An essential difference between two-dimensional and three-dimensional models of cycloidal rotors, the presence of unsteady trailing cycloidal vorticies in the wake, was studied. The velocity induced by these vorticies is the primary mechanism producing flow retardation for low span/radius ratio, finite blade number rotors. Results of using idealized rigid wake model of finite blade cycloidal rotors to investigate some cycloidal rotor problems are presented

Young, B. J.

English

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THE VELOCITY FIELD OF A SYSTEM OF UNSTEADY CYCLOIDAL VORTICIES
Bernard J. Young
Young Energy Systems
15 Cedar Lane; RFD #2
Dennisport, _ 0_639
ABSTRACT
An essential difference between two-
dimensional and three-dimensional model_
of eysloidal rotors is the presence of un-
steady trailing cycloldal vortlcles in the
wake. The velocity induced by these vor-
ticies is the primary mechanism producing
flow retardation for low span/radius
ratio, finite blade number rotors. An
idealized rigid wake model of finite blade
cycloidal rotors is used to investigate
s_e cycloidal rotor problems.
INTR OD UCT ION
During the past decade a rekindled
interest in energy conversion has stimula-
ted the development of aerodynamic theories
and numerical methods for analyzing rotors
whose blades make cycloidal paths through
the incident stream. Momentum theories
and vortex theories appear in the litera-
ture. In the case of vortex theories,
some noteworthy models of the wake have
been presented by Holme and by Wilson in
References 1 and 2.
Missing from these developments is
information about the velocity seen by the
blades of a three-dimensional, finite
bla_e number cycloidal rotor as it moves
about its orbit. Also missing is the
answer to the question: "What is the
opti_n_m orbital load distribution?" There
is good reason the latter question has
gone u_answered: the problem is complica-
ted. in spite of research spanning seven
decades, no theory or method having engin-
eering useDalness has emerged for cycloidal
propulsors. The power of present compu-
ters m_y provide the means for pursuing
this question further; it is hoped that
this work may be a step in that direction.
DISCUBSION OF THE PROBLEM
The load distribution on the blades
of a oycloidal rotor has components due
to:
_flat plate angle of attack
_camber
_unsteady angle of attack (pitch)
_unsteady motion normal to stream (heave)
_circulation induced by bound vorticity
fl_om other blades
-circulation induced from shed vorticity
in the wake.
Two-dimensional theory can be used to de-
termine the load due to these components
once the velocity at each orbital position
and the character of the wake are determln-
ed_
An essential difference between the
two-dimensional and three-dimenslonal
models of cycloidal rotors is the presence
of unsteady cycloidal vorticies in the
wake. The velocity induced by these vor-
ticies is the primary mechanism producin_
flow retardation for finite and low span/
radius ratio, finite blade number rotors.
It is for this reason that the author has
chosen to investigate the velocity field
of unsteady rigid cycloidal vorticies.
The presence of unsteady bound and shed
vorticity in the field between these vor-
ticles is recognized. The computation of
the velocity induced by unsteady bound
and shed vorticity is not within the scope
of this paper. A comparison of the results
of this paper with the results of the two-
dimensional method of Mendenhall and Span-
gler (Ref. 3) shows that the velocity in-
duced by trailing cycloidal vortlcies is
several times that of the bound and shed
vorticies. _
The wake _s assumed to be convected
downstream at a constant velocity. It is
further assumed that it does not deform.
While such assumptions are not as defensi-
ble for cycloldal rotors as they are for
helicoidal rotors, it can be argued that
the induced velocity seen by the blade is
primarily influenced by the portion of the
cycloidal vortex filament which is nearest
the blade.
The validity of the model inves_Igated
may not prove useful for the complete
range of tipspeeds and rotor loadings.
One would expect it to be valid for very
low tipspeed ratios where the model re-
duces to a multielement airfoil system in
rectilinear flow. One would also expect
it to be valid for lightly loaded rotors.
It should be less satisfactory for a
cycloidal propulsor whera a contracting
wake will cause blades in the downstream
half of the orbit to cut rolled vortex
sheets originating in the upstream half of
the orbltj The expanding wake of the
turbine should minimize such effects. As
with many other developments in the theory
of fluid mechanics, the ultimate useful-
ness can only be established when the com-
plete induced velocity field is computed
(including bound and shed vorticity effects)
and when correlation with experimental
data has been acheived.
THE INDUCED VELOCITY
We have in mind a discrete vortex
lifting line model of a straight blade
cycloidal turbine. We seek to evaluate
the velocity induced at the blades of the
B5
https://ntrs.nasa.gov/search.jsp?R=19820015813 2020-03-21T16:59:10+00:00Z
rotor by a systemof cycloidal vortlcies
originating at different spanwiselocations
on the blades. Onesuchvortex filament
ant the coordinate systemfor this discus-
sion are shownin Figure I. The cycloidal
curves followed by vorticies trailing fromthe i-th blade under rigid wakeassumptions
are parameterisedin time by:
xi : R cos(O-_i+_t) + ut
: R sin(@-_i+wt)Yl
zi=z
wherethe scalar quantity R is the radius
of the blade orbit; u is a characteristic
velocity through the rotor;
_i = (i-I) 2_/N
is the angular position of the i-th blade
of an N-blad4rotor whenblade I is at
position @= 0;_ is the rotor angular
velocity; and t is time (less than zero)
representing conditions which orinigated
in the past.
According to the Biot-Savart law the
velocity induced by a vortex filament is
given by the following integral:
V = I/_ R x dl
where p is^the circulation along the vortex
filament; R is the vector from the point
(x ,y ,z ) where the velocity is computed
to°th8 p_int (x_,y_,z_) on the vortex fil-
ament; dl is th_u_it_tangent vector along
the vortex filament directed away from the
bound vortex.
The circulation is a function of the
spanwise position z, orbital position 9,
and blade number i:
[i =A(z)B n cos(n@ -Yi +_t + _n )
where Bn and __ give the variation in
circulation aro_und the orbit and A(z) is
a spanwise load function.
The velocity induced by this system
has streamwise and sidewise components
given by the following integrals:
APPLICATION OF THE MODEL
A straight bladed cycloidal rotor
having blade span/radius of 4/3 was div-
ided into ten equal segments. A trailing
cycloidal vortex system was modeled by
eleven cycloidal vortex filaments origina-
ting at the ends of these segments. The
velocity induced at the midpoints of these
segments was computed for a local tipspeed
ratio R_/u = 2.85. The circulation on the
segments was taken as the integrated aver-
age over the length of the segment for an
elliptical spanwise load distribution.
The orbital load distribution was cosinu-
soidal (Bn, _n = 0 for n_l).
The velocity induced at the midpoints
of the two segments nearest the midspan
of a one bladB cycloidal rotor
is shown in _igure 2. Sor the middle 60%o
os the span the induced velocity is nearly
constant. Near the tips it is not for two
reasons. First, the discrete vortex lift-
ing line model with equal segments does
not yield the correct induced velocity
near the tips. In steady, rectilinear flow
models the spacing is ususlly modified near
the tips. Secondly, _here is no apriori
guarantee that the circulation specified
will result in the same pitch for the
cycloidal vorticies that was used in com-
puting the induced velocities. In fact,
one must modify the pitch of the cycloidal
vorticies and the repeat the calculation.
Such iteration is also required in solving
the helicoidal rotor problem. The second
iteration has not been caried out in the
work reported here.
Another question arising from these
results is: "How do we determine the char-
acteristic velocity at which the wake
moves downstream?" We want, of course, to
make a choice which gives the most accurate
results. This question remains open. but
in this work the velocity at @ = 180 _ has
been used. This is done because we will
eventually want to use more terms in the
cosine series for orbital circulation in
the attempt to produce a more uniform
induced velocity in the wake and the a
more optimal rotor.
N o
d_@)=i___l-ACz)P_/hicosC@-_i+_t+@)Cz-zo)COS(@-Yi4_t) dt
[(X-Xo )2 + (y-yo)2 + (Z-Zo)213/2
N o
_@ )=i___l-A(z )Rw/_f cos (@- _i+_t+_) (Z-ZQ) (sin(@-_i+_G) -u/R_Idt
= I_(X_Xo)2 + (y-yo)2+ (,-,o)2_3/_
For a straight blade rotor parallel to the
z-axis we are not interested in the span-
wise velocity component. These integrals
have been evaluated by numerical integra-
tion.
36
The results in Figure 2 form the
basis on whichthe strength of the circu-
lation and/or the incident streammaybe
Ja__edto producethe local velocity at
the rotor (u(180°)) whic_matchesthe
assumedlocal tipspeed ratio. If both
ar_ varied, a range of powercoefficients
is obtained. The largest then mouldbe
the optimumpowercoefficient for that
local tipspeed ratio, or pitch, for that
orbital andspanwisecirculation distribu-tion.
Figure 3 showsi_land3 blade rotor
results of varying the circulation and
adjusting the in_flowvelocity until a
locsl tipspeed ratio of 2.8_ wasacheived.
Thesepowercoefficients are not quantita-
tively exact since they do not accountfor induced velocities from shedand
_ou2_vorticity, and are not derived from
a completediteration of spanwiseloading
andwakepitch. They do exhibit qualita-
tive results showingan increase in obta-
inable p_er coefficient with increasedblade n_nloerand showthat there is an
optimal loading for eachblade number.
CONCLUDINGREMARKS
This methodwouid be enhancedif
..... meansof evaluating the induced
velocity integral were found. Asymptotic
series and a regression analysis of com-
puted values seem reasonable approaches.
The induced velocity components
for bound vorticity and shed vorticity
_st be included in the analysis.
The performance computation requires
modifying the pitch of the trailing
vorticies so that the wake motion is con-
sis$_ut with the prescribed inflow condi-
tions and the induced velocities at the
appropriate location. An iteration is
requi_ed_
The results of the inviscid theory
must be reckonne@ with the reallty of the
viscous fluid effects. Correlation with
experimental data is tbm ultimate test of
a fluid dynamic theory or method.
The method should be exercised in fln
attempt at the solution of the optimal
orbital load distribution problem by in-
cluding further terms in the circulation
series. Assuming we can find the optimal
load distribution, we must then see if it
can be obtained by a technically feasible
lifting surface.
With minor extensions, this method
could be applied to curved blade rotors
of the Darrleus type. This is appealing
because of the greater number of units
which exist and the abundant theoretical
and experimental date which is available
for comparison.
REFERENCES
I. Holme, 0.: A Contribution to the
Aerodynamic Theory of the Vertical-
Axle Wind Turbine, International
Symposium on Wind Energy Systems,
BHRAFluid Engineering, Cranfield,
England, Sept. 1976
Wilson, R. E.: Vortex Sheet Analysis
of the Giromill, J. of Fluids Eng.,
Vol lO0, Sept. 1978, pp 340-342
. Mendenhall, Michael R. and 3. B.
Spangler: Theoretical Analysis of
Cycloidal Propellers, Nielsen Engin-
eering and Research, Rept. TR 53,
June 1973
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Power Coefficient vs.
Circulation Parameter
for 1 and 3 Blades
4O


The velocity field of a system of unsteady cycloidal vorticies

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