Calculation of the Optimum Pitch Distribution of a Propeller with Sweepback Citation DeGroff, Harold M. (1948) Calculation of the Optimum Pitch Distribution of a Propeller with Sweepback. Engineer's thesis, California Institute of Technology. doi:10.7907/6PJQ-Q596. https://resolver.caltech.edu/CaltechETD:etd-12192008-151056 Abstract The principal problem dealt with is the derivation of a method for computing the optimum pitch distribution of a lightly loaded, sweptback propeller. The method is based on an extension to the work of Glauert as presented in Volume IV of the Durand series. A complete calculation is carried out for a swept propeller in which the resultant velocities along the blade vary through the transonic regime. The analytical work is simplified by the use of the calculus of variations. A survey of the German literature on this topic is included along with a complete discussion of the aerodynamic parameters to be considered in making of such a design. Item Type: Thesis (Engineer's thesis) Subject Keywords: Aeronautical Engineering Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Aeronautics Thesis Availability: Public (worldwide access) Research Advisor(s): Unknown, Unknown Group: GALCIT Thesis Committee: Unknown, Unknown Defense Date: 1 January 1948 Record Number: CaltechETD:etd-12192008-151056 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-12192008-151056 DOI: 10.7907/6PJQ-Q596 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 5081 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 15 Jan 2009 Last Modified: 21 Mar 2023 21:51 Thesis Files Preview PDF (Degroff_h_1948.pdf) - Final Version See Usage Policy. 28MB Repository Staff Only: item control page

CALCULATION OF THE OPTIMUM PITCH DISTRIBUTION OF A PROPELLER WITH SWEEPBACK Thesis by Harold DeGroff in Partial Fulfillment of the Requirements for the Degree of Aeronautical Engineer California Institute of Technology Pasadena, California 1948 CONTENTS Abstract. Definition of Symbols, I. Introduction 1,1 -A sweptback propeller-pre and con. 1.2 Survey of the German Literature, 1.9 Statement of a hypothetical swep Seprope tier design problem. II, Aerodynamic parameters in the problem. 2el Blade section. 2.2 Activity factor, 2.5 Interference effects. 204 FPlanform. Zoo Optimum design Cy variation. IIT Determination of analytical expresions for calculating the optimum pitch distribution. Sel Sources of the method. Development by Glauert. 5,2 Extension of Glauvuert's analysis to the swept propeller. a» Coordinate system, Geometry. b, Derivation of the equations, ec, Optimum circulation for an ideal swept propeller. a4. Effect of profile drag. Soe Further extension of the analysis, a» Restrictive assumptions lifted. 10 15 18 20 20 22 22 28 Ko €) oe 35 introduction of the calculus of variations,38 be. The effect of a finite number of blades, Co Evaluation of the constant A*t, d. Effect of the drag-lift ratio upon - optimum distribution of circulation, eé, To calculate the magnitude of the blade correction factor K, £5. Summary of the steps in the calculation of the optimum pitch distribution. 44 48 49 o1 52 CONTENTS (continued) IV, The hypothetical examples Ve Conclusions. References» Appendix I» Application of the calculus of variations to pitch distribution calculations, 1. Description and statement of the purpose of the calculus of variations. Ze Nota tions Se Optimum pitch distribution by calculus of variations. Graphs » 61 635 65 69= ABSTRACT The principal problem dealt with is the deriva- tion of a method for computing the optimum pitch distribu- tion of a lightly loaded, sweptback propeller. The method is based on an extension to the work of Glauert as presented in Volume IV of the Durand series. A complete calculation is ‘carried out for a swept proneller in which the resultant veloc- ities along the blade vary through the transonic regime. The analytical work is simplified by the use of the calculus of variations. <A survey of the German literature on this topic is included along with a complete discussion of the aerody- namic parameters to be considered in making such a design. DEFINITION OF SYMBOLS JT dv S135 &Ep cv 4B dE of Eo Jf thrust of a blade element of span dr at r. (2 in pounds) number of propneller blades blade element circulation air density in slugs ver cu. foot. rotational velocity of propeller (rad/sec) induced radial velocity of slipstream (rad/sec) radius of a blade element dr or/and df torque of a blade element of span dr at r ( Q in lbs-feet) translational velocity at a blade element (ft/sec) forward speed of the airplane (ft/sec) enerey loss of a blade element of span dr at r ( E in ft-lbs per second). Zi v* radial coordinate factor = circulation factor = =(16 | —— Vv profile energy loss of a blade element of span’ drat r. (E, in ft-lbs per second) resultant velocity at a blade element = Vuze (ar ® ( in feet per second) blade element drag coefficient blade element lift coefficient advance angle = faa! drag-lift ratio = Goldstein tip loss reduction factor SYMBOLS (continued) 3 “a or woe I SB 2, (0. DAO coordinate along the peak pressure line of the swept profile | sweepback angle radius of the swept propeller measured perpen= dicular to the blade center line (feet) values of thrust, torque, etc,, of the blade element of span dy at Pe W cos B rotational velocity ® (Q- ¥)r (ft/sec) blade chord measured perpa@idicular to bs (feet) wu cos 6 RF cos pf x /e (y+Z )/2 coefficients in the eubic in ps ° Power coefficient #* _P_ PN? D advance ratio e Vp propeller diameter in feet, induced angle of attack of the blade element I. INTRODUCTION lel A Sweptback vropeller--Pro and Con. "Why should a propeller with sweepback be consider= ed?" Such a proposal leads to complications, both aerodynamic and structural, which can be solved completely only by exten= sive and expensive testing vrograms. Before such a procedure is undertaken its logic must be proven and as much theoretic= al work as possible should be put forward to indicate that gains may be expected by such an unconventional design. Introducing sweepback into a propeller is merely a continuation of the fight which propeller engineers have been making for some time against the losses of compressibility. One step in that fight was the simple expedient of limiting the rotational speed of the propeller. However, with high speed and cruising speed of flight moving nearer to Mach one the limitation of rotational speed has reached a definite barrier, Solidity ratios, prop diameters and gear ratios may no longer be altered sufficiently to prevent the operation of some part of the blade above critical Mach number. There- fore, if the high efficiencies ordinarily associated with pro- pellers are to be maintained at these high speeds it is nec= essary to utilize some other, more drastic, design details. A ease in point is the turboprop. This installa- tion is especially applicable to airplane designs which re- quire long range, low specifie fuel consumption, and a high cruising speed. if the basic propeller efficiency can be main= tained in:-excess of 85% for airplane Mach numbers approaching - 5 «= 0.9 then the propeller driven aircraft will fulfill the above design conditions. The fuel consumption of jet driven air- planes is still much too high to fulfill the above require- ments in their present state of advancement . Sweepback is just one of the features which may contribute to the estab- lishment of the turboprop installation. “The physical way in which sweeping back the planform of the propeller will delay compressibility losses is simply explained and is well understood from similar proposals for sweeping back wings of high speed airplanes. With sweepback the propeller sections will be subjected to an effective vel- ocity which is the product of the resultant velocity and a factor proportional to the cosine of the sweepback angle. Thus the propeller will be operating in a reduced velocity field and will not experience compressibility losses as soon as would conventional plenform propellers. Drag and lift divergence at the tip sections will be delayed and the overall efficiency of the propeller will be correspondingly increased. 1.2 Survey of the German Literature Some work has been done on the design and building of sweptback propellers. Such a propeller was given a full scale test flight by the Curtis-Wright Propeller Division but there has been no published report on the results of that test. The principal impetus to the idea of propellers of this config- uration was originated in Germany. However, there is no indi- cation in the literature that they ever got beyond the model testing stage in their designs. Reports of these tests are contained in several captured German documents. (See refer= ences 1, 2, and 3). These papers are very brief and almost all design and theoretical details have been ommitted,. The substance of these references will be given here. The tests were made in wind tunnels of relatively low velocity. Tip Mach numbers of approximately 0.98 were ob= tained in a flow of 157 mph. This gave a ratio of tip speed to forward speed of about 1.0 3: 0.15. The geometrical angle of attack was 135 degrees on the pressure side of the blade sections. The following series of planforms was proposed. LEADING EDGE (A) — (8B) (¢c) (D) (5) Fig. lel Proposed propeller planforms. The straight planform prop tested had a diameter of 5.6 feet. For the curved planform the curvature was obtained by displacing the blade elements. The curvature was in a plane defined by the axis of the straight blade and the zero lift direction at 0.7 radius, Sweepback at the tips was 45 degrees. The wind tunnel tests were actually only made on the first three configurations shown above, The results of these tests are indicated in the following eraph (Fig. 1-2) where the efficiency indicated corresponds to the optimum operating con= dition of the model propellers, 0.9 | | ie Om moe — : o°8 “Sen 4h TU) a (ay Sa. 75 < ae $ \ 0.7 BN ex (B—T 0.6 0.5 0.6 0.9 0.8 0.9 1.0 Mn be Fig, l-2 Optimum efficiency for various planforms. In addition to-the results indicated in the above figure there was noted qualitatively that for the sweptback planform the thrust coefficient increased and the torque coef= ficient decreased with increasing Mach number. No attempt was made to evaluate the scale effect in these tests. A further interesting statement in the discussion was that the trailing edge was found to be the region of highest stress, Referring to Figure 1-2 it is noted that the advan- “tage shown by the sweptback blade is not evident with the swept forward blade. It must be concluded that the gain is not due solely to the delay of the compression shock. More signifi- cant are the radial pressure gradients due to the different directions of sweep, These radial pressure gradients are independent of shock wave development and react differently on the flow of the boundary layer. It must be concluded that this effect is in addition to the action of centrifugal force on the boundary layer air. Further, the inclined compression shock plane diverts the streamlines outwards in the case of the sweptback airfoil and inwards in the case of the sweptforward airfoil. This is because only the velocity componemt perpen= dicular to the plane of the shock suffers a discontinuous re= duction at the shock while the component parallel to the shock plane remains unchanged. Sweptback Planform Sweptforward planform Fig. 1-3 Velocity gradients due to inclined shock fronts. The above leads to changes in induced drag. The sweptback tip gives an effective increase in diameter and re= duction in induced drag and the opposite to this effect is obtained with the sweptforward tip. Further, these flow con-= ditions influence the boundary layer in such a way as to favor the sweptback propeller. 1.3 Statement of a Hypothetical Swept Propeller Design Problem Let us assume a hypothetical propeller design pro= blem.e It is required to design a propeller incorporating sweepback which will provide high basic efficiencies (of the order of 85%) at the following operating conditions: Ae Airplane velocity 550 mph B, Altitude - - 85,000 feet CG. Prop rpm | | 1260 De. Prop diameter 13 feet E. Engine BHP ~ gg00 &’. Four bladed propeller This particular design condition imposes a Mach number variation along the blade ranging from sub- to supersonic. A reasonable amount of sweepback will reduce the entire velocity field to an effective value below the critical of the blade sections, (See Figure 1-4). In this paper we will only be concerned with the acrodynamic design of the propeller. The structural pro- blems are present and are extensive but will not be treated in this material. | In designing the propeller aerodynamically we may assume complete freedom as to blade section, blade chord, ac=- tivity factor, ete The only physical features of the pro- peller which are prescribed are those on the diameter and num= ber of blades, It. AERODYNAMIC PARAMETERS IN THE PROBLEM Having established the fact that the blade is to operate at transonic speeds, and being aware of the detrimen- tal effects incurred in this regime, it is now necessary to investigate the design parameters which will delay or reduce these effects. In general, the following factors must be con= sidered: A. Blade Section B. ‘Activity Factor ¢. Interference Effects D. Planform E. Optimum Design C_ Variation GL These factors will now be discussed in the above order. 2.1 Blade Section This division of the problem has had some attention in reference #6. The efforts in this reference have been, in the main, experimental and the results are tabulated therein. Since this paper is not fundamentally concerned with the inves- tigation of airfoil sections and their characteristics we will settle upon the sections as recommended in the reference. Yet, at the same time, it would also seem advisable to understand the mechanics by which these particular airfoils have a lower Mop! Therefore, the particular sections tested by the NACA will be examined In the light of existing information on this problem. The sections being considered are designated as the NACA 16-series. -ll- The critical Mach number has been defined as corre= sponding to the free stream velocity when some local velocity on the airfoil reaches the local speed of sound. Associated with this critical Mach number is an increasingly large posi- tive change in the drag coefficient, This drag increase is attributed to shock loss which occurs when the local veloci- ty at any point on the airfoil reaches sonic velocity, i.ee, when Mor is reached. However, it has been shown theoretically that the fact that some local velocity is sonic, while a nec» essary condition for shock loss, is not necessarily a suffic- jent condition, reference #4. However, such a situation is thought to be an unstable one, and within engineering accur= acies the two phenomena can be thought of as occurring sim=- ultaneously. A very convenient method is available for theore etically predicting the Mey of any section if “Ryo » the pressure coefficient for low velocities, is known, reference #4. This method involves the assumption of small perturba= tions, i.¢., the local velocity, u, is assumed small such that its relation to the free stream velocity, V, at any point “along the surface is of the order as yi Such an assumption cannot hold very near the leading edge of an airfoil unless the radius of the leading edge is vanishing= ly small, Or, another way of expressing this criterion is that the thickness of the section must be small as compared -12- with the chord. In accordance with the above assumptions the following relation may be derived. Mee = I- $b ty Cay] ° where, M oe is the critical Mach number y is the ratio of the specific heats CP, is the maximum negative pressure coef= ficient on the airfoil. Por example, from reference #6, Cp, for the NACA 16-009 sece tion is -.225. Hence, . x M. ee $ [- 4S! (-. 225) ° = 0.791 And for the NACA 07-009 section, CP, is -.210 for zero angle of attack for which ws, | Mee = I~ & [- Ht t-210)] = 0,800 Experimental data as presented in reference #6 make possible the comparison with the theoretical results as cal- culated aboves Thus for a 16-009 section the experimental value of M,, is given.as 0,802 for a variance of about 1.5% between the theoretically calculated and the experimental valu6s» From a perusal of the theoretical calculations above 4t would appear that) the NACA 07-009 section has a higher Mor then the 16- series section NACA 16-009. However, this is - not supported experimentally, reference #6. The explanation seems to bé in the fact that the 07-section has a more full olde pressure diagram near the leading edge where the compressi- bility effects are more critical. Thus the critical Mach number is not determined uniquely by the value of (C_ ) .. Po min and the form of the pressure distribution is important, at least insofar as the configuration near the leading edge is eoncerned. In the general case this would lead to a question as to which would bear the greater weight, a low value of (Cp dmin OF low leading edge negative pressures. We may generalize to the extent that “Ry must be minimized and with as little expense near the leading edge as possible. This means a very slight leading edge radius and small thickness. It is evident that the local velocities corres- ponding to a given free stream velocity, V, can be reduced and, hence, Moe inereased by three means, 1. Decrease the thickness 2. Decrease the leading edge radius 5. Maintain the lowest possible design Cy (minimun camber) Before concluding these remarks on blade section characteristics, it is necessary to discuss the desirability @ blade sections in order to prevent a cr Pete O se) ao of using symmetri decrease in thrust at high values of M. This effect with cambered sections has been shown oxperimentally at about M = 0.8, and its result is to decrease the angle of zero lift with increasing Mach number. The extent of this change can be severe enough to incur negative thrust at Gambered tip sections. German data sustain this contentions, This statement is based upon a personal conversa= tion with the German aerodynamicist Goethert. It is further substantiated by CGD 483 wherein reference is made to measure= ments of this effect by Blankenburg (UM 1319), Of course, the desirable way to handle this in the design would be to utilize cambered sections with allowances in angle of attack to come pensate for the change. Two things make this approach diffi-# cult. First, there are no design data available which speci- fically define the shift of the curve and, secondly, allow= ance for this effect at high speed would lead to excessive blade angles at the other operating conditions, Therefore, at higen forward design speeds it would appear that symmetrical sections would be desirable, However, it must be remembered that the M,,, of a blade section is a function of the peak minimum pressure, And, while a symme= trical section will have a full pressure curve at ¢= 0°, it will peak seriously at the leading edge at angles of attack greater than zero. And it follows that the extent of this peak will increase with angle of attack. The writer has made theoretical calculations by the method of reference #5 which show that for the l6-series symmetrical sections angles of at= tack above 3° will result in an advance pressure peak, For these sections an angle of attack of 3° gives Cy yalues between eo and .4 , The design condition for this blade is the high speed condition which infers a lightly loaded propeller. ' Therefore, angles of attack in excess of 3° will not be re= quired. Hence, it appears that when both the effect of the =15= shift of the lift curve slope and the fact that small angles of attack are required are taken into account, the sym- metrical section is the logical choices In a further effort to reduce compressibility ef fects it is important that the section thickness be kept to a ‘minimum. This thickness variation will be dictated by struc= tural considerations, 206 Activity Factor The activity factor is a quasi-empirical factor based upon the blade area distribution of the propeller, It is defined by the relation, Lok 3 AF = £00 000 S (B NE ) Ws) 0.48 where, GC is the chord at the radius r D is the prop diameter Ris the radius # D/2 100,000 / 16 merely gives the AF a convenient magnitude» | An examination of the above relation shows that for a given D, AF will be increased by increasing the chord length © along the blade, Therefore, increased AF of a propeller will correspond to decreased aspect. ratio, In evaluating this statement it must be recalled that propeller theory is derived on the basic assumption of two dimensional flow. Therefore the several paragraphs which follow are intended only to indicate qualita- tively the changes in power absorbtion associated with changes in AF at near sonic. Mach number operations = 16= We are accordingly interested in determining the variation of Mor with aspect ratios (<2) a Ing AR 200 a) ae det} oor Fig, 2-1 Change of Lift curve slope with M and AR, Cc ; , AR = 1 M Pigs 2-2 Change of C, with Mach number and ad, AR 200 , AR 214 Fig, 2-5 Change of C with Mach number and AR » (2), s) clel fa) AR Fig, 2-4 Lift curve slope as a function of AR for a given Mach numbers for AR =s-l up to M = This variation is as shown in the following figures, Figure 2+1 shows the variation of the slope of the lift curve with change in Mach number with aspect ratio as the parameter, Thus for a given slope at M0, the var= lation with aspect ratio will be as follows, For any given Mach number, M, the slope of the lift curve will inerease with increas= ing aspect ratio, See figure 24, Qualitatively this graph is the same as for the case in which compressibility effects are ig-= noreds And in this same respect, experimental results show no measurable effect of M on 4 Ay, 0.80, Figure 2-2 is of pri+ mary interest in that it clear- ly shows the delay in drag rise for the lower AR planforms., Thee oretically there is no limit to which this effect may be carried to advantage. However, other -1L7e considerations, ¢@.ge, Skin friction and boundary layer exten= sion will limit this reasoning, Also, Figure 2-3 shows that cL decreases with decrease in AR so that the most efficient section does not necessarly correspond to the lowest AR which is structurally feasible. High Mach number operation causes an extension of laminar flow over high speed sections which is character= ized by the curve in Figure 2-5. It is to be noted that this "drag bucket" occurs for one small range of C With some 1° sections, viz., the 1lé-sections, this effect is not indicated in Design Cy the test data available, How-— | C ever it is felt that such a phen- emenon would appear and that ine Figs 2-5 The drag-lift curve in the “critical” vestigation is necessary if we Tange o . are to rationally select design oc, 's in the range above Mar? ‘As mentioned before, this bucket occurs over a very limited range of lift coefficients, The qualitative conclusion to the above is that from a standpoint of drag rise in the compressibility range the AF should be as high as is structurally feasible. However, this will not lead to the most efficient planform since the lift of the planform will begreatly decreased. in summary it is seen that an increasing AF leads to the following effecte: = IB « l. Higher thrust coefficient (incompressible) 2. Higher power coefficient (incompressible) 3, Loss of section lift coefficient (com- pressible) 4. Decrease in section drag coefficient (compressible), i.e., the value of M S praised. [+e 5. Little change in lift curve slope from compressibility considerations under M = O.8. 2 Thus it is seen that an optimum AF must exist. However, its empirical definition precludes the possibility of arriving at such an optimum analytically. Therefore the procedure will have to be one of cut and try. 2.5 Interference Effects There are indications that interference effects at high subsonic and transonic speeds may be of importance in determining the overall characteristics of aerodynamic per- formance in these speed ranges. There is no available material to indicate that this effect has been considered in the case of propellers. However, it is possible that interference of large proportions can occur both between the blade and the spinner and between the blade and the fuselage. This is es- pecially so in the case of the swept propeller, which enters the spinner at either a swept forward or swept back angle and which curves bask over the fuselage. In general, the result @19@ of this interference could be either favorable or unfavorable depending upon the configuration. There is no theoretical method available for predicting these effects, which means that resort must be made te experiment and that a number of configurations would have to be tested in order to evaluate the Pesults.e Little light can be thrown on this problem by ¢on= sideration of interference effects on other parts of the air= plane, for two reasons, The principal one is that very little is known about the problem theoretically and little or no experimental information is available at this time, Such ine formation may become available later, and at that time can be closely studied for analogies with the propeller interference problem, However, in this consideration we come upon the second difficulty, which is that the propeller interference problem is somewhat unigue, That is, it will bear little ree » semblance to gher interfering parts of the airplane, If quale itative comparisonscan be obtained by considering interferencé, say at the wing and fuselage, such should be taken into account put it cannot be hoped that anything quantitative will be obtained, =@20e 20.4% 2.5 Planform and Optimum Design C, Variation Introduction of sweepback into the planform of the propeller is the basic factor underlying the presentae tion of this paper, The purpose and principles applicable to the use of sweepback have been discussed in the introduc= tion and will not be treated further here, In attempting to define the optimum design Cy, variation along a swept propeller, it becomes evident that this is one aspect of the problem which is amenable to some theoretical investigation. Accordingly, the principal part of this work is the presentation of 9 means whereby the | piteh distribution of a swept propeller may be calculated so as to give optimum efficiency at a given operating condition, At this point we will merely list the differences to be expected underlying the optimum pitch distribution of a conventional planform propeller versus a swept planform propeller. le The field of induced velocities of the propeller Will be altered by compressi- pility and aweepback. 25 Profile drag coefficient will be decreased by aweepback, 3. Section thrust and energy loss will be aLterede oo Bes 4, All such effects will be expressable analytically. Section III is concerned with the analytical. treatment of | this problem, Ill DETERMINATION OF ANALYTICAL EXPRESSIONS FOR CALCULATING THE OPTIMUM PITCH DISTRIBUTION Sel Sources of the Method. Develovment by Glauert. There is a wealth of material which may be searched in an effort to find a framework upon which to build a method for computing the optimum pitch distribution of a sweptback propeller, Up to the present nothing has been published covers ing such an extension. Thus the writer was completely free from "the power of suggestion” in the choice of the basic framework, British writers have done a great deal of work on the optimum propeller problem and these works appear in the R&M publications, Several are mentioned in the references at the emd of this paper, All of the British works are adap= tations of the Golastien theory into tabular and graphical methods, Prepeller engineers in this country also use Gold= stein's work as a basis but in the large have developed their own methods of computation, All of these papers have as their purpose the procurement of the answer with minimum labor and as a result the original parameters in such developments are obscured, At the other extreme is the work of Glauert in Volume IV of the Durand series, There the problem is presen= ted much more clearly and concisely. Therefore it was de= cided to use the fundamentals of this reference as a framework and to expand it to include a computing procedure which should not be unnecessarily complicated, As the method was enlarged upon it was found that the algebra became unwieldy and so the approach was generalized slightly to include methods of the a PA os calculus of variations. This step was inspired by references #8 and #9 and its adaptability will be evident upon inspection of the section in this paver which is devoted to it. A brief outline of the reference by Glauert follows after which this method will be applied to the swept propeller problem. The problem dealt with in Glauert's paper is the de-~ the optimum serodynamic design of a prop of a given diameter operating ‘ +L 8 he 2 x, such a condition was firs a4 & aa oe Jen h a ~ S ppl specified that for highe an 4 ' wee ae i 7 _ y ana Le . “~ ~ “ i ay ee foo propeller should move axially aft Prom the propeller as rigid screw surfaces after undergoing an initial deformation. One of each such surfaces thereby corresponding to each blade. This condition is general but its calculation in specific cases is so difficult that simplifying assumptions must be made to make solutions tractable. The first such assumptions that may be made are that the propeller does not experience profile drag, has an infinite number of blades, and is lightiy loaded. Then the velocities induced by the propeller will be small in compari- son with the translational and rotational velocity of the pro- peller. At a padius rv we have the following expressions for the thrust and torque of a blade element of span dr = erp(a-¥)r and. where the circulation has a value defined by 6r=20wr* The preceeding equation arises from the assumption of an ine finite number of blades so that the vortex sheets lie close togethers Then if we combine with the above equations an expression for the thrust from momentum considerations, dT _ ar = amr pu (u-v) and neglect second order terms in the induced velocities, the following relation for the axial velocity at the propeller dise is obtained, u=ve gre And, accordingly, the thrust is g- arpar Now the energy loss for a frictionless propeller is dE. dQ a dr 2 dr —Vor And in the order of this approximation dE _ p*r’pn| ar Vv dr 47 |V ar Now define ern war 2 eee. a = fs U* orve an X="v . Then 4. ‘ve dE _ ZITPV" 2 1#Xx or RQ -~ KX -25~ and vt = LEY" yy dr Now the basis upon which the Betz condition was derived is that the ratio of the increments of VT and E due to an arbie trary increase inf at any station should be independent of the radial coordinate x , Sur condition for optimum perfor- mance in this case then becomes z x? ory TAN or TAT A is an arbitrary constant, The effect of varying its magni«- tude is clearly defined by the power operating condition which is given in our problem, The problem is now extended to account for the effect of the profile drag of the blade elements, It is evie dent. that the efficiency will be altered and by examining the effect in detail we see that the expression for optimum distri- bution of circulation along the blade is also changed, In particular, the additional energy loss associated with profile drag is ae eee W Gr = FePwlo And, if we use the fact that, OT! pg pe et ar Bc pW (C, cos # - Cy sin $) the expression for profile drag loss may be written as dE. T dr df =» 2G ws This relation contains several additional simplifying assump- tlons, namely, il. Effect of profile drag on the thrust is ignored. a p is assumed to be small. 5S. W is approximately equal to Slr. We will find in extending this theory to the swept back pro- peller that these assumptions are too restrictive; especially so are 2 and 3, and accordingly such assumptions wiil not be made at that time. But, for this present case, we assume that such conditions hold and the expression for the energy loss may then be written in the form dE arpa |r v | + carpe dr A V ALY} The expression for thrust is unaltered and we ob- tain for the work and loss of each element the following re- lations in terms of the parameters ¥ and x. 4 JT _ £UPV Vv apa dr dE 2m pm*| 2 14x" 2 = 41 PM \52 + EBX dr af) ZX And, applying the optimum condition as before, we obtain a z L4X Z CA-€X) X r) X tEX = AX or, et a This relation could almost have been deduced directly. it was stated that A is defined by the power input. . Then the above equation merely states that a portion of the power goes into the drag loss. Such loss must be proportional to the a2 Ta drag-lift ratio and to the relative velocity at each blade element . Since the velocity is proportional to the radius parameter, the validity of the above form is apparent. | | All of that which has been said in the preceeding paragraphs applies only to a propeller of an infinite number ‘of blades. We may now see the effect of considering our pro= peller to be comprised of a finite number of blades, The obvious effect of considering only a finite > number of blades is that the air between the vortex sheets will experisnce an important radial induced velocity which is not experienced by a propeller of an infinite number of blades » in addition, it is also apparent that this effect will become more pronounced near the blade tips. This consideration has led to the so»called tip loss correction factors (to be dis» tinguished from compressibility Loss) » An approximate correc= tion has been obtained by Framdtl and a more exact analysis by Goldstein, Goldstein's solution refers only to lightly loaded propellers, Neither approach will be given in detail here, It is sufficient at this point to say that the axial velocity is reduced and that this reduction requires that a factor be entered in the momentum equation for thrust. It is this reduction factor that is given by Prandtl and Goldstein in their analyses» The corrected form of the momentum equa= tion then is dT = = Uu - ar 7 40 (u WK K is a function of the advance amgle ¢$ . This functional -28- relation is treated later. Then since the axial induced vele ocity has been decreased by a factor K the optimum condition for the case of the frictionless, lightly loaded propeller is x*K 14x” y=A And, similarly for a propeller with drag, the optimum 7 is given by (AK-EX) x* 1#X* y= The reduction factor K is discussed in more detail in section 29d Ds 3.2 Extension of Glauert's Analysis to the Swept Propeller 3e2a Coordinate system, Geometry. Before carrying out the actual analysis it will be necessary to define some additional coordinates, Let us cone sider the following as a plan view of the propeller as seen if one were able to develop a helical surface into a plane. ¥ (locus of maximum f pressure points.) is the local angle of sweepback W is the resultant a we errno r v* +(an* _ _ | | -¢ Fig, 3-1 Coordinate System -29- Im addition, a great deal cam be said about the mag= nitude of the effective velocity which will be experienced by the blade element dz . | Since our calculations are to be carried out on the basis of a strip theory the problen is two dimensional and may be likened to a wing of infinite span in side slip, Therefore, if the viscous nature of the flow is neglected, the aerody= namic characteristics of a section will clearly be completely defined by the component of the resultant velocity which is normal to d » By such reasoning the resultant velocity is given by W cos @. Certainly this gives the correct lift of an element since any velocity along the blade in the spanwise direction cannot have a pressure gradient. By the same reason= ing the induced drag will also be correctly described but not the friction drag. However, we will use the cosine factor here keeping in mind that it is not an exact description of what takes place. If a more satisfactory expression for the effec= tive velocity is developed at any time, it will be a fairly simple matter to alter this analysis accordingly. in the light of the above assumption covering the magnitude of the relative velocity, a few words should be said about the plane in which the sweepback is to be accom= plished, We propose here that the blade should be ‘twisted. in such a fashion that the sweepback is accomplished in the plane of the helix which corresponds to the desired operating condition. The reason for this is best described by several sketches, . Fig. 3-2 Planes of sweepback. oaO= Sweepback in the fore and aft. plane: The dotted line corres= ponds to reduced resultant vel- ocity and we notice that very large sweep back will be ree quired to reduce the resultant velocity W by any satisfactory amount » Sweepback in the rotational plane: It is obvious that here we would also need large sweepback to accomplish the desired pure poses The logical solution to this pro= blem then is to effect the sweep back in the plane of the helix. Then the sweepback is directly effective upon the resultant velocity and the forward and rotational velocities also have as their effective values V cos SL cos p eeeb Derivation of the Equations. Upon the basis of the preceeding conditions and as- sumptions we will now proceed to derive the equations for -3l- thrust, torque, etc., which will be required in the course of the analysis. For a prepeller of straight planform the thrust may be written dl. - op 7 4ETpuc(u-v) This deseribes the momentum of the fluid in an annular element of radius r and width dr, Then the equivalent expression for a blade element dz is | at dy = 41 pu (u-Vv) cos B An expression for the circulation, [ , experienced by any blade clement will be needed, The value for a propeller of conventional planform is Br=zurtwr) = Zur" Now this expression describes the circulation about a closed circular path of circumference er corresponding toa veloci- ty wr, And, since this will be the value of the circulation in the flow irrespective of the type of planform the expres- sion will be carried over unchanged, Accordingly, for any element dz the circulation is given by Fr Br =ZTor S=(1) Another expression for thrust is obtained by application of the Kut ta-Joukowsky equation which, for conventional planform propellers, gives dT _ w -32— Recalling that we are applying the cosine of the sweepback angle as a measure of the reduction of the velocity compon= ents then it is elear that the equivalent expression for the swept propeller is dv w =—= BFO(N-$) l cos 3=(2) And by similar reasoning, the expression for the torque which for conventional planform is dQ = Brpur dr Brp will, in the case of the swept propeller become, dQ at = BF pur cos B The relation between the rotational and axial vel-= ocity should be unaltered by these ‘transformations’ over that obtained for the straight planfom, This may be checked by equating the expressions for thrusts Alrpu(u-V) cos B = Brp(R- S)r cos 8 Sinee the cos B factor cancels it is clear that the condition is sustained, Accordingly, the axial velocity will be obtaine ed from the relation u(u-V) = + [a- 2] wr’ 3e2e Optimum Circulation for an Ideal Swept Propeller The analysis will now be made under the assumptions, l. Flow is frictionless 2, The propeller has an infinite number of blades» So» The propeller is ligtily loaded, Assumption S will be expressed by using ( u = V )<<¥ and neglecting terms of w in comparison to 1. , Then the equa= tion defining the axial velocity may be written as, From equation 3=(1) we have that 2 en = OL Therefore, the axial velocity is _ Brn a= "4nV 4+V Also, equation 3-(2) for the thmst becomes dT _ a Bp Lr cos B And, to the same approximation, the equation for the torque is dQ _ . ,| Fn dy DIP cos BU = Brpr cos p |S=8 tv] Since the flow has been assumed frictionless the energy loss is described by gE, aQ_y dT | Lowwi+¥ rR WV 7 BT Proosp |-2(u-v) + ad and since, w _ Br “% ~ ¢mr® -d4= we obtain, dE | Brpn [2 ¥] cos dz 4 Using the notation of Glauert, we now write Brit | LLLP - 2 a = an znVve ang ke'y So that ya - Zier ¥X cosB and rr dE 2M PV X*+] a Lap x 15 7) cos B As mentioned seviously. "he Betz condition for op= timum operation is satisfied when the ratio of increments of VT and E caused by increasing the circulation is independent of the radius, 4 ave “A cos B x|(v 44%) - x] dE amp" x4] 2 V2 a os B 2X ctl(s taxi 3°] Therefore, the optimum condition is given by x* TF X® J= A It should be noticed that this is the same relation that was obtained for the conventional propeller under the same assump- tions, This result was to be expected, Since the flow is frictionless and the propeller has an infinite number of blades it is obvious that both the thrust and energy will be altered by the same factor, vizs., cos e » and it follows that in forme ing the ratio of increments of these two factors that the cosine function is cancelled, However, it cannot be expected that this will be the case when the analysis is extended to inelude profile drag. That, then is the next step. Se2d Effect of Profile Drag We will make am approximate check upon the effect of profile drag by including the following assumptions, il, Neglect the effect of drag on the thrust 2. Assume that the advance angie @ is small So Let the resultant velocity W ®flr cos @ The energy loss will now be given by, dE dQ dv dEo — = =e Vo $$ Se dz ft d¥ v dj oS where the energy loss due to profile drag is Fe 7 = BCE Wie = BOLT’ cos’p The thrust may be expressed in the conventional manner as dT a . =— =—BcPW |G cos$-Cysing and, if we write for the drag-lift ratio e= £2 e=-¢ then ET ban. prtcostg dy ZOPC HT cos’ B é for small gp and W= fr cos 6 Lt follows that AT d§ 3 CALL cos =3G= Then we can write the total energy loss as dF. gp @ yt et = 7 8 vy i-ex dy tas Vag OO cos B) Aa Vixl cos) If this is written in terms of the parameters 8 and x it becomes | ; dE _ zpVv 2 X*4#l dt hk cos | 3 2x + YEX* cos f| and the thrust force remains 4 di ZI p\ V 5 ; cos 6 ( ) Applying the condition for optimum performance as previously defined, we find that the desired ratio in this ease is Al st) XAy dE\ ZyAv(ItX’) z o( 2X +t AXEX cos B Thus the optimum distribution of circulation in this simpli fied case where profile drag is included is given by _ (A-€X cos B) x” / 14x? v Note that the corresponding equation for a propeller of conven= tional planfom was _ (A-€x)x* fex? Thus it is seen that the optimum distribution of circulation is altered when one includes the effect of profile drag. This effect, too, could have been deduced rather than. derived for the energy loss due to this drag will be a funce -37e tion of 6 » The chief interest here is the power to which this factor enters into the relations Clearly, then, the next step to be taken is to ex= tend the analysis with the simplifying assumptions withdrawn so that the results will be applicable to calculations, Also it will be necessary to include the tip effect in the equa tions, But before doing that it is of some interest to look at the preceeding results graphically. let us assume that the constent A = 0.5; that the drag-lift ratio € is constamt along the blade @ 0,053 and tht the sweepback angle is constant ® 60°, For these values the following results are obtained, Table S-1, and are plotted in figure 3-3, Table 51 Optimum Circulation A=0.5 €#0.05 6 60° K=“"A, | 0.5 | 1.0 | 2.0 | 4.0 | 6.0 | 820 | 10.0 ye AK 8.100 | 0.250 | 0.400 | 0.471 | 0.487 | 0.492 | 0.495 y Hee 0.095 | 0,225 | 0.320 | 0.284 | 0.195 | 0.099 | 0.0 yA ease 02097 | 06258 | 0.360 | 0.577 | 06340 | 0.296 | 0.250 Note. that when the profile drag is considered, sweeping the propeller allows for higher loadings on the outboard sections of the blade» This result is to be expected since the ener= ey loss is reduced in relation to the effective velocity and the amount of this reduction is increased as one passes to -48- the outboard sections of the blade. By inspection of Figure 3-3 we would conclude that the effects would be very slight for propellers operating at low Lt Vv ratios. However it must be remembered that these results were based upon the assump= tion of low advance angle (high <l/ V ratio) 60 that conelus= ions in the high forward velocity range must be withheld until the analysis is extended beyond the present restrictive assumptions, This is done next. 4.4 Further Extension of the Analysis Os0e Restrictive Assumptions Lifted. Introduction of the Calculus of Variations. Let us first summarize the assumptions under which the preceeding analysis was made. They were that 1. The sweepback is accomplished in the plane of the relative velocity. 2. the propeller is lightly loaded 3. the effect of drag on thrust could be neg- lected 4, the advance angle ¢ is small 5. the relative velocity can be written W =fLr cos 6 6. and, that the drag-lift ratio is constant along the blade. Now assumption 1 will be retained for the reasons put forth on page 30 . Also, this entire paper is meant to apply to . propellers operating at high eruising speed conditions so that assumption 2 is applicable. By looking at a velocity diagram -39= of a blade element it is easily seen that assumptions 3, 4, and 5 must be cancelled at high forward speeds. (Fig 3-4) dL U cos 6 0 Xp (aE )reosp Fig 3-4 Blade element forces Finally, it will be necessary to allow for changes in blade thickness and chord along the blade so that the drag lift ratio cannot sensibly be considered constant. Thus, assumption 6 must also be lifted and the analysis will be car= ried out on this basis, For the present the propeller will be assumed to consist of an infinite number of blades and the correction for a finite number of blades will be applied later. We now proceed to calculate the values of energy loss and thrust. In order to include the effect of drag the thrust will be described in the following manner, Denote the effective velocities in the forward, rotational and resultant directions as Us» RL , and Wo respectively. Then ~ Ve “1 U B= cos! Me = sp! TE | We We We have, dT _ di “2 Bc pwe Cy, (cos # - € sin f) and at Wee C, = so that we can write for the thrust, dT. ay = BPWe I (cos $- € sim $) or, substituting for cos @ and sin # their values in terms of the velocity components, we obtain AT dF The velocity components may be expressed in our dimensionless Br (ky,-EUe) = EPI (ky ~ EX) cos p parameters X and x as epearhe wv and u= ore ty = Lest2) so that Finally, the expression for the work obtained from the pro peller is Te IBY yoy g [y- £082] Va FBX eZ The corresponding energy loss will be derived from the equae tion SE _ SQ oT , dbs a ae ar 3=( 3) We will use expressions for torque input and work output in the forms, dQ Bi ly = BPI cos Bp c= +y| 3=( 4) and JT | . w _ V wr = 6rpr cos 6 (1L- as ) 3~( 5) It will be necessary to derive a suitable expression for the energy loss due to drag, This loss is given by, SEL, ——@2 = — But, also AT aE? > 8c) We "[G cos $ - Gp sin @| These equations may be combined to give a3 (fug - ies cos B Then if we substitute the translation and rotational velocity components for We and recall that ue < (s+z) and ky = XV the expression for the energy loss due to drag becomes +2 JE, | x? t <= "| cos 6 aR IP MS cos BV x 5 It will be convenient to write € z X TAL e7¢ and z -F Then the energy loss due to drag can be written as, dé, _ ZmpVv* (xX? 4") cos B dF —s- 2 J~ pA The expression for the entire energy loss is obtained by 5=(6) JX cos B substituting equations 3+(4), 3-(5), and 3=(6) into 3+(3) to give 4 2 2,,/% / + dE _ ZITPV cos ye Mtl 4 yy (CHU) cos 6 Az 2 ZX 6 = & more convenient expression is obtained by using the variable io throughout, Then dE _ ZITPV de 4 Zp ye Poa y2yp oye cos 6 ex} ae + cos B fe are Cl And, in the same notation, the thrust work is vit, 2mpv 2 a cos B [mM (x4e) - Eu? =x] With these two expressions in hand we can calculate the ratio of increments of ve and a caused by an arbitrary increase in the circulation. Our optimum condition states that this ratio shall be a constant independent of the radial co-~ ordinate x, However, to do this in exactly the same manner as in the previous examples leads to algebra which is hopeless= ly complicated. The expression for optimum comes out in the form of a quartic with complicated coefficients involving the basic parameters. But this complication can largely be cir= cumvented by resort to calculus of variations. Appendix I restates the problem in terms of this ealculus and refers to other propeller calculations which have been made in this -43— fashions in this way the optimum condition is expressed by 4(vs) = as = {\' 2 (#) ~? or | a The ~reciprocal of the constant is taken so as to give it a more convenient magnitude for computing purposes. The next step then is to carry out the indicated differentiation, Upon doing this and then forming the ratio as indicated above we obtain | 4 ZHpY (cos B)ZE Liss) -Z4] 0 4 _ 2 2, v2 . 2 Zz aly V cos 8) ZX oh Ki Noe + cos B ie $xOUL-1) + Sf £ pete } a a _> or x © (d=)? é- Zmpv- " ‘s The term Z 7] 058 eancels identically, S=(7) Accordingly, the expression for optimum circulation is obtained in the form of a cubic, p+ Butch + D =0 3=(8) where ¥+Z a a and where the coefficients of the cubic are z@ —=-(Sd#1) -Ad [z(zsel) x — cos B (38+) = 2@ z [i +as(A@ - sf] : | 2 x*4/ Z5(2541) +ZAS [(s+#2) xe 7 08 6] C= Zz, Zz [ tag (2H - cos) | é _ - $+) -Ad [287 cies cos (3-i)| Zz [ tag (XH cos 6) ~4he A solution in this form means that calculations are going to be somewhat tedious. We can anticipate the results by inspecting equation 3+(7) and comparing with the similar expressions obtained for the simplified cases, The numerator which expresses the change in propulsive force with change in circulation has the additional factor €( 1-2). This is a small quantity of the order of € so that only for the smaller values of x will it have an effect upon the results. We also see that for large x the term in the denominator, which des= eribes the change in profile drag energy, is reduced to€x cos p This agrees exactly with the result obtained from the simpli-~ fied analysis for all x, So it is seen that the corrections included in the more exact expression are of second orders Their contribution is negligible for higher values of x, But for high forward speeds x will, in general, lie below 1 and thus it becomes necessary to retain the correction terms, Same ple calculations have been made by equation 3-(8) and the re= sults are indicated in Table 52 and Figure 5-3, It is seen at x = 0.5 that the optimum circulation as calculated here is about 25% below that indicated by the simplified solution. For higher values of x the results coincide for all practical purposes, S$.0b0 The effect of a Finite Number of Blades, All of the preceeding theory has assumed the proe peller to have an infinite number of blades, We want now to consider the changes which must be made in the analysis in ore der to account for the fact that the propeller actually has a Table 32 Variation of Optimum Circulation with x A'2i1/A2 2.0 € ® 0,05 p= 60° x B c D iM 3 o5 | © 21,65 133,08] = 116 | 1,036 0.072 1] = 44,60 583.9 | = 590 | 1,102 0.202 2| = 94.76 2,781, | = 3,109 | 1,164 | 0.328 4) «197,02 | 12,322 -14,242 | 1.178 0.356 6 | 298.31 | 28,395 #32,625 | 1.167 0,334 8 | 399,03 | 50,884 =57,608 | 1.142 0.284 10 | 499,51 | 79,814 -86,732 | 1.120 0.240 finite number of blades, The physical fact is that the slip= stream is composed. of series of vortex sheets, one for each blade of the propeller. By the Biot Savart Law thereare vele- ocities induced by these vortices. Thinking of the propeller as composed of an infinite number of blades allowed us to sub= stitute for this complex vortex pattern a geometrically much simpler pattern. We then could envisage a bound vortex upon the disc of the propeller with the horseshoes elements of this vortex passing backwards along the axis of the propeller and the eylinder generated by the blade tips. In such a system we were able to obtain expressions for induced velocities in the axial and rotational directions. There were no radial veloci- ties induced by this vortex system. Passing to the actual fact of a finite number of blades, hence a finite number of vortex sheets, introduces a radial flow of the air between the -4G= vortex sheets. Immediately it is seen that this effect is such as to reduce the axially induced velocity and hence to reduce the thrust as computed from momentum considerations, There= fore, qualitatively, the effect is very clear but to carry out a quantitative analysis is not quite so straight forward. The problem has been solved by several people, chiefly Prandtl and Goldstéén. Prandtl solves the problem in an approximate fash= ion by replacing the vortex sheets by a series of parallel lines, . Goldstéén, on the other hand, has solved the potential problem exactly. This solution exactly represents the Betz optimum condition, By reason of the complicated nature of the pro= blem it is obvious that the exact solution must also be very complicated, Thus Goldstein was only able to obtain a solution for lightly loaded propellers, Both the analyses of Prandtl and Goldstein arrive at a reduction factor which must be applied to the thrust as calculated by momentum considerations, This factor which we shall denote by K, is a function of the station radius, the advance ratio d » and the number of blades, Gold= stein'’s analysis must be completely reformulated for each dif= ferent number of blades while the approximate theory of Prandtl results in a general expression into which we need only substi- tute the parameters including the number of blades, Since it is exact, and since our problem fits the assumption of light loading, we will use the reduction factor as derived by Goldstein. Table 5-3 and Figures Se5, o-6, and o=7 in the Appendix II give values of K for 2, 5, and 4 bladed propellers, This is a convenient point at which to consider the -AG~= effect of compressibility upon the equations which we have been writing» Both momentum theory and vortex theory have been used in this analysis and both»are based upon an incompressible fluid. Tsien and Lees in reference #10 have evaluated the ef+= fect of compressibility upon the induced velocities and hence upon the thrust and torque for lightly loaded propellers. Upon such an assumption they applied the approximate Biot-Savart Law for compressible fluids and found that the axially induced velocities as calculated for an infinite number of blades is unaffected by compressibility. This is a direct consequence of the assumption that induced velocities are sMall and accor= dingly that density variations may be neglected in a first approximation, When one considers a finite number of blades the correction involved is increased by a factor proportional to the ratio Tae e it has been found that the correc tion to the axially induced velocity by consideration of a finite number of blades is of major importance only for a propeller of a small number of blades (2) and then only on the outer portions of the blade. Thus for a four bladed pro- peller considerations of compressibility involve only a minor correction to a minor correction and therefore will be neg- lected here, It is important to note, however, that the section characteristics to be used must correspond to the effective ‘Mach number at which the blade section will operate. It should also be remembered that the above statements will only apply within the assumptions of the Biote-Savart equation and for a lightly loaded propellers om AB ‘We now introduce the reduction factor K into the relations which define the optimum distribution of circulation along the blade. The expression for the propulsive work be= comes 4 dT _ £NPV Vas - Ae cos B [ZN EX - Ze (u-N)] The expression for the energy will be unchanged since both the torque and the thrust are reduced by the factor K, Then in our cubie which defines the optimum circulation fe + BM Ch +D=0 5-(9) the coefficients are given by _ = [ecanss) +il- As lz C4 Zz SKN? - cos 8 GS tN) B= Z fl 4 AS [(x*+1)/X? - cos B]} c: Zd(SCEK) + I) + ZASVSAZ)OCtI)/X* ~ cos 6] . kN FAST CX4i)/x* — cos 6] oe -S(EK +N) - AS ZEEE — Kcos B)(S-1)] : Z{I+AS [CX41)/X* — cos 6} S,o¢ Evaluation of the constant At The signifigance of A‘* and the means of determining its value can be shown by the following considerations. The power absorbed by the propeller is given by fag _ empvi— zy p=) 2 D>. x8 cosp(Z +!) ref Now, 2k 4. Pp | (20V y y ~* pnio*® ~ D ‘no! x 8 cos (2 +!) S-( 10) rfe -A9- Or, this relation may very conveniently be written as a _J 5 Cp = D X¥ cos B(> +!) F=f, where _ Vv aay; and where ¥Y must define the optimum circulation as previous= ly obtained as a function of A, K, and the radius factor X. Cs is known numerically for a given operating condition. Thus for a given distribution of € and cos 6 » the optimum 8 is determinable as a function of A. Accordingly _ the summation is a definite number for each A and this constant ean be adjusted to give the desired value of Cre Having de= termined the correct A in this manner the desired range of is known, The method of incorporating this detail in the cal= culation procedure is outlined in section 5.5f,. 3,564 Effect of the Drag-Lift Ratio Upon Optimum Distribution of Circulation, It has been tacitly assumed that the optimum cire- culation does not depend upon the drag lift ratio;that is, in establishing the optimum be >» € has been treated as a constant. Actually, of course, there is a dependence and to be strictly correct the derivative of € with respect to the circulation should have been included. However, the operating blade sec= tion characteristics are the unknowns in this problem and to include the derivative gE in a general way leads to compli= or cations which are excessive. In fact, such a consideration is -50- largely unnecessary as may be seen by violating a mathematical principle and allowing the € to vary in the solutions prev- iously derived upon the basis of its being constant. Let us consider the approximate solution corresponding to the case of the sweptback propeller with drag. The expression for opti«+ mum distribution of circulation in that instance is (A- €X cos 8) x* Ix? oF in addition, we will restrict this investigation to values of x <1 since this is the speed range of primary interest here. Then for constant sweepback of 60° and A= 0.5 the following variation is found between ¥ and € » Values of ¥ are in the body of the table, Note that the variation of ¥ 6 x 0,5 120 becomes of important magnitude sOL 0,099 0,247 : only as € exceeds 0,05 and 202 0,099 0.245 then only for x—~1, Propeller 905. 0,098 .| 0,242 sections seldom operate at such 005 0,097 0,257 . a high drag lift ratio and when ok 0,095 0.225 their coefficients do approach ae 0,090 0,200 such a ratio it is at the thick= er shank sections of the blade where the dependence of ZY upon € is very slight. Also, in actual cases, A will, in general, exceed 0,5, This diminishes the effect of € percentagewise. Therefore, the computational procedure will be to assume a reasonable range of values of € along the blade, calculate the corresponding distribution of circulation, ob- -5l- tain the actual distribution of € along the blade to fit the desired dimensions and force coefficients, and then to check for the necessity of correction due to this change, These statements are included in the outline of computational pro= eedure in section 3.5f. 3,e To Calculate the Magnitude of the Blade Correction Fac= Figure 5-5 gives the factor K as a function of the advance angle (pro, ) with the blade stations as parameter. L is the induced angle corresponding to the induced axial and rotational velocities, immediately it is seen that our calculations must include successive approximations. That is sing is first calculated upon the basis of the forward vel= ocity Vand the rotational velocity flr. The proper values of K at each blade station are then selected and the computation of optimum circulation is carried out, This will then give a value of the axially and rotational induced velocities from which (¢ #o:.) may be determined. Having these, the corrected values of K are determined and the calculations repeated. It is sufficient to take the second computation as the final re=- sult since for lightly loaded propellers the convergence in the approximation procedure is very rapid. These steps are included in the outline of the computational procedure in section 3.éf, S,of Summary =5P = of Steps in the Calculation. (For determina= tion of le 2 Se 4» Ss Geo 7s 8 Qo 10, ll» Optimum Circulation for Lightly Loaded Operation) Determine the Mach number range along the blade corresponding to the desired operating conditon, Assume a sweepback curve such that all blade stations are reduced to operation at a desired effective Mach number (say 0,8). Assume a reasonable distribution of € along the blade, keeping in mind the high Mach number op= eration, The range of 0.015 € S 0,05 should normally be sufficient, Calculate K at each of the blade stations, Determine Cp and J corresponding to the desired operating condition. Calculate the coefficients in the cubie equation 3-(9) for representative values of A. (Two rather widely syaced will suffice). Calculate the optimum jb from equation 3<(9) for each of the values of A. The summation in the equation for C,, » equation 5-(10) may next be evaluated graphically and the proper A determined. The distribution of pe corresponding to this A is then calculated and hence Y is determined, Calculate Cex amv’ x £0 BR We From Cre the proper values of € along the -blade ean be chosens -55= 12. oO, is calculated using the relation, Vv k= (8tz) 183, Proper values of K are chosen. 14. The solution is completed for this new value of Ke Note: The solution should finally be checked for the effect of variation of € (step 11) amd to see that the proper value of C, has been maintained, Bho Iv Tht HYPOTHETICAL EXAMPLE ~ The following operating conditions were assumed: V = 550 mph BS 4 h = 25,000 ft. REM = 1260 D= 15 ft. P = 2800 HP Then Cp= O.4L2 and J =k. 96 l. Aetual Mach number variation, The range of Mach number along the blade for the above operating condition is shown in Figure l-4, Values of M at the desired blade stat= tions are given in Table 41, row (2). 2. Calculation of sweepback angles. The effective Mach number is reduced to 0.8 at each station. (rows 3 and 4) 3, Assumed variation of € along the blade. These figures (row 5) are based upon a thickness variation from 12% at $0.3 to bat § = 0,95, It is assumed that 16 series symmetrical airfoils are used throughout the length of the blade, See reference #6. 4. Calculation of the tip factor K (lst approxima=- tion). It is first assumed that the advance angle @ is deter= mined uniquely by the forward velocity of flight and the rota= tional speed of the propeller. Values of K are taken from Figure 3-5, (see row 9) 5, Determination of the coefficients in the cubic. Several values of A are assumed and the coefficients evaluated for these values of A. Values of 6 and cos B as determined =55- above are used. (see rows 10, 11, and 12) 60 p&(the loading factor) is now determined for those values of the coefficients as obtained in step 5, The corresponding ¥ is also obtained. (see rows 13, 14, and 15) Values of ¥ versus & (blade station) are plotted in Figure 4=le 7e Calculation of C,o The finite sum in equation 3-(10) is evaluated mechanically for each of the three values of A» Values of the integrand at each blade station and for each of the three values of A are given in row 16, The inte« grand is plotted versus blade station in Figure 4-2, fe? is plotted versus A, Figure 4-3, 8. Determination of the proper A. Our power condie tion gives a power coefficient of Cy = 0.422, The correspond= Co D CoD ing value of aT is 0.0715, From the plot of ar versus A it is found that we need an A = 4,1 9, Calculation of ¥ for Cp 80.422 (A # 4.1) With A = 4,1 the coefficients in the cubie are calculated (row 17). YX is determined by solving for m from the cubic. (row 18). Note: A check on Cp showed that the proper value has been attained. | 10, The blade element loading (c,e) is calcula= ted from Zz eV ¥ = = 194 8 Values are in row 19 and are plotted in Figure 44, lil, Check on tip correction factor K, The in= duced velocity is computed from u=¥ (942) or Uu-V= ve = 403.78 Its value, in feet per second, is given in row (20). The new sin d and K follow in rows (21) and (22). Note how little K has changed which is a direct result of the light loading of the propeller, (To the order of accuracy of which these calculations have been made it is not necessary to correct the solution for the change in K) “O7=, Table 4«L Numerical Example Ls Station ors) 0.45 0,60 0,70 0,80 0.90 0.95 2. |M (operating) |.830 | .878 |.942 (.993 |2,045 |1.100 | 1.132 3. | eosp 2963 | 910 |.850 |.806 |.765 |.727 | 2706 4, | 86 15°22"/ 24°30"|31°47"/ 56°18"| 40°" | 48°22") 45°15! 5e | € 005 204 208 008 o0L =| 01 oOL 6. |r (ft/sec) 257 386 515 600 686. 792 815 7. | tané="ar 3.14 |2.09 |1.568 11.346 |1.178 |1.045 | 0.990 8. | sin ¢ 0955 «| 6902 | .843 '|.803 |.762 |.723 | 704 9.) K 1.172 |.917 |.736 | .625 |.507 |.358 | .254 10,) Coeff. , A=. 1: B -1421| -26.32| -47.16| -82 98] -191.4| -217.6] -250.8 C 58.81 | 200.7 |654.5 |2041 | 10838 | 14301 | 16162 D @46 981) 1852] 657.5) 2142 | -11695| -15173| ~16832 1l.| Coeff., A = Se to B ~14 416] -26 .30| -47.21] 83,12] -191.7| -218.2) -231.4 Cc 58.42 |199.7 |652.6 |2038 | 10837 | 14311 | 16178 DO ~45 247/176 8] -620.0| -2010 | =10975) -14426| ~16147 12. Coeff., A = 5: B 14 514| 26 »28| -47.23) -85 .16) -191,8] -218.3/] 231.6 ¢ 58.55 |199.4 |652.3 | 2038 | 10836 | 14311 | 16181 D #45 621) -175.1| -612.4) -1984 | =10830| 14275} -16087 13,| Opt. Values,A=1: fee 1.037 | 1.066 |1.088 | 1,098 /1.100 |1.079 | 1,057 y 0074 | 2132 | 0176 | 2196 | .200 | .158 | 2114 14.| Opt. Values, A®3: Me 1,006 |1.016 |1.024 |1.029 | 1.0315/1.024 | 1,013 ¥ 2012 | .032 | .048 | ,058 | .063 | .048 | .086 15. Opt. Values ,A=53 Me 1,0008/1.005 |1.011 }1,015 |1,0178)1.015 |1,009 y 20016] 0090] .022 | .030 | .0356| .026 | .018 16.2/6,, integrand: . er 00256 | 0613 | 1040 | .1292 | .1434 |.1184 | .086l A®8 00037 | 0142 | 0267 | 0858 | .0424 | ,0342 | ,o1e9 A 25 00005 | .0039 | .0121 | .0183 | .0235 | 0184 | ,0130 17.|Coeff., A = 4413 B -14 916) -26 629 | -47.22| -85.17| 191.7) -218,2|=231,5 c 58.57 |199.6 |652.5 |2038 |10836 |14310 |16180 D #45 929) +17507/=61469|=1993 | -10877/ -14324|=16128 Table 4] ( continued) 1. |Station 0.5 | 0.45] 0.60] 0.70] 0.80] 0.90] 0.95 18.,)/Opt. Values,A=4.1? be 1.002 | 1.009 | 1.016 | 1,020 | 1.022} 1.017 | 1.012 x 0004} 018} 032] 040) 044} 034] .022 192] Gye 00765 | 544 | .612 | .765 | .842 | .650 | .421 20.) (u-V) 1062 | 7.27 | 12.92 |16.2 | 17.8 | 13.7 | 8.9 Zle| sin (@#at;) | 953 | .904 | .847 | 808 | .769 | .729 | 708 22» K o917 | .735 | 622 | .504 | .356 | 2253 1.172 Ve CONCLUSTONS In this paper we have developed a means for calcu= lating the pitch distribution of a propeller of sweptback plan» form so as to give peak efficiency of operation. It would be advantageous to be able to calculate the characteristics of such a propeller by a strip analysis and thus to indicate that our purpose has been achieved, However, such a calculation would necessarily be based upon the same simplifying assump= tions under which the pitch distribution was computed, Accord= ingly the most that could be expected from such.a check would be the substantiation of the algebra, What is needed are the actual test data from a propeller which has been designed aerodynamically by this method and, of course, such data are not available, Substantiation of calculations such as the one put forward in this paper will have to await the appear= ance of test data for such configurations and the more complete understanding of the basic aerodynamic characteristics of swept= back airfoils. -60- REPERENCES le Quick, A., Tests on Airscrews with Swept Curved Blades, CGD #488. | Be Seiferth, R., Investigation of an Airscrew with Sweptback Blade Tips at High Tip Speed, cGD #500. 3, Modern Problems of Airscrew Aerodynamics, CGD #483 4. Liepmann, H. and Puckett, A., Aerodynamics of a Compressi= ble Fluid, pp» 142, 247-252 5» Theodorsen, T., Theory of Wing Sections of Arbitrary Shape, NACA TR #411. 6 Stack, Jo, Tests of Airfoils Designed to Delay the Com= pressibility Burble, NACA TN #976 7. Glauert, H., Volume IV, Durand Series, pp. 251-269 8. Lock, C., and Pankhurst, R., Determination of the Opti- mum Twist of an Airscrew Blade by the"Calculus of Variation”, R&M #2088, Jan. 1942. 9o Haines, As, and Diprose, K., The Application of the Cal culus of Variations to Propeller Design with Particular Reference to Spitfire VII with Merlin 61, RaM #2083 May, 1943. 10sTsien, H. and Lees, L., The Glauert-Prandtl Approximation for Subsonic Flows of a Compressible Fluid. JAS, Vol 12, No. 2, pp. 186-188. -Gle= APPENDIX I. APPLICATION OF THE CALCULUS OF VARIATIONS TO PITCH DISTRIBUTION CALCULATIONS | (1) Deseription and Statement of the Purpose of the Calculus of Variations. The following description may be found in many standard mathematical texts, What follows immediately here will be general in nature amd will not apply directly to this particular problem, However, the calculus of variations lends itself so well to the calculation of propeller operating char= acteristics that it is worthwhile to include a brief, general description, in this paper.» | Suppose we desire to find the form of a function, f, such that if y= f(x), then Xy J OKG.g dK Xo shall be a maximum or a minimum where B(KY,Y') is amy desired function of these variables, y* me » and AIX X, and x, are any desired values of x, Thus we are looking 1 for a definite integral Xy rf P(X, g, YD dx Xe which corresponds to the form of a curve giving an I which is greater than or less than any other curve which we may draw between the emd points x, and x,. Now let y ® f(x} be the ° desired function, or curve, and y = F(x) be any other contin- =62= uous curve joining the same emd points, Then if, Y, aK) = F(X) - £ (x) Cc y= FOX) = f(x)#¢4) we will have. a r % 4) Y= fn) #0 CK) representing the same curve as v gr FX) Now consider the expression Y= FO) + GX) where o& is a parameter independent of x. This function then represents an entire family of curves in which a) when € 20, y = f(x) b) when of #1, y * f(x)+q (x) * Mx) Then if we take of small enough, the function of%(X) for this o and for all smaller of‘s » and for all values of x between X_q and Ey» will be less then any previously chosen 5 a For these values of of the curve Y= FOO FAUX) is said to be in the neighborhood of y = f(x). Then, if y = f(x) and y = F(x} are given, and if by definition %, r= [ G(X, 4,4") dX Xe then I for amy one of the of family of curves is given by, x, r(t)=f b [x, yreqon, y/t0q'(X) | dx x, =-63= where I (of) is a function of of alone. Then if y ® f(x} is the curve of the family for which I is to be a minimum it is readily apparent that I (4) must be a minimum when ¢® 0, A necessary condition for this is that 4 ' 4 rt) | =O This condition is expressed by the equation / I'(0)=0 (2}) Notation a) Variation of y (dy ): For any x between x, and x, U(X) = F(X) ~ £(X) ‘Thus 7 (x) is an increment in y produced by changing f(x) ine to F(x) without change in x, %(x) is the variation in y (re= presented by dy). dy is, in general, an arbitrary function of Xe b) Variation of yt (dy'): The slope at any point of the curve y = (x) is the difference between the slopes of the curves y * F(x) and f(x) for the corresponding x. Thus oS a'(Xx) = ox (9) is the change produced in y’ by changing f(x) into F(x), e) Variation of y't (dy!) By extension of (b) we have that (= G4= u =f a X)=— (d 4°) = (44) is the change produced in y'* by changing f(x) into F(x). Now let us denote a given function of y as ¢ (y) and increase y by r) yo Then 5 | — |P(y) | d aly ¢ q! 7 will be an approximation of the inerement in Ply)s the order of the approximation depending upon the magnitude of dy. If 6 y is an infinitesimal then the approximation is in error by an infinitesimal of order higher then Os This approximate inerement is called the variation of bly)e Phus — Variation of Aly) = socy)= Zoey) éy Also, if y and y’ are treated as independent variables, the variation in a function #(y, y') due to an increment dy? of y* and éb y of y will be Variation of dé(y, y') = t Oy t ee sy Similarly, “) ker, Variation of d¢(y"*, y'') 8 dy + ELEY gyn og" All of the above variations are supposed to be caused by varying the f in y ® f(x) without changing x, So, JP(K. FY) Sy + SPO! gy Jb (x, y,y') = - oy! =-65< Note that df is an approximate increment in @ caused by chang= ing x without changing @ while d¢ is an approximate increment in vs) caused by changing @ without changing x» Hence, with this new notation, our condition that I(&) be a minimum when ¢& 0 is that X, | SH(X,Y,Y") =0 Ry And let it be defined that the variation of a definite inte= gral is the integral of the variation of the integrand, i1.¢., af bax - [ dbax Xe Xe Se we may write our minimum condition as é[=0 (3) Optimum Pitch Distribution by Calculus of Variations The acknowledgment here is to references #8 and #9. In applying the calculus of variations the British writers make use of the power loss and torque grading coefficients which are seldom used in this country, Therefore, rather then to outline the reference her we will merely consider the general ideas involved. It is assumed that all factors are fixed except the blade setting @ ; that is, that determination of the op~ timum will involve investigating only a single variable, This restriction is not necessary and this entire procedure may be generalized to any desired number of variables. In general =66- @ will vary along the radius r in a fashion which it is the pur- pose of this analysis to determine, Then we let @= f(r) tfQ(r) wher f(r) is a function to be determined, “@(r) is an arbitrary function, and o& is a parameter which will allow the determina- tion of f(r) so that certain integrals involving @ are a mine= imum. These certain integrals may define the torque input and power losses of the propeller. By the reasoning of part (1) of this discussion it is seen that the integrals will be funce tions of the ‘parameter & alone and when oO is zero that the desired functions will be determined. To make of zero is to write the variation of the integrals equal to ZELO» One such minimum integral will exist for the torque input and one for the power loss, In order for both integrals to be a minimum the ratio of the two integrands will have to be related by a con- stant. | That is the general principle of this method and to | check its utility we will apply it to the simpler cases of section III, Using the work output and power loss as the quan tities of interest we apply the above principles and state ‘that we want [# Lae) sp yr =O for all forms of d/ which make J(v Mr) _ J yO a O =-67= The only way that this can be true is for or ~ or Applying this criterion to a simple case, page 26, gives Vr) _ ZTpv" oy or ~ Ah of and o(%nr) _ empy* az] 4x as or 7 x 7 &% so that the condition of optimum circulation satisfies the relation Je x? Xx x +t EX” = AX The other cases treated in this paper could be checked in a similar fashion, This discussion mekes it appare ent that nothing has been added to our analysis method by uti- lizing the calculus of variations. However, it puts the entire procedure on a more formal basis and allows a great deal of algebraic simplification when we treat the more exact equa- tions, It is also apparent that the great many similar pro blems which exist in propeller aerodynamics may very nicely be treated by the calculus of variations, =68- Table 3-3 Goldstein Factor (K) for 2, 3, and 4 Bladed Propellers sin (¢td) 2005 210 220 peel) 240 260 280 290 1.00 x 8 43 1,000 1,000 1.000 1,000 1,000 1,000 0.994 0,997 0.998 0.978 0.992 | 0.996 0.958 0.984 0.991 0.950 0.975 02986 0,916 1,010 1,055 0.935 1.061 1.105 1,012 1,185 1.256 x ® ,45 1.000 1.000 1.000 0.998 0.999 1.000 0.991 02995 0,997 0,959 0.987 0.994 0.906 0.966 0.984 0.784 02909 0.954 0.677 0.849 0.924 0.657 0.832 0.917 0,633 0,832 00925 x # ,60 1,000 1,000 1,000 0.997 0.999 1,000 0.961 0,998 0.995 0.874 0.955 0.984 0.783 0.902 0 6950 0.608 0.771 0.855 0.502 0.661 0.756 0.464 0.622 0.713 0.425 0,588 0.681 x = ,70 0.999 1,000 1,000 0.988 0.998 1.000 0.901 0.964 0.989 0.774 0,892 0.945 0.665 0,809 0.883 0.492 0.650 02745 0.598 0,533 0.627 0.560 0,494 0,581 0.325 0.457 0,543 x = 75 0.997 0,999 1,000 0.971 0.994 0.999 0.852 0.955 0.973 0.709 0.843 0,909 0.595 0.746 0.803 0.435 0.582 0.678 0,547 0.470 0,562 0.311 02429 0.516 0.279 0.593 0.477 x = ,80 0,994 0,998 0.999 0.957 0.980 0.993 0,784 0.884 0.940 0,634 0.774 0,852 0.520 0.670 0.759 0,576 0.508 0.601 02297 0,404 0.490 0.265 0.567 0.447 0.258 0.535 0.412 x = 085 0.985 0.990 0.998 0.877 0.948 0.985 0,694 0.810 0,882 0.548 0.684 0.774 0.442 0.581 0.671 0.516 0.429 0.517 0.247 0.557 02413 0.220 0.504 0,575 0.197 0.278 0.345 x @ ,90 0.950 0.973 0,995 0.775 0,872 0.945 02578 0.695 0,777 0.444 0.566 0.651 0,551 0.471 0.554 02249 0.541 0.414 0.193 0.265 0.529 0.172 0.259 0.298 0,154 0.218 0.272 x 2 ,95. & Oo | Peto | Bei | BEA | mH GTD9 |) BOND) Bw tod] Pm CID | wm CA D9 0.780 0.865 0.945 0.586 0.692 0.770 0.415 0.512 0.590 0.508 0.406 0.476 0,243 0.351 0.596 0.171 0 »256 0.290 0.131 0,182 0.228 0.117 0.164 0.205 05105 0.149 0.187 Pp : Supe neeel nem belted War feu f mane 4 af td +. t