Three-Dimensional Theory of Ducted Propellers

Abstract

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 ...

General Harmonic Solutions for the Ducted Propeller

“…The distribution selected for the propeller circulation is the typical distribution which was previously used into a form with a kernel which has a similar Cauchy singularity, we can utilize the same technique. Fortunately, we found such a way to decouple these equations earlier 1 and so have a solution in effect for the higher harmonics.…”

Shrouded Propeller Test Program Data Analysis. Volume 1

“…3 should be considered to achieve maximum net thrust. Depending, then, upon the application and upon the performance compromise situation area ratios both less than and greater than those tested in this program warrant further investigation (2). In order to achieve area ratios greater than 1.3 the added diffusion necessitates either lengthening the shroud or locating the propeller further forward in the shroud to allow more length for efficient diffusion.…”

“…3-2. The association of ri 2 and ro 2 with static tap location for the various shroud models is shown in Table 10. 3-I.…”

The unsteady pressure field of a ducted impeller

“…This is a Fredholm integral equation of the first kind in which the kernel exhibits a weak Cauchy singularity. Solution procedures for this equation are described by Ordway et al (1960), Dang (1972Dang ( , 1975a, b), Dang & Norrie (1975) and Ordway & Greenberg (1961).…”