# Full text of "Screws, Propellers and Fans Based on a Mobius Strip"

## See other formats

^. AMAA AIAA/CEAS 98-2260 SCREWS, PROPELLERS AND FANS BASED ON A MOBIUS STRIP M. GILINSKY Hampton University, Hampton, VA J.IVI. SEiNER, and F.D. BACKLEY 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 SCREWS, PROPELLERS AND FANS BASED ON THE MOBIUS STRIP MikhaU Gilinsky^ Hampton University, Hampton, Virginia 23668 John M. Seinert, and Floyd D. Backley* NASA Langley Research Center, Hampton, Virginia 23681 ABSTRACT 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. 1. INTRODUCTION 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the significant effect is observed in the small scale devices with screws like Mobius strip: coffee-beans, mixers (the unpublished experimental re- sults conducted in Russia (1991) by Prof.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- ter. 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 [9,10]. 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. II. MOBIUS-SHAPED EMBODIMENTS 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- proximation. 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 = l/2poo^^R^ (1) Cd = T r r '->r Jo JRi 1 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 (3) 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- cations. 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: (5) (6) {X/aT + {Y/bT = 1 (4) 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: G 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 (9a) (96) (10a) (105) 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. III. CONCLUSIONS 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. IV. ACKNOWLEDGEMENTS 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 suggestions. V. REFERENCES 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, 1996. 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. LIMITED CASE 2: FRONTAL ROTATION OF THE RING MAXIMAL DRAG Figure 2 MOBIUS STRIP Figure 3 TWO-PETAL ELLIPTIC SCREW 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- plane. MOBIUS SCREW MODIFICATIONS 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. MOBIUS-SHAPED SCREW FOR MIXERS (ellipse-parabola contour) Super-elliptic Mobius-Shaped Screw b) 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. 9 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.