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AIAA/CEAS 98-2260 


Hampton University, Hampton, VA 

NASA Langley Research Center 

Hampton, VA 

4th AIAA/CEAS Aeroacoustlcs Conference 

June 2-4, 1998 /Toulouse, France 

For permission to copy or republish, contact the Confederation of European Aerospace Societies 


MikhaU Gilinsky^ 

Hampton University, Hampton, Virginia 23668 

John M. Seinert, and Floyd D. Backley* 

NASA Langley Research Center, Hampton, Virginia 23681 


A Mobius strip concept is intended for improving 
the working efficiency of propellers and screws. Appli- 
cations involve cooling,boat propellers, mixing in appli- 
ance, blenders, and helicopters. Several Mobius shaped 
screws for the average size kitchen mixers have been 
made and tested. The tests have shown that the mixer 
with the Mobius shaped screw pair is most efficient, and 
saves more than 30'/. of the electric power by comparison 
with the standard. The created video film about these 
tests illustrates efficiency of Mobius shaped screws. 


Several new methods for improving the working ef- 
ficiency of propellers and screws and the mixing of ex- 
haust jets with ambient air have proposed by the two 
first authors of this paper in 1993. The new concept 
for improving the working efficiency of propellers and 
screws was formulated in final version in the patent ap- 
plication to NASA [1]. NASA Technical Briefings has 
published the note [2] in 1996, which presents the short 
description of this invention. All these proposals are 
based on the untraditional designs which provide more 
efficient mixing of two flows. In general, this also fa- 
vors noise reduction and thrust augmentation, and can 
be applied for the rocket, aviation, domestic and other 
industries. The theoretical and experimental research 
results for several nozzle designs (Bluebell, Chisel, Tele- 
scope shaped nozzles, corrugated Co-Annular nozzles, 
Co-Annular nozzle with Screwdriver Centerbody) are 
promising (see, for example, the papers [3-5]). There 
is only the preliminary theoretical and experimental re- 
sults of efficiency of the Mobius shaped propellers and 
screws. In particular, the significant effect is observed 
in the small scale devices with screws like Mobius strip: 
coffee-beans, mixers (the unpublished experimental re- 
sults conducted in Russia (1991) by Prof. Veniamin 
Maron, Mikhail Gilinsky with colleagues). The same 
general idea unites "A Bluebell nozzle" and "A Mobius 
strip" concepts: to prefer curvilinear surfaces with sig- 
nificant change of a curvature along a flow by compari- 
son with the traditional axisymmetric or plane surfaces. 
Therefore some research results for the first designs can 

t Research Professor, Senior Member AIAA 

t Senior Research Scientist, Associate Fellow AIAA 

* Engineering Technician 

be used for improving of the second and vice versa. 

The new concept for improving the working effi- 
ciency of propellers and screws has proposed in the in- 
vention disclosure in 1993. NASA Technical Briefings 
has published the note [2], 1996, which presents the 
short description of this invention. The last version of 
this invention is in NASA patent application [1]. The 
concept is based on the Mobius strip-one-side surfaces. 
There are several embodiments of such shapes. Each of 
them can be optimal for different application in differ- 
ent surrounding media. 

Mainly, in this paper the perspective resea,rch plan 
is presented which should be fulfilled as a part of a long 
term projects under several NASA and CRDF grants 
for Hampton University. These grants are directed for 
creation of a Fluid Mechanics and Acoustics Labora- 
tory at Hampton University for research, training and 
teaching minority students to enhance the motivation 
of students within scientific areas and problems impor- 
tant for NASA, aviation and domestic industries. These 
projects are conducting within a Partnership between 
Hampton University and NASA Langley Research Cen- 

The research includes an analysis of the Mobius el- 
liptical screw modification in the air flows. Definition 
of the optimal parameters provided meiximum of thrust, 
lift, mixing, or minimal noise should be obtained by the 
analytical theory, numerical simulation and experimen- 
tal tests. These research results will be also applied for 
sport goals by improving of the sail or oar shapes. The 
final conclusions are drawn by numerical simulation of 
2D or 3D flow around the rotated screws or blades and 
experimental tests in the NASA wind tunnels and in 
the natural conditions using small scale models (toys, 
sport vehicles etc.). In the theory we will use a perfect 
gas model, Euler or Navier-Stokes equations with the 
several turbulent models and several packages of CFD 
codes, CFL3D [6], ADPAC [7], or ROTOR [8], involv- 
ing the specialists-users of these codes in joint research 
for solution of these complicated 3D problems. Also we 
will use approximate methods for rotating blades using 
the big experience in this field described in the reports 

The similar research for some screw modifications 
in liquids will be also analyzed. These liquids are water, 

glycerine, viscous-plastic medium, which can be applied 
to enhance efficiency of the screws for boats and mixers. 
And, of course, it is very important and interesting an 
analysis of application such shapes in dispersive and dry 
medium like sand, cement powders, coffee-beans and 
different mixtures. 

It is possible to apply other corrugated shapes with 
more complicated analytical representation than ellip- 
tical or super-elliptical Mobius shapes. However, ellip- 
tical shapes are very convenient for theoretical descrip- 
tion of this phenomenon. 

2.1 Mobius Strip Definition and Motivation 

We have generalized Mobius strip concept and pro- 
posed the several new modifications for different appli- 
cations. The main goal of the proposed design is to 
reduce a rotated element drag and, simultaneously, to 
increase the area for capture of the still medium without 
increasing the power needed for rotation. Such surface 
can be the Mobius strip which is one-sided surface. The 
Mobius strip has been proposed as the basis for opti- 
mally shaped airplane and boat propellers, fans, heli- 
copter rotors, mixing screws, coffee grinders, and con- 
crete mixers. The ambient medium can be air, water, 
concrete, coffee beans etc. Conventional (non-Mobius) 
devices of this type consist mostly of two-sided blades, 
which are not always optimal. 

In Figures 1 and 2 the two limited cases are con- 
sidered for a rotating round ring with an infinitesimal 
thickness around Z-axis. In the first case a ring has a 
cylindrical surface with X-£ixis as an axis of symmetry 
(a lateral rotation). In the second case a ring is located 
in meridional plane which revolves around Z-axis to- 
gether with a ring (a frontal rotation). In the first case 
a medium drag is minimal as well as is minimal medium 
capture that provides mixing efficiency, and, vise versa, 
in the second case a medium drag and mixing efficiency 
are maximal. So, a drag is ziro at the lateral rota- 
tion with neglecting of viscous effects. At the frontal 
rotation a drag can be calculated using rough approx- 
imate ainalytical approach or numerical methods. For 
example, a drag ring can be calculated the simple ap- 

Let us assume that at the frontal steady rotation 
all kinetic flow energy (in an inverse motion) completely 
transfer to a potential energy at the ring surface, i.e. to 
a total flow pressure. In other words, we will neglect of 
velocity squared by comparison with pressure. A pres- 
sure behind frontal ring side is assumed equal constant 
pressure in surrounding medium, poo- This corresponds 
assumption that this is a cavity region with constant 
pressure as in the case of blunt bodies moving in an 
incompressible liquid with a big enough speed. Than, 
introduce a drag coefficient, cj, as a ratio between a full 

force, F, to the ring area, Sr = 7r(i?^ - Rj) and a maxi- 
mal dynamic force to the frontal ring area, \/2pooV^Sr, 
which is produced by flow with the majcimal linear ring 
velocity, Voo — CiR2- Here, Ri and R2, are internal and 
external ring radii, poo is medium density, Q is rota- 
tional ring speed. We have the following relationships: 

Cp{R,4>) = koip{R,<i>)-Poo),ko = 




= T r r 

'->r Jo JRi 


CpRdRd(P = -(1 + Rl/Ri) (2) 

where we have used that p — Poo = p 00^^ R"^ si'n? <)> . On 
the face on it the last formula (2) for drag coefficient, c^, 
is paradoxical: with increase of internal radius, i?i, i.e. 
with decrease of ring wide, a drag coefficient increases. 
This effect is connected with decrease of ring area and 
increase of average flow velocity. Recall that a local 
flow velocity increases linearly along a radius, in the 
limited case, iii = i?2. a drag coefficient, cj = 1/2, that 
equal the drag coefficient for a round frontal located 
disk in hypersonic flow in accordance with hypersonic 
Newton theory. It is not small wonder because we used 
the same assumption that in this theory. Justification 
of such approach can be analytical Conor's theory [15] 
which has shown that Newtonian impact formula for 
drag coefficient, 

Cp = 1 — k^cos6 


is exact even for incompressible inviscid flows around 
some blunt bodies such as an elliptical cylinder, ellipsoid 
of revolution, and three-axial ellipsoid. Here, ^ ia a 
local angle of flow velocity vector with a tangent plane 
to the surface in considered surface point. Comparison 
of exact numerical solutions with the solution using the 
formula (3) are shown a very good agreement for wide 
class of other bodies. 

The analogical formula with (2) can be given in 
the quadratures for elliptic ring but this formula is more 
cumbersome, and we omit it in this paper, as well as the 
formula for a drag coefficient of a Mobius screw which 
shown in Figure 4. Calculation of this coefficient is con- 
ducted using simple single integration numerically. This 
is a subject of other paper. Here note only, that drag of 
a round Mobius-shaped element AqAnB^Bo in Figure 
4 (i.e. one petal of the Mobius screw located in the first 
quarter of the Cartesian coordinat system OXYZ) has 
a drag less than frontal rotating ring over ~40'/. for the 
same rotation speed. This can be also explained by less 
common frontal area for Mobius element by compari- 
son with frontal ring. This comparison was made for 
the same area of both designs. 

Thus there is some intermediate ring shape which 
provide compromising values for medium drag and cap- 
ture. One such possible shape is a Mobius Strip shown 

in Figure 3. This strip at the Z-axis in the its top has 
a shape close to frontal rotating ring, and lower transit 
to the lateral rotating ring. There are many different 
modifications of Mobius similar surfaces with different 
analytical representations. Below some of its will be 
illustrated. - 

A Mobius strip is made by giving a half twist to 
a strip of elastic material, then joining the ends to ob- 
tain a smooth surface. This design is one-side in the 
sense that in principle, one can trace out a continuous 
line along the strip from any point on its surface to any 
other point on the surface, without leaving (for exam- 
ple, through a border) or penetrating the surface. The 
one-sided, smooth shape of a Mobius strip provides a 
large capture area while generating the least possible 
turbulence in three-dimensional flow, and thus maxi- 
mized working efficiency. 

Propeller shapes based on the Mobius strip, and 
their orientations with respect to axes of rotation, can 
be varied and extended to suit the requirements of spe- 
cific applications. For example, a propeller or fan blade 
could be made as a single basic Mobius strip shape as in 
Figures 5 and 6, where the strip can be rotated around 
Z axis. Figure 6 shows another example, in which the 
ends of a single strip were twisted and joined at an axis 
of rotation to form two fan blades, each of which is the 
equivalent of a single Mobius strip that has been folded 
and joined to the axis at the fold. Other potential varia- 
tions include multiple strips fastened in the same plane 
or different planes, rotating about the same axis or dif- 
ferent axes; and strips made wholly or partly of circular 
(Figure 5), elliptical (Figure 6), super-elliptical (Figure 
10) or otherwise curved sections (Figure 9). Corrugated 
front or back edges of the Mobius strip (or both edges) 
can be applied to enhance mixing (Figure 11). Shown 
above screws in Figures 6 and 12 are preferable for pro- 
pellers and mixers. If we will use only upper part of 
these embodiments then the screws are preferable for 
producing thrust and applicable for boat-screw, fans 
and others. The two elliptic Mobius shaped modifica- 
tion are shown in Figures 7 and 8. Of course, these ex- 
amples are only illustrative pictures, which clarify our 
approach. In reality, instead of an infinitesimal thin 
screw surface should be applied a wing shaped screw. 
The optimal geometrical parameters of the screw as well 
as its cross section can be different for different appli- 

2.2 The Main Design Description 

The simpliest Mobius shaped surface can be de- 
scribed by super-elliptical equation. Recall the super- 
elliptical 2D contour is described, for example, in the 
plane XY by the equation: 

where a,b are extremal values of radius-vector modulus 
of this curve (known as half-axes). For n=2 the super- 
ellipse is the usual ellipse, and when exponent n increase 
to infinity the super-ellipse transforms to rectangular. 
A Mobius super-elliptical screw modification can 
be constructed following method. Our description will 
explain with the references to Figure 4. Let we draft el- 
lipse (or circle) with origin O of a Cartesian coordinates 
XYZ in the plane Z=0. Call this ellipse as "based el- 
lipse" with the half-axes ao,bo. Let the interval AqAn 
of Z-£ixis is divided by the N equal sub-intervals, and 
some a''-arc of the based ellipse BqBn also is divided 
by the same number N of the equal sub-arcs. The sur- 
face of the screw can be made from super-ellipses which 
join in consecutive order end-points Ai (i=0,l,...N) of 
the vertical sub-intervals on the Z-axis with the end- 
points Bi (i=0,l,..N) of the based ellipse sub-arcs on 
the horizontal plane Z=0. The screw surface can be 
described in Cartesian coordinates: 

"/r," _„"/K"^l/" 

z = Ml-^7a"-I/o"/n 

. , bN -b„ y 

bi = Oo -I arctan - 

y'max X 

where n=n(^) or in cylindrical coordinates r, if, z: 


{X/aT + {Y/bT = 1 


z - [bo+{bN-bo)- ][l-r (— ^— +-r;r~)J "J 

V'max "o "o 

where b^ = \OBo\,bN = \OBn\, and ipmax is the maxi- 
mal azimuthal angle of the petal, which can be changed 
along the petal edge, for example, sinusoidaly as for a 
Bluebell nozzle design. In sipliest case this change can 
be described by the formula: 


ymax = (Vmax)o[l + 6cOs{nc 7r/2)] (8) 

where s,Sm are the current and maximal length of the 
super-elliptic initial petal edge (i.e. the curve AnBn in 
Figure 4), or the corresponding values of the areas sec- 
tors under this curve, or square root of this areas. This 
allows to form corrugated edge uniformly then using 
usual dependence of the polar angle in the plane XOZ. 
Note, the exponent n also can depend of the azimuthal 
angle, which provides to change super-ellipticity down- 
stream of the screw surface. 

Thus in the case of an invariable edge and super- 
ellipticity a surface shape depends from six parameters: 
two half-axes, ao, bg, and exponent n, two end-point co- 
ordinates bo,b]\f and the length of the arc /;, = | ^-- 
BoB!^\. Let Uo-l, i.e. a^ is a characteristic length. 
Then we can characterize screw geometry by four main 
parameters: a ratio of the general axes of the super- 
ellipse -c=a,/6,, exponent n, the vertical Z-interval length, 

h=\AoAN\, and a'-arc (or its length). Below we sup- 
pose that a based ellipse is a circle, so that a,, = 60 = 1. 
In this surface the vertical oriented region at the Z-axis 
smoothly transits to horizontal oriented region, and a 
flow at this solid surface also follows of such change. 

We can use several designs as described above which 
are mounted to the same vertical interval (in practice, 
to the same part of the axial cylindrical shaft). Sym- 
metrical located several designs around axis of rotation 
Z produce the thrust or lift as in the case of a usual 
wing or propeller. Several Mobius screw modifications 
with the three and four symmetrical elements (petals) 
illustrate Figures 7 and 8. If, for example, to mount two 
2-petal screws antisymmetrical relatively of XOY plane, 
then a screws as are shown in Figures 14a,f doesn't pro- 
duce the thrust, but enhances mixing. In this case the 
design can be applied in different mixers. Possible also 
a set of the elements located along the axis of rotation 
Z and mounted at the consecutive intervals as well as 
symmetrically and anti-symmetrically. 

In Figure 9a almost arbitrary complicated Mobius- 
shaped screw for mixers is shown. Such a surface can 
be constructed and analytical described using different 
combinations of smooth 3D surfaces based on the set of 
space lines. We construct these hnes using ellipse and 
parabola as a projection of this space curve to the coor- 
dinate plane XOY. So that giving the coordinates of the 
fixed extremal points of the needed surface, A{0,0,za), 
M{xM,yM,ZM), and B(a;B,0,0) (see Figure 9b), we can 
get this space curve equation in parametric shape as: 

y = I/Af[l-(^-^M)"/a?]^/" 
X = xmII + V^-v/vm] 

ifO<X<XM, ZA<2 <ZM 

y = ywU-C^AM-l)^] 
x = XM+b,[l-y"'/y'^Y''" 

ii XM < X < XB, 0< Z < ZM 



where a^ = za - zm, bx = xb - xm, and n,m— 2,4,.... 
For continuation of this curve in the antisymmetric quar- 
ter of a Cartesian coordinate system {x > 0,y <Q,z < 
0) we use antisymmetric reflection relative bend point 
B. Based on this analytical representation several dif- 
ferent screws for kitchen mixers were made and tested 
at the NASA LaRC. 

We expect to obtain essential working efficiency of 
propellers and screws, and for different applications to 
obtain noise reduction and thrust augmentation in a 
wide region of Mach and Reynolds and numbers for avi- 
ation, domestic and other industries. 

2.3 Mobius-shaped screws for mixers. 

All industrial companies, who are interested in the 
invention [1], agree to participate in funding of research, 
development and joint marketing. However, these com- 
panies require the preliminary experimental and nu- 
merical simulation proofs, that it can be effective and 
adopted to the company's product. Large scale experi- 
ments with the proposed design are very expensive for 
us in this time. As yet, we have only been able to con- 
duct cheap and simple tests. Several Mobius shaped 
screws for the middle class of the kitchen mixers have 
made and tested. Some of them are shown in Fig. 14. 
These pairs are mounted in the mixer, which was es- 
tablished on the top of the vessels with water and small 
plastic particles on the bottom. The screws and cor- 
responding equipment are shown in Fig.l4a-f for the 
small vessel (b,c) and bigger vessel (d,e). The rotation 
speed of the screws increased smoothly by the voltage 
regulator, and power expenditure was measured by the 
Digital Wattmeter. These are shown on the right and 
left of the vessel respectively in Figure 14b-d. When 
rotation speed mount to the definite value the particles 
acquire a motion, go up and involve in vortex motion 
with water, and mixing process arises. The power value 
corresponding this mixing start characterizes eflSciency 
of the mixer. The tests have shown that the mixer with 
the Mobius shaped screw pair (right in Figure 14a or left 
in Figure 14f) is most efficient, and saves more than 30'/, 
of the electric power by comparison with the standeird 
(left in Figure 14a). The video film about these tests 
was created and it clearly demonstrate Mobius-shaped 
screws eflBciency. It can be used with the scientific- 
popular goals, for teaching, and with commercial goals. 
In particular, the discovered effect can be applied in the 
manufacture of liquid semiconductors. 


New Mobius strip concept is intended for improv- 
ing the working eflBciency of propellers and screws. Ap- 
plications involve cooling.boat propellers, mixing in ap- 
pliance, blenders, and helicopters. Several Mobius shaped 
screws for the average size kitchen mixers have been 
made and tested. The tests have shown that the mixer 
with the Mobius shaped screw pair is most eflRcient, and 
saves more than 30*/, of the electric power by comparison 
with the standard. The created video film about these 
tests illustrates eflSciency of Mobius shaped screws. 


We would like to acknowledge the Jet Noise Team 
support and help, to thank Dr. Dennis Bushnell for 
his attention and interest to our research and useful 


1. Gilinsky, M.M. and Seiner, J.M., Blade of a Ro- 
tary Machine Having Improving Efficiency, Patent Ap- 
plication, NASA Case #LaRC-15146-l, June, 12, 1996. 

2. Propellers and Fans Based on the Mobius Strip, 
NASA Tech Brief, 1996, pp.72-73. 

3. Gilinsky, M.M. and Seiner, J.M., Corrugated 
Nozzles for Acoustic and Thrust Benefits of Aircraft En- 
gines, CEAS/AIAA Paper #96-1670, 2nd CEAS/AIAA 
Aeroacoustics Conference, State College, PA, May 6-8, 


4. Seiner, J.M. and Gilinsky, M.M., Nozzle Thrust 
Optimization while Reducing Jet Noise, AIAA J, No. 
3, 1996, pp. 420-427. 

5. Gilinsky, M.M., Kouznetsov, V.M., and Nark, 
D.M., Acoustics and Aeroperformance of Nozzles with 
Screwdriver-Shaped and Axisymmetric Plugs, AIAA Pa- 
per # 98-2261, 4th AIAA/CEAS Aeroacoustics Confer- 
ence, June 2-4, 1998, Toulouse, France. 

6. Krist, S.L., Biedron, R.T., and Rumsey, C.L., 
1996, CFL3D User's Manual (Version 5.0), NASA Lan- 
gley Research Center, 3 lip. 

7. Hall, E., Topp, D., and Delaney, R., 1996, AD- 
PAC User's Manual, NASA CR 195472. 

8. Rai, M., Three-Dimensional Navier Stokes Sim- 
ulation of Turbine Rotor-Stator Interaction; Parts I and 
II, 1989, Journal of Propulsion and Power, Vol.5, No. 3, 
pp. 307-319. 

g.Johnson, W., 1980, Helicopter Theory, Princeton 
University Press. 

10. Chopra I, 1985, Notes on Helicopter Dynamics, 
University of Maryland, College Park. 310p. 

ll.Hayes, W.D., and Probstein, R.F., 1966, Hyper- 
sonic flow theory, V. I.-N.Y.-London: Acad. Press. 

12.Gonor, A.L., 1976, Flow Field Determination on 
the Surface of Some Bodies in an Incompressible Fluid 
Stream, Fluid Dynamics, V.ll, No 2, pp. 330-333. 



Figure 2 


Figure 3 


Fig. 1 Limited case 1: Lateral rotation of the round ring at the Z-axis, Drag and mixing efficiency are minimal 
Fig. 2 Limited case 2: Frontal rotation of the round ring at the Z-axis. Drag and mixing efficiency are maximal. 
Fig. 3 The Round Mobius-shaped ring rotating at the Z-axis. Drag and mixing efficiency are intermediate. 
Fig. 4 The two-petal elliptic screw with rotation at the Z-axis. The surface is constructed by ellipses which join 
the points, Au of the Z-axis interval, [^o,^^], with the points, S;, of the arc, (B„,5^), of the ellipse in XOY- 


Fig. 5 The round Mobius-shaped screw with a cylindric holder for mixers. 

Fig. 6 The elliptic Mobius-shaped propeller with the half-axes a/=4, and 6/ = l. 

Fig. 7 The 3-petal Mobius-shaped round screw for motor-boats or propeller for airplanes creating thrust. 

Fig. 8 The 4-petal Mobius-shaped round screw for motor-boats or propeller for airplanes creating thrust. 


(ellipse-parabola contour) 

Super-elliptic Mobius-Shaped Screw 


Figure 10 

Fig. 9 The 2-petal MSbius-shaped screw with a cylindric holder for mixers. The contours are constructed using 
an ellipse parabola combination for space curve projection to YOZ and XOY planes, a) A common view. 
b) The schematics draft; A, M, and B are extremal points of the space curve described by the equations (9-10). 
Fig.lO The super-elliptic Mobius-shaped screw with a cylindric holder for mixers. The contours are constructed 
using equations (7,8) with n=5, a=b=l, 6„=2.5, and 6,v=3.0. a) 2-petal screw (m=2). b) 4-petal screw (m=4). 

Fig. 11 The corrugated 2-petal round Mobius-shaped screw with a cylindric holder for mixers. The lip-line edge 

has a sinusoidal shape and rotation is around the Z-ajcis. 

Fig.l2 The 2-petaI Mobius-shaped blade for fans. The blade is made from a thin steel strip and this design was 

tested at the NASA LaRC. Only preliminary results aire currently available. 

Fig.l3 The super-elliptic sail with the based circle. The sail can be turned aJong the based circle around the 

cylindrical holder for an optimal thrust depending on wind direction. 


Fig. 1* -Mobius-Shaped Screws for Mixers. 

a)Picture of the three pair tested screws. Standard pair 
is on the left; b)-Equipment for tests: voltage regula- 
tor (left), mixer over the small vessel with water, plastic 
particles, and the 1st Mobius screw pair (center), digital 
wattmeter (right). c)-The same as in b) but with the 2nd 
Mobius screw pair. d)-The same as in b) but with bigger 
vessel. e)-Big vessel with the Ist Mobius screw pair during 
the test, f ) Screws tod a standard mixer used in tests.