mph
increases.
Fork drag
d
forces with
w
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Americcan Institute off Aeronautics aand Astronautiics
the Zipp 10
080 wheel con
nfiguration are illustrated in Figure
F
9b. Witth the Reynoldds fork, we see almost no chaange in
the drag fo
orce as the yaw
w angle increasses. We do how
wever see a geeneral increasee in the drag foorce amplitude as the
yaw increaases. With the Blackwell fork, a decrease in the drag withh increasing yaaw, similar to w
what was noteed with
the Zipp 40
04 wheel, was seen. And, ag
gain, the amplittude of the dragg force was seeen to increase w
with increasingg yaw.
The fo
ork drag forrces
with the HED
H
H3 TriSpo
oke
wheel configuration is
shown in Figure 9c. We
W
observed a very stro
ong
dependencce on the fork
fo
drag correesponding to the
passage off the spokes. For
F
the case of
o the Reyno
olds
fork we noted a drag
d
maximum//minimum pairr at
low yaw angles occurrring
just beforre and after the
spoke passsage. At higher
yaw anglees an interestting
transition to a dual peeak
maximum (occurring just
j
before and
d just after the
spoke passsage) was notted.
The averag
ge drag force was
w
seen to be relativ
vely
Figure
F
9b: Forrk Drag Forcee vs. Yaw Anggle (Zipp 1080) at 20mph
constant over all yaw
y
angles. Fo
or the case of the
Blackwell fork, a very
dominant maximum drag
was obserrved to occur at
the mom
ment of spo
oke
passage beetween the forrks.
This maxiimum peak was
w
seen to be highest at thee 0o
yaw case, and was seen
n to
decrease slightly with
w
increasing yaw.
The prrimary differen
nce
between th
he Blackwell fork
fo
and the Reynolds
R
fork
k is
the presencce of several sllots
along each
h side of the fo
ork.
Streamlinees, in Figure 10,
illustrate how
h
the slots in
the fork arre used to defllect
air flow away from the
wheel. Although
A
this is a
potentially
y desirable efffect
Figure
F
9c: Forrk Drag Forcee vs. Yaw Anggle (Trispoke) at 20mph
it accompllishes this at the
expense of
o increasing the
drag on th
he fork due to the
action of reedirecting the flow.
f
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Americcan Institute off Aeronautics aand Astronautiics
Figure 10: Streamlin
nes depicting flow
f
for the Ziipp 1080 and Blackwell Forrk
ved Turning Moments
M
for DDES
D
Calculattions on all W
Wheels
D. Resolv
Time history
h
plots, an
nd the correspo
onding whiskeer plots, for thee turning mom
ments computedd from the inteegrated
side forcess, for all 9 desig
gn configuratio
ons, for all yaw
w angles, at a sppeed of 20mphh are illustratedd in Figures 111a, 11b
and 11c. No
N experimentaal data for turn
ning moments were
w available for comparisoon.
Figure
F
11a: Ziipp 404 Turniing Moment vvs. Yaw Angle at 20mph
In each off these three fig
gures, the wheeel only configuration is show
wn at far left; the Reynolds fork is shown in the
middle, and the Blackwell fork is show
wn at far right. The overall treends and depenndence on timee with respect tto yaw
angle at 30
0mph are very similar to thosee observed at 20mph,
2
and connsequently theese results are nnot shown heree.
For thee Zipp 404, a maximum turn
ning moment was
w observed at 10o yaw. T
This matched our previous R
RANS
analysis2 for
f this wheel.. A slight dam
mping effect on
o the maximuum turning m
moment was obbserved for booth the
Reynolds and
a Blackwell forks. At the highest yaw angle,
a
20o, the damping effecct being providded by the forkks was
14
Americcan Institute off Aeronautics aand Astronautiics
not significcant. Addition
nally, the ampllitude of the tu
urning momentt was seen to iincrease signifficantly at the higher
yaw angless. However, th
he average turn
ning moment was
w seen to deccrease at these hhigher yaw anggles.
Figure
F
11b: Zip
pp 1080 Turn
ning Moment vvs. Yaw Anglee at 20mph
Figurre 11c: HED H3
H TriSpoke Turning
T
Momeent vs. Yaw A
Angle at 20mph
h
g moments forr the Zipp 108
80 were seen to
t increase in terms of maggnitude with inncreasing yaw angle.
Turning
Turning moments
m
for thee Zipp 1080 were
w
noted to act
a in a directiion opposite too those observeed for the Zippp 404.
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Americcan Institute off Aeronautics aand Astronautiics
The directtion of the turrning moment matched our previous RAN
NS analysis2 ffor this wheel,, however, thee local
o
maximum turning momeent at 14 yaw was not seen with
w the DDE S results. Thee amplitude off the turning m
moment
fluctuation
ns were seen to increase with increasing yaw
w angles.
Turning
g moments forr the HED H3 TriSpoke weree seen to reachh a positive maaximum at 5o tto 10o yaw, andd were
then seen to
t become negative as the yaaw angle increaased. This trennd matched our
ur previous RA
ANS analysis2 ffor this
wheel very
y well. Time history plots showed
s
that th
he fluctuationss in turning mooment matchedd the passage of the
spokes bettween the fork
k gap. It is no
otable that at 15
1 o yaw, the tuurning momennts were evenlyy oscillating abbout a
mean valuee near to zero.
E. Resolv
ved Torque - Validation
V
Casse
Torquee contours, obtaained for a RA
ANS simulation
n for the simplle validation caase (spinning ddisc in a (practtically)
infinite meedium) are illusstrated in Figurre 12, for the nominal
n
rotatioonal speeds corr
rresponding to 20mph and 30mph.
Fiigure 12: Torq
que Contours for the Spinniing Disc Valid
dation Case
The torrque, calculateed for the frontt and back of the
t spinning diisc for the nom
minal speed off 20mph ( = 26.45)
was 0.0131
18 N·m, essen
ntially the samee for both sidees. This comppared very withh the range off 0.01264 to 0..01323
N·m, estim
mated from the experimental results32. At the
t higher nom
minal speed of 30mph ( = 339.68), the calcculated
torque wass 0.02522 N·m
m (again, essen
ntially the sam
me for both siddes) which alsoo compared w
with the experim
mental
range of 0..02552 to 0.026
689 N·m.
ved Aerodynam
mic Torque fo
or DDES Calcu
ulations on alll Wheels
F. Resolv
Time history
h
plots, an
nd the correspo
onding whiskeer plots, for thee turning mom
ments computedd from the inteegrated
side forcess, for all 9 desig
gn configuratio
ons, for all yaw
w angles, at a sppeed of 20mphh are illustratedd in Figures 133a, 13b
and 13c. No experimen
ntal data for aerodynamic
a
to
orque were avvailable for com
mparison. In each of thesee three
he wheel only configuration is shown at far left; the R
Reynolds fork is shown in the middle, annd the
figures, th
Blackwell fork is shown at far right. Th
he overall tren
nds and dependdence on time w
with respect to yaw angle at 330mph
are qualitattively similar to
t those observ
ved at 20mph, and
a consequenntly these resultts are not show
wn here.
Torquee variations witth respect to time tended to be
b small for alll of the designn points studiedd. For the Zippp 404,
the torque was seen to reach
r
a maxim
mum at 5o yaw, and decreasee with increasiing yaw after tthat point. Foor yaw
o
o
angles in the
t range of 10
1 to 15 , thee presence of the
t Reynolds ffork led to a reduction in toorque, and a ffurther
reduction was
w seen with the
t Blackwell fork.
For thee Zipp 1080, th
he torque depeendence with reespect to yaw was seen to bbe very similar to that observved for
the Zipp 40
04, with a max
ximum value occurring
o
at 5o yaw, and decrreasing thereaft
fter with increaasing yaw anglee. For
this deeperr rim, the preseent of either forrk did not havee any significannt effect on torrque.
Of all three
t
wheels sttudied, the torq
que on the HE
ED H3 TriSpokke was seen to be the lowest.. Fluctuations in the
torque werre also seen to be
b quite small,, and were nearrly constant ovver the range off yaw angles sttudied.
16
Americcan Institute off Aeronautics aand Astronautiics
Figure 13
3a: Zipp 404 Torque
T
vs. Yaw
w Angle at 200mph
Figure 13b
b: Zipp 1080 Torque
T
vs. Yaaw Angle at 200mph
17
Americcan Institute off Aeronautics aand Astronautiics
Figure
F
13c: HED H3 TriSpo
oke Torque vss. Yaw Angle at 20mph
G. Resolv
ved Power Req
quirements forr all Wheel/Fo
ork combinatiions
Time history
h
plots, an
nd the correspo
onding whiskerr plots, for the resolved poweer for both of tthe forks, conffigured
for each wheel,
w
for all yaaw angles, at speeds
s
of both 20mph and 300mph are illusttrated in Figurees 14a, 14b annd 14c.
The total power
p
resolved
d for each desiign configuratiion was calcullated by summ
ming the powerr contribution ffor the
Figure 14a: Power vs. Yaw Angle (Z
Zipp404) at 200mph (left) an
nd at 30mph (rright)
wheel (inccluding the tiree, rim, hub and
d spokes) using
g equation (4) and the poweer needed to addditionally oveercome
mph is substanntially higher tthan that for 220mph.
the fork drrag. For all co
onfigurations, the
t power requ
uirement at 30m
In general, the power neeeded to increase speed from 20mph
2
to 30mpph went up by a factor of 3X.
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Americcan Institute off Aeronautics aand Astronautiics
For thee Zipp 404, illu
ustrated in Figu
ure 14a, it can be seen that th
the overall pow
wer requiremennt is highest with the
Blackwell fork. For the case of the Reynolds fork, th
he power at botth speeds is seeen to increase with increasinng yaw
angle. Thee amplitude off oscillation abo
out the mean power
p
is seen tto be fairly eveenly distributedd, and increasees with
increasing yaw angle. Fo
or the case of the
t Blackwell fork, a slight m
minimum poweer requirementt is observed frrom 5o
o
w.
to 10 yaw
The Zip
pp 1080 powerr requirements are shown in Figure 14b. T
The power requuirements for thhe Reynolds foork are
still lower than those neeeded for the Blaackwell fork, however
h
the di fference betweeen the two connfigurations is not as
great as waas observed forr the Zipp 404 wheel.
Figure 14
4b: Power vs. Yaw Angle (Z
Zipp1080) at 220mph (left) an
nd at 30mph ((right)
Figure 14
4c: Power vs. Yaw
Y Angle (T
TriSpoke) at 2 0mph (left) an
nd at 30mph ((right)
The HE
ED H3 TriSpok
ke power requiirements are sh
hown in Figuree 14c. Again, tthe Blackwell fork requires ggreater
power than
n the Reynoldss fork. Peak power
p
requirem
ments occur whhen a spoke paasses through tthe fork slot foor both
forks. In the
t case of thee Blackwell forrk, this peak vaalue is far aboove the mean vvalue, and reacches a maximuum at a
15o yaw an
ngle.
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Americcan Institute off Aeronautics aand Astronautiics
H. Frequeency Resolutio
on for all Wheeels
Power spectral densitties were comp
puted using Fou
urier transform
ms for the resolved wheel draag force (see F
Figures
8a, 8b and 8c), the resolv
ved fork drag force
f
(see Figu
ures 9a, 9b andd 9c), the compputed turning m
moment (see F
Figures
11a, 11b an
nd 11c) and th
he fully resolveed power (see Figures
F
14a, 144b and 14c). IIn most but noot all cases, dom
minant
peaks weree observed for the different operating
o
condiitions of speedd and yaw anglee. The dominaant Strouhal nuumber,
St, where St
S = fD/V peak
k for each configuration, for each of the reesolved propertties noted are iillustrated in F
Figures
15a thru 15
5d.
Figurre 15a: Strouh
hal Number (b
based on Wheeel Drag Forcee) vs. Yaw Anggle
Figure 15b: Strouh
hal Number (b
based on Fork
k Drag Force)) vs. Yaw Angle
In Figu
ure 15a, we obsserved a peak Strouhal
S
numb
ber at 10o yaw, which then deecreased with iincreasing yaw
w angle
for the Zip
pp 404 wheel. There was little difference no
oted between aany of the confi
figurations. Strrouhal numberrs were
also rough
hly the same fo
or both speeds. A single peaak Strouhal nuumber was obseerved for the Z
Zipp 1080 at 5o yaw.
Similar to the Zipp 404, the Strouhal number
n
was gen
nerally seen too decrease withh increasing yaaw angle. Therre was
no consisttent differencee between the different con
nfigurations annd speeds studdied. In com
mparing the Sttrouhal
numbers fo
or the Zipp 404
4 and the Zipp 1080 with resp
pect to yaw anngle, the latter w
was seen to connsistently exhiibit the
lower frequ
uency. This observation
o
sug
ggests that flow
w is remainingg attached to thhe Zipp 1080 w
wheel longer reelative
to the Zip
pp 404. For the HED H3
3 TriSpoke, th
he Strouhal nnumber showed no dependeence on yaw angle,
configuratiion or speed. The frequency
y observed maatched that of the passage off the spokes. Neither of thee other
wheels sho
owed dominan
nt peaks relatin
ng to the spok
ke passage; thee periodic flucctuations weree due entirely to the
shedding of
o fluid from th
he wheels durin
ng their rotation
n.
20
Americcan Institute off Aeronautics aand Astronautiics
Figurre 15c: Strouh
hal Number (b
based on Turn
ning Moment)) vs. Yaw Anggle
Figure 15d: Strouhal
S
Num
mber (based on
n Power) vs. Y
Yaw Angle
In Figu
ure 15b, we rep
port the Strouh
hal number for drag on each oof the forks foor the various w
wheels. Notabbly, the
results for the forks tend
d to match tho
ose observed for
f the drag cooming from thhe wheels. Thhis suggests thhat the
periodic flu
uctuation of drrag on the fork
ks is dominated
d by the sheddiing of fluid com
ming from the wheels. Withh either
the Zipp 404
4 or Zipp 10
080 present, wee note the sam
me lack of depeendence of Strrouhal numberr on configurattion or
speed. Wiith the HED H3
H TriSpoke, we
w plotted several strong seccondary peaks in addition too the dominantt ones.
These seco
ondary peaks occur
o
at harmon
nic frequencies, suggesting tthat there may be strong interactions betweeen the
fork and th
he characteristic wide spokess for the HED
D H3 TriSpoke wheel relatedd to their passaage between thhe fork
gap.
Strouhaal number calcculations based
d on the turnin
ng moment (ssee Figure 15cc) and the pow
wer (see Figuree 15d)
match thosse observed forr the wheel draag force with respect
r
to yaw
w angle, configuuration and speed. This reinnforces
the concep
pt that shedding
g from the wheeels is a strong
gly periodic phhenomenon whhich is dependeent on the yaw
w angle
and appearrs to be a charracteristic of th
he wheel itselff. In our previious work2, thiis same observvation was notted for
several ‘wh
heel-only’ stud
dies carried outt at a single yaw
w angle of 10o .
21
Americcan Institute off Aeronautics aand Astronautiics
IV. Discussion
In this work CFD was used to explore the complex and unsteady nature of air flow around several commercially
available front bicycle wheel configurations. Extending our previous work1,2, this study examined more realistic
front wheel geometry, adding the front fork, top tube, head tube, down tube, caliper and brake pads to the modeling
domain: Three wheels, namely the Zipp 404, Zipp1080 and HED H3 TriSpoke were considered. In addition, two
commercially available front fork designs, the Reynolds Carbon fork and the Blackwell Bandit slotted fork, were
also studied. This led us to consider a total of nine different design configurations. An approach, previously
presented1, was used to model each wheel with accurate spoke rotation, using a realistic ground contact. Unsteady
calculations were run with the commercial solver, AcuSolveTM using the DDES turbulence model. Simulations were
run for each of the nine design configurations over a combination of 2 speeds and 5 yaw angles, producing a total of
90 unique design points. This again extends our previous work2 where transient runs were carried out on only 6
unique design points (6 different ‘wheel-only’ configurations examined at one speed (20mph) and one yaw angle
(10o)). For all 90 design points, 256 time steps were analyzed, resulting in the need to review 23,040 individual time
steps. To make the analysis and management of this amount of data possible and practical, two new methodologies
were introduced. First, a co-processing approach was developed between AcuSolveTM and FieldView to generate
substantially reduced, yet sufficient and accurate representations of the solver data for every time step. Second,
several FieldView FVXTM scripts were developed and subsequently run in a remote, concurrent, batch mode of
operation to automate and significantly reduce the time needed to carry out the postprocessing presented in this
study. These FieldView FVXTM postprocessing scripts calculated i) the resolved forces (drag, side and vertical),
broken into pressure and viscous contributions for each of the major components (tire and rim, hub, spokes, fork and
caliper & brake pads), ii) the turning moments, iii) the aerodynamic torques acting on the rotating wheels, iv) the
power requirements for each design configuration studied, and v) the power spectral densities for selected variables
of interest. The workflow used here was further streamlined and automated by customizing FieldView FVXTM
scripts to generate spreadsheet ready data output for seamless transfer into other software packages.
The co-processing approach between AcuSolveTM and FieldView delivered three significant benefits. First, disk
space requirements needed to store the solver data were substantially reduced by writing out the solver results for
each time step in a reduced form. The typical size of a time step result (for any of the 9 design configurations) was
on the order of approximately 50MB. In our previous studies1,2, we needed to store the full volume data for each
time step, and additionally, export the data to the FieldView format to complete the postprocessing. For each time
step, disk space requirements were approximately 1.7GB. Co-processing reduced disk space requirements by more
than a factor of 30. Second, the time needed to postprocess each time step was significantly reduced. Elapsed times
needed to calculate resolved forces and aerodynamic torques were recorded and compared when working with the
full volume export versus the co-processing data. On average, a 15-fold to 20-fold reduction in elapsed
postprocessing time was realized when working with the co-processing data. Numerical results were identical when
using either the full volume data or the co-processing data. Third, the memory requirements needed for loading and
postprocessing were reduced by roughly a factor of 2 when working with the co-processing data. The computational
overhead introduced by having to write the co-processing files was observed to be roughly 3% of the total elapsed
solver time. Normally, the native solver volume data would be written at each time step. The time spent by writing
the co-processing files instead was observed to be offset by not having to write the native solver volume data. We
believe that benefits of the co-processing methodology introduced here will be of considerable value, particularly for
transient analyses.
The CFD predictions for the drag coefficient and trend behavior with respect to yaw angle were seen to be in
relatively good agreement with the experimental drag force data, kindly provided by Zipp Speed Weaponry
(Indiana, USA), along with several other published sources6,10-13,22,23 for the Zipp 404 and the HED H3 TriSpoke
wheels. For these wheels, the predicted drag coefficients tended to be lower than those recorded from the wind
tunnel. We speculate that the primary source of these differences arises from the use of different boundary
conditions and domain extents used in our simulations relative to the actual wind tunnel conditions. Agreement
between the predicted drag coefficients and the wind tunnel data for the Zipp 1080 was poor. Notably, the minimum
drag at 15o yaw, which was observed experimentally, and correctly captured in our previous work using RANS
modeling2, was not seen. Very large oscillations in the drag force on the wheel at higher yaw angles lead us to
speculate that the predicted flow may be prematurely detaching from, and quickly re-attaching to, the surface of this
wheel in a physically unrealistic way. Although the surface area of the rim and tire for this wheel (0.487m2) is
higher than the Zipp 404 (0.319m2) or the HED H3 TriSpoke (0.396m2), the surface mesh density and y+ values
were comparable between all three wheels studied. Grid refinement, particularly on the rim and the tire, and a
review of the DDES turbulence modeling methodology should be considered.
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American Institute of Aeronautics and Astronautics
It was expected that the presence of the front fork near the rotating bicycle wheel would have a significant impact
on the resolved forces and turning moments for the wheels themselves, and preliminary unpublished RANS analyses
on these design configurations reinforced this expectation. Surprisingly, the presence of either fork was seen to have
no significant effect on the wheel drag for any of the wheels studied. A slight damping effect on the turning
moment was seen on the Zipp 404 for intermediate yaw angles only; otherwise, no significant effects were observed.
However, significant differences between forks were seen. The Blackwell fork exhibited much higher drag forces
when compared to the Reynolds fork. Drag forces on the Reynolds fork were seen to increase slightly with
increasing yaw angle when used with the Zipp 404, and remained relatively constant for all yaw angles when used
with either the HED H3 TriSpoke or Zipp 1080 wheel. In contrast, the drag forces on the Blackwell fork were seen
to generally decrease with increasing yaw angle. With the Blackwell fork/HED H3 TriSpoke combination, a very
strong peak drag force, well in excess of the average fork drag, was observed as the wide spoke made the passage
through the fork gap.
Turning moment trends with respect to yaw angle were in very good agreement with previous RANS results2 for
each of the wheels studied here. The Zipp 404 exhibited a maximum turning moment at a 10o yaw angle for both
speeds studied. Turning moments for the Zipp 1080 acted in the opposite direction relative to the Zipp 404.
Previously2, it was observed that as the rim depth increased, the balance of side forces shifted from the trailing half
of the wheel to the leading half. This observation holds true for the DDES results as well. The HED H3 TriSpoke
also demonstrated a maximum turning moment at a yaw angle of 10o, similar to that observed for the Zipp 404. Of
greater relevance to stability however is the amplitude of the turning moment. The Zipp 404 was observed to have a
very low turning moment amplitude for all yaw angles except for 20o, suggesting that it should be a relatively stable
wheel to ride. The Zipp 1080 however had a very large turning moment range for yaw angles above 10o suggesting
that it might have control and stability issues in cross winds. Although the HED H3 TriSpoke was observed to
exhibit a smaller turning moment range compared to the Zipp 1080, this range was relatively high and constant for
yaw angles above 5o. While the average turning moment for the HED H3 TriSpoke was near zero at a 15o yaw
angle, this may be misleading in terms of stability since the oscillations from the average value were significant.
The contributions to overall power requirements coming from the aerodynamic torque are calculated, and are
believed to be presented for rotating wheels in contact with the ground for the first time in this work. We observed
significant differences between wheels and significant variations with respect to yaw angle for both the Zipp 404
and Zipp 1080. As expected, the highest aerodynamic torque was predicted for the Zipp 1080 since it is the wheel
with the greatest surface area. Although the HED H3 TriSpoke has a higher surface area relative to the Zipp 404, it
exhibited lower aerodynamic torque values. Anecdotally, the contribution of aerodynamic torque to overall power is
believed by the cycling community at large to be very small, on the order of a few percent at most4. In this work,
the contribution of aerodynamic torque to the total power requirement ranged from 9% to 10% for the HED H3
TriSpoke, 12% to 18% for the Zipp 404, and 17% to 24% for the Zipp 1080. Differences in the aerodynamic
torque for each wheel were expected. Unpublished data from Zipp indicates that the ‘wattage-to-spin’ requirement,
or the power needed to turn a test wheel during wind tunnel experiments, varies significantly between wheels.
To obtain a sense of the relative performance merits for the design configurations studied here, the overall power
requirements were calculated for each combination of front wheel and fork; wheel only configurations were not
included here. In general, it was observed for all cases that the power requirement needed to increase the speed
from 20mph to 30mph went up by a factor slightly greater than three. This result is consistent with the convention
that the increase in power goes up with the cube of the increase in speed. The highest power requirement was
observed for the combination of the Zipp 1080 and the Blackwell fork, however, we re-iterate at this point that the
drag predictions for the Zipp 1080 were not in good agreement with wind tunnel results. Notably the Blackwell fork
had a higher power requirement than the Reynolds fork when used in combination with any of the wheels studied.
The HED H3 TriSpoke exhibited the lowest power requirement relative to the other two wheels. But, the peak
power requirement, which occurs during the transit of the spoke through the fork gap, is well in excess of the
average value particularly for the case of the Blackwell fork configuration. Under practical riding conditions, if the
rider is unable to generate the power needed to ‘push thru the peaks’, a lower average speed will be achieved. We
suggest that power fluctuations, and not average power requirements, associated with the combination of a specific
front wheel and fork should be of primary relevance to riders.
The wheel drag, fork drag, turning moment and power all exhibited strong periodic behavior with significant
differences between wheels (but not configurations) and yaw angles. Of the three wheels studied here, only the
HED H3 TriSpoke demonstrated a correlation, for all variables examined, between the Strouhal Number (=0.895)
and the rotational motion of the three spokes. This observation was consistent for all yaw angles examined. A
comparison of the Strouhal numbers based on the fork drag and either of the wheel drag, turning moment or power,
suggest that the periodic fluctuations are being driven by the wheel for the case of the Zipp 404 and the Zipp 1080.
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American Institute of Aeronautics and Astronautics
Previously2 we proposed that the dominant periodic behavior comes from the periodic shedding of fluid off of the
rim and tire. Comparing the Zipp 1080 with the Zipp 404, we expect the flow to stay attached to the Zipp 1080
longer since it has a deeper rim, and this is consistent with the Strouhal number predictions (for the same yaw
angle). Starting at a 0o yaw angle, we observed the Strouhal number to increase for both the Zipp 404 and Zipp1080
(when periodic behavior was present) with increasing yaw angle. This suggests that the flow was initially attached
and started to become detached as the yaw angle increased. Interestingly, for nearly all cases, a maximum Strouhal
number was observed at 10o yaw. Beyond this 10o yaw angle, the Strouhal number was seen to decrease. In our
previous work1, we identified a vortex structure on the suction side of the wheel for the case of the Zipp 404. This
vortex structure started forming in the upper quadrant of the wheel, and eventually extended along the leading edge
of the wheel to a point just ahead of the ground contact. We speculate that as this structure grows with increasing
yaw angles, the rate of shedding slows down, as evidenced by the observation of the reduced Strouhal number.
Further analysis of the data should be conducted to confirm this speculation.
The primary objective of this work was to more realistically model the performance of a commercial bicycle
wheel, first by including more of the components around the front wheel (such as the fork, frame, calipers and brake
pads), and second by running transient instead of steady CFD simulations. The new co-processing and concurrent
postprocessing methodologies developed for this work make the analysis of many wheel/fork/component
combinations with AcuSolveTM and FieldView both possible and practical. Several critical issues such as the wheel
rim depth and cross-sectional profile, wheel diameter (650c or 700c), and development of an integrated front
fork/frame/caliper assembly, optimized to work with a specific wheel, still remain open. We believe that CFD is
now in a position to fully explore wind tunnel discoveries, and advance the understanding of the critical design
changes which ultimately lead to the performance improvements sought after by competitive and amateur cyclists
and triathletes.
Acknowledgments
The authors would like to express their thanks to David Greenfield, President of Elite Bicycles, Inc., for his time
spent discussing the current trends and technologies used in the design and production of present day bicycle racing
wheels, and for allowing us to measure some of the wheels and forks modeled in this work.
The authors would also like to express their thanks to Josh Poertner, Category Manager – Technical Director of
Zipp Speed Weaponry, for providing us with the production wind tunnel data that was used in this work.
Finally the authors would like to acknowledge the kind assistance from Philippe Spalart, Steve Allmaras and
James McLean, Boeing and Alan Egolf, Sikorsky who helped us to understand and appreciate the importance of
aerodynamic torque effects acting on bicycle wheels.
References
1
Godo, M.N, Corson, D. and Legensky, S.M., 2009, “An Aerodynamic Study of Bicycle Wheel Performance using CFD”, 47th
AIAA Aerospace Sciences Annual Meeting, Orlando, FL, USA, 5-8 January, AIAA Paper No. 2009-0322.
2
Godo, M.N, Corson, D. and Legensky, S.M., 2010, “A Comparative Aerodynamic Study of Commercial Bicycle Wheels using
CFD”, 48th AIAA Aerospace Sciences Annual Meeting, Orlando, FL, USA, 4-7 January, AIAA Paper No. 2010-1431.
3
Garcia-Lopez, J., Rodriguez-Marroyo, J.A., Juneau, C.-E., Peleteiro, J., Martinez, A.C. and Villa, J.G., 2008, “Reference values
and improvements of aerodynamic drag in professional cyclists”, J. Sports Sci., 26(3), pp. 277-286.
4
Kyle, C.R, “Selecting Cycling Equipment”, 2003. In High Tech Cycling, 2nd Edition, Burke, E.R, ed. Human Kinetics, pp. 1-48.
5
Lukes, R.A., Chin, S.B. and Haake, S.J., “The Understanding and development of Cycling Aerodynamics”, 2005, Sports
Engineering, 8, pp. 59-74.
6
Greenwell, D.I., Wood, N.J., Bridge, E.K.L and Add, R.J. 1995, “Aerodynamic characteristics of low-drag bicycle wheels”,
Aeronautical J., 99 (983), pp. 109-120.
7
Zdravkovich, M.M., 1992, “Aerodynamics of bicycle wheel and frame”, J. Wind Engg and Indust. Aerodynamics, 40, pp. 5570.
8
Sargent, L.R., “Carbon Bodied Bicycle Rim”, U.S. Patent 5,975,645, Nov, 2, 1999.
9
Hed, S.A and Haug, R.B., Bicycle Rim and Wheel”, U.S. Patent 5,061,013, Oct. 29, 1991.
10
Tew, G.S. and Sayers, A.T., 1999, “Aerodynamics of yawed racing cycle wheels”, J. Wind Eng. Industr. Aerodynamics, 82,
pp. 209-222.
11
Kyle, C.R. and Burke, E. 1984, “Improving the racing bicycle”, Mech. Eng., 106, pp. 34-35.
12
Kyle, C.R., 1985, “Aerodynamic wheels”, Bicycling, pp. 121-124.
13
Kyle, C.R., 1991, “New aero wheel tests”, Cycling Sci. pp. 27-32.
24
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14
Fackrell, J.E. and Harvey, J.K., 1973, “The Flowfield and Pressure Distribution of an Isolated Road Wheel”, Advances on
Road Vehicle Aerodynamics”, Stephens, H.S., ed., BHRA Fluid Engineering, pp. 155-165.
15
Fackrell, J.E., 1974, “The Aerodynamics of an Isolated Wheel Rotating in Contact with the Ground”, Ph.D. Thesis, University
of London, London, U.K.
16
Fackrell, J.E. and Harvey, J.K., 1975, “The Aerodynamics of an Isolated Road Wheel”, in Pershing, B. ed., Proc. of 2nd AIAA
Symposium of Aerodynamics of Sports and Competition Automobiles, LA, Calif., USA, pp. 119-125.
17
Wray, J, 2003, “ A CFD Analysis into the effect of Yaw Angle on the Flow around an Isolated Rotating Wheel”, Ph.D. Thesis,
Cranfield University, U.K.
18
McManus, J. and Zhang, X., 2006, “A Computational Study of the Flow Around an Isolated Wheel in Contact with the
Ground”, J. Fluids Engineering, 128, pp. 520-530.
19
Spalart, P.R. and Allmaras, S.R., 1992, “A one-equation Turbulence Model for Aerodynamic Flows”, 30th AIAA Aerospace
Sciences Annual Meeting, Reno NV, USA, 6-9 January, AIAA Paper No. 92-0439.
20
ShihLiou, W.WShabbir, A.Yangand Zhu, J. “ew Eddy Viscosity Model for High Reynolds
Number Turbulent Flows”, Comput. Fluids, 24(30), pp. 227-238.
21
AcuSolve, General Purpose CFD Software Package, Release 1.8, ACUSIM Software, Mountain View, CA, 2009.
22
Prasuhn, J., 2007, “Revolutionary Speed: Zipp’s Wind Tunnel Test”, Triathlete, June 2004, pp 192-195.
23
Kuhnen, R., 2005, “Aerodynamische Laufrader: Gegen den Wind”, TOUR Magazine, 9, pp. 24-31.
24
GAMBIT, Grid Generation Software Package, Version 2.4.6, ANSYS Inc., Pittsburgh, PA, 2008.
25
TGrid, Unstructured Grid Generation Software Package, Version 5.0.6, ANSYS Inc., Pittsburgh, PA, 2009.
26
Hughes, T.J.R., Franca, L.P. and Hulbert, G.M., 1989, "A new finite element formulation for computational fluid dynamics.
VIII. The Galerkin/least-squares method for advective-diffusive equations", Comp. Meth. Appl. Mech. Engg., 73, pp 173-189.
27
Shakib, F., Hughes, T.J.R. and Johan, Z., 1991, "A new finite element formulation for computational fluid dynamics. X. The
compressible Euler and Navier-Stokes equations", Comp. Meth. Appl. Mech. Engg., 89, pp 141-219.
28
Johnson, K and Bittorf, K., “Validating the Galerkin Least Square Finite Element Methods in Predicting Mixing Flows in
Stirred Tank Reactors”, Proceedings of CFD 2002, The 10th Annual Conference of the CFD Society of Canada, 2002.
29
Jansen, K. E. , Whiting, C. H. and Hulbert, G. M., 2000, "A generalized-alpha method for integrating the filtered NavierStokes equations with a stabilized finite element method", Comp. Meth. Appl. Mech. Engg.", 190, pp 305-319.
30
Lyons, D.C., Peltier, L.J., Zajaczkowski, F.J., and Paterson, E.G., 2009 “Assessment of DES Models for Separated Flow From
a Hump in a Turbulent Boundary Layer”, J. Fluids Engg, 131, 111203-1 – 111203-9.
31
Spalart, P., Deck, S., Shur, M., Squires, K., Strelets, M. and Travin, A., 2006 "A New Version of Detached-Eddy Simulation,
Resistant to Ambiguous Grid Densities", J. Theoretical and Computational Fluid Dynamics, 20, 181-195.
32
Schlichting, H. and Gersten, K., “Boundary-Layer Theory”, 8th Edition, 2000, Springer, Berlin, pp123.
33
FieldView, CFD Postprocessing Software Package, Version 12.3, Intelligent Light, Rutherford, NJ, 2009.
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