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This study treats the ducted propeller with finite blade number at zero angle of attack in a uniform, incompressible, inviscid flow. The approximation* of a lightly loaded propeller and of thin airfoil theory are made. In the absence of thickness efiects, appropriate vortex distributions represent the blades, the shroud and their respective shed vo:rtices. By means of Fourier analysis of the velocity field in piopeller fixed coordinatcs, the problem for an arbitrary, radial blade circulation distribution is reduced to a form similar to the ring wing integral equations of J. Weissinger. The kernels are not the same for the two cases except for the zeroth harmonic. The equation for this harmonic, which corresponds to a generalized actuator disk formulation, is Ldentical to that of an equivalent, axisymmetric ring wing. The effect of blade number, blade circulation profile and strength, propeller advance ratio and tip clearance, and location of the propeller plane are found and may be evaluated over a range of parameters from tables provided. A numerical example is given. i-i 16 1.6 Formulation of Basic Equation for-y CHAPTER TWO-Derivation of Shroud Vortex Distribution 2.1 Periodic Expansion of Shroud Vortex Distribution 2.2 Reduction of Shroud Contributions 2. 2.3 Decomposition of Governing Equation for-y 28 2.1I Splitting of Governing Complex Equation 2.5 Discussion of Coupled Equations 36 2.6 Shroud and Propeller Loading CHAPTER THREE-Solutions for Duct Leading 3.I Fundamental Solution 43 3.2 Effect of Parameters or the Fundamental Load 47 3.3 Limiting Case of Infinite Advan, 1,1atio 3,14 Infinite Blade uNumber 64 iii CHAPTER THREE 3.5 General Solution for Higber Harmonics 3.6 Example REFERENCES iv Am. .,. amplitudes of sine term of Fourier harmonic 3 A. ... amplitudes of :osne term of Fourier harmonic CU. ... amplitudes of complex Fourier harmonic D distance between vortex element and field point FY, ... functions of complex kernel of intermediate form of governing equations Gn(2) Riegels function 1h, ,ld vector influence functions for axial, radial and circumferential vortex elements respectively IA* ... integ'zl a of chozdwise vortex distribution and decoupled kernel imaginary part of complex function J propeller advance ratio, U/ p Kkernel of final governing equations N blade number P "fcrcin% function" of higher propeller harmc ics Qn_(W) Le%/ndre function of second kind and half order R radius of the shroud camber surfacm in the propeller plane used as reference length real part of complex function R p propeller semi-diameter Sn (Qn+l/2+"n-3/2) Tn (on+1I2-on-312) U uniform free stream V c shroud chord gmN'hmN functions dependent on higher harmonics of shed vorticity 46,j_,k unit vectors in the Cartesian system (xyz) 1-,0i_ unit vectors in the cylindrical system (xr,O) i 4j dummy summation index £blade index number m rank of Fourier harmonic n order of Legendre function p static pressure qfluid velocity t time variable x,r,e cylindrical coordinates fi: ed in propeller x,,rs s cylindrical coordinates of n.t ...

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A DETAILED NUMERICAL EVALUATION OF SHROUD PERFORMANCE FOR FINITE-BLADED DUCTED PROPELLERS * lA GOVERNMENT RESEARCH REPORT rU.S. DEPARTMENT OF COMMERCE OFFICE OF TECHNICAL SERVICES distributes this and thousands of similar reports in the interest of science, industry, and the public-for which research and new products mean better health, better living, and a stronger economy. HOW TO GET OTHER REPORTS The Office of Technical Services is the Nation's clearinghouse for reports of research supported by the Army, Navy, Air Force, Atomic Energy Commission, and other Government agencies. Abstracts of new reports available are published twice a month in U. S. GOVERNMENT RESEARCH REPORTS ($15 a year domestic). Selected Reports of particular interest to small business are described monthly in TECHNICAL REPORTS NEWSLETTER ($1 a year domestic). Translations of foreign technical material are also available from the Office of Technical Services and other sources. These are listed or abstracted semimonthly in TECHNICAL TRANSLATIONS ($12 a year domestic). The above periodicals may be ordered from Superintendent of Documents, U. S. Government Printing Office, Washington, D. C., 20402, or through a U. S. Department of Commerce Field Office. Inquiries about the availability of reports and translations on any particular subject may be directed toOfficeofTechnical Services, U. S.