I , / ;• NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS REPORT No. 777 THE THEORY OF PROPELLERS Ill-THE SLIPSTREAM CONTRACTION WITH NUMERICAL VALUES FOR TWO-BLADE AND FOUR-BLADE PROPELLERS By THEODORE THEODORSEN 1944 For sale by the Superintendent of Documents - U. S. Government Printing Office - Washington 20, D. C. - Price 20 cents -S.- AERONAUTIC SYMBOLS 1. FUNDAMENTAL AND DERIVED UNITS Metric English Symbol Unit Abbrevia- Abbrevia- uunit t tion Length 1 meter--------------------m foot (or mile) --------- -ft (or mi) Time -------- --t second ----------------- --s second (or hour) ------- --see (or hr) Force---------- F weight of 1 kilogram kg weight of 1 pound-------lb Power ------- - P horsepower (metric) ----- --------— horsepower-------------hp Speed v f kilometers per hour--------kph miles per hour ---------- mph imeters per second ------- -mps feet per second--------- fps 2. GENERAL SYMBOLS W Weight=mg V Kinematic viscosity g Standard acceleration of gravity=9.80665 rn/s 2 p Density (mass per unit volume) or 32.1740 ft/sec' Standard density of dry air, 0.12497 kg-m-'-s' at 15° 0 W M Mass= — and 760 mm; or 0.002378 lb-ft-4 sec' g Specific weight of "standard" air, 1.2255 kg/ml or I Moment of inertia=mk2. (Indicate axis of 0.07651 lb/cu ft radius of gyration k by proper subscript.) Coefficient of viscosity 3. AERODYNAMIC SYMBOLS S Area ill Angle of setting of wings (relative to thrust line) S. Area of wing it Angle of stabilizer setting (relative to thrust 6' Gap line) b Span (7 Resultant moment C Chord Resultant angular velocity V vi A Asp, ect ratio, B Reynolds number, p— where 1 is a linear dimen- V True air speed sion (e.g., for an airfoil of 1.0 ft chord, 100 mph, 1 q Dynamic pressure, 2p standard pressure at 15° C, the corresponding Reynolds number is 935,400; or for an airfoil in L Lift, absolute coefficient (YL= L of 1.0 chord, 100 mps, the corresponding gS Reynolds number is 6,865,000) D Drag, absolute coefficient a Angle of attack Angle of downwash D0 Profillrag, absolute coefficient D Angle of attack, infinite asjiect ratio qS at Angle of attack, induced D1 Inducirag, absolute coefficoefficient C=- a Angle of attack, absolute (measured from zero- lift position) 0=D,, D Parasi drag, absolute coefficient '1 Flight-path angle 0 Cross- ud force, absolute coefficient qS ERRATUM NACA REPORT No. 777 THE THEORY OF PROPELlERS III - THE SLIPSTREAM CONTRACTION WITH NUMERICAL VALUES FOR NO-BLADE AND FOUR BLADE PROPELLERS By Theodore Theoclorsen 194 Page 18, figure 7(a): The bottom part of the lowest curve In tIre lower left-hand corner of the figure should. be - - Instead. of - J/1 w1 vw REPORT No. 777 THE THEORY OF PROPELLERS Ill-THE SLIPSTREAM CONTRACTION WITH NUMERICAL VALUES FOR TWO-BLADE AND FOUR-BLADE PROPELLERS By THEODORE THEODORSEN Langley Memorial Aeronautical Laboratory Langley Field, Va. I t ft LM National Advisory Committee or A&onauüi' IIwlpIa1trs, 1500 New Hampshire Avenue NW., Washington . P. C. Created by act of Congress approved Mareli 3. 1915, for the supervision and dir'Cl ion of ti ientific stud v of the problems of flight (1]. S. Code, title 49, sec. 241). Its membership \\'tis ierea'i1 to 1. v act approv'd March 2, 1929. The members are appointed by the President, and serve as -uch without CompisJtion. JEROME C. IIcSSAKER, Sc. I)., Cambridge, Mass.. Ci',non LYMAN J. BRIGOS, Ph. D., Fire Chairman, Director. National AUIIREY W. FITCH, ViceAdiniral, United States Nav y . Deputy Bureau of Standards. Chief of Operations (Air). Navy Depart iiient. CHARLES G. ARBOT, Sc. D., Vice Chairman, Executive Committee, WILLIAM LITTLEWOOD, M. E.. Jackson IJeihts, Long Island. N.Y Secretary, Smithsonian Institution. HENRY H. ARNOLD, General, United States Army, Commanding FRANCIS W. REICHELDERFER, Sc. D.. (hief, Ututel states General, Arm y Air Forces, War Department. Weather Bureau. WILLIAM A. M. BURDEN, Special Assistant to the Secretar y of LAWRENCE B. RICHARDSON, Rear Admiral United States Navy, Commerce. Assistant Chief, Bureau of Aeronautic- Nav y DepartIII('nt VANNE VAR BUS H, Sc. D., Director, Office of Scientific Research EDWARD WARNER, Sc. D., Civil Aeronain cs Board, Washing- and Development, Washington, D. C. ton, D. C. WILLIAM F. DIJRAND, Ph. D., Stanford Universit y , California. ORVILLE WRIGHT, Sc. P., Dayton, Ohio. OLIVER P. ECH0I.S, Major General, United States Arm y , Chief of Maintenance, Materiel, and Distribution, Arm y Air Forces, THEODORE P. WRIGHT, Sc. D., Administrator of Civil Aero- War Department. nautics, Department of Commerce. GEORGE W. LEWIS, Sc. D., Director of Aeronautical Research JOHN F. VICTORY, LL. M., Secretary HENRY J. E. REID, Sc. D., Engineer-in-Charge, Langley Memorial Aeronautical Laboratory, Langley Field, Va. SMITH J. DEFRANCE, B. S., Engineer-in-Charge, Ames Aeronautical Laboratory, Moffett Field, Calif. EDWARD R. SHARP, LL. B., Manager, Aircraft Engine Research Laborator y, Cleveland Airport, Cleveland, Ohio CARLTON KEMPER, B. S., Executive Engineer, Aircraft Engine Research Laborator y, Cleveland Airport, Cleveland, Ohio TECHNICAL COMMITTEES AERODYNAMICS OPERATING PROBLEMS POWER PLANTS FOR AIRCRAFT MATERIALS RESEARCH COORDINATION AIRCRAFT CONSTRUCTION Coordination of Research Needs of Military and Civil Aviation Preparation of Research Programs Allocation of Problems Prevention of Duplication LANLiLY ML\IORIAL,AERONAITICAL LABORATORY ÂMES AER0\ \ITICAL LABORATORY Langley Field, Va. MoTh r Field. Calif. ,('RAFT ENGINE RESEARCH LABORATORY, Cleveland Airport, Cleveland, Ohio unified control, for all agencies, of scientific research on the fundamental problemsI of jug/if IFICE OF AERONAUTICAL INTELLIGENCE, Washington, P. C. I st a aipilcif ion, and dissemination of scientific and technical information on aeronautics II REPORT No. 777 THE THEORY OF PROPELLERS Ill-THE SLIPSTREAM CONTRACTION WITH NUMERICAL VALUES FOR TWO-BLADE AND FOUR-BLADE PROPELLERS By THEODORE THEODORSEN SUMMARY x ratio of radius of element to tip radius of vortex As the conditions of the ultimate wake are of concern both sheet (r/R) theoretically and practically, the magnitude of the slipstream yR radial velocity contraction has been calculated. It will be noted that the con- V advance velocity of propeller traction in a representative case is of the order of only 1 percent w rearward displacement velocity of helical vortex surface of the propeller diameter. In consequence, all calculations - w need involve only first-order effects. Curves and tables are given w=V for the contraction coefficient of two-blade and four-blade pro- number of blades pellers for* various valves of the advance ratio; the contraction p coefficient is defined as the contraction in the diameter of the mass coefficient wake helix in terms of the wake diameter at infinity. The contour lines of the wake helix are also shown at four values of x (F7rV?KU) the advance ratio in comparison with the contour lines for an F circulation at radius infinite number of blades. fpFw- INTRODUCTION circulation function for single rotation K (x) 2xVw Since reference is often made to the wake infinitely far behind the propeller, it is desirable to establish certain angular velocity of propeller, radians per second relationships between the dimensions, of the propeller and due to a douhlet'element at those of the wake helix at infinity. The present paper con- y' radial velocity' at point P O,x except for a-constant factor siders the relationship of the propeller diameter and the wake diameter, or the problem of the slipstream contraction. ( [0 cos (0+r) —sin (0+ T)] [1-2x2+X202+x cos (0+r)] The discussion is restricted to a consideration of first-order [1+x2+X202 -2x cos (O+r)] effects, that is, to the determination of the contraction per K(x).. unit of loading for infinitely small loadings only. It will be y2 =—y1 wherenr=O, 1, 2, . pl seen that the contractions are indeed very small, of the order of a few percent of the propeller diameter, and that the high-order terms are therefore not of concern. The inter- • Y1 =f y2 dx ference velocity accordingly • is neglected as small compared with the stream velocity. The wake helix lies on a perfect Y2 angle of contraction, except for a constant cylinder and the pitch angle is everywhere the same. It is noted that the assumption of zero loading corresponds to factor (2=J Y1 do) that used by Goldstein for a different purpose. • contour line of contraction, except for a constant SYMBOLS R tip radius of propeller factor'(Yi=fYido) r radius of element of vortex sheet r0 '- Ic3 X3 r contraction - total contractioni n terms of radius - \KI r0 total contraction or contraction at =O Pc • /x37 angle between starting point of spiral line and point Rc3- contraction coefficient (-4 11 3) H pitch of spiral 0 angular coordinate on vortex sheet = sin {tan(x - x )E(k) + x32[F(k) -E(k)IJ hH— - X advance ratio () wi =[(j_ k) F(k)_-yE(k)]' 1 2 REPORT NO. 777—NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS THEORY By introducing the nondimensional quantities The radial velocity is obtained by using the Blot-Savart H law and integfating over the entire surface of discontin- IrR uity. If ArG is the total contraction, the problem is to deter- (2) - mine the ratio for various numbers of blades at several F advance ratios. Simple expressions referring to zero load- ing are used throughout. in the Biot-Savart law, the following expression is obtained The radial inward velocity dvR' at the point P is calculated. for the radial inward velocity dvR' due to an element on (See figs. 1' and 2.) This velocity results from an element the wake helix of strength f: ü cos (O+r) —sin (o +7-) (3) 4ir R [1+x2+X282-2x cos (0+T)] By differentiatiiig equation (3) with respect to x, the field of a doublet element on the helical vortex sheet is obtained, the doublet element consisting of two neighboring singlet elements each of strengthf.